UMRK Help File
FIELD: g_data.g
SYNOPSIS: Each row in the table g_data corresponds to a particular (isomorphism class of) a
simple Lie algebra of rank <= 8. The field g_data.g holds the Cartan type (one of A1,...,A8,
B2,...,B8,C2,...,C8,D4,...,D8,E6,E7,E8,F4,G2) of the Lie algebra.
REFERENCES:
N. Bourbaki, "Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York, 2002.
J. Humphreys, "Reflection Groups and Coxeter Groups", Cambridge Studies in Adv. Math. 29, Cambridge Univ. Press, Cambridge, UK, 1990.
FIELD: g_data.cartan_class
SYNOPSIS: Each row in the table g_data corresponds to a particular (isomorphism class of) a
simple Lie algebra of rank <= 8. The field g_data.g holds the Cartan class (i.e, A,B,C,D,E,F or G)
of the Lie algebra.
REFERENCES:
N. Bourbaki, "Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York, 2002.
J. Humphreys, "Reflection Groups and Coxeter Groups", Cambridge Studies in Adv. Math. 29, Cambridge Univ. Press, Cambridge, UK, 1990.
FIELD: g_data.rank
SYNOPSIS: This field holds the rank of a Lie algebra (the dimension of any maximal commutative subalgebra).
FIELD: g_data.ebase
SYNOPSIS: The field g_data.ebase provides a simple base of the root system of a Lie algebra, following
the conventions of Bourbaki. Each simple root is expressed as linear combinations of standard basis
vectors e1,e2,... of a Euclidean space.
REFERENCES:
N. Bourbaki, "Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York, 2002.
FIELD: g_data.fund_wgts
SYNOPSIS: Let B be the base of simple roots for a reduced root system and let C be
the Cartan matrix of B:
C[i,j] = 2*(B[i],B[j])/(B[i],B[i]) , 1 <= i,j <= Rank(g)
and let CI be its matrix inverse.
The fundamental weights (corresponding to B) are given by
w[i] = sum_j CI[i,j]*B[j] (sum over 1 <= j <= Rank(g))
For a given Lie algebra g, the field g_data.fund_wgts holds an ordered list of vectors
(presented as linear combinations of standard Euclidean basis vectors e1,e2,....) that
are the fundamental weights corresponding to the simple roots in g_data.ebase.
REFERENCES:
N. Bourbaki, "Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York, 2002.
J. Humphreys, "Reflection Groups and Coxeter Groups", Cambridge Studies in Adv. Math. 29, Cambridge Univ. Press, Cambridge, UK, 1990.
FIELD: g_data.cartan_matrix
SYNOPSIS: Let B be a base of simple roots for a reduced root system with inner
product ( , ). The Cartan matrix for B is Rank(g) x Rank(g) matrix with entries
C[i,j] = 2*(B[i],B[j])/(B[i],B[i]) , 1 <= i,j <= Rank(g) .
For a given Lie algebra g, the field g_data.cartan_matrix holds the entries
of the Cartan matrix for B = gdata_ebase. It is presented as a list of lists of
integers; with the i^th list of integers containing the entries of the i^th row
of the Cartan matrix.
REFERENCES:
N. Bourbaki, "Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York, 2002.
J. Humphreys, "Reflection Groups and Coxeter Groups", Cambridge Studies in Adv. Math. 29, Cambridge Univ. Press, Cambridge, UK, 1990.
FIELD: g_data.inv_cartan_matrix
SYNOPSIS: For a given Lie algebra, the field g_data.inv_cartan_matrix contains
the matrix inverse of the Cartan matrix for g. (See g_data.cartan_matrix.)
FIELD: g_data.pos_roots
SYNOPSIS: For a given Lie algebra g, the field g_data.pos_roots holds the positive
roots of the root system of g with respect to the simple base g_data.ebase. Each
positive root is presented as a linear combination of the standard Euclidean vectors
e1,e2,....
REFERENCES:
N. Bourbaki, "Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York, 2002.
J. Humphreys, "Reflection Groups and Coxeter Groups", Cambridge Studies in Adv. Math. 29, Cambridge Univ. Press, Cambridge, UK, 1990.
FIELD: g_data.highest_root
SYNOPSIS: For a given Lie algebra g, the field g_data.highest_root holds the highest
root with respect the simple base g_data.ebase. This root is presented as a linear
combination of the standard Euclidean vectors e1,e2,....
REFERENCES:
N. Bourbaki, "Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York, 2002.
J. Humphreys, "Reflection Groups and Coxeter Groups", Cambridge Studies in Adv. Math. 29, Cambridge Univ. Press, Cambridge, UK, 1990.
FIELD: g_data.rho
SYNOPSIS: For a given Lie algebra g, the field g_data.rho holds (1/2) times the vector
sum of the positive roots of g with respect to the simple base g_data.ebase. This vector
is presented as a linear combination of the standard Euclidean vectors e1,e2,....
REFERENCES:
N. Bourbaki, "Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York, 2002.
J. Humphreys, "Reflection Groups and Coxeter Groups", Cambridge Studies in Adv. Math. 29, Cambridge Univ. Press, Cambridge, UK, 1990.
FIELD: g_data.exponents
SYNOPSIS: For a given Lie algebra, g, let S be the simple reflections corresponding to
a base B of simple roots, and let W be the Weyl group (generated by S). A Coxeter element
in W is a product s_1*s_2*...*s_r of r = Rank(g) distinct simple reflections. Such Weyl
group elements are all conjugate in W. The common order of these Coxeter elements is
the Coxeter number of g. The Coxeter elements, thought of as orthogonal transformations of
the vector space spanned by the roots of g, also share the same set of eigenvalues. These
eigenvalues are always h^th roots of unity where h is the Coxeter number of g. If we write
the particular eigenvalues that occur as
z^m_1,z^m_2, ... , z^m_r
where z = exp(2i*pi/h) and r = Rank(g), then the integers m_1, ... , m_r that occur are
called the exponents of g.
Alternatively, let H be a Cartan subalgebra for g, and let P(H)^W be the ring of W-invariant
polynomials on H. By a theorem of Chevalley, P(H)^W is generated by r homogeneous polynomials
of degrees m_1+1, m_2+1 , .... , m_r + 1.
REFERENCES:
N. Bourbaki, "Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York, 2002.
J. Humphreys, "Reflection Groups and Coxeter Groups", Cambridge Studies in Adv. Math. 29, Cambridge Univ. Press, Cambridge, UK, 1990.
FIELD: g_data.coxeter_number
SYNOPSIS: For a given Lie algebra, g, let S be the simple reflections corresponding to
a base B of simple roots, and let W be the Weyl group (generated by S). A Coxeter element
in W is a product s_1*s_2*...*s_r of r = Rank(g) distinct simple reflections. Such Weyl
group elements are all conjugate in W. The common order of these Coxeter elements is
the Coxeter number of g.
REFERENCES:
N. Bourbaki, "Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York, 2002.
J. Humphreys, "Reflection Groups and Coxeter Groups", Cambridge Studies in Adv. Math. 29, Cambridge Univ. Press, Cambridge, UK, 1990.
FIELD: g_data.dual_root_system
SYNOPSIS: If B[1], .., B[r] is a base for an irreducible root system R, the vectors
B^[i] = 2/(B[i],B[i]) * B[i]
also form a base for an irreducible root system; it is called the root system
dual to R. For a given Lie algebra g, the field g_data.dual_root_system holds the
Cartan type of the root system dual to root system of g.
FIELD: g_data.connection_index
SYNOPSIS: Let w_1, ... , w_r be the fundamental weights
of a Lie algebra g. The weight lattice of g is the set of all Z-linear combinations of
the fundamental weights. For any element w in the weight lattice one necessarily has
(w, alpha) is an integer
for all simple roots alpha. More generally, a weight is said to integral if its inner product
with any simple root is an integer. The set of integral weights is a lattice that contains the
weight lattice. The connection_index of g is the index of the weight lattice inside the lattice
lattice of integral weights.
REFERENCES:
N. Bourbaki, "Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York, 2002.
FIELD: g_data.order_of_weyl_group
SYNOPSIS: For a given Lie algebra g, the field g_data.order_of_weyl_group holds the
cardinality of the Weyl group of g.
FIELD: g_data.idx_of_coxeter_elt
SYNOPSIS: For a given Lie algebra g, let S be the simple reflections corresponding to
a base B of simple roots, and let W be the Weyl group (generated by S). A Coxeter element
in W is a product s_1*s_2*...*s_r of r = Rank(g) distinct simple reflections. Such Weyl
group elements are all conjugate in W. The field g_data.idx_of_coxeter_elt holds
the cw_idx of this conjugacy class (i.e. the integer
John Stembridge's Coxeter/Weyl
program uses to indicate this conjugacy class).
FIELD: g_data.idx_of_longest_elt
SYNOPSIS: For a given Lie algebra g, with Weyl group W, the field g_data.idx_of_longest_elt
holds the cw_idx of the conjugacy class containing the
longest element of W (i.e. the integer John Stembridge's Coxeter/Weyl program uses to indicate
this conjugacy class).
FIELD: g_data.admissible_graphs
SYNOPSIS: Fix a Lie algebra g of rank r with root system R and Weyl group W. A set T of r
linearly independent roots is said to be an admissible set if the roots in T are
mutually obtuse root:
(t1,t2) <= 0 for all distinct t1,t2 in T .
An admissible graph of g is a W-conjugacy class of such an admissible set. These were
introduced and used by R. Carter in his classification of the conjugacy classes of Weyl
groups. For a given Lie algebra g, the field g_data.admissible_graphs holds a complete list of
representatives of admissible graphs. A given representative is presented as a two item list.
The first item is a list of the integers, each integer representing a particular positive root
(following the ordering of the positive roots in g_data.pos_roots).
The second list item is Carter's label for the corresponding admissible diagram (N.B., Carter's
labeling of admissible diagrams, while closely resembling (typographically) the Bala-Carter
labeling of nilpotent orbits, has nothing to do with nilpotent orbits).
REFERENCES:
R. Carter, "Conjugacy classes in the Weyl group", Comp. Math. 25 (1972), 1-59.
FIELD: orbit_data.id
SYNOPSIS: The field orbit_data.id holds the primary keys for the table orbit_data. In other words,
the field orbit_data_id holds a unique identifier for each orbit that is contained in the table.
These identifiers were formed by simplying concatenating the Cartan type of the underlying
Lie algebra g with 'orb' and then with an integer that labels the nilpotent orbits of g.
Thus, orbit_data.id holds an expression like "D4orb3".
FIELD: orbit_data.g
SYNOPSIS: The field orbit_data.g holds the Cartan type of the simple Lie algebra g in which
an orbit resides.
FIELD: orbit_data.idx
SYNOPSIS: The field orbit_data.idx holds an integer that labels and distinguishes the various
nilpotent orbits for a given group. For the exceptional groups, this integer corresponds to the
place in which the orbit in tables of Collingwood and McGovern. For the classical groups this
integer corresponds to the place in which the partition corresponding to the orbit appears in
the lexicographical ordering of the partitions that parameterize the orbits.
REFERENCES:
D. Collingwood and W. McGovern, "Nilpotent Orbits in Semisimple Lie Algebras", Van Nostrom Reinhold, New York, 1993.
FIELD: orbit_data.common_name
SYNOPSIS: The field orbit_data.common_name gives (at least a
comprehensible variation of) the label usually ascribed to a
nilpotent orbit in the literature. For the exceptional groups this is a
label derived from the Bala-Carter specification of the orbit (where an
orbit O is specified by prescribing the Cartan type of a minimal Levi
subalgebra L that intersects the orbit O and a designation a1,b1,.. to
indicate the (distinguished) nilpotent orbit in L that coincides with
the intersection of O and L.) There are cases, however, where the Cartan type
alone is not sufficient to identify a particular conjugacy class of Levi subalgebras.
In such cases, some additional notational embellishments are used to distinguish
distinct conjugacy classes of Levis (such adding tildes to factors whose simple
roots are short roots in g, or using primes and double primes to distinguish pairs
of isomorphic but non-conjugate Levis).
Nilpotent orbits for classical groups are commonly labeled by partitions
of n+1 (for type A_n), certain partitions of 2n (for types C_n and D_n),
or certain partitions of 2n+1 (for type B_n). For type D_2k this
labeling is ambiguous when a partition is very even (consisting only of
even parts with even multiplicity). Usually one appends labels I and II
to these very even partitions in order to distinguish the two nilpotent
orbits that can be attached to a very even partition. We, however, use
the convention that if a orbit for D_2k corresponds to a very even
partition
[m_1,m_1,m_2,m_2,....,m_k,m_k]
with numeral I, then its common_name is also
[m_1,m_1,m_2,m_2,....,m_k,m_k]
while the orbit with same partition and numeral II has as its common_name
[m_1,m_1,m_2,m_2,....,m_k,m_k,0]
as the 0 appended to the original partition has a (suitably) innocuous
effect on computations based on partition of an orbit.
REFERENCES:
D. Collingwood and W. McGovern, "Nilpotent Orbits in Semisimple Lie Algebras", Van Nostrom Reinhold, New York, 1993.
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
FIELD: orbit_data.cbc_parameters
SYNOPSIS: For a given nilpotent orbit in a Lie algebra g, the field
orbit_data.cbc_parameters holds the "combinatorial Bala-Carter parameter" from
which the orbit can be constructed directly. A cbc_parameter consists of a subset
Gamma of the simple roots (denoted as a subset of the integers from 1 to Rank(g)
following Bourbaki's labeling of simple roots), and a "distinguished" subset
gamma of Gamma. Here "distinguished" is a very simple combinatorial condition:
let Delta_Gamma and Delta_gamma be the root systems generated by the simple roots
respectively in Gamma and gamma (so Delta_gamma is a Levi subsystem of Delta_Gamma).
By definition, gamma is a "distinguished" subset of Gamma if
#Delta_gamma + #Gamma = #{positive roots lambda in Delta_Gamma of the form
lambda = mu + alpha where mu is in Delta_gamma
union {0} and alpha is in Gamma - gamma}
where, for a set A, #A denotes the cardinality of A.
The field orbit_data.cbc_parameters specifies a pair [Gamma,gamma] where
both Gamma and gamma are presented as lists of integers between 1 and Rank(g),
each integer corresponding to a simple root of g following the ordering
conventions of Bourbaki.
REFERNCES:
D. Collingwood and W. McGovern, "Nilpotent Orbits in Semisimple Lie Algebras", Van Nostrom Reinhold, New York, 1993.
B. Binegar, "Subsystems, Nilpotent Orbits and Weyl Group Representations",OSU Lie Groups Seminar, 11/18/2009.
FIELD: orbit_data.weighted_dynkin_diagram
SYNOPSIS: Suppose x is a nilpotent element of a simple Lie algebra g. By a theorem
of Jacobson-Morozov, one can attach to x a standard sl(2) triple {x,h,y}. One can
then conjugate the semisimple element, h, of this triple by G so that it lies
in the dominant chamber of a Cartan subalgebra of g. When this is done, the eigenvalue
of ad(h) on any simple root is 0, 1, or 2. In this manner, for a given x, a certain weight
(0, 1, or 2) can be ascribed to each simple root. (These weights are, in fact, independent
of the choices made in their construction.) Adopting an ordering alpha_1, .... ,alpha_r
of the simple roots, one can then, for any given x, form a list WDD(x) = [m_1,...,m_r] of
the corresponding weights. It turns out that if x,x' live in the same nilpotent orbit, then
WDD(x) = WDD(x'); and if x,x' live in different nilpotent orbits, then WDD(x) and WDD(x')
are distinct. Thus the "weighted Dynkin diagram" WDD(x) is a complete invariant of the
orbit of x.
For a given nilpotent orbit, the field orbit_data.weighted_dynkin_diagram presents
the weights as a list of Rank(g) integers, ordered in such a way that the
i^th integer in the list is the weight corresponding to the i^th simple
root with respect to Bourbaki's ordering of the simple roots.
FIELD: orbit_data.springer_rep
SYNOPSIS: In a seminal paper ["A construction of representations of Weyl groups", Inv. Math. 44
(1978), 279-293], T.A. Springer attaches to each nilpotent orbit of a semisimple Lie algebra g,
a particular irreducible representation of the Weyl group of g. For a given nilpotent orbit
O, the field orbit_data.springer_rep holds an integer that corresponds to the value of the field
wrep_data.cw for this particular Weyl group representation. (Thus, this field links the orbit_data
table to the wrep_data table.)
FIELD: orbit_data.dim
SYNOPSIS: For a given nilpotent orbit, the field orbit_data.dim hold the dimension of the orbit.
FIELD: orbit_data.pi1
SYNOPSIS: For a given (adjoint) nilpotent orbit O, the field orbit_data.pi1 holds the isomorphism class of
the fundamental group of O. We use the notation to Z2 (resp. Z3, etc.) to indicate the additive group of integers
modulo 2 (resp. modulo 3, etc). S3, S4, and S5 indicate the symmetric groups on 3, 4 or 5 letters. For certain
nilpotent orbits in type B_n and D_n, the fundamental groups are actually central extensions of a product of
Z2's by another Z2. Lacking a means to properly articulate such central extensions, in such cases, the field
orbit_data.pi1 only provides a value like cext(Z2^2,Z2) which indicate at least the form of the fundamental group
(even if it does not speciify the actual central extension).
FIELD: orbit_data.A
SYNOPSIS: Let O be a nilpotent orbit in a simple Lie algebra g and let x be a point in O. Let G^x denote the
stability subgroup of x inside the adjoint group of g and let (G^x)^o be the connected component of G^x containing
the identity element. Then the quotient group G^x/(G^x)^o is a finite group, usually called the "component group"
(of the stabilizer) of x, and denoted as A(O). For nilpotent orbits in classical groups A(O) is either trivial or
always a product of Z2's (integers mod 2 under addition), while for the exceptional groups it is one of symmetric
groups 1=S1, S2, S3,S4, or S5.
For a given nilpotent orbit, the field orbit_data.a holds the isomorphism class of its component group.
FIELD: orbit_data.Abar
SYNOPSIS: For a given nilpotent orbit 0, the field orbit_data.abar holds the ismomorphism class of Lusztig's
canonical quotient of the component group of stability subgroup of O. This group plays a critical role in
Lusztig's classification of the unipotent representations of finite groups of Lie type.
REFERENCES:
G. Lusztig, "A class of irreducible representations of a Weyl group I, Indag. Math 41 (1979), 323-335.
G. Lusztig, "A class of irreducible representations of a Weyl group II, Indag. Math 44 (1982), 219-226.
G. Lusztig, "Characters of Reductive Groups over a Finite Field", Annals of Mathematics Studies, Vol. 107,
Princeton Univ. Press, Princeton, NJ, 1984).
E. Sommers, "Lusztig's Canonical Quotient and Generalized Duality", J. Alg 243 (2001), 790-812.
FIELD: orbit_data.orbits_immed_above
SYNOPSIS: In the table orbit_data, for a given Lie algebra g, the nilpotent orbits are indexed by
integers held in the field orbit_data.idx. For a given nilpotent orbit O, the field orbit_data.orbits_immed_below
holds the indices of the nilpotent orbits immediately above O in the (algebraic-geometrical) closure relations
amongst the set of nilpotent orbits. (An orbit O' being immediately above O means that O resides in the
Zariski closure of O' and there exists no other orbit in the boundary of O' whose closure contains O.
FIELD: orbit_data.orbits_immed_below
SYNOPSIS: In the table orbit_data, for a given Lie algebra g, the nilpotent orbits are indexed by
integers held in the field orbit_data.idx. For a given nilpotent orbit O, the field orbit_data.orbits_immed_below
holds the indices of the nilpotent orbits immediately below O in the (algebraic-geometrical) closure relations
amongst the set of nilpotent orbits.
FIELD: orbit_data.spaltenstein_dual_orbit
SYNOPSIS: For a given nilpotent orbit O in a simple Lie algebra g, the field orbit_data.spaltenstein_dual_orbit holds
the index (see orbit_data.idx) of the Spaltenstein dual orbit in g. For classical groups,
this is a special nilpotent orbit that is related to original orbit by a combinatorial operation on partition parameters
(the first step being taking the transpose of a partition). For the exceptional groups, the dual of an orbit
was defined by Spaltenstein precisely so that the duality map would closely mimic the situtation for classical groups,
especially with respect to orbit induction.
REFERENCES:
N. Spaltenstein, "Classes Unipotentes et Sous-Groupes de Borel", Lec. Notes in Math. 946, Springer-Verlag, New York, 1982.
D. Collingwood and W. McGovern, "Nilpotent Orbits in Semisimple Lie Algebras", Van Nostrand Reinhold, New York, 1993.
FIELD: orbit_data.barbasch_vogan_dual_orbit
SYNOPSIS: For a given nilpotent orbit O in a simple Lie algebra g, the field orbit_data.barbasch_vogan_dual_orbit holds
the id (see orbit_data.id) of the Barbasch-Vogan dual orbit in the dual Lie algebra g^.
This BV_dual orbit is defined as follows. Start with a representive element x of O. Form a standard sl(2) triple {x,h,y}.
Re-interprete h/2 as an infinitesimal character for representations of the dual Lie algebra g^ and then define the BV dual
of O to be the nilpotent orbit in g^ corresponding to the associated variety of the maximal primitive ideal in the universal
enveloping algebra of g^ with infinitesimal character h/2.
REFERENCE:
D. Barbasch and D. Vogan, Jr. "Unipotent representations of complex semisimple groups", Ann. Math. 121 (1985), 41-110.
FIELD: orbit_data.special_orbit_above
SYNOPSIS:
FIELD: orbit_data.producing_levi
SYNOPSIS: For a given nilpotent orbit O, the field orbit_data.producing_levi contains the index (see levi_data.idx of the (conjugacy class of the)
minimal Levi subalgebra L that has non-trivial intersection with O. (Thus, O will be the G-saturation of a distinguished orbit of L)>
FIELD: orbit_data.producing_psilevis
SYNOPSIS: The "extended" simple roots of a simple Lie algebra g is the union of the simple roots of g with the lowest root.
A standard pseudo-Levi subalgebra of g is a subalgebra generated by a Cartan subalgebra H together with the root vectors corresponding
to a proper subset of the extended simple roots. A pseudo-Levi subalgebra of g is a subalgebra of g that is conjugate to a standard
pseudo-Levi subalgebra.
For a given nilpotent orbit O, the field orbit_data.producing_psilevis contains the indices (see psilevi_data.idx of the (conjugacy classes of)
pseudo-Levi subalgebras L such that L intersect O is distinguished in L (so O is not contained in any proper Levi subalgebra of L).
FIELD: orbit_data.inducing_levis
SYNOPSIS: Let l be a levi subalgebra of a semisimple Lie algebra g and let p = l + u be any extension of l to a parabolic subalgebra of g.
The space Ad(g)(u) is a union of nilpotent orbits of g and within this there is unique Zariski dense orbit R(l), which is independent of
the choice of parabolic extension p. R(l) is the Richardson orbit attached to the Levi subalgebra l. Put another way, the Richardson orbit
corresponding to a levi subalgebra l is the nilpotent orbit obtained via induction of nilpotent orbits from the trivial nilpotent orbit inside
l. We remark that if sigma is the irreducible representation of W attached to R(l) by the Springer correspondence, then sigma is also the
Macdonald representation attached to the root subsystem corresponding to l.
For a given nilpotent orbit O, the field orbit_data.inducing_levis yields a list of the indexes (see levi_data.idx) of the conjugacy classes
of Levi subalgebras whose associated Richardson orbit is O.
REFERENCES:
R.W. Richardson, "Conjugacy classes in parabolic subgroups of semisimple algebraic groups", Bull. London Math. Soc. 6 (1974), 21-24.
G. Lusztig and N. Spaltenstein, "Induced Unipotent Classes", J. London Math. Soc. 19 (1979), 41-52.
D. Collingwood and W. McGovern, "Nilpotent Orbits in Semisimple Lie Algebras", Van Nostrand Reinhold, New York, 1993.
FIELD: orbit_data.orbit_special_yn
SYNOPSIS: A nilpotent orbit O is special if one of the following equivalent conditions holds:
(i) O lies in the image of the Spaltenstein duality map
(ii) O lies in the image of the Barbasch-Vogan duality map
(iii) The irreducible representation of W corresponding to O under the Springer correspondence
is a special representation of W.
(iv) O is the associated variety of a primitive ideal with regular integral infinitesimal character.
For a given nilpotent orbit O, the field orbit_data.orbit_special_yn is equal to 1 or 0 according to
whether the orbit is special or not.
REFERENCES:
N. Spaltenstein, "Classes Unipotentes et Sous-Groupes de Borel", Lec. Notes in Math. 946, Springer-Verlag, New York, 1982.
D. Collingwood and W. McGovern, "Nilpotent Orbits in Semisimple Lie Algebras", Van Nostrand Reinhold, New York, 1993.
FIELD: orbit_data.richardson_yn
SYNOPSIS: Let l be a levi subalgebra of a semisimple Lie algebra g and let p = l + u be any extension of l to a parabolic subalgebra of g.
The space Ad(g)(u) is a union of nilpotent orbits of g and within this there is unique Zariski dense orbit R(l), which is independent of
the choice of parabolic extension p. R(l) is the Richardson orbit attached to the Levi subalgebra l. Put another way, the Richardson orbit
corresponding to a levi subalgebra l is the nilpotent orbit obtained via induction of nilpotent orbits from the trivial nilpotent orbit inside
l. We remark that if sigma is the irreducible representation of W attached to R(l) by the Springer correspondence, then sigma is also the
Macdonald representation attached to the root subsystem corresponding to l.
For a given nilpotent orbit O, the field orbit_data.richardson_yn is equal to 1 or 0 according to whether O is a Richardson orbit for some
levi subalgebra or not.
REFERENCES:
R.W. Richardson, "Conjugacy classes in parabolic subgroups of semisimple algebraic groups", Bull. London Math. Soc. 6 (1974), 21-24.
G. Lusztig and N. Spaltenstein, "Induced Unipotent Classes", J. London Math. Soc. 19 (1979), 41-52.
D. Barbasch and D. Vogan, Jr. "Unipotent representations of complex semisimple groups", Ann. Math. 121 (1985), 41-110.
D. Collingwood and W. McGovern, "Nilpotent Orbits in Semisimple Lie Algebras", Van Nostrand Reinhold, New York, 1993.
FIELD: wrep_data.id
SYNOPSIS: The field wrep_data.id holds the primary keys for the table wrep_data. In other words,
the field wrep_data_id holds a unique identifier for each irreducible W-representation contained
in the table. These identifiers were formed by simplying concatenating the Cartan type of the underlying
Lie algebra g with 'wrep' and then with an integer that labels the particular irreducible representation
of the Weyl group of g. Thus, if the field wrep_data.id holds an expression like "D4owrep3" it means that
the W-rep corresponding to this row in the table is the third W-rep of D4 (the ordering used is that of
used by John Stembridge's Coxeter/Weyl package).
FIELD: wrep_data.g
SYNOPSIS: For a given W-representation, the field wrep_data.g holds the Cartan type of the Lie algebra underlying W.
FIELD: wrep_data.cw_idx
SYNOPSIS: For the Weyl group W of a fixed Lie algebra g, the individual irreducible representations of W are indexed
by an integer which prescribes where the character of a representation occurs in the list of irreducible characters
of W that is produced by John Stembridge's Coxeter/Weyl package.
FIELD: wrep_data.common_name
SYNOPSIS: For the Weyl groups of classical Lie algebras, irreducible representations are naturally parameterized either
by partitions (for type A_n), or pairs of partitions (for types B_n, C_n and D_n). For the exceptional Lie algebras, there
are several naming conventions, the most uniform of which is that used in Carter's book. Here a W-rep of an exceptional
Weyl group is specified by giving its dimension D and the degree d in which it first occurs in the symmetric algebra of the
reflection representation of W. Together the two invariants, D and d, nearly split the irreducible representations into singlets.
The exceptions occur in types G2 and F4, and so here some additional demarcation is needed.
For a given irreducible representation of a classical Weyl group, the value of the field wrep_data.common_name will be
a partition of n+1 (if W is of type A_n), or a pair of partitions [mu,nu] where |mu| + |nu| = n (for types B_n, C_n, and D_n).
For type D_n, and additional embellishment is needed when mu=nu, as will be two irreducible W-reps corresponding to such a duplicate
pair of partitions. These will be separated, notationally, by adding a 1 (corresponding to a label I in the literature) or a 2
(corresponding to a label II in the literature) to partition pair. Thus, for example the two irreducible representations of the Weyl
group of D8 corresponding to the partition pair [[2,2],[2,2]] will be denoted as [[2,2],[2,2],1] (numeral I) and [[2,2],[2,2],2]
(numeral II).
For a given irreducible representation of an exceptional Weyl group, the value of the field wrep_data.common_name will be
an expression of the form phi[D,d] where D is the dimension of the representation and d is its degree (as defined above);
so long as the values D and d are sufficient to distinguish the representation. In the exceptional cases, we supplement the
invariants D and d with a 1 or 2 which correspond to the primed and double-primed representations in Carter's notation. Thus,
phi[6,6,1] <---> Carter's $\phi_{6,6}^{\prime}$
phi[6,6,2] <---> Carter's $\phi_{6,6}^{\prime \prime}$
(here we have resorted to Tex code in order to render Carter's notation in text form).
REFERENCE:
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
FIELD: wrep_data.springer_parameters
SYNOPSIS: According to a construction of T.A. Springer, to each irreducible representation of a Weyl group W, one can attach
a certain nilpotent orbit O of the underlying Lie algebra and a certain irreducible representation sigma of the component group
A(O) of O. (See orbit_data.a for a definition of A(O).) The datums (O,sigma), in fact, are distinct
for distinct irreducible representations of W and so can be used to parameterize the irreducible representations of W. Such datums
are called Springer parameters.
For a given irreducible W-representation, the field wrep_data.springer_parameters contains the representations Springer parameters.
These are presented as a pair [O,sigma] here O is the common name of a nilpotent orbit and sigma
is list of integers that indicates a particular representation of A(O).
If W is of classical type, the groups A(0) will be certain quotients of groups isomorphic to (Z2)^n (Z2 being the group of integers mod 2).
(Exactly which quotient depends on the type of W, but for any given classical W the passage from (Z2)^n to A(O) is explicitly described,
e.g., in Carter's book.) Thus, for classical W, the irreducible representations of A(0) can be prescribed by describing how the generators
of each Z2 factor in its extenstion to (Z2)^n act. Thus, for classical W, when A(0) is a quotient of (Z2)^n the irreducible representations
of A(0) will be prescribed by a sequence of n 1's or -1's (which, respectively, indicate the trivial and non-trivial representations of Z2).
We again refer the reader to Carter's text in order get a better grasp of exactly which representation of A(O) such a list indicates.
The particular case of the trivial representation of A(0), however, is easily recognized; it will always be presented as a sequence
of 1's (no -1's).
When W is of exceptional type, the groups A(O) will always be one of the symmetric groups 1=S1, S2, S3,S4, or S5; and so in these cases
a partition parameterization is available for the irreducible representations of A(O). Thus, in the exceptional cases, the field
wrep_data.springer_parameters will hold a pair [O,p] where O is the common name of a nilpotent orbit of the underlying Lie algebra and
p is a certain partition. We remark that, in the exceptional cases, trivial representation of A(O) will always be indicated by a
partition with exactly one part.
REFERENCES:
T.A. Springer, "A construction of representations of Weyl groups", Inv. Math. 44 (1978), 279-293.
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
FIELD: wrep_data.orbit
SYNOPSIS: For a given irreducible representation sigma of a Weyl group W, the field wrep_data.orbit holds the "number" of the
nilpotent orbit (of the Lie algebra underlying W) that corresponds to sigma under the Springer correspondence. This number is
the value of the field orbit_data.id of the Springer related nilpotent orbit.
REFERENCES:
T.A. Springer, "A construction of representations of Weyl groups", Inv. Math. 44 (1978), 279-293.
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
FIELD: wrep_data.local_system
SYNOPSIS:
SYNOPSIS: The Springer correspondence attaches to each irreducible representation of a Weyl group a certain nilpotent orbit O in
the Lie algebra underlying W, and a certain irreducible representation of the component group A(O) of O. For a given irreducible
representation sigma of a Weyl group W, the field wrep_data.local_system describes the irreducible representation of the component
group A(O) under the Springer correspondence. See wrep_data.springer_parameters for
an explanation as to how this representation is specified.
REFERENCES:
T.A. Springer, "A construction of representations of Weyl groups", Inv. Math. 44 (1978), 279-293.
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
FIELD: wrep_data.dim
SYNOPSIS: For a given irreducible representation of a Weyl group, the field wrep_data.dim holds the dimension of the representation.
FIELD: wrep_data.lowest_degree_of_fdp
SYNOPSIS: Let V be the reflection representation of a Weyl group W (the reflection representation being the irreducible representation
for W carried by the span of the roots of the underlying Lie algebra). Let S(V) be the symmetric algebra of V and let S(V)_k be the subspace
spanned by symmetric tensor products with exactly k factors. Each subspace S(V)_k is completely reducible as a W-module and each irreducible
representation of W occurs in finitely many S(V)_k's.
For a given representation sigma of an irreducible representation of a Weyl group, the field wrep_data.lowest_degree_of_fdp give the
minimal value of k for which S(V)_k contains sigma as a subrepresentation. (See also wrep_data.fake_degree_poly_coeffs.)
FIELD: wrep_data.fake_degree_poly_coeffs
SYNOPSIS: Let V be the reflection representation of a Weyl group W (the reflection representation being the irreducible representation
for W carried by the span of the roots of the underlying Lie algebra). Let S(V) be the symmetric algebra of V and let S(V)_k be the subspace
spanned by symmetric tensor products with exactly k factors. Each subspace S(V)_k is completely reducible as a W-module and each irreducible
representation of W occurs in finitely many S(V)_k's. If sigma is an irreducible W-rep, then it's fake degree polynomial is an expression of
the form
p(q) = Sum ( m(sigma,k) * q^k ) (sum over k)
where m(sigma,k) is the multiplicity of sigma in the W-representation carried by S(V)_k.
For a given irreducible representation of a Weyl group, the field wrep_data.fake_degree_poly_coeffs is a list of integers
holding the multiplicities m(sigma,0), msigma(sigma,1), .... , m(sigma,k) , where k is the greatest integer such that m(sigma,k)
does not equal 0.
REFERENCES:
W.M. Bayen and G. Lusztig, "Some numerical results on the characters of exceptional Weyl groups", Math. Proc. Cambridge Phil. Soc. 84 (1978),
417-426.
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
E.M. Opdam, "A remark on the irreducible characters and fake degrees of finite real reflection groups", Inv. Math. 120 (1995), 447-454.
FIELD: wrep_data.generic_degree_poly_coeffs
SYNOPSIS: For each finite field F_q (q being a power of a prime number) and each root system g, there is an associated finite group
of Lie type G_q. Let B_q be a Borel subalgebra of G_q and let I_q be the representation of G_q induced from the trivial representation
of B_q. The irreducible representations of G_q that occur in I_q are indexed by the irreducible representations of Weyl group of g.
Let d(sigma,q) denote the dimension of the component of I_q corresponding ot sigma. Then it turns out, for fixed sigma, d(sigma,q) depends
polynomially on q. More precisely, for each prime power q, d(sigma,q) coincides with the evaluation of particular polynomial with rational
coefficients at q. This polynomial is the "generic degree polynomial" for sigma.
For a given representation sigma of a Weyl group, the field wrep_data.generic_degree_poly_coeffs holds coefficients of the generic degree
polynomial of sigma, up to the highest non-vanishing coefficient.
REFERENCES:
C.T. Benson, "The generic degrees of the irreducible characters of E8", Comm. Alg. 7 (1979), 1199-1209.
G. Lusztig, "A class of irreducible representations of a Weyl group", Indag. Math. 41 (1979), 323-335.
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
FIELD: wrep_data.dual_rep
SYNOPSIS: Suppose sigma is irreducible representation of a Weyl group W. Then the tensor product of sigma with the sign
representation is another irreducible representation of W. Generally speaking, this operation maps special representations
of W to special representations of W. However, for one pair of representations of the Weyl group of E7 and two pairs of
representations of the Weyl group of E8, special representations are mapped to non-special representations and vice-versa.
his papers on special representations of Weyl groups, Lusztig defined another involution i of W^ (that is trivial on all
but the aforementioned six representations) such that i composed with tensoring by sign is an involution that maps special
representations to special representations.
FIELD: wrep_data.special_rep_yn
SYNOPSIS: According to Lusztig's original definition, an irreducible representation of a Weyl group is "special" if the
degree of the lowest order term of the representation's fake degree polynomial is the same as the degree of the lowest
order term of its generic degree polynomial. Such representations are attached to special nilpotent orbits (see
orbit_data.special_orbit_yn). An alternative, W-intrinsic characterization
of special representations can be found in Subsytems, Nilpotent Orbits, and Weyl Group Representations.
For a given irreducible representation sigma, the field wrep_data.special_rep_yn has value 1 if sigma is special, and value 0 otherwise.
FIELD: wrep_data.orbit_rep_yn
SYNOPSIS: An irreducible representation sigma of a Weyl group is called an orbit representation (also a "Springer" representation), if
its Springer parameters (O,chi) consist of an nilpotent orbit O and the trivial representation of the its component group A(O).
FIELD: wrep_data.macdonald_rep_yn
SYNOPSIS: Let g be a simple Lie algebra, R its root system, and W its Weyl group. Let R' be a root subsystem of R and let p_W'
be the product of the positive roots in R' regarded as a homogeneous function on on the underlying vector space V of R. Under
the natural action of W on functions on V, the span of the polynomials w(p_W'), w in W, carries is an irreducible W-module.
The corresponding irreducible representation is the Macdonald representation corresponding to W' (or R').
Alternatively, let W' be the reflection subgroup of a Weyl group W generated by the reflections corresponding to a base of a root
subsystem of R' of g, and let sgn(W') be the sign representation of W'. The Macdonald representation corresponding to R' is the
unique summand of the induced representation
Ind_{W'}^{W)} ( sgn(W') )
of lowest degree (with the "degree" of an irreducible representation meaning the degree of the lowest order term of its fake degree
polynomial).
For Weyl groups of types A_n, B_n and C_n every irreducible representation is realizable as a Macdonald representation.
This, however, is not true in general. For a given irreducible representation sigma of a Weyl group the field
"wrep_data.macdonald_rep_yn" holds the integer 1 if is realizable as a Macdonald representation, and 0 if not.
REFERENCES:
I.G. Macdonald, "Some irreducible representations of Weyl groups", Bull. London Math. Soc. 4 (1972), 148-150.
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
FIELD: wrep_data.inducing_subsystems
SYNOPSIS: Let g be a simple Lie algebra, R its root system, and W its Weyl group. Let R' be a root subsystem of R and let p_W'
be the product of the positive roots in R' regarded as a homogeneous function on on the underlying vector space V of R. Under
the natural action of W on functions on V, the span of the polynomials w(p_W'), w in W, carries is an irreducible W-module.
The corresponding irreducible representation is the Macdonald representation corresponding to W' (or R').
Alternatively, let W' be the reflection subgroup of a Weyl group W generated by the reflections corresponding to a base of a root
subsystem of R' of g, and let sgn(W') be the sign representation of W'. The Macdonald representation corresponding to R' is the
unique summand of the induced representation
Ind_{W'}^{W)} ( sgn(W') )
of lowest degree (with the "degree" of an irreducible representation meaning the degree of the lowest order term of its fake degree
polynomial).
For a given irreducible representation sigma of a Weyl group the field "wrep_data.macdonald_rep_yn" holds a list of integers, with
each interger corresponding to the (see id of a root subsystems Gamma such that sigma is the Macdonald
representation corresponding to the reflection subgroup generated by the reflections corresponding to Gamma.
REFERENCES:
I.G. Macdonald, "Some irreducible representations of Weyl groups", Bull. London Math. Soc. 4 (1972), 148-150.
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
FIELD: wrep_data.irred_character
SYNOPSIS: A character for a Weyl group W can be specified by giving its values on each conjugacy class in W. In the UMRK database,
we have ordered W conjugacy classes as they appear inJohn Stembridge's Coxeter/Weyl program.
For a given irreducible representation sigma, the field wrep_data.irred_character lists the values of the irreducible character corresponding to
sigma on the conjugacy classes of W.
FIELD: wrep_data.cell_id
SYNOPSIS: Each irreducible representation of a Weyl group W belongs to one or more (generally reducible) cell representations. Each cell representation
has a unique constitutent that is a special representation of W (in the terminology of Lusztig). If a given irreducible representation X appears in two
cell representations, say, cellrep1 and cellrep2, then cellrep1 and cellrep2 also share the same special representation of W. In this way, every irreducible
representation of W is attached to a unique special representation of W. The cell_id's listed in the table Wrep_Data are integers that provide this correspondence;
albeit indirectly. A cell_id integer is actually the index of the special nilpotent orbit corresponding (via the Springer correspondence) to the attached special
orbit. Thus, if cell_id = 3, then the corresponding special representation is the special representation corresponding to the third special orbit that occurs in
a listing of the special orbits that follows the listing of nilpotent orbits in Orbit_Data.
REFERENCES:
D. Kazhdan and G. Lusztig, "Representations of Coxeter groups and Hecke algebras", Inv. Math. 51 (1979), 165-184.
G. Lusztig, "Left cells in Weyl groups", Lec. Notes in Math 1024, Springer, Berlin, 1983.
G. Lusztig, "Characters of Reductive Groups over a Finite Field", Annals of Math Studies 107, Princeton Univ. Press, Princeton, 1984.
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
FIELD: ss_data.id
SYNOPSIS: For a given root subsystem of the root system of a simple Lie algebra g, the field ss_data.id holds a primary key labeling
the corresponding row in the table ss_data. This id is constructed by concatenating the name of the Lie algebra with the string "ss" and
the value of the field
FIELD: ss_data.g
SYNOPSIS: For a given (W-conjugacy class of) root subsystem(s), the field ss_data.g holds the Cartan type of its parent (irreducible) root system.
FIELD: ss_data.ss_idx
SYNOPSIS: For a given root subsystem, the field ss_data.idx holds an integer which provides a unique label to the
various (W-conjugacy classes of) root subsystems for a given irreducible root system. These integers labels were chosen
so that an ordering of root subsystems by their rank and cardinality would be consistent with the natural ordering of
the corresponding integers.
FIELD: ss_data.ibase
SYNOPSIS: If S is a root subsystem of an irreducible root system g, it will be W-conjugate to the root system generated by
a simple base consisting of positive roots of g (with respect ot some fixed positive system on g).
For a given W-conjugacy class of root subsystems, the field ss_data.ibase holds the indices of the positive roots (these
indices are just integers indicating the position of a root in the fixed ordering of positive roots occurring the field
g_data).
FIELD: ss_data.ebase
SYNOPSIS:
SYNOPSIS: If S is a root subsystem of an irreducible root system g, it will be W-conjugate to the roots system generated by
a simple base consisting of positive roots of g (with respect ot some fixed positive system on g).
For a given W-conjugacy class of root subsystems, the field ss_data.ibase holds a list of positive roots (presented as linear
combinations of the standard Euclidean basis vectors e1,e2,...) that provide a simple base for a particular root subsystem
in this W-conjugacy class.
FIELD: ss_data.positive_roots
SYNOPSIS: For a given W-conjugacy class of root subsystems, the field ss_data.positive_roots holds a listing of the positive
roots (each presented as a linear combination of the standard Euclidean basis vectors e1,e2,...) corresponding to a particular
representive root subsystem with a positive system compatible with that of the parent irreducible root system>
FIELD: ss_data.cartan_type
SYNOPSIS: For a given W-conjugacy class of root subsystems, the field ss_data.cartan_type holds its Cartan type (presented as
a linear combination of the basis Cartan types A1,...,E8).
FIELD: ss_data.dim
SYNOPSIS: For a given W-conjugacy class S of root subsystems, the field ss_data.dim holds the dimension of the corresponding Lie
subalgebra (or if the subsystem is not closed, the sum of the cardinality and rank of S).
FIELD: ss_data.coxeter_conj_class
SYNOPSIS: Suppose B' and B" are bases for two root subsystems of an irreducible root system g. Let c' and c" be Coxeter elements
of corresponding to B' and B" in their respective Weyl groups (thus, c' is the product of the simple reflections corresponding to
B' and c" is the product the simple reflections corresponding to B"). Then the root subsystems generated by B' and B" are W-conjugate
(W being the Weyl group of g) if and only if c' and c" are conjugate in W. Thus, the W-conjugacy classes of Coxeter elements of
subsystems provide a simple means to tell whether or not two subsystems are W-conjugate (and so for most practical purposes
equivalent).
For a given subsystem, the field ss_data.coxeter_conj_class holds the index (see wcc_data.idx) of the
particular conjugacy class in W to which its Coxeter element belongs.
(Remark: since we're only identifying objects up to conjugacy, and trying to be brief, we have been a bit cavalier in definition of
a Coxeter element of a subsystem. However, given an ordered base of subsystem, there is a more-or-less canonical choice of a
corresponding Coxeter element - just the product of the simple reflections written in the same order).
FIELD: ss_data.macdonald_rep
SYNOPSIS: Let g be a simple Lie algebra, R its root system, and W its Weyl group. Lie R' be a root subsystem of R and let p_W'
be the product of the positive roots in R' regarded as a homogeneous function on on the underlying vector space V of R. Under
the natural action of W on functions on V, the span of the polynomials w(p_W'), w in W, carries is an irreducible W-module.
The corresponding irreducible representation is the Macdonald representation corresponding to W' (or R').
Alternatively, let W' be the reflection subgroup of a Weyl group W generated by the reflections corresponding to a base of a root
subsystem of R' of g, and let sgn(W') be the sign representation of W'. The Macdonald representation corresponding to R' is the
unique summand of the induced representation
Ind_{W'}^{W)} ( sgn(W') )
of lowest degree (with the "degree" of an irreducible representation meaning the degree of the lowest order term of its fake degree
polynomial).
For a given root subsystem R' of a Weyl group the field "ss_data.macdonald_rep" holds the index (see, wrep_data.idx of the
irreducible representation of W corresponding to the Macdonald representation attached to R'.
REFERENCES:
I.G. Macdonald, "Some irreducible representations of Weyl groups", Bull. London Math. Soc. 4 (1972), 148-150.
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
FIELD: ss_data.levi_idx
SYNOPSIS: For a given root subsystem S, the field ss_data.levi_idx holds the index (see levi_data.idx of the Levi subsystem
that is W-conjugate to S, or if there is no such Levi subsystem ss_data.levi_idx holds the value 0.
FIELD: ss_data.psilevi_idx
SYNOPSIS: For a given root subsystem S, the field ss_data.levi_idx holds the index (see psilevi_data.idx of the pseudo-Levi subsystem
that is W-conjugate to S, or if there is no such pseudo-Levi subsystem ss_data.levi_idx holds the value 0.
FIELD: ss_data.closed_subsystem_yn
SYNOPSIS: A root subsystem S of a irreducible root system (associated to a simple Lie algebra) g is closed if the following condition holds:
if s,t are in S and (s+t) is in g, then (s+t) is in S
W-conjugacy classes of closed subsystems correspond to Ad(g) conjugacy classes of semisimple subalgebras of g.
The field ss_data.closed_subsystem_yn holds the value 1 or 0 depending on whether or not the corresponding subsystem is closed.
FIELD: ss_data.distinguished_subsets
SYNOPSIS: Let Gamma be a simple base for a root subsystem of an irreducible root system g. A subset gamma of Gamma is "distinguished"
subset if the following cardinality condition holds.
#{roots in Delta_gamma} + #{roots in Gamma} = #{number of positive roots in Delta_Gamma of the form mu + alpha where
mu is in Delta_gamma union {0} and alpha is in Gamma \ gamma}
(Here Delta_Gamma and Delta_gamma are the root systems generated by Gamma and gamma.) The W-conjugacy classes of pairs (Gamma,gamma)
with Gamma a subset of the simple roots and gamma is a distinguished subset of Gamma are in a natural (and constructible) one-to-one
correspondence with nilpotent orbit in the Lie algebra corresponding to g. The W-conjugacy classes of pairs (Gamma,gamma), where Gamma
is a subset of the extended simple roots (the simple roots with the lowest root adjoined) and gamma is a distinguished subset of Gamma,
are in a one-to-one correspondence with the conjugacy classes of components groups A(O) attached to nilpotent orbits of g.
For a given root subsystem with base B = ss_data.ibase, the field ss_data.distinguished subsets gives a list of the subsets of
B that are distinguished in B.
REFERENCES:
E. Sommers, "A generalization of the Bala-Carter Theorem for Nilpotent Orbits", Int. Math. Res. Notices 1998, no. 11, 539-562.
FIELD: ss_data.bc_g_orbits
SYNOPSIS: Suppose Gamma is a simple base for a root subsystem of the root system of a simple Lie algebra g. Let gamma be a
distinguished subset of Gamma (see ss_data.distinguished_subsets). If Gamma is
a closed subsystem, then the Lie algebras g', g" generated by, respectively, the root vectors corresponding to Gamma, gamma
will be a semisimple subalgebras of g, and the Ad(g)-saturation of the Richardson orbit of g' corresponding to the subalgebra
g" (a Levi subalgebra of g') will be a particular nilpotent orbit of g. We refer to this orbit as the bc_orbit attached to the
pair (Gamma,gamma).
For a closed subsystem with base Gamma, the field ss_data.bc_g_orbits holds the indexes (see orbit_data.idx of the nilpotent
for the g-orbits attached to its distinguished subsets (ordered in a way that corresponds to the ordering of distiguished subsets in
ss_data.distinguished_subsets).
For a non-closed subsystem (for which the above construction of a nilpotent orbit is not applicable), the field ss_data.bc.g_orbits simply holds
the value [].
REFERENCES:
B. Binegar, "Subsystems, Nilpotent Orbits and Weyl Group Representations", OSU Lie Groups Seminar, 11/18/2009.
FIELD: ss_data.bv_wreps
SYNOPSIS: Barbasch and Vogan have defined a duality map that attaches to any nilpotent orbit in a semisimple Lie algebra g a special nilpotent
orbit the Lie algebra dual to g. If the Springer correspondence is used to attach a special representation of W to the image orbit and then
to pull back this special representation to a special orbit in g, then the Barbasch-Vogan map agrees with the usual Spaltenstein duality map.
If one begins with a distinguished pair (Gamma,gamma) where Gamma is a base for a root subsystem and gamma is a distinguished subset of gamma
(see ss_data.distinguished_subsets), then an analogous (but more general) map can be formulated that
sends (Gamma,gamma) to a particular irreducible representation of W. This map goes as follows
Psi : (Gamma,gamma) --> TrInd_{W_Gamma^}^{W}( sgn(W_Gamma} x TrInd_{W_gamma}^{W_Gamma} (sgn(W_gamma)) )
where TrInd is truncated induction, W_Gamma^ is the subgroup of W generated by the base dual to W_Gamma (corresponding to an interchange of
long and short roots). When Gamma is a subset of the simple roots, then the pair (Gamma,gamma) will map to a W-representation that is the
special representation whose corresponding special nilpotent orbit is the Barbasch-Vogan (or Spaltenstein) dual of the nilpotent orbit
with combinatorial Bala-Carter parameters (see orbit_data.cbc_parameters) (Gamma,gamma). In fact, when the domain of Psi is restricted to the combinatorial
Bala-Carter parameters of nilpotent orbits, the image is the set of special representations of W. Moreover, when the domain of Psi is restricted to
the set of distinguished pairs (Gamma,gamma) where Gamma is a proper subset of the extended simple roots (the simple roots of g plus the lowest
root), then the image of Psi is the set of orbit (Springer) representations.
For a given base Gamma of a root subsystem of g, the field ss_data.bv_wreps is a list of the indices (see "wrep_data.cw_idx) of
the W-representations obtained when the map Psi is applied to (Gamma,gamma), gamma being a distinguished subset of Gamma. The order in which these
W-representations are listed correlates to the order of the distinguished subsets listed in ss_data.distinguished_subsets).
REFERENCES:
D. Barbasch and D. Vogan, Jr. "Unipotent representations of complex semisimple groups", Ann. Math. 121 (1985), 41-110.
B. Binegar, "Subsystems, Nilpotent Orbits and Weyl Group Representations", OSU Lie Groups Seminar, 11/18/2009.
FIELD: ss_data.admissible_diagrams
SYNOPSIS: Admissible diagrams are a combinatorial construct due to R. Carter used to parameterize the conjugacy classes in Weyl group. See
admissible_diagrams.
For a given root subsystem Gamma, the field ss_data.admissible_diagrams provides a list of the admissible diagrams for Gamma obtained by
collecting together the admissible diagrams of its irreducible components.
REFERENCES:
R. Carter, "Conjugacy classes in the Weyl group", Comp. Math. 25 (1972), 1-59.
FIELD: ss_data.res_matrix
SYNOPSIS:
Let g be a simple Lie algebra with root system R. If Gamma is a base for a closed root subsystem of F, then it will correspond to
a regular subalgebra g' of g (generated by the root vectors corresponding to Gamma and -Gamma). The restriction matrix for this subalgebra
enables one to compute how a given finite-dimensional representation of g decomposes, upon restriction, as a representation of g'.
If a subsystem Gamma is closed, and g' is the corresponding subalgebra, the field ss_data.res_matrix contains an expression of the form
m_1*X[w_1] + .... + m_k*X[w_k]
in which a particular term m_i*X[w_i] indicates that when the adjoint representation of g is restricted to g' a representation of g' with
highest weight w_i (expressed in terms of its components with respect to a set of fundamental weights ordered as are the simple roots of g'
expressed by Gamma) with multiplicity m_i.
If a subsystem Gamma is not closed, then the field ss_data.res_matrix will contain an expression like the above, except that some of
the coefficients m_i will be negative (and so ss_data.res_matrix will look like a virtual representation of g', except there will be no
such regular subalgebra of g).
REFERENCES:
M.A.A. van Leeuwen, A.M. Cohen and B. Lisser, "The LiE Manual", http://www-math.univ-poitiers.fr/~maavl/LiEman/manual.pdf.
FIELD: ss_data.adjoint_rep
SYNOPSIS: Suppose Gamma is a base for subsystem of the root system R corresponding to a simple Lie algebra g. Suppose Gamma is split
into subsets such that each subset is a base for an irreducible component of the root subsystem generated by Gamma. (This is done
in the field ss_data.split_base). Then one can can express the adjoint representation of the Lie
algebra corresponding to R (which actually corresponds to a subalgebra of g only when Gamma corresponds to a closed subsystem) as the
sum of the adjoint representations of the individual components. Knowledge of this "adjoint representation" is useful in making sense
of the field
FIELD: ss_data.lie_type
SYNOPSIS: For a root system Gamma of a simple Lie algebra g, the field ss_data.split_base) holds a list of
subsets of Gamma that constitute bases for the irreducible components of the root system generated by Gamma. The field ss_data.lie_type is
a concatentation of the Cartan types of these irreducible components, reflecting the way such a Lie algebra or root system would be declared
in the software package LiE.
REFERENCES:
M.A.A. van Leeuwen, A.M. Cohen and B. Lisser, "The LiE Manual", http://www-math.univ-poitiers.fr/~maavl/LiEman/manual.pdf.
FIELD: ss_data.split_base
SYNOPSIS: For a root system Gamma of a simple Lie algebra g, the field ss_data.split_base holds a list of
subsets of ss_data.ebase that constitute bases for the irreducible components of the root system generated by Gamma.
The field ss_data.lie_type is a concatentation of the Cartan types of these irreducible components, reflecting
the way such a Lie algebra. Thus, for example, if
ss_data.ebase = [e1-e2,e2-e3,e4-e5,e5]
then
ss_data.split_base - [[e1-e2,e2-e3],[e4-e5,e5]] .
FIELD: levi_data.id
SYNOPSIS: For a given Levi subsystem of the root system of a simple Lie algebra g, the field levi_data.id holds a primary key labeling
the corresponding row in the table levi_data. This id is constructed by concatenating the name of the Lie algebra with the string "levi" and
the value of the field
FIELD: levi_data.g
SYNOPSIS: For a given (conjugacy class of) Levi subsystems, the field Levi_data.g holds the Cartan type of its parent (irreducible) root system.
FIELD: levi_data.levi_idx
SYNOPSIS: For a given Levi subsystem, the field levi_data.idx holds an integer which provides a unique label to the
various (W-conjugacy classes of) Levi subsystems for a given irreducible root system. These integers labels were chosen
so that an ordering of Levi subsystems by their rank and cardinality would be consistent with the natural ordering of
the corresponding integers.
FIELD: levi_data.ibase
SYNOPSIS: If S is a Levi subsystem of an irreducible root system g, it will be W-conjugate to the root system generated by
a subset of the simple roots of g (with respect to some fixed positive system).
For a given W-conjugacy class of Levi subsystems, the field levi_data.ibase holds the indices of these simple roots (these
indices are just integers indicating the position of a simple root in the standard Bourbaki ordering of the simple roots of g).
FIELD: levi_data.ebase
SYNOPSIS: If S is a Levi subsystem of an irreducible root system g, it will be W-conjugate to a root system generated by
a simple base consisting of a subset of the simple roots of g.
For a given W-conjugacy class of Levi subsystems, the field levi_data.ibase holds a list of positive roots (presented as linear
combinations of the standard Euclidean basis vectors e1,e2,...) that provide a simple base for a Levi subsystem
in this W-conjugacy class.
FIELD: levi_data.dim
SYNOPSIS: For a given W-conjugacy class S of Levi subsystems, the field levi_data.dim holds the dimension of the corresponding Levi
subalgebra.
FIELD: levi_data.cartan_type
SYNOPSIS: For a given W-conjugacy class S of Levi subsystems, the field levi_data.cartan_type holds the Cartan type of the corresponding Levi
subalgebra.
FIELD: levi_data.coxeter_conj_class
SYNOPSIS: Suppose B' and B" are bases for two root subsystems of an irreducible root system g. Let c' and c" be Coxeter elements
of corresponding to B' and B" in their respective Weyl groups (thus, c' is the product of the simple reflections corresponding to
B' and c" is the product the simple reflections corresponding to B"). Then the root subsystems generated by B' and B" are W-conjugate
(W being the Weyl group of g) if and only if c' and c" are conjugate in W. Thus, the W-conjugacy classes of Coxeter elements of
subsystems provide a simple means to tell whether or not two subsystems are W-conjugate (and so for most practical purposes
equivalent).
For a given Levi subsystem S, the field levi_data.coxeter_conj_class holds the index (see wcc_data.idx) of the
particular conjugacy class in W to which any Coxeter element of the Weyl subgroup generated by S belongs.
FIELD: levi_data.macdonald_rep
SYNOPSIS: Let g be a simple Lie algebra, R its root system, and W its Weyl group. Lie R' be a root subsystem of R and let p_W'
be the product of the positive roots in R' regarded as a homogeneous function on on the underlying vector space V of R. Under
the natural action of W on functions on V, the span of the polynomials w(p_W'), w in W, carries is an irreducible W-module.
The corresponding irreducible representation is the Macdonald representation corresponding to W' (or R').
Alternatively, let W' be the reflection subgroup of a Weyl group W generated by the reflections corresponding to a base of a root
subsystem of R' of g, and let sgn(W') be the sign representation of W'. The Macdonald representation corresponding to R' is the
unique summand of the induced representation
Ind_{W'}^{W)} ( sgn(W') )
of lowest degree (with the "degree" of an irreducible representation meaning the degree of the lowest order term of its fake degree
polynomial).
For a given Levi subsystem R' of an irreducible root system R, the field "levi_data.macdonald_rep" holds the index (see, wrep_data.idx) of the
irreducible representation of W corresponding to the Macdonald representation attached to R'.
REFERENCES:
I.G. Macdonald, "Some irreducible representations of Weyl groups", Bull. London Math. Soc. 4 (1972), 148-150.
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
FIELD: levi_data.distinguished_subsets
SYNOPSIS: Let Gamma be a simple base for a root subsystem of an irreducible root system g. A subset gamma of Gamma is "distinguished"
subset if the following cardinality condition holds.
#{roots in Delta_gamma} + #{roots in Gamma} = #{number of positive roots in Delta_Gamma of the form mu + alpha where
mu is in Delta_gamma union {0} and alpha is in Gamma \ gamma}
(Here Delta_Gamma and Delta_gamma are the root systems generated by Gamma and gamma.) The W-conjugacy classes of pairs (Gamma,gamma)
with Gamma a subset of the simple roots and gamma is a distinguished subset of Gamma are in a natural (and constructible) one-to-one
correspondence with nilpotent orbit in the Lie algebra corresponding to g. The W-conjugacy classes of pairs (Gamma,gamma), where Gamma
is a subset of the extended simple roots (the simple roots with the lowest root adjoined) and gamma is a distinguished subset of Gamma,
are in a one-to-one correspondence with the conjugacy classes of components groups A(O) attached to nilpotent orbits of g.
For a given Levi subsystem with base B = levi_data.ibase, the field levi_data.distinguished subsets gives a list of the subsets of
B that are distinguished in B.
REFERENCES:
P. Bala and R. Carter, "Classes of unipotent elements in simple algebraic groups, I", Math. Proc. Camb. Phil. Soc. 79 (1976), 401-425.
P. Bala and R. Carter, "Classes of unipotent elements in simple algebraic groups, II", Math. Proc. Camb. Phil. Soc. 80 (1976), 1-18.
D. Collingwood and W. McGovern, "Nilpotent Orbits in Semisimple Lie Algebras", Van Nostrand Reinhold, New York, 1993.
B. Binegar, "Subsystems, Nilpotent Orbits and Weyl Group Representations", OSU Lie Groups Seminar, 11/18/2009.
FIELD: levi_data.bc_g_orbits
SYNOPSIS: Suppose Gamma is a simple base for a root subsystem of the root system of a simple Lie algebra g.
Let gamma be a distinguished subset of Gamma (see levi_data.distinguished_subsets). If Gamma is a closed subsystem, then
the Lie algebras g', g" generated by, respectively, the root vectors corresponding to Gamma, gamma will be a semisimple subalgebras
of g, and the Ad(g)-saturation of the Richardson orbit of g' corresponding to the subalgebra g" (a Levi subalgebra of g') will be a
particular nilpotent orbit of g. We refer to this orbit as the bc_orbit attached to the pair (Gamma,gamma).
For a Levi subsystem with base Gamma, the field levi_data.bc_g_orbits holds the indexes (see orbit_data.idx of the nilpotent
Ad(g)-orbits attached to its distinguished subsets (ordered in a way that corresponds to the ordering of distiguished subsets in
levi_data.distinguished_subsets).
REFERENCES:
P. Bala and R. Carter, "Classes of unipotent elements in simple algebraic groups, I", Math. Proc. Camb. Phil. Soc. 79 (1976), 401-425.
P. Bala and R. Carter, "Classes of unipotent elements in simple algebraic groups, II", Math. Proc. Camb. Phil. Soc. 80 (1976), 1-18.
D. Collingwood and W. McGovern, "Nilpotent Orbits in Semisimple Lie Algebras", Van Nostrand Reinhold, New York, 1993.
B. Binegar, "Subsystems, Nilpotent Orbits and Weyl Group Representations", OSU Lie Groups Seminar, 11/18/2009.
FIELD: levi_data.richardson_orbit
SYNOPSIS: Let Gamma be a base for a Levi subsystem of a semisimple Lie algebra g, and let l be the corresponding Levi subalgebra.
The nilpotent orbit of g induced from the 0-orbit of l is called the Richardson orbit attached to l (or Gamma).
The field levi_data.richardson_orbit holds the orbit_data.cw_idx)
REFERENCES:
R.W. Richardson, "Conjugacy classes in parabolic subgroups of semisimple algebraic groups", Bull. London Math. Soc. 6 (1974), 21-24.
D. Collingwood and W. McGovern, "Nilpotent Orbits in Semisimple Lie Algebras", Van Nostrand Reinhold, New York, 1993.
FIELD: levi_data.positive_roots
SYNOPSIS: In the UMRK database, representative bases of (conjugacy classes of) Levi subsystems have been chosen in such a way that the
corresponding sets of positive roots (for the Levi subsystems) are also positive roots for the parent simple Lie algebra g.
For a given Levi subsystem l, the field levi_data.positive_roots holds the indices of the positive roots of l as they appear in the list
of positive roots of g (see g_data.pos_roots).
FIELD: psilevi_data.id
SYNOPSIS: Let g be a simple Lie algebra and R its root system. A standard pseudo-Levi subsystem is a root subsystem of R that is generated
by a subset of the extended simple roots (the set of simple roots together with the lowest root). More generally, a pseudo-Levi subsystem
is a subsystem that is W-conjugate to a standard pseudo-Levi subsystem. Such conjugacy classes are in a one-to-one correspondence with
Ad(g) conjugacy classes of pseudo-Levi subalgebras of g. The table psilevi_data holds data for each conjugacy class of pseudo-Levi
subsystem.
For a given pseudo-Levi subsystem of a simple Lie algebra g, the field psileiv_data.id holds a primary key labeling the corresponding row
in the table psilevi_data. This id is constructed by concatenating the name of the Lie algebra g with the string "psilevi" and
the value of the field
FIELD: psilevi_data.g
SYNOPSIS: For a given (conjugacy class of) pseudo-Levi subsystem(s), the field psilevi_data.g holds the Cartan type of its parent (irreducible) root system.
FIELD: psilevi_data.levi_idx
SYNOPSIS: For a given pseudo-Levi subsystem, the field psilevi_data.idx holds an integer which provides a unique label to the
various (W-conjugacy classes of) pseudo-Levi subsystems for a given irreducible root system. These integers labels were chosen
so that an ordering of pseudo-Levi subsystems by their rank and cardinality would be consistent with the natural ordering of
the corresponding integers.
FIELD: psilevi_data.ss_idx
SYNOPSIS: For a given (conjugacy class of) pseudo-Levi subalgebra(s), the field psilevi_data.ss_idx holds an integer that labels the
corresponding pseudo-Levi subsystem in the table ss_data. (See ss_data.idx.)
FIELD: psilevi_data.ibase
SYNOPSIS: If S is a Levi subsystem of an irreducible root system g, it will be W-conjugate to the root system generated by
a proper subset of the extended simple roots of g (the set of simple roots of g together with the lowest root).
For a given W-conjugacy class of Levi subsystems, the field levi_data.ibase holds the indices of these "simple" roots (these
indices are just integers in 0,1,...,Rank(g) indicating either the position of a simple root in the Bourbaki ordering of the simple roots of g occurring the field
g_data) or, if 0, the indicating the lowest root.
FIELD: psilevi_data.ebase
SYNOPSIS: If S is a psi-Levi subsystem of an irreducible root system g, it will be W-conjugate to the roots system generated by
a simple base consisting of a proper subset of the extended simple roots of g.
For a given W-conjugacy class of pseudo-Levi subsystems, the field psilevi_data.ibase holds a list of positive roots (presented as linear
combinations of the standard Euclidean basis vectors e1,e2,...) that provide a simple base for a pseudo-Levi subsystem
in this W-conjugacy class.
FIELD: psilevi_data.dim
SYNOPSIS: For a given W-conjugacy class S of pseudo-Levi subsystems, the field psilevi_data.dim holds the dimension of the corresponding
pseudo-Levi subalgebra.
FIELD: psilevi_data.cartan_type
SYNOPSIS: For a given W-conjugacy class S of pseudo-Levi subsystems, the field psilevi_data.dim holds the dimension of the corresponding pseudo-Levi
subalgebra.
FIELD: psilevi_data.coxeter_conj_class
SYNOPSIS: Suppose B' and B" are bases for two root subsystems of an irreducible root system g. Let c' and c" be Coxeter elements
of corresponding to B' and B" in their respective Weyl groups (thus, c' is the product of the simple reflections corresponding to
B' and c" is the product the simple reflections corresponding to B"). Then the root subsystems generated by B' and B" are W-conjugate
(W being the Weyl group of g) if and only if c' and c" are conjugate in W. Thus, the W-conjugacy classes of Coxeter elements of
subsystems provide a simple means to tell whether or not two subsystems are W-conjugate (and so for most practical purposes
equivalent).
For a given pseudo-Levi subsystem S, the field psilevi_data.coxeter_conj_class holds the index (see wcc_data.idx) of the
particular conjugacy class in W to which any Coxeter element of the Weyl subgroup generated by S belongs.
FIELD: psilevi_data.macdonald_rep
SYNOPSIS: Let g be a simple Lie algebra, R its root system, and W its Weyl group. Lie R' be a root subsystem of R and let p_W'
be the product of the positive roots in R' regarded as a homogeneous function on on the underlying vector space V of R. Under
the natural action of W on functions on V, the span of the polynomials w(p_W'), w in W, carries is an irreducible W-module.
The corresponding irreducible representation is the Macdonald representation corresponding to W' (or R').
Alternatively, let W' be the reflection subgroup of a Weyl group W generated by the reflections corresponding to a base of a root
subsystem of R' of g, and let sgn(W') be the sign representation of W'. The Macdonald representation corresponding to R' is the
unique summand of the induced representation
Ind_{W'}^{W)} ( sgn(W') )
of lowest degree (with the "degree" of an irreducible representation meaning the degree of the lowest order term of its fake degree
polynomial).
For a given pseudo-Levi subsystem R' of an irreducible root system R, the field "psilevi_data.macdonald_rep" holds the index
(see, wrep_data.idx) of the irreducible representation of W corresponding to the Macdonald representation attached to R'.
REFERENCES:
I.G. Macdonald, "Some irreducible representations of Weyl groups", Bull. London Math. Soc. 4 (1972), 148-150.
R. Carter, "Finite Groups of Lie Type", John Wiley and Sons, New York, 1985.
FIELD: psilevi_data.distinguished_subsets
SYNOPSIS: Let Gamma be a simple base for a root subsystem of an irreducible root system g. A subset gamma of Gamma is "distinguished"
subset if the following cardinality condition holds.
#{roots in Delta_gamma} + #{roots in Gamma} = #{number of positive roots in Delta_Gamma of the form mu + alpha where
mu is in Delta_gamma union {0} and alpha is in Gamma \ gamma}
(Here Delta_Gamma and Delta_gamma are the root systems generated by Gamma and gamma.) The W-conjugacy classes of pairs (Gamma,gamma)
with Gamma a subset of the simple roots and gamma is a distinguished subset of Gamma are in a natural (and constructible) one-to-one
correspondence with nilpotent orbit in the Lie algebra corresponding to g. The W-conjugacy classes of pairs (Gamma,gamma), where Gamma
is a subset of the extended simple roots (the simple roots with the lowest root adjoined) and gamma is a distinguished subset of Gamma,
are in a one-to-one correspondence with the conjugacy classes of components groups A(O) attached to nilpotent orbits of g.
For a given root pseudo-Levi subsystem with base B = ss_data.ibase, the field psilevi_data.distinguished subsets gives a list of the subsets of
B that are distinguished in B.
REFERENCE:
E. Sommers, "A generalization of the Bala-Carter Theorem for Nilpotent Orbits", Int. Math. Res. Notices 1998, no. 11, 539-562.
B. Binegar, "Subsystems, Nilpotent Orbits and Weyl Group Representations", OSU Lie Groups Seminar, 11/18/2009.
FIELD: psilevi_data.bc_g_orbits
SYNOPSIS: Suppose Gamma is a simple base for a root subsystem of the root system of a simple Lie algebra g. Let gamma be a
distinguished subset of Gamma (see psilevi_data.distinguished_subsets). If Gamma is
a closed subsystem, then the Lie algebras g', g" generated by, respectively, the root vectors corresponding to Gamma, gamma
will be a semisimple subalgebras of g, and the Ad(g)-saturation of the Richardson orbit of g' corresponding to the subalgebra
g" (a Levi subalgebra of g') will be a particular nilpotent orbit of g. We refer to this orbit as the bc_orbit attached to the
pair (Gamma,gamma).
For a pseudo-Levi subsystem with base Gamma, the field psilevi_data.bc_g_orbits holds the indexes (see orbit_data.idx of the nilpotent
for the g-orbits attached to its distinguished subsets (ordered in a way that corresponds to the ordering of distiguished subsets in
psilevi_data.distinguished_subsets).
REFERENCES:
E. Sommers, "A generalization of the Bala-Carter Theorem for Nilpotent Orbits", Int. Math. Res. Notices 1998, no. 11, 539-562.
B. Binegar, "Subsystems, Nilpotent Orbits and Weyl Group Representations", OSU Lie Groups Seminar, 11/18/2009.
FIELD: psilevi_data.d
SYNOPSIS: In his paper on a generalized Bala-Carter theorem for nilpotent orbits, E. Sommers attaches to any proper subset J of the extended simple roots
a certain integer d_J defined as follows. Let c_i , i=1,...,Rank(g) be the coefficient of the i^th simple root when the highest root of g is written as
a linear combination of the simple roots. The integer d_j is defined to be the greatest common divisor of those c_i for which the corresponding simple root
does not belong to J.
REFERENCE:
E. Sommers, "A generalization of the Bala-Carter Theorem for Nilpotent Orbits", Int. Math. Res. Notices 1998, no. 11, 539-562.
FIELD: realform_data.id
SYNOPSIS: For a given real form of a simple Lie algebra g, the field realform_data.id holds a primary key
that identifies the real form uniquely (amongst all real forms in the table realform_data). This key is
formed by concatenating the Cartan type of g with the string "rf" and the integer realform_data.idx
which corresponds to the way the Atlas program orders the real forms of g. (The compact real form is always
indicated by the integer 1, subsequent "less compact" real forms by increasing integers until one arrives at the most noncompact,
split, real form.)
REFERENCE:
Atlas for Lie Groups Homepage, www.liegroups.org
FIELD: realform_data.idx
SYNOPSIS: Each real form of a simple Lie algebra is assigned an integer in such a
way that the integer corresponding to the compact real form is 1, and subsequent less and less compact
real forms by the integers 2, 3, ... until one reaches the most compact, split real form. For a given
real form this integer index is held in the fields realform_data.idx.
FIELD: realform_data.g
SYNOPSIS: For a given real form of a complex group g, the field realform_data.g holds the Cartan type of g.
FIELD: realform_data.g_R
SYNOPSIS: For a given real form of a simple complex Lie algebra g, the field realform_data.g_R holds its
common name. When the underlying Lie algebra g is of classical type, these common names typically refer to a
standard matrix realization of the real form. For example, the "common names" of the real forms of the simple
complex Lie algebra of type A3 are
A3rf1 <-> su(4)
A3rf2 <-> su(1,3)
A3rf3 <-> su(2,2)
A3rf4 <-> su(2,H)
A3rf5 <-> sl(4,R)
When G = G2, F4, E6, E7, or E8, we use the following conventions: The compact real form will be denoted by
the lowercase version of its Cartan types, thus the common names of the compact real forms will be g2, f4,
e6, e7, and e8. The other real forms will be indicated as follows. Suppose g = k + p is the Cartan decomposition
of a real form and let d := dim(p) - dim(k). The "common name" for this real form will then be G(d), where G is the
Cartan type of g.
REFERENCES:
S. Helgason, "Differential Geometry and Symmetric Spaces", Academic Press, New York, 1962.
FIELD: realform_data.K
SYNOPSIS: For a given real form gR of a simple complex Lie algebra g, the field realform_data.K holds the Cartan type
of any maximal compact subalgebra of gR.
FIELD: K_R
SYNOPSIS: For a given real form, the field realform_data.K_R holds the common name (see realform_data.g_R) of
any maximal compact subalgebra.
FIELD: realform_data.Krep_p
SYNOPSIS: For a real form with Cartan decomposition g = k + p, the field realform_data.Krep_p describes the representation of k carried by
p. The following notation is used, since k is always reductive, its irreducible finite-dimensional representations can be prescribed in terms
of their highest weights, and these highest weights can, in turn, be prescribed by their components with respect to the basis of fundamental
weights with the ordering following that of the simple roots. Of course, if k has a non-trivial center, then a weight with respect to that center
must also be prescribed. The following examples should make our notation clear.
The compact part k of the real form E6(-14) of E6 has Cartan type D5T1 =~ D5 x T1 =~ so(10) + R . The representation of k carried by the
noncompact part p is
X[0,0,0,1,0,1] + X[0,0,0,0,1,-1]
This means that this representation is the sum of two irreducible finite-dimensional representations of k; one with highest weight [0,0,0,1,0] on
the so(10) factor and weight 1 on the toroidal factor, and another with highest weight [0,0,0,0,1] on the so(10) factor and weight -1 on the
toroidal factor. Note that when k is reductive, one can refer to realform_data.K in order to properly identify the
the highest weights with respect to the simple or toroidal factors of k. Thus, a highest weight for a realform_data.K = A1A1D4 would be
the concatenation of a integer specifying a highest weight for the first A1 factor, with an integer specifying a highest weight for the
second A1 factor, and a quadruple of integers specifying a weight for the D4 factor.
FIELD: realform_data.dim_p
SYNOPSIS: For a given real form g_R, the field realform_data.dim_p holds the dimension of the noncompact part of g_R.
FIELD: realform_data.Krep_k
SYNOPSIS: For a real form with Cartan decomposition g = k + p, the field realform_data.Krep_k describes the representation of k carried by
k, (which, of course, is just the adjoint representation of k). See realform_data.Krep_p for a
description of the notation used to prescribe representations of k.
FIELD: realform_data.dim_k
SYNOPSIS: For a given real form g_R, the field realform_data.dim_p holds the dimension of any maximal compact subalgebra of g_R.
FIELD: realform_data.vogan_diagram
SYNOPSIS: A Vogan diagram is a annotated Dynkin diagram used to prescribe a particular real form. This basis idea behind
this notation is as follows. Given a real form, one can choose a maximally compact Cartan subalgebra in such a way that the
Cartan involution theta either preserves the root space of a simple root or it permutes it with the root space of another simple root.
If theta acts by 1 on a the root space of simple root, then the corresponding node of the Dynkin diagram is given a label c, if instead
it acts by -1, then the corresponding node is given a label nc. If theta permutes two simple root spaces, then the corresponding nodes
of the Dynkin diagram are given a common label of the form C# where # is some integer used to distinguish this particular pair of simple
roots from similarly behaving pairs.
For a given real form, the field realform_data.vogan_diagram holds a list of the labels of the Vogan diagram corresponding to the real form,
the individual labels (c,nc,C1,C2, etc.) are ordered in a way that follows the standard Bourbaki ordering of the simple roots.
REFERENCE:
A. Knapp, "Lie groups: beyond an introduction", Progress in Mathematics 140 (2nd ed.), Boston, Birkhauser Boston, 2002.
FIELD: realform_data.satake_diagram
SYNOPSIS: A Satake diagram in an annotated Dynkin diagram used to prescribe a particular real form. These diagrams were born out of
a method of classification of real forms developed by S. Araki that utilizes a maximally split Cartan subalgebra. (Vogan diagrams
use a maximally compact Cartan subalgebra).
For a given real form of a semisimple Lie algebra g, a Satake diagram is a labeling of the nodes of the Dynkin diagram by black dots
and open circles and possible some arcs drawn between pairs of open circles. For a given real form, the field realform_data.satake_diagram
is an ordered list of labels in {b,w,w1,w2,...}. The ordering of these labels correlates to the ordering of the simple roots in the
standard Bourbaki ordering. A label b (resp., w) in the i^th place indicates the i^th simple root is represented as a black node
(resp., open circle) in the corresponding Satake diagram. The labels w#, # some integer, will occur in pairs and these will correspond to
open circles connected by arcs in the actual Satake diagram.
REFERENCES:
S. Araki, "On root systems and an infinitesimal classification of irreducible symmetric spaces", J. Math. Osaka City Univ. 13 (1962), 1-34.
S. Helgason, "Differential Geometry, Lie Groups, and Symmetric Spaces", Academic Press, New York, 1978.
FIELD: realform_data.rel_root_sys
SYNOPSIS: If a is a maximal noncompact abelian subalgebra of g, then weights of g with respect to a will form a root system.
For a given real form, the field realform_data.rel_root_sys will hold the Cartan type of this root system.
FIELD: realform_data.real_rank
SYNOPSIS: For a given real form g_R with Cartan decomposition g_R = k_R + p_R, the field realform_data.real_rank will hold the
dimension of any maximal abelian subalgebra of p_R.
FIELD: realform_data.inner_class
SYNOPSIS: If the Vogan diagram (see realform_data.vogan_diagram) of a real form
contains pairs of nodes labeled by C1,C2 etc, then the corresponding Cartan involution "comes from" a non-trivial outer involution
of the Lie algebra. If this is the case, the field realform_data.inner_class will hold the value 2. If, on the other hand, the
Cartan involution corresponds to an inner automorphism, then the field realform_data.inner_class will hold the value 1.
FIELD: realform_data.compact_yn
SYNOPSIS: The field realform_data.compact_yn will hold the value 1 if the corresponding real form is compact, and 0 otherwise.
FIELD: realform_data.split_yn
SYNOPSIS: The field realform_data.split_yn will hold the value 1 if the corresponding real form is split, and 0 otherwise.
FIELD: realform_data.quasisplit_yn
SYNOPSIS: The field realform_data.compact_yn will hold the value 1 if the corresponding real form is quasi-split, and 0 otherwise.
FIELD: realform_data.equal_rank_y
SYNOPSIS: The field realform_data.compact_yn will hold the value 1 if the corresponding real form is equal rank (meaning rank(k) =
real rank of g), and 0 otherwise.
FIELD: realform_data.complex_yn
SYNOPSIS: The field realform_data.complex_yn will hold the value 1 if the corresponding real form is of complex type, and 0 otherwise.
By complex type, we mean that the real form corresponds to the (real) Lie algebra obtained from a simple complex Lie algebra by restriction
of scalars.
FIELD: realform_data.hermitian_symmetric_yn
SYNOPSIS: The field realform_data.hermitian_symmetric_yn will hold the value 1 if the corresponding real form is of hermitian symmetric type,
and 0 otherwise. Real forms of hermitians symmetric type may be characterized by the fact that their maximal compact subalgebras contain a
central torus.
FIELD: realform_data.str_orth_seq
SYNOPSIS: Let g = k + p be a Cartan decomposition of a complex semisimple Lie algebra g corresponding to a particular real form. Choose
a Cartan subalgebra h = t + a of g compatible with the Cartan decomposition of g and such t is a Cartan subalgebra of k. Let Delta(p,t) be the
weights with respect to t of the representation of k carried by the noncompact part p. Two weights w1,w2 in Delta(p,t) are said to be
strongly orthogonal if neither w1+w2 nor w1-w2 are roots of k with respect to t. (This implies, in particular, that elements of the corresponding
weight spaces commute.) A gamma_1, gamma_2 , .... , gamma_k be a sequence of elements of
Delta(p,t) such that
- gamma_1 is the highest root in Delta(p,t)
- each partial sum gamma_1 + gamma_2 + ... + gamma_i (1 <= i <= k ) is a dominant weight (has non-negative inner product each simple root of k)
- each distinct pair gamma_i , gamma_j in the sequence are strongly orthogonal.
Such a sequence of p-weights is called a strongly orthogonal sequence of noncompact weights. Such sequences are useful in the study of the K_C
orbits in the nilpotent cone in p (they correspond to spherical nilpotent orbits).
The field realform_data.str_orth_seqs holds the possible (maximal) sequences of strongly orthogonal noncompact weights for the corresponding
real form. The individual weights are expressed in terms of their coordinates with respect to the fundamental weights of k. (Proper interpretation
of these coordinates will require knowing the ordering of the fundamental weights - but this can be inferred from the ordering of the simple
factors held in the realform_data.K).
REFERENCES:
B. Binegar, "On a class of multiplicity-free nilpotent K_C-orbits", J. Math. Kyoto Univ. 47, 735-766 (2007).
FIELD: realform_data.seqs_of_prim_Ktypes
SYNOPSIS: Let gamma_1, gamma_2 , ... , gamma_k be a strongly orthogonal sequence of noncompact weights for a real form g_R of a complex
simple Lie algebra g. (See realform_data.str_orth_seqs.) Each subsequence gamma_1 , .... , gamma_i
(i <= k) will correspond to a particular spherical nilpotent O_i orbit in p (the -1 eigenspace of the Cartan involution of g). Moreover, the
partial sums
omega_1 = gamma_1
omega_2 = gamma_1 + gamma_2
:
omega_i = gamma_1 + gamma_2 + ... + gamma_i
will be dominant weights that provide a set of generators for the monoid of K-types that occur in the regular functions on the orbit O_i.
For a given strongly orthogonal sequence of noncompact weights, the sequence omega_1, ... , omega_k is called the corresponding sequence
of primitive K-types.
For a given real form, the field realform_data.seqs_of_prim_Ktypes holds the sequences of primitive K-types corresponding to the sequences
of strongly orthogonal noncompact weights held in the field realform_data.str_orth_seqs.
REFERENCES:
B. Binegar, "On a class of multiplicity-free nilpotent K_C-orbits", J. Math. Kyoto Univ. 47, 735-766 (2007).
FIELD: realform_data.res_matrix
SYNOPSIS: Suppose k is a reductive subalgebra of a semisimple Lie algebra g. Choose a Cartan subalgebra t of k and extend it to a Cartan
subalgebra h of g. Then there will be an associated restriction matrix M that provides a simple means for converting weights (with respect
to h) of a finite dimensional representation of g to weights (with respect to t) of finite dimensional representations of t:
t-weight = M x h-weight
The t-weight so obtained will be precisely the restriction of the h-weight to the subalgebra t of h. Thus, a restriction matrix allows one
to determine from the weights and multiplicities of a representation of g the set of weights and multipicities of its restriction to k.
This latter data together with e.g., the Kostant multiplicity formula allows for a straightforward decomposition of the restriction into a
sum of irreducible k-modules.
For a given real form, the field realform_data.res_matrix gives an expression for restriction matrix M that can be utilized as follows. First of all,
the matrix itself is expressed as an ordered list of its row vectors. Secondly, to apply it one first expresses a weight in terms of its coordinates
with respect to a base of fundamental weights (ordered in standard Bourbaki order). After multiplying from the left by M, one will obtain a
vector that will list the coefficients of the restriction with respect to the fundamental weights of k (and/or characters of central torii).
FIELD: wcc_data.id
SYNOPSIS: For a given conjugacy class within a Weyl group W of a simple complex Lie algebra g, the field wcc_data.id will hold a primary key
which uniquely identifies the corresponding row in the table wcc_data. This key is constructed by simply concatenating the Cartan type of
g with the string "wcc" and an interger index (see wcc_data.cw_idx that distinguishes the various conjugacy
classes of W.
FIELD: wcc_data.cw_idx
SYNOPSIS: For a given conjugacy class within a Weyl group W of a simple complex Lie algebra g, the field wcc_data.cw_idx will hold an
integer used to distinguish this particular conjugacy class from other conjugacy classes of W. The actual value of cw_idx just
FIELD: wcc_data.class_rep
SYNOPSIS: For a given conjugacy class within a Weyl group W of a simple complex Lie algebra g, the field wcc_data.class_rep holds a
representive element w of conjugacy class. This element is expressed as a list of integers between 1 and Rank(g) which in turn provides
a prescription for constructing w as a (minimal) product of simple reflections (an integer i representing the simple reflection corresponding
to the i^th simple root in the standard Bourbaki ordering of simple roots).
FIELD: wcc_data.length
SYNOPSIS: The length of a Weyl group element w is the minimal number of factors needed to express w as a product of simple reflecttions. This
natural number also turns out to be the number of positive roots that get mapped to negative roots by the action of w. Within a conjugacy class
all elements have the same length.
For a given conjugacy class c, the field wcc_data.length holds the common length of its elements.
REFERENCES:
FIELD: wcc_data.admissible_diagrams
SYNOPSIS: The length of a Weyl group element w is the minimal number of simple reflections needed to realize w as a product of simple reflections.
The Weyl groups elements within a single conjugacy class all have a common length.
For a given conjugacy class c within a Weyl group W of a simple complex Lie algebra g, the field wcc_data.length will hold the common length of
the elements of c.
FIELD: wcc_data.reflection_word_length
SYNOPSIS: A reflection in a Weyl group is a Weyl group element that belongs to the conjugacy class of a reflection by a simple root;
or even more simply, a reflection is a Weyl group element corresponding to a reflection by a (not necessarily simple) root. Each Weyl
group element may be expressed a product of reflections. The reflection word length of an element w is the minimal number of reflections
needed to express w as a product of reflections. It is an invariant of the conjugacy class of w.
For a given conjugacy class c within a Weyl group W of a simple complex Lie algebra g, the field wcc_data.reflection_word_length holds the
common reflection word length of elements of c.
FIELD: wcc_data.class_size
SYNOPSIS: For a given conjugacy class c within a Weyl group W of a simple complex Lie algebra g, the field wcc_data.class_size will hold the
number of elements of c.
FIELD: wcc_data.elt_order
SYNOPSIS: The order of a Weyl group element w is the smallest integer n such that w^n = 1 (the identity element in W). Each element of
a conjugacy class has the same order.
For a given conjugacy class within a Weyl group W of a simple complex Lie algebra g, the field wcc_data.elt_order will hold the common
order of its elements.
FIELD: wcc_data.characteristic_poly_coeffs
SYNOPSIS: Each Weyl group element w acts on the reflection representation (the action of W on the vector space spanned by the simple
roots) and so corresponds to a certain Rank(g) by Rank(g) matrix M_w. The characteristic polynomial of w is the characteristic polynomial p_w
of this matrix
p_w(q) = det(1-q*M_w)
This polynomial is an invariant of the conjugacy class of w.
For a given conjugacy class c within a Weyl group W of a simple complex Lie algebra g, the field wcc_data.characteristic_poly_coeffs will hold
the coefficients of the powers of indeterminant q in the common characteristic polynomial of the elements of c.