This package contains functions for computing the decomposition matrices for Iwahori--Hecke algebras of the symmetric groups. As the (modular) representation theory of these algebras closely resembles that of the (modular) representation theory of the symmetric groups --- indeed, the later is a special case of the former --- many of the combinatorial tools from the representation theory of the symmetric group are included in the package.
These programs grew out of the attempts by Gordon James and myself [JM1] to understand the decomposition matrices of Hecke algebras of type A when <q>=-1. The package is now much more general and its highlights include:
The modular representation theory of Hecke algebras
The ``modular'' representation theory of the Iwahori--Hecke algebras of type A was pioneered by Dipper and James [DJ1,DJ2]; here we briefly outline the theory, referring the reader to the references for details. The definition of the Hecke algebra can be found in Chapter~Iwahori-Hecke algebras; see also Hecke.
Given a commutative integral domain R and a non--zero unit q in
R, let <H>=<H>_{<R>, <q>} be the Hecke algebra of the symmetric
group Sym_n on n symbols defined over R and with parameter
q. For each partition mu of n, Dipper and James defined a
Specht module S(mu). Let rad S(mu) be the radical of
S(mu) and define D(mu)=S(mu)/rad S(mu). When R
is a field, D(mu) is either zero or absolutely
irreducible. Henceforth, we will always assume that R is a field.
Given a non--negative integer i, let [i]_q=1+q+ldots+q^{i-1}. Define e to be the smallest non--negative integer such that [<e>]_q=0; if no such integer exists, we set e equal to 0. Many of the functions in this package depend upon e; the integer e is the Hecke algebras analogue of the characteristic of the field in the modular representation theory of finite groups.
A partition mu=(mu_1,mu_2,ldots) is e--singular if there
exists an integer i such that
mu_i=mu_{i+1}=cdots=mu_{i+<e>-1}>0; otherwise, mu is
e--regular. Dipper and James [DJ1] showed that
D(nu)ne(0) if and only if nu is e--regular and that
the D(nu) give a complete set of non--isomorphic irreducible
H--modules as nu runs over the e--regular partitions of n.
Further, S(mu) and S(nu) belong to the same block if and
only if mu and nu have the same e-core [DJ2,JM2]. Note
that these results depend only on e and not directly on R or q.
Given two partitions mu and nu, where nu is e--regular, let
d_{munu} be the composition multiplicity of D(nu) in
S(mu). The matrix D=(d_{munu}) is the decomposition
matrix of H. When the rows and columns are ordered in a way
compatible with dominance, D is lower unitriangular.
The indecomposable H-modules P(nu) are indexed by e-regular
partitions nu. By general arguments, P(nu) has the same
composition factors as sum_{mu} d_{munu} 'S'(<mu>) ; so these
linear combinations of modules become identified in the Grothendieck
ring of H. Similarly, 'D'(<nu>) = sum_{mu} d_{numu}^{-1}
'S'(<mu>) in the Grothendieck ring. These observations are the
basis for many of the computations in Specht.
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Two small examples
Because of the algorithm of [LLT], in principle, all of decomposition matrices for all Hecke algebras defined over fields of characteristic zero are known and available using Specht. The algorithm is recursive; however, it is quite quick and, as with a car, you need never look at the engine:
gap> H:=Specht(4); # e=4, 'R' a field of characteristic 0 Specht(e=4, S(), P(), D(), Pq()) gap> InducedModule(H.P(12,2)); P(13,2)+P(12,3)+P(12,2,1)+P(10,3,2)+P(9,6)
The [LLT] algorithm was applied 24 times during this calculation.
For Hecke algebras defined over fields of positive characteristic the major tool provided by Specht, apart from the decomposition matrices contained in the libraries, is a way of ``inducing'' decomposition matrices. This makes it fairly easy to calculate the associated decomposition matrices for ``small'' n. For example, the Specht libraries contain the decomposition matrices for the symmetric groups Sym_n over fields of characteristic 3 for n<15. These matrices were calculated by Specht using the following commands:
gap> H:=Specht(3,3); # e=3, 'R' field of characteristic 3 Specht(e=3, p=3, S(), P(), D()) gap> d:=DecompositionMatrix(H,5); # known for $n\<2e$ 5
|1 4,1
|. 1 3,2
|. 1 1 3,1^2
|. . . 1 2^2,1
|1 . . . 1 2,1^3
|. . . . 1 1^5
|. . 1 . . gap> for n in [6..14] do > d:=InducedDecompositionMatrix(d); SaveDecompositionMatrix(d); > od;
The function InducedDecompositionMatrix contains almost every trick
that I know for computing decomposition matrices. I would be very
happy to hear of any improvements.
Specht can also be used to calculate the decomposition numbers of the q--Schur algebras; although, as yet, here no additional routines for calculating the projective indecomposables indexed by e--singular partitions. Such routines will probably be included in a future release, together with the (conjectural) algorithm [LT] for computing the decomposition matrices of the q--Schur algebras over fields of characteristic zero.
In the next release of Specht, I will also include functions for
computing the decomposition matrices of Hecke algebras of type B,
and more generally those of the Ariki--Koike algebras. As with the
Hecke algebra of type A, there is an algorithm for computing the
decomposition matrices of these algebras when R is a field of
characteristic zero [M].
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Credits
I would like to thank Gordon James, Johannes Lipp, and Klaus Lux for their comments and suggestions.
If you find Specht useful please let me know. I would also appreciate hearing any suggestions, comments, or improvements. In addition, if Specht does play a significant role in your research, please send me a copy of the paper(s) and please cite Specht in your references.
Andrew Mathas (Supported in part by SERC grant GR/J37690)
a.mathas@ic.ac.uk
Imperial College, 1996.
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References
[A] S. Ariki, On the decomposition numbers of the Hecke algebra of G(m,1,n), preprint~(1996).
[B] J. Brundan, Modular branching rules for quantum GL_n and the Hecke algebra of type A, preprint 1996.
[DJ1] R. Dipper and G. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3), 52 (1986), 20--52.
[DJ2] R. Dipper and G. James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3), 54 (1987), 57--82.
[G] M. Geck, Brauer trees of Hecke algebras, Comm. Alg., 20 (1992), 2937--2973.
[Gr] I. Grojnowski, Affine Hecke algebras (and affine quantum GL_n) at roots of unity, IMRN~5 (1994), 215--217.
[J] G. James, The decomposition matrices of GL_n(q) for n le 10, Proc. London Math. Soc.,~60 (1990), 225--264.
[JK] G. James and A. Kerber, The representation theory of the symmetric group, 16, Encyclopedia of Mathematics, Addison--Wesley, Massachusetts~(1981).
[JM1] G. James and A. Mathas, Hecke algebras of type A at q=-1, J. Algebra (to appear).
[JM2] G. James and A. Mathas, A q--analogue of the Jantzen--Schaper Theorem, Proc. London Math. Soc. (to appear).
[K] A. Kleshchev, Branching rules for modular representations III, J. London Math. Soc. (to appear).
[LLT] A. Lascoux, B. Leclerc, and J-Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. (to appear).
[LT] B. Leclerc and J-Y. Thibon, Canonical bases and q--deformed Fock spaces, Int. Research Notices (to appear).
[M] A. Mathas, Canonical bases and the decomposition matrices of Ariki--Koike algebras, preprint~1996.
GAP 3.4.4