Frank Lübeck

Elementary Divisors of Gram Matrices for Specht Modules of Symmetric Groups and Jantzen Filtrations

Introduction

Let S_n be the symmetric group on n letters.

The irreducible complex representations of S_n have a natural parameterization via the partitions of n. Such a partition is a decreasing sequence of positive integers (n₁ >= n₂ >= ... >= n_k > 0 ) whose sum equals n. In our notation the partition (n) corresponds to the trivial character of S_n. (Note that some authors use the "dual" notation, i.e., the trivial character is parameterized by (1,1,...,1).)

For each partition (n₁, n₂, ..., n_k) the corresponding representation can be realized via an explicit module for the integral group ring ZS_n, which is constructed as a submodule of the permutation module for a Young subgroup described by the partition. This is called the Specht module S(n₁, n₂, ..., n_k).

The standard scalar product with respect to the natural basis of the permutation module induces a nondegenerate bilinear form on the Specht module. In the tables below we give for many Specht modules of small dimension the list of elementary divisors of the Gram matrix of this bilinear form. This contains interesting information about the modular representations of the symmetric groups, in particular:

The irreducible representations of S_n over the finite field GF(p), p a prime, are parameterized via the partitions of n which contain each number at most p-1 times (these are called p-regular). The dimension of such a module equals the number of elementary divisors of the Gram matrix of the Specht module with the same parameter, which are not divisible by p (in other words the rank of that Gram matrix reduced modulo p).
A Specht module S with bilinear form <., .> as above has for each prime p the Jantzen filtration of submodules S = S(0) >= S(1) >= ... >= S(m) > 0, where S(i) consists of the x in S with <x, y> in pⁱZ for all y in S. The dimension of S(i) equals the number of elementary divisors of the Gram matrix for S which are divisible by pⁱ.

There is a theorem, due to Jantzen and Schaper, which gives a combinatorial formula for the determinant of the Gram matrix, i.e., the product of all elementary divisors. This formula can be evaluated for partitions of n with n a few hundreds, say. But for computing the exact distribution of the elementary divisors no method seems to be known except computing the whole Gram matrix directly. In the lists below we give the results for all Specht modules for which n < 13 or whose dimension is less than some given bound (currently this bound is about 11700).

More precisely, the Jantzen sum formula expresses for each prime p the sum of the composition factors of the modules S(i) in the Jantzen filtration of a Specht module as linear combination of Specht modules (the coefficients are integral but not necessarily positive). For many of the Specht modules in our lists the p-modular decomposition numbers are known. In such a case one can also express this linear combination as a linear combination of simple p-modular modules. In many cases this information together with the known elementary divisors of the Gram matrix determines uniquely the simple composition factors of each single module S(i) reduced modulo p. We include these data in our tables below.

Notation

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We write multiple entries in partitions with an exponent, e.g., (4,2³,1) instead of (4,2,2,2,1).

The elementary divisors are given in their prime factorization and their multiplicity is denoted by an exponent. Example:

[1]³⁵ [3]⁸ [3*5]² [3*5*7]¹⁹ means that the matrix is of rank 64 = 35+8+2+19 and that the elementary divisors are 35 times 1, 8 times 3, 2 times 15 = 3*5 and 19 times 105 = 3*5*7.

Recall that we denote with (n) the trivial representation and with (1,1,...,1) the sign representation of S_n.

For the tables of Jantzen filtrations we denote the simple module in characteristic p corresponding to a p-regular partition lambda by D(lambda).

We explain the notation for the Jantzen filtrations by an example, we consider S(4,4,2,1) in characteristic 3: The Jantzen filtration is:

[D(4²,2,1)]⁰ +[D(4²,3)+D(6,2²,1)+2*D(6,5)]¹ +[D(7,2²)+D(9,2)]² +[D(7,4)]³

This means that the composition factors of S(i) are given by the sum of those modules in square brackets [...] whose formal exponent is at least i. In particular, for i = 0, summing up all expressions in square brackets we recover the decomposition of the Specht module S(4,4,2,1) modulo 3.

Some extremal cases

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For some types of partitions of n we can describe the elementary divisors in a generic way. These cases are left out in the tables below. The formulae are extrapolations from the computed cases. So, I don't claim here to have a proof for them, but only that they are checked within the range covered by the tables.

We write b(n, k) for the n,k-th binomial coefficient (i.e., n choose k).

Partition	Specht degree	elementary divisors
(n)	1	[1]
(n-k, 1^k), 0 < k < n	b(n-1, k)	[k!]^{b(n-2, k)} [n k!]^b(n-2, k-1)
(n-2,2), n odd	n(n-3)/2	[1]^(n²-5n+2)/2 [n-2]^n-2 [(n-1)(n-2)/2]
(n-2,2), n even	n(n-3)/2	[1]^(n²-5n+2)/2 [(n-2)/2] [n-2]^n-3 [(n-1)(n-2)]
(2²,1^n-4), n odd	n(n-3)/2	[4(n-4)!] [2(n-1)(n-4)!]^n-2 [2(n-1)(n-2)(n-4)!]^(n²-5n+2)/2
(2²,1^n-4), n even	n(n-3)/2	[2(n-4)!] [2(n-1)(n-4)!]^n-3 [4(n-1)(n-4)!] [2(n-1)(n-2)(n-4)!]^(n²-5n+2)/2
(1ⁿ)	1	[n!]

The partitions of form (n-k, 1^k) for 0 <= k <= n are called hook partitions. The formula above was known and proved by G. James and A. Mathas (not published).

Comparing Cases of Dual Partitions

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For a partition (n₁, n₂, ..., n_k) the dual partition is (m₁, m₂, ..., m_l) with m_i being the number of n_j which are smaller or equal to i.

The dimension d of the Specht module S = S(n₁, n₂, ..., n_k) is equal to the dimension of S' = S(m₁, m₂, ..., m_l). Let e₁ <= ... <= e_d be the elementary divisors of the Gram matrix for S and f₁ <= ... <= f_d those for S'.

Then the products d e_i f_d+1-i equal n! for all 1 <= i <= d.

This has been proved by M. Künzer (but it is not yet published).

Methods and References

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A general reference for the symmetric group and its representations, which contains the definitions of Specht modules and their Gram matrices, is

G. James and A. Kerber, The representation theory of the symmetric group, Addison Wesley, Reading, Massachusetts (1981)

The elementary divisors for n < 13 were already printed in

A. Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, American Mathematical Society, University Lecture Series 15, (1999)

This book also contains a discussion of the Jantzen filtration in a more general setting (see Chapter 5).

Most known decomposition numbers for the symmetric groups can be obtained with the Specht share package for GAP 3, written by Andrew Mathas. He also provided a program which decides if it is possible to find and then computes the Jantzen filtrations from the decomposition numbers and the elementary divisors.

I have written the programs which compute the Gram matrices for Specht modules with GAP and an external program written in C. The biggest challenge is to compute the elementary divisors of matrices of rank a few thousands. This is done with functions provided by the authors GAP package EDIM. The algorithm used there uses a combined modular and p-adic approach. The elementary divisors are computed without using the actually known primes dividing the determinant of the matrices. This allowed to check the found determinant with the one computed by the Jantzen-Schaper theorem.

The tables

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Comments and suggestions (Email) are welcome.

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Last updated: Mon Jul 12 23:23:20 2004 (CET)