### Elementary Divisors of Gram Matrices of Certain Specht Modules

M. Künzer, G. Nebe
2003 Communications in Algebra
The elementary divisors of the Gram matrices of Specht modules S λ over the symmetric group are determined for two-row partitions and for two-column partitions λ. More precisely, the subquotients of the Jantzen filtration are calculated using Schaper's formula. Moreover, considering a general partition λ of n at a prime p > n − λ 1 , the only possible non trivial composition factor of S λ Fp is induced by the morphism of Carter and Payne, as shown by means of Kleshchev's modular Related work
more » ... lar Related work Grabmeier used Schaper's analoguous formula for the Weyl modules over the Schur algebra as an ingredient to determine the graduated hull of p-adic Schur algebras [4, 11.13]. Kleshchev and Sheth [12, 3.4], and independently, Reuter [26, 4.2.22], described the submodule structure of S (n−m,m) Fp . 4 0.3 Results 0.3.1 Two-row partitions Let n 1, let 0 m n/2 and let p be a prime. Since the decomposition numbers of S (n−m,m) Fp are in {0, 1} by James' formula, the Jantzen filtration may be calculated by means of Schaper's formula. Theorem (2.5). The multiplicities of the simple modules in the subquotients of the Jantzen filtration of S (n−m,m) Fp are determined. In particular, the elementary divisors of S (n−m,m) are calculated. Moreover, combining arguments of Plesken  and Wirsing , we show that if m 3, then S (n−m,m) Q does not contain a unimodular ZS n -lattice, that is, a lattice X satisfying X X * (2.13). For m ∈ {1, 2}, unimodular lattices do occur and have been classified by Plesken [25, p. 98 and II.5]. 0.3.2 At a large prime Suppose given a partition λ of n and a prime p > n − λ 1 . Using the theorem of Carter and Payne [3, p. 425], the direction of the Carter conjecture proven by James and Murphy [9, p. 222], as well as Kleshchev's modular branching rule [11, 0.6], the Jantzen filtration of S λ Fp may be calculated. Theorem (3.5). If p does not divide a first row hook length in the range [1, λ 2 ], then S λ Fp is simple. If p divides the first row hook length h t of the node (1, t), t ∈ [1, λ 2 ], then [S λ Fp ] = [D λ Fp ] + [D λ[t] Fp ], where λ[t] is the partition arising from λ by the according Carter-Payne box shift. The constituent [D λ[t] Fp ] lies in the v p (h t )th Jantzen subquotient. Explicit diagonalization The results mentioned so far are based on Schaper's formula, so that no diagonalizing bases can be deduced. In general, an explicit diagonalization seems to be complicated. For hook partitions, however, it is easier to diagonalize directly (5.5) than to apply Schaper's formula, as has already been remarked by James and Mathas [unpublished].