Elementary Divisors of Gram Matrices of Certain Specht Modules

Abstract

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 − λ₁,the only possible non trivial composition factor of is induced by the morphism of Carter and Payne,as shown by means of Kleshchev's modular branching rule. This enables the Jantzen filtration to be calculated in this case as well.

Key Words:

• Symmetric group
• Specht modules

Acknowledgments

We thank F. Lübeck for access to his immense computations (Lübeck,[2001]),which served as our indispensable guideline. We thank A. Mathas for help with several details. We thank A. Kleshchev for help with modular branching (3.5). We thank E. Wirsing for help with Pell equations (2.13). We thank J. Müller for making (Müller,[2000b]) available to us. The first author would like to thank the IGD Lyon for kind hospitality during the first part of this project.