Abstract :
In this paper, we will give a proof of the complete submodule structure of Specht modules corresponding to 2-part partitions for the general linear group GL(n,q) in characteristic p coprime to q (in non-defining characteristic).
The multiplicities in the Specht module S(n−l,l) being at most 1, we introduce a partial order on the set of composition factors. Let e be the lowest integer such that p 1+q+ +qe−1. We explicitly construct the j, such that D(n−j,j) is a composition factor of S(n−l,l), by looking at well-defined sets I of exponents of p whose coefficients in the p-adic expansion of (l−j)/e are relevant. We thus parametrise μI(l)=(n−j,j). The family of such sets I then forms a partially ordered set under inclusion. We show that this is isomorphic to the poset of composition factors in S(n−l,l).
The results in this paper are not to be confused with the results for 2-column partition Specht modules in the defining characteristic of the general linear group, obtained in [A.M. Adamovich, PhD thesis, Moscow State University, 1992] and discussed in [A. Kleshchev, J. Sheth, J. Algebra 221 (1999) 705–722].
If we were to take q≡1modp, we would find the same submodule structure as in the corresponding Specht modules of the symmetric group (cf. [A. Kleshchev, J. Sheth, J. Algebra 221 (1999) 705–722]).
