, Volume 34, Issue 1, pp 141-162,
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Date: 30 Nov 2010

Permutation resolutions for Specht modules

Abstract

For every composition λ of a positive integer r, we construct a finite chain complex whose terms are direct sums of permutation modules M μ for the symmetric group \(\mathfrak{S}_{r}\) with Young subgroup stabilizers \(\mathfrak{S}_{\mu}\). The construction is combinatorial and can be carried out over every commutative base ring k. We conjecture that for every partition λ the chain complex has homology concentrated in one degree (at the end of the complex) and that it is isomorphic to the dual of the Specht module S λ . We prove the exactness in special cases.