Permutation resolutions for Specht modules of Hecke algebras

TL;DR

This paper extends a combinatorial chain complex construction from symmetric groups to Iwahori Hecke algebras, proving partial exactness results and advancing the understanding of Specht module resolutions.

Contribution

It generalizes the permutation resolution chain complex to Iwahori Hecke algebras and proves partial exactness, building on prior symmetric group results.

Findings

Extended chain complex definition to Iwahori Hecke algebra

Proved partial exactness results for the complex

Connected to recent proofs of the symmetric group case

Abstract

In [Boltje,Hartmann: Permutation resolutions for Specht modules, J. Algebraic Combin. 34 (2011), 141-162], a chain complex was constructed in a combinatorial way which conjecturally is a resolution of the (dual of the) integral Specht module for the symmetric group in terms of permutation modules. In this paper we extend the definition of the chain complex to the integral Iwahori Hecke algebra and prove the same partial exactness results that were proved in the symmetric group case. A complete proof of the exactness conjecture in the symmetric group case was recently given by Santana and Yudin, Adv. in Math. 229 (2012), 2578-2601.