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.
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R. Boltje’s research partially supported by NSF Grant DMS 0200592.
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Boltje, R., Hartmann, R. Permutation resolutions for Specht modules. J Algebr Comb 34, 141–162 (2011). https://doi.org/10.1007/s10801-010-0265-1
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DOI: https://doi.org/10.1007/s10801-010-0265-1