Characteristic-free resolutions of Weyl and Specht modules

Abstract

We construct explicit resolutions of Weyl modules by divided powers and of co-Specht modules by permutational modules. We also prove a conjecture of Boltje-Hartmann on resolutions of co-Specht modules.Comment: 31 page

Homological properties of quantised Borel–Schur algebras and resolutions of quantised Weyl modules

“…These complexes are lifting to the q-setting of the corresponding RΣ r -module complexes described in [BH11]. It was proved in [SY12], that ‹ C λ * is a permutation resolution of the co-Specht modules Hom R (S λ , R) for q = 1 and λ a partition of r. In this section we will prove a similar result for any invertible q in R.…”

“…The original motivation of the present paper was to extend the results of [SY12] to the context of quantised Schur algebras and Hecke algebras. This is easily achieved once one has a quantised version of the generalization of Woodcock's theorem given in [SY12].Fix positive integers n and r, a commutative ring R and an invertible element q in R. Consider the quantised Schur algebra S R,q (n, r) and the quantised positive Borel-Schur algebra S + R,q (n, r). For each partition λ = (λ 1 , .…”

Boltje-Maisch resolutions of Specht modules

“…In module category of the group algebra RS r over an arbitrary commutative ring R, Hartmann and Boltje constructed a finite chain complex in [4] for any composition λ of a positive integer r. Almost all factors are constructed by restricted subsets of homomorphisms between permutation modules except the last one, which is the dual of Specht module S λ . Partial exactness results of position -1 and 0 about this complex were already achieved in Hartmann and Boltje's work [4] and a full proof of the exactness was obtained recently in [24] with help of Bar resolution in homology theory and the construction using the Schur functor. In [1] [2] [3] [8] [9] [23] [26] [30] [27] [31] and etc., some other permutation resolutions of Specht modules have been established.…”

The multistep homology of the simplex and representations of symmetric groups

“…The generator for D (m+1,m) in Finally we note that there is an extensive theory of resolutions of (dual) Specht modules by Young permutation modules, beginning with [4]; the authors' conjectured resolution was proved to be exact in [21] using the Schur algebra. Even in the two-row case, the terms in these resolutions are sums of multiple Young permutation modules.…”