Extensions of modules over Schur algebras, symmetric groups and Hecke algebras.
Schur-Weyl duality gives a well-known relationship between the representation theory of finite symmetric groups (\(\Sigma_d\)) and general linear groups (\(\text{GL}_n(k)\)) over a field \(k\). The Schur algebra \(S(n,d)\) arises from this relationship and its representations are certain polynomial representations of \(\text{GL}_n(k)\). Motivated by this relationship, the authors use homological tools (specifically spectral sequences) on Schur algebras to obtain explicit relationships between extensions for symmetric groups and those for general linear groups. The spectral sequences are constructed using two adjoint functors to the Schur functor \({\mathcal F}\colon S(n,d)\to k\Sigma_d\). These adjoint functors were studied earlier by \textit{M. Auslander} [Commun. Algebra 1, 177-268 (1974; Zbl 0285.16028)]. The authors work in a more general setting by taking an arbitrary finite-dimensional algebra \(A\) over a field \(k\) with an idempotent element \(e\). Then the ``Schur functor'' maps \(A\) to \(eAe\). The adjoint functors give rise to two Grothendieck spectral sequences relating extensions over \(A\) to extensions over \(eAe\). These spectral sequences along with study of the adjoint functors allow the authors to deduce several explicit relationships between extensions. The authors then specialize to the case when \(A\) is a quasi-hereditary algebra. For example, \(A\) could be a Schur algebra or \(q\)-Schur algebra. (In the latter case, modules for the \(q\)-Schur algebra correspond to modules for a quantum general linear group and \(eAe\) is a Hecke algebra.) The general results lead to even more precise relationships involving \(\text{Ext}^1\)- and \(\text{Ext}^2\)-groups over \(A\) and \(eAe\), from which some known results are quite nicely recovered. A vanishing result and further relationships are given. Then the authors focus on the case when \(A\) is a \(q\)-Schur algebra of the form \(S_q(d,d)\) (which can be reduced to the ordinary Schur algebra \(S(d,d)\) by setting \(q=1\)). This leads to explicit formulas relating extensions for a Hecke algebra to extensions for a quantum general linear group (or specializing, between \(k\Sigma_d\) and \(\text{GL}_d(k)\)). In the case when \(A=S(d,d)\), the image of simple modules under one of the adjoint functors is given, information about which is crucial to further computations with these methods. Lastly, the authors give several examples to show how known cohomology groups for general linear groups can be used to compute cohomology groups for finite symmetric groups and vice versa. While a number of the results presented in this work were previously known, this new approach leads to quite elegant and simple proofs. Further, the techniques have led to a number of other results, for example in work of Kleshchev-Nakano, Doty-Nakano, and Bendel-Nakano-Pillen (for finite Chevalley groups).
scientific article; zbMATH DE number 2153663