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Projective resolutions of representations of \(\text{GL}(n)\). (English) Zbl 0859.20034

We compute the global dimension of the Schur algebra \(S(n,r)\) over any field, for \(n\geq r\). This algebra can be defined as the algebra of endomorphisms of \(V^{\otimes r}\), for a vector space \(V\) of dimension \(n\), which commute with the action of the symmetric group \(S_r\). Modules over this algebra are equivalent to polynomial representations of \(\text{GL}(n)\) of degree \(r\); so our computation gives the optimal upper bound for the length of a projective resolution of a polynomial representation of \(\text{GL}(n)\) of degree \(r\).
The result is that, for \(n\geq r\), the global dimension of \(S(n,r)\) over a field of characteristic \(p\) is 2 times (\(r\) minus the sum of the base-\(p\) digits of \(r\)). The proof uses some explicit exact sequences, coming from the bar resolution in homological algebra, which relate symmetric powers, exterior powers, and divided powers. We also compute the global dimension of the Schur algebra \(S(n,r)\) over the integers for \(n\geq r\). For \(n<r\), we give a good upper bound for the global dimension of the Schur algebra.
Reviewer: B.Totaro (Chicago)

MSC:

20G05 Representation theory for linear algebraic groups
20C30 Representations of finite symmetric groups
20G10 Cohomology theory for linear algebraic groups
18G10 Resolutions; derived functors (category-theoretic aspects)
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