On certain subalgebras of the Schur algebra
Let k be an infinite field of arbitrary characteristic, and let n, f be positive integers, E be an n-dimensional vector space over k. The well- known action of the group \(G=Gl_ n(k)\) on the f-fold tensor product \(E^ f\) determines a k-algebra homomorphism \(T_ f: kG\to End_ k(E^ f)\). For a subgroup \(H\leq G\) let \(S(H)=T_ f(kH)\). The algebra S(G) is called the Schur algebra for k, n, f and is isomorphic to the algebra introduced by I. Schur in 1901. Let B and \(B^+\) be the Borel subgroups of G consisting of the lower and upper triangular matrices, respectively, and let \(\Lambda\) and \(\Lambda^+\) be the sets of weights and dominant weights of the G-module \(E^ f\). Convenient bases of the algebras S(G), S(B) and \(S(B^+)\) are given, formulas for the product of two basis elements are obtained. The basis for S(G) under consideration contains a complete system of orthogonal idempotents \(\xi_{\lambda}\) (\(\lambda\in \Lambda)\). It is proved that \(S(G)=\sum_{\mu \in \Lambda^+}S(B)\xi_{\mu}S(B^+)\) (Theorem 5.2). This allows to obtain ``one half'' of the Borel-Weil theorem for G: \(Ind^ G_ B(k_{\lambda})=0\) if \(\lambda\in \Lambda\) is not dominant. The proof of Theorem 5.2 is entirely combinatorial and uses no representation theory. It is also proved that S(B) and \(S(B^+)\) are quasi-hereditary in the sense of L. Scott [see \textit{E. Cline}, \textit{B. Parshall} and \textit{L. Scott}, J. Reine Angew. Math. 391, 85-99 (1988; Zbl 0657.18005)]. The results and machinery of the author's book [Polynomial representations of \(GL_ n\) (Lect. Notes Math. 830, 1980; Zbl 0451.20037)] are heavily used throughout the paper.
scientific article; zbMATH DE number 4177377
scientific article