Combinatorics and the Schur algebra. (English) Zbl 0794.20053
The author studies the structure of the Schur algebra \(S_ R(n,r)\) over a commutative ring \(R\). A basis of codeterminants for this algebra is given which is analogous to the one constructed by G.-C. Rota [Théorie combinatoire des invariants classiques, Sér. Math. Pures Appl. 1\(\setminus\)S-01 (IRMA, Strasbourg, 1976/1977)] for the \(R\)-module \(A_ R(n,r)\) of all homogeneous polynomials of degree \(r\) in \(n^ 2\) indeterminates over \(R\). Towards the end of the paper (§8) a connection is made with the Weyl and Schur modules for the general linear groups.
Reviewer: A.Khammash (Makkah)
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 20C30 | Representations of finite symmetric groups |
| 16R30 | Trace rings and invariant theory (associative rings and algebras) |
| 05E10 | Combinatorial aspects of representation theory |
Keywords:
bideterminants; Weyl modules; Schur algebras; codeterminants; homogeneous polynomials; Schur modules; general linear groupsReferences:
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