Combinatorics and the Schur algebra (Q686025)

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 \textit{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 (\S8) a connection is made with the Weyl and Schur modules for the general linear groups.