A generalization of Weyl's denominator formulas for the classical groups.
Let \(1<k<n\). We consider an \(n\choose k\) by \(n\choose k\) matrix \(M\) whose rows are indexed by the Young diagrams that fit inside the diagram \((n-k)^k\) and whose columns are indexed by \(k\)-element subsets \(\{i_1,\dots,i_k\}\) of \(\{1,\dots,n\}\). The entry at the intersection of the column whose index is \(\lambda\) with the row of index \(\{i_1,\dots,i_k\}\) is defined to be \(s_\lambda(x_{i_1},\dots,x_{i_k})\), where \(s_\lambda\) is a Schur function and \(x_1,\dots,x_n\) are variables. One also orders the rows and columns appropriately. The first theorem is then that the determinant of \(M\) is a power of the Weyl denominator for \(\text{GL}_n\). For instance, if \(k=1\), we have \(s_\lambda(x_i)=x_i^{|\lambda|}\) and one gets the usual expression of the Weyl denominator as a Vandermonde determinant. The Schur functions are characters of general linear groups and thus have analogues for symplectic groups, orthogonal groups, and spin groups. The authors prove in all these cases that the determinant is a power of the Weyl denominator.
scientific article; zbMATH DE number 5078149