On complexes relating the Jacobi-Trudi identity with the Bernstein- Gelfand-Gelfand resolution. II
In two earlier papers [Part I, J. Algebra 117, No. 2, 494-503 (1988; Zbl 0668.20033); Contemp. Math. 88, 209-217 (1989; Zbl 0721.20023)] the author constructed complexes \(Y(\lambda)\), \(Y'(\lambda)\) of \(GL_ n(K)\)- modules, where \(K\) is a field of characteristic zero and \(\lambda= (\lambda_ 1, \lambda_ 2,\dots, \lambda_ m)\) is a sequence of non- negative integers. The complex \(Y(\lambda)\) gives a realization of the Jacobi-Trudi identity. In this paper he describes the boundary maps of \(Y(\lambda)\) and \(Y'(\lambda)\) in terms of certain Shapovalov elements in the enveloping algebra \(U({\mathfrak N}^ -)\), where \({\mathfrak N}^ -\) is the Lie algebra of strictly lower triangular \(m\times m\) matrices. He also describes certain subcomplexes of \(Y(\lambda)\) and \(Y'(\lambda)\) that can be obtained recursively from complexes associated with sequences of positive integers whose sum equals \(m\).
scientific article; zbMATH DE number 130489