Standard Bases for Tensor Products of Exterior Powers

Abstract

For n >= a >= b, the tensor product V = Lambda(a)(C-n) circle times Lambda(b)(C-n)has a natural filtration 0 = Vm+ 1 subset of V-m subset of ... subset of V-2 subset of V-1 subset of V-0 = V of Gl(n) submodules where m = min(n - a,b) and V/V-1 is the Cartan product. For each 1 <= u <= m, we construct a basis for V-u and a basis for the quotient V/V-u. The elements in the basis for V-u can be regarded as a generalization of the quadratic relations, and the elements in the basis for V/V-u are parametrized by a set of skew tableaux satisfying a condition that cleanly extends the well known semistandardness condition defining a basis for V/V-1.