Standard Bases for Tensor Products of Exterior Powers

For n ≥ a ≥ b, the tensor product \(V=\bigwedge ^{a}(\mathbb {C}^{n})\otimes \bigwedge ^{b}(\mathbb {C}^{n})\) has a natural filtration 0 = Vm+ 1 ⊆ Vm ⊆⋯ ⊆ V2 ⊆ V1 ⊆ V0 = V of Gln submodules where m = min(n − a,b) and V/V1 is the Cartan product. For each 1 ≤ u ≤ m, we construct a basis for Vu and a basis for the quotient V/Vu. The elements in the basis for Vu can be regarded as a generalization of the quadratic relations, and the elements in the basis for V/Vu are parametrized by a set of skew tableaux satisfying a condition that cleanly extends the well known semistandardness condition defining a basis for V/V1.