Decomposition of the canonical representation of \(W(1)\times End[m]\) on \(\Lambda(m)\) (Q1127113)

Let \(W(n)\) be the superalgebra of all (super)derivations of a Grassmann algebra \(\Lambda(n)\). By definition it has the representation on \(\Lambda(u)\), a tautological one. The author considers the tensor powers of this representation \(\bigotimes^m\Lambda(n)\). Now we can define also an action of \(\text{End} [m]\) on \(\bigotimes^m\Lambda(n)\) where \(\text{End} [m]\) is a semigroup of maps of \(\{1,2,\dots, m\}\) into itself. These actions commute so we get the \(W(n)\times \text{End} [m]\) action. In the paper, the case \(n=1\) is considered. Of course, \(\bigotimes^m\Lambda(1)\simeq \Lambda(n)\) explaining the title of the article. The author describes natural subspaces in \(\Lambda(m)\) and the factors of the resulting filtrations using Young diagrams and tensor products of representations. The results generalize the classical Schur-Weyl \(\text{sl}_n\)-\(S_m\) duality.