Maximal submodules for \(W\times A\) (Q5945618)

Excerpt from the introduction: This work arose from an attempt to generalize certain results of \textit{K. Nishiyama} [Schur duality for the Cartan type Lie algebra \(W_n\), J. Lie Theory 9, 233-248 (1999; Zbl 0999.17038)] on irreducible quotients of the \(W_n\times \tau _m\) module \(\otimes ^mP\). Here \(\otimes ^mP\) is the \(m\)-fold tensor product of the polynomial algebra \(k[x_1,\dots .x_n]\) in \(n\) variables over a field \(k\) of characteristic zero, \(W_n\) is the Lie algebra of derivations of \(P\), and \( \tau _m\) is the semigroup of all maps of \(\{1,2,\dots ,m\}\) to itself. For \( n\geq m\), Nishiyama relates each irreducible \(W_n\times \tau _m\) quotient \dots to an irreducible quotient \dots of \(gl(n,k)\times \mathfrak{S}_i\) module \( \otimes ^ik^n\) (for each \(i\leq n\)). In this way, he obtains a correspondence between irreducible \(W_n\) quotients of \(\otimes ^{\substack{ m \\ }} P\) and irreducible quotients of \(\tau _m\) quotients of \(\otimes ^mP\). \dots In this paper, we obtain a relation between maximal \(W_n\times \tau _m\) submodules (and hence irreducible quotients) of \(\otimes ^mP\) and maximal \( gl(n,k)\times \mathfrak{S}_i\) submodules of \(\otimes ^ik^n\) under much more general conditions, where \(k\) can be replaced by an arbitrary commutative ring with unit \(R\). (Reviewer's note: here \(\mathfrak{S}_i\) denotes the symmetric group of \(i\) letters).