Covariants of the symmetric group and its analogs in Weyl algebras (Q1813416)

We call the algebra \(E(V)=S(V)\otimes\Lambda(V)\), \(S(V)\) the symmetric algebra, \(E(V)\) the exterior algebra for a given \(G\)-module \(V\), the Weyl algebra. For the Poincaré series \(P^ G_ \pi(t,s)=\sum_{p,q} m_{p,q} (\pi) t^ p s^ q\) (where \(\pi\) is an irreducible representation of the group \(G\), \(m_{p,q}(\pi)\) is the multiplicity of it in \(E^{p,q} (V)=S^ p(V)\otimes\Lambda^ q(V)\)), a superanalog of the expression of \(P^ G_ \pi(t,0)\) in the cases where \(V=\mathbb{R}^ n\) and \(G\) is either the symmetric group \(S(n)\) or \(C(n)=S(n)\times Z^ n_ 2\) is proved and a combinatorial interpretation of the multiplicities is given.