Symmetry classes of polynomials associated with the ‎direct ‎product of permutation groups

Abstract

Let $G_{i} $ be a subgroup of $ S_{m_{i}} , 1 leq i leq k$. Suppose $chi_{i}$ is an irreducible complex character of $G_{i}$. We consider $ G_{1}times cdots times G_{k} $ as subgroup of $ S_{m} $, where $ m=m_{1}+cdots +m_{k} $. In this paper, we give a formula for the dimension of $H_{d}(G_{1}times cdots times G_{k}, chi_{1}timescdots times chi_{k})$ and investigate the existence of an o-basis of this type of classes.

Keywords

• Symmetric polynomials
• symmetry class of polynomials‎
• ‎orthogonal basis
• ‎permutaion groups‎
• ‎complex characters