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

‎Let G_{i} be a subgroup of S_{m_{i}}‎ ,‎\ ۱ \leq i \leq k‎. ‎Suppose \chi_{i} is an irreducible complex character of G_{i}‎. ‎We consider G_{۱}\times \cdots \times G_{k} as subgroup of S_{m} ‎, ‎where m=m_{۱}+\cdots‎ +‎m_{k} ‎. ‎In this paper‎, ‎we give a formula for the dimension of H_{d}(G_{۱}\times \cdots \times G_{k}‎, ‎\chi_{۱}\times\cdots \times \chi_{k}) and investigate the existence of an o-basis of this type of classes‎.

کلیدواژه ها:

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