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Symmetry classes of polynomials associated with the direct product of permutation groups. (English) Zbl 1330.05159
Summary: 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 \(H_{d}(G_{1}\times \cdots \times G_{k}, \chi_{1}\times\cdots \times \chi_{k})\) and investigate the existence of an o-basis of this type of classes.

MSC:
05E05 Symmetric functions and generalizations
15A69 Multilinear algebra, tensor calculus
20C15 Ordinary representations and characters
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