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Zonal polynomials and domino tableaux. (English) Zbl 0759.05098
Let \(H\) be a subgroup of a finite group \(G\). One can define the idempotent element \(\theta\) of the group algebra \({\mathcal A}(G)\) by \(\theta=\Sigma_{h\in H}h/| H|\). If \((H,G)\) is a Gelfand pair then the decomposition of \(\theta\) into minimal idempotents yields a basis for the Hecke algebra \({\mathcal H}(H,G)\). The authors prove that the Fourier transform of the minimal idempotents is supported by standard domino tableaux. Moreover, they present a multiplication algorithm for zonal polynomials and relate the expansion coefficients to the Littlewood-Richardson’s coefficients.

MSC:
05E10 Combinatorial aspects of representation theory
05E05 Symmetric functions and generalizations
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