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Products of Geck-Rouquier conjugacy classes and the Hecke algebra of composed permutations

Abstract : We show the $q$-analog of a well-known result of Farahat and Higman: in the center of the Iwahori-Hecke algebra $\mathscr{H}_{n,q}$, if $(a_{\lambda \mu}^ν (n,q))_ν$ is the set of structure constants involved in the product of two Geck-Rouquier conjugacy classes $\Gamma_{\lambda, n}$ and $\Gamma_{\mu,n}$, then each coefficient $a_{\lambda \mu}^ν (n,q)$ depend on $n$ and $q$ in a polynomial way. Our proof relies on the construction of a projective limit of the Hecke algebras; this projective limit is inspired by the Ivanov-Kerov algebra of partial permutations.
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Pierre-Loïc Méliot. Products of Geck-Rouquier conjugacy classes and the Hecke algebra of composed permutations. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.921-932. ⟨hal-01186272⟩

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