Kazhdan constants of some colored permutation groups. (Q1888811)

For a group \(G\) generated by a finite set \(S\), denote by \(\widetilde G\) the set of all equivalence classes of unitary representations of \(G\) in separable Hilbert spaces. The unitary dual \(\widehat G\) of \(G\) is the subset of \(\widetilde G\) consisting of all irreducible representations. For every representation \(\pi\) with a representation space \(V_\pi\) of \(G\), define the `Kazhdan constant' of \(\pi\) with respect to \(S\) by \(K_G(S,\pi)=\inf_{\xi\in S(V_\pi)}\max_{s\in S}\|\pi(s)-s\|\), where \(S(V_\pi)=\{\xi\in V_\pi:\|\xi\|=1\}\). Also, \(\widetilde K_G(S)=\inf\{K_G(S,\pi)\mid\pi\in\widetilde G\) without nonzero fixed points\} and \(K_G(S)=\inf\{K_G(S,\pi)\mid\pi\in\widehat G^*\}\), where \(\widehat G^*=\widehat G-\{\chi_0\}\) and \(\chi_0\) is the trivial character. Generalizing results of \textit{R. Bacher} and \textit{P. de la Harpe} [J. Algebra 163, No. 2, 495-515 (1994; Zbl 0842.20013)], the author determines the Kazhdan constants for the Coxeter groups of types \(B\) and \(D\) and for colored permutation groups. In the proofs he uses combinatorial representation theory, in particular double partitions and double standard Young tableaux.