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Reflection length with two parameters in the asymptotic representation theory of type B/C and applications. (English) Zbl 1520.43003

Summary: We introduce a two-parameter function \(\phi_{q_+ , q_-}\) on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length. We show that this signed reflection function \(\phi_{q_+ , q_-}\) is positive definite if and only if it is an extreme character of the infinite hyperoctahedral group and we classify the corresponding set of parameters \(q_+, q_-\). We construct the corresponding representations through a natural action of the hyperoctahedral group \(B(n)\) on the tensor product of \(n\) copies of a vector space, which gives a two-parameter analog of the classical construction of Schur-Weyl. We apply our classification to construct a cyclic Fock space of type B generalizing the one-parameter construction in type A found previously by M. Bozėjko and M. Guţă [Commun. Math. Phys. 229, No. 2, 209–227 (2002; Zbl 1001.81041)]. We also construct a new Gaussian operator acting on the cyclic Fock space of type B and we relate its moments with the Askey-Wimp-Kerov distribution by using the notion of cycles on pair-partitions, which we introduce here. Finally, we explain how to solve the analogous problem for the Coxeter groups of type D by using our main result.

MSC:

43A35 Positive definite functions on groups, semigroups, etc.
20F55 Reflection and Coxeter groups (group-theoretic aspects)
46L53 Noncommutative probability and statistics

Citations:

Zbl 1001.81041
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References:

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