2021/04/29 by Bożejko, Marek, Dołęga, Maciej, Ejsmont, Wiktor +1
#Combinatorics (math.CO) #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2104.14530
We introduce a two-parameter function ϕq+,q- on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length keeping track of the long and the short reflections separately. We show that this signed reflection function ϕq+,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 Bożejko and Guta. 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.