Reflection length with two parameters in the asymptotic representation theory of type B/C and applications
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866918439034028032 |
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| author | Bożejko, Marek Dołęga, Maciej Ejsmont, Wiktor Gal, Światosław R. |
| author_facet | Bożejko, Marek Dołęga, Maciej Ejsmont, Wiktor Gal, Światosław R. |
| contents | We introduce a two-parameter function $ϕ_{q_+,q_-}$ 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 $ϕ_{q_+,q_-}$ 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_14530 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Reflection length with two parameters in the asymptotic representation theory of type B/C and applications Bożejko, Marek Dołęga, Maciej Ejsmont, Wiktor Gal, Światosław R. Representation Theory Combinatorics Operator Algebras Probability We introduce a two-parameter function $ϕ_{q_+,q_-}$ 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 $ϕ_{q_+,q_-}$ 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 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. |
| title | Reflection length with two parameters in the asymptotic representation theory of type B/C and applications |
| topic | Representation Theory Combinatorics Operator Algebras Probability |
| url | https://arxiv.org/abs/2104.14530 |