Reflection length with two parameters in the asymptotic representation theory of type B/C and applications

[2104.14530] Reflection length with two parameters in the asymptotic representation theory of type B/C and applications Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Representation Theory arXiv:2104.14530 (math) [Submitted on 29 Apr 2021 ( v1 ), last revised 29 Jun 2021 (this version, v2)] Title: Reflection length with two parameters in the asymptotic representation theory of type B/C and applications Authors: Marek Bożejko , Maciej Dołęga , Wiktor Ejsmont , Światosław R. Gal View a PDF of the paper titled Reflection length with two parameters in the asymptotic representation theory of type B/C and applications, by Marek Bo\.zejko and 3 other authors View PDF HTML (experimental) Abstract: We introduce a two-parameter function $\phi_{q_+,q_-}$ 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 $\phi_{q_+,q_-}$ 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 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. Comments: 31 pages, 9 figures, comments are welcome; v2: 36 pages, 9 figures; new results in Section 4.2.2; Appendix, where we discuss the analogous situation in type D (which completes the picture of all infinite Coxeter groups of Weyl type) Subjects: Representation Theory (math.RT) ; Combinatorics (math.CO); Operator Algebras (math.OA); Probability (math.PR) Cite as: arXiv:2104.14530 [math.RT] (or arXiv:2104.14530v2 [math.RT] for this version) https://doi.org/10.48550/arXiv.2104.14530 Focus to learn more arXiv-issued DOI via DataCite Journal reference: J. Funct. Anal., 284 (5), 109797, 2023 Related DOI : https://doi.org/10.1016/j.jfa.2022.109797 Focus to learn more DOI(s) linking to related resources Submission history From: Maciej Dołęga [ view email ] [v1] Thu, 29 Apr 2021 17:47:24 UTC (47 KB) [v2] Tue, 29 Jun 2021 13:06:53 UTC (53 KB) Full-text links: Access Paper: View a PDF of the paper titled Reflection length with two parameters in the asymptotic representation theory of type B/C and applications, by Marek Bo\.zejko and 3 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.RT < prev | next > new | recent | 2021-04 Change to browse by: math math.CO math.OA math.PR References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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