On representation theory of partition algebras for complex reflection groups

Mishra, Ashish¹ ; Srivastava, Shraddha²

¹ Instituto de Ciências Exatas e Naturais Universidade Federal do Pará Belém Pará Brazil
² The Institute of Mathematical Sciences Chennai India

Algebraic Combinatorics, Tome 3 (2020) no. 2, pp. 389-432.

Résumé

This paper defines the partition algebra, denoted by ${𝒯}_k(r,p,n)$, for complex reflection group $G(r,p,n)$ acting on $k\text{-}\mathrm{fold}$ tensor product ${\left({ℂ}^n\right)}^{\otimes k}$, where ${ℂ}^n$ is the reflection representation of $G(r,p,n)$. A basis of the centralizer algebra of this action of $G(r,p,n)$ was given by Tanabe and for $p=1$, the corresponding partition algebra was studied by Orellana. We also define a subalgebra ${𝒯}_{k+\frac{1}{2}}(r,p,n)$ such that ${𝒯}_k(r,p,n)\subseteq {𝒯}_{k+\frac{1}{2}}(r,p,n)\subseteq {𝒯}_{k+1}(r,p,n)$ and establish this subalgebra as partition algebra of a subgroup of $G(r,p,n)$ acting on ${\left({ℂ}^n\right)}^{\otimes k}$. We call the algebras ${𝒯}_k(r,p,n)$ and ${𝒯}_{k+\frac{1}{2}}(r,p,n)$ Tanabe algebras. The aim of this paper is to study representation theory of Tanabe algebras: parametrization of their irreducible modules, and construction of Bratteli diagram for the tower of Tanabe algebras

$${𝒯}_0(r,p,n)\subseteq {𝒯}_{\frac{1}{2}}(r,p,n)\subseteq {𝒯}_1(r,p,n)\subseteq {𝒯}_{\frac{3}{2}}(r,p,n)\subseteq \cdots \subseteq {𝒯}_{\lfloor \frac{n}{2}\rfloor }(r,p,n).$$

We conclude the paper by giving Jucys–Murphy elements of Tanabe algebras and their actions on the Gelfand–Tsetlin basis, determined by this multiplicity free tower, of irreducible modules.

Détail

Reçu le : 2018-12-22
Révisé le : 2019-08-11
Accepté le : 2019-09-18
Publié le : 2020-04-01

DOI : 10.5802/alco.97

Classification : 05E10, 20F55, 20C15
Mots-clés : Complex reflection groups, Tanabe algebras, Schur–Weyl duality, Jucys–Murphy elements

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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