On representation theory of partition algebras for complex reflection groups

Mishra, Ashish; Srivastava, Shraddha

doi:10.5802/alco.97

Mishra, Ashish ¹; Srivastava, Shraddha ²

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

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

Abstract

This paper defines the partition algebra, denoted by $𝒯_{k} (r, p, n)$ , for complex reflection group $G (r, p, n)$ acting on $k - fold$ tensor product ${(ℂ^{n})}^{\otimes k}$ , 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 ${(ℂ^{n})}^{\otimes k}$ . 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 \dots \subseteq 𝒯_{⌊ \frac{n}{2} ⌋} (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.

Received: 2018-12-22
Revised: 2019-08-11
Accepted: 2019-09-18
Published online: 2020-04-01

DOI: 10.5802/alco.97

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

Author's affiliations:

Mishra, Ashish ¹; Srivastava, Shraddha ²

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Mishra, Ashish; Srivastava, Shraddha. On representation theory of partition algebras for complex reflection groups. Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 389-432. doi : 10.5802/alco.97. https://alco.centre-mersenne.org/articles/10.5802/alco.97/

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