# ALGEBRAIC COMBINATORICS

On representation theory of partition algebras for complex reflection groups
Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 389-432.

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

 ${𝒯}_{0}\left(r,p,n\right)\subseteq {𝒯}_{\frac{1}{2}}\left(r,p,n\right)\subseteq {𝒯}_{1}\left(r,p,n\right)\subseteq {𝒯}_{\frac{3}{2}}\left(r,p,n\right)\subseteq \cdots \subseteq {𝒯}_{⌊\frac{n}{2}⌋}\left(r,p,n\right).$

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.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.97
Classification: 05E10,  20F55,  20C15
Keywords: Complex reflection groups, Tanabe algebras, Schur–Weyl duality, Jucys–Murphy elements
@article{ALCO_2020__3_2_389_0,
author = {Mishra, Ashish and Srivastava, Shraddha},
title = {On representation theory of partition algebras for complex reflection groups},
journal = {Algebraic Combinatorics},
pages = {389--432},
publisher = {MathOA foundation},
volume = {3},
number = {2},
year = {2020},
doi = {10.5802/alco.97},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.97/}
}
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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