This paper defines the partition algebra, denoted by ${\mathcal{T}}_{k}(r,p,n)$, for complex reflection group $G(r,p,n)$ acting on $k\text{-}\mathrm{fold}$ tensor product ${\left({\u2102}^{n}\right)}^{\otimes k}$, where ${\u2102}^{n}$ 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 ${\mathcal{T}}_{k+\frac{1}{2}}(r,p,n)$ such that ${\mathcal{T}}_{k}(r,p,n)\subseteq {\mathcal{T}}_{k+\frac{1}{2}}(r,p,n)\subseteq {\mathcal{T}}_{k+1}(r,p,n)$ and establish this subalgebra as partition algebra of a subgroup of $G(r,p,n)$ acting on ${\left({\u2102}^{n}\right)}^{\otimes k}$. We call the algebras ${\mathcal{T}}_{k}(r,p,n)$ and ${\mathcal{T}}_{k+\frac{1}{2}}(r,p,n)$ 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
$${\mathcal{T}}_{0}(r,p,n)\subseteq {\mathcal{T}}_{\frac{1}{2}}(r,p,n)\subseteq {\mathcal{T}}_{1}(r,p,n)\subseteq {\mathcal{T}}_{\frac{3}{2}}(r,p,n)\subseteq \cdots \subseteq {\mathcal{T}}_{\lfloor \frac{n}{2}\rfloor}(r,p,n).$$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.
Revised: 2019-08-11
Accepted: 2019-09-18
Published online: 2020-04-01
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}, publisher = {MathOA foundation}, volume = {3}, number = {2}, year = {2020}, pages = {389-432}, doi = {10.5802/alco.97}, language = {en}, url={alco.centre-mersenne.org/item/ALCO_2020__3_2_389_0/} }
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/item/ALCO_2020__3_2_389_0/
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