Alladi Ramakrishnan Hall
On representation theory of partition algebras for complex reflection groups
Ashish Mishra
Universidade Federal do Para
In this talk, we define the partition algebra, denoted by
$\mathpzc{T}_k(r,p,n)$, for complex reflection group $G(r,p,n)$ acting
on $k$-fold tensor product $(\mathbb{C}^n)^{\otimes k}$, where
$\mathbb{C}^n$ 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
$\mathpzc{T}_{k+\frac{1}{2}}(r,p,n)$ such that $\mathpzc{T}_k(r,p,n)
\subseteq \mathpzc{T}_{k+\frac{1}{2}}(r,p,n) \subseteq
\mathpzc{T}_{k+1}(r,p,n)$ and establish this subalgebra as partition
algebra of a subgroup of $G(r,p,n)$ acting on $(\mathbb{C}^n)^{\otimes
k}$. We call the algebras $\mathpzc{T}_k(r,p,n)$ and
$\mathpzc{T}_{k+\frac{1}{2}}(r,p,n)$ as Tanabe algebras. Our aim is to
study the representation theory of Tanabe algebras: parametrization of
their irreducible modules by studying the decomposition of
$(\mathbb{C}^n)^{\otimes k}$ as a $G(r,p,n)$-module, and construction
of Bratteli diagram for the tower of Tanabe algebras
$$\mathpzc{T}_0(r,p,n) \subseteq \mathpzc{T}_{\frac{1}{2}}(r,p,n)
\subseteq \mathpzc{T}_1(r,p,n) \subseteq
\mathpzc{T}_{\frac{3}{2}}(r,p,n) \subseteq \cdots \subseteq
\mathpzc{T}_{\lfloor \frac{n}{2}\rfloor}(r,p,n).$$
We also describe Jucys-Murphy elements of Tanabe algebras and their
actions on the Gelfand-Tsetlin basis, determined by this multiplicity
free tower, of irreducible modules. This is joint work with Shraddha
Srivastava.
Done