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.