##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## John Hall

#### UCSD (Visiting Professor)

## Deformations of the full transformation semigroup

##### Abstract:

The Full Transformation Semigroup $T_n$ is the semigroup of all maps from a set of size $n$ to itself. The representation theory of $T_n$ is closely tied to that of the Symmetric Group $S_n$, which it contains. However the usual questions are considerably more difficult to answer because the associated semigroup algebra ${\cal C} T_n$ is not semisimple. \vskip .1in \noindent In this talk we define a deformation of ${\cal C} T_n$, introducing a parameter with the aim of making the algebra generically semisimple. We show that the irreducible modules of the deformed algebra are in fact isomorphic to those of the Rook Monoid, which have a simple combinatorial description due to Cheryl Grood. We derive character formulas, and discuss the specialization to certain "bad" values of the parameter.

Host:

### November 15, 2005

### 3:00 PM

### AP&M 7321

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