# Mathematics Colloquia and Seminars

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### Crystal graphs and Hecke algebras

**Algebra & Discrete Mathematics**

Speaker: | Monica Vazirani, UC Berkeley |

Location: | 693 Kerr |

Start time: | Thu, Apr 11 2002, 3:10PM |

What parameterizes the irreducible modules of the symmetric group $S_n$? In characteristic 0, it is partitions of $n$, and in characteristic $p$, it is the $p$-regular partitions of $n$.

The $p$-regular partitions also parameterize a basis of the basic representation of the affine Lie algebra ${\widehat{\mathfrak sl}_p}$.

This is no coincidence: a theorem of Grojnowski gives a conceptual reason for the two to be the same and also describes the crystal graph of the basic representation in terms of the representation theory of $S_n$ in characteristic $p$.

One ingredient needed for this is a partial branching rule that describes the socle of restriction of simple $S_n$-modules.

All of this is true more generally for certain $q$-deformations of $S_n$, the ``cyclotomic Hecke algebras.''

I will explain these results, and more recent theorems which describe how the affine Lie algebra controls the combinatorics of non-semisimple representations of these Hecke algebras.

(Coffee & cookies @ 2:45 in 550 Kerr )