The University of Sydney
School of Mathematics and Statistics  
Algebra Seminar  
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University of Sydney Algebra Seminar

Bob Howlett (University of Sydney)

Friday 29th April, 12.05-12.55pm, Carslaw 275

Specht modules and W-graphs

Given a partition of n it is possible to construct a representation of the symmetric group of degree n, or the corresponding Iwahori-Hecke algebra, on a space with basis parametrized by the standard tableaux corresponding to the given partition. The method that is simplest to describe produces the so-called seminormal form of the corresponding representation of the symmetric group. Unfortunately this yields matrices that are not integral. Using instead the Murphy basis produces matrices that are integral, though harder to compute. The transition matrix relating these two bases is upper triangular. A third equivalent version of the module will be described, related to the others by another upper triangular basis change. This basis is constructed by a variant of the Kazhdan-Lusztig W-graph algorithm; the upshot is that it is now possible to determine the structure of the left cells of the symmetric groups by means of an algorithm that just works with tableaux of a fixed shape.