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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.
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