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Seminario del 2014
2014
07 ottobre
Eric Marberg
Seminario di algebra e geometria
Let (W,S) be a Coxeter system and let * be an involution of the
corresponding Coxeter diagram, that is, a self-inverse automorphism of
W which preserves the set of simple generators S. Many fundamental
properties of elements in a Coxeter group, such as the notion of a
reduced expression, the length function, and the exchange and deletion
principles have interesting, nontrivial analogues for just the subset
of *-twisted involutions in W, by which we mean the elements w in W
with w^* = w^{-1}. The first part of this mini-course will give an
introduction to such combinatorial properties of twisted involutions,
as developed by Hultman, Richardson, Springer, and others. As
motivation, we will review along the way some of the applications of
this theory to the study of symmetric varieties. The remainder of the
course will be a survey of recent progress and open questions related
to some problems in combinatorics and representation theory in which
the twisted involutions of a Coxeter group play a central role. Topics
will include among the following, as time allows: Lusztig and Vogan's
recent study of the "twisted" analogue of the regular representation
of an Iwahori-Hecke algebra; Rains and Vazinari's theory of
"quasiparabolic sets," of which conjugacy classes of twisted
involutions serve as important motivating examples; connections
between certain variants of the Poincaré series defined for twisted
involutions and q-analogues of orthogonal polynomials as studied by
Cigler and others; and reduced expression counting problems for
twisted involutions. The course should be accessible to graduate
students with some prior exposure to the study of Coxeter groups, or
at least with a little knowledge of finite group theory.