Richard Green (CU) The classification of a(2)-finite Coxeter groups

Sep. 19, 2017 3pm (MATH 350)

Topology

Sebastian Bozlee Stacks

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We explore various aspects of the problem of representing a finite lattice as the congruence lattice of a finite algebra or an interval in the subgroup lattice of a finite group. We describe constructive methods that yield concrete representations, as well as some nonconstructive ways to prove existence of a representation. Minimal representations are discussed including computer programs to find them. These methods enable us to prove that every lattice with at most seven elements, with only one possible exception, has a representation as a congruence lattice of a finite algebra. This is joint work with Ralph Freese (U Hawaii) and Peter Jipsen (Chapman U).

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Lusztig's a-function is an integer-valued function on the elements of a Coxeter group, and it is defined in terms of the structure constants of the Kazhdan--Lusztig basis of the associated Hecke algebra. We call a Coxeter group "a(k)-finite" if it has finitely many elements with a-value equal to k. A Coxeter group element has a-value 1 if and only if it is "rigid", meaning that it has a unique reduced expression. The classification of a(1)-finite Coxeter groups is known, and this talk will describe the classification of a(2)-finite Coxeter groups.

This is joint work with Tianyuan Xu (Queen's University).