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).
Representing finite lattices as congruence lattices of finite algebras
Sep. 19, 2017 2pm (MATH 350)
Lie Theory
Richard Green (CU)
X
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).