Algebraic & Enumerative Combinatorics - Adrien Segovia-The dimension of semidistributive extremal lattices

Abstract: The order dimension of a partially ordered set (poset), which is often difficult to compute, is a measure of its complexity. Dilworth proved that the dimension of a distributive lattice is the width of its subposet on its join-irreducible elements. We generalize this result by showing that the dimension of a semidistributive extremal lattice is the chromatic number of the complement of its Galois graph (see Section 3.5 of arXiv:2511.18540). We apply this result to prove that the dimension of the lattice of torsion classes of a gentle tree with n vertices is equal to n. No advanced background is required to follow the talk.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm in MC 5417.