UW-PIMS Mathematics Colloquium

Sponsored by the UW Department of Mathematics and the Pacific Institute for the Mathematical Sciences.

February 10, 2017 at 2:30pm
SMI 102
Patricia Hersh
North Carolina State University
Poset topology meets combinatorial representation theory

Back in the 1960's, Gian-Carlo Rota recognized the importance of an interpretation for the coefficients in inclusion-exclusion counting formulas as reduced Euler characteristics of associated simplicial complexes. We will begin by discussing some of the history of this powerful link between combinatorics and topology. Then we will turn to the example of weak Bruhat order. This partial order on permutations captures much of the structure of how permutations may be written as products of adjacent transpositions. The relations satisified by the adjacent transpositions, the braid relations, may be shown to control a poset invariant called the Moebius function -- the aforementioned reduced Euler characteristic in disguise. A topological result of Quillen, the Quillen Fiber Lemma, enables key properties of weak Bruhat order to be transferred to posets of current interest in the representation theory of Kac-Moody algebras, the so-called crystal graphs of highest weight representations. However, computer computations of the Moebius function for crystal graphs led us also to discover unexpected new relations amongst the crystal raising and lowering operators. This work on crystal graphs is joint work with Cristian Lenart.

March 13, 2017 at 2:30pm
Chelsea Walton
Temple University


March 31, 2017 at 2:30pm
John Urschel
Massachusetts Institute of Technology


April 7, 2017 at 2:30pm
SMI 102
Qing Han
University of Notre Dame


April 21, 2017 at 2:30pm
Robert McCann
University of Toronto


April 28, 2017 at 2:30pm
Maciej Zworski
University of California, Berkeley


May 12, 2017 at 2:30pm
Candice Price
University of San Diego


May 26, 2017 at 2:30pm
Piper Harron
University of Hawaii at Manoa


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