Algebra and Combinatorics Seminar

Date: November 11, 2022

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Oliver Pechenik, U of Waterloo

Title: Geometry of quasisymmetric functions

Abstract: The combinatorics of symmetric function theory plays a central role both in combinatorial representation theory (of symmetric and general linear groups) and in enumerative geometry (through the cohomology of Grassmannians). The latter connection yields "K-analogues" of the classical symmetric function bases and their combinatorics by enriching the cohomology of Grassmannians to their K-theory rings. Quasisymmetric functions (QSym) are analogues of symmetric functions introduced by Stanley and Gessel in the 70s for primarily enumerative reasons, but also with a key role in the representation theory of 0-Hecke algebras. However, analogous connections to geometry and topology have been missing. In particular, although there has significant interest in "K-analogues" of quasisymmetric functions, there has been no known space whose K-theory they describe. We build on work of Baker & Richter (2008) to identify a loop space with a cellular cohomology basis corresponding to a classical basis of QSym. We then introduce an instance of "cellular K-theory," yielding the first geometrically-interpreted K-basis of QSym. Our polynomials are similar to ones introduced by Lam & Pylyavskyy (2007) and yet are new. This is joint work with Matt Satriano (arXiv:2205.12415).