MIT-Harvard-MSR COMBINATORICS SEMINAR

Speaker:  Joshua Swanson  (University of Southern California)

Title:  Higher coinvariant algebras, q-Stirling numbers, and Coxeter-like complexes

Abstract:

Super coinvariant algebras and the generalized coinvariant algebras of Haglund--Rhoades--Shimozono were both introduced to provide representation-theoretic models for the t=0 case of the combinatorial Delta Conjecture (now Theorem). The generalized coinvariant algebras are well-understood but do not generalize beyond t=0. The super coinvariant algebras are very natural and do generalize, though they remain relatively mysterious and few of their conjectured properties have been proven.

We will begin by summarizing the current state-of-the-art knowledge regarding super coinvariant algebras. This will quickly lead us to very natural enumerative considerations involving new q-Stirling numbers and set partition-like objects associated to the full monomial groups G(m, 1, n). Outside of the real cases of types A and B, we will see that the super coinvariant algebras and generalized coinvariant algebras (introduced by Chan--Rhoades) diverge. We will describe how the super coinvariant algebras are (conjecturally) governed by the combinatorics of a variation on the Coxeter complex, which is topologically a sphere, while the Chan--Rhoades generalized coinvariant algebras are (provably) governed by the combinatorics of the Milnor Fiber Complex, which is topologically a wedge of spheres.

Joint work with Nolan Wallach and Bruce Sagan.