# OIST Representation Theory Seminar

### Recordings of talks will be available on this page and here if the speaker agrees to have their talk recorded.

### Tuesday 26th October 2021, 9:30–10:30am JST (UTC+9), online on Zoom

### George Seelinger, University of Michigan

### Title: Diagonal harmonics and shuffle theorems

**Abstract:** The Shuffle Theorem, conjectured by Haglund, Haiman, Loehr, Remmel and Ulyanov, and proved by Carlsson and Mellit, describes the characteristic of the $S_n$-module of diagonal harmonics as a weight generating function over labeled Dyck paths under a line with slope −1. The Shuffle Theorem has been generalized in many different directions, producing a number of theorems and conjectures. We provide a generalized shuffle theorem for paths under any line with negative slope using different methods from previous proofs of the Shuffle Theorem. In particular, our proof relies on showing a "stable" shuffle theorem in the ring of virtual GL_l-characters. Furthermore, we use our techniques to prove the Extended Delta Conjecture, yet another generalization of the original Shuffle Conjecture.

#### Meeting ID: 96692955672, password TBA.

### Tuesday 9th November 2021, 9:30–10:30am JST (UTC+9), online on Zoom

### Arik Wilbert, University of South Alabama

### Title: Real Springer fibers and odd arc algebras

**Abstract:** Arc algebras were introduced by Khovanov in a successful attempt to lift the quantum sl2 Reshetikhin-Turaev invariant for tangles to a homological invariant. When restricted to knots and links, Khovanov’s homology theory categorifies the Jones polynomial. Ozsváth-Rasmussen-Szabó discovered a different categorification of the Jones polynomial called odd Khovanov homology. Recently, Naisse-Putyra were able to extend odd Khovanov homology to tangles using so-called odd arc algebras which were originally constructed by Naisse-Vaz. The goal of this talk is to discuss a geometric approach to understanding odd arc algebras and odd Khovanov homology using Springer fibers over the real numbers. This is joint work with J. N. Eberhardt and G. Naisse.