Workshop On Representation Theory, Combinatorics, And Geometry: Speaker Presentations.pdf

Representation Theory, Combinatorics, and Geometry

This workshop, held October 19-21 in Charlottesville, Virginia, brought together researchers to explore intersections between representation theory, combinatorics, and geometry.

Key Highlights:

Crystal Bases and Queer Superalgebras: Anne Schilling (UC Davis) presented a characterization of crystal bases for the quantum queer superalgebra, addressing open questions related to Assaf & Oguz's work. The talk introduced new graph representations and provided counterexamples to existing conjectures.

Double Affine Bruhat Order: Mandy Welch (Virginia Tech) discussed the double affine Weyl semigroup for irreducible, simply-laced root systems. She classified cocovers and covers of elements in this order, using both geometric and combinatorial approaches.

Multiparameter Schur Algebras and Quantum Symmetric Pairs: Chun-Ju Lai (University of Georgia) explored multiparameter Schur algebras of type B/C and their relationship to quantum symmetric pairs with unequal parameters. They established favorable properties using Lusztig's bar-invariant basis and developed a two-parameter upgrade for certain cases.

Saturation Conjecture in Type D: Joshua Kiers (UNC Chapel Hill) delved into the saturation conjecture for type D Lie groups, focusing on the relationship between tensor cones and saturated tensor cones. He discussed recent progress and presented formulas for extremal rays of the saturated cone.

Coherent Sheaves on Hilbert Schemes: Ben Webster (University of Waterloo) provided insights into coherent sheaves on Hilbert schemes using the Coulomb lens. He highlighted connections to Bezrukavnikov and Kaledin's perspective and discussed tilting generators.

Cohomology of Grassmannians: Voula Collins (University of Connecticut) presented work on the cohomology of the cotangent bundle to a Grassmannian, introducing "puzzles" as a tool for computing structure constants. She also proposed modifications to apply to general Grassmannians using R-matrices.

Stability of Group Algebra Centers: Jinkui Wan (Beijing Institute of Technology) reported on stability properties of centers in group algebras of general linear groups, leading to a universal stable center with positive integer structure constants.

Other Presentations:

Additional talks covered:

Traces of tensor product categories by Michael Reeks (University of Ottawa).
Vanishing of Littlewood-Richardson polynomials and their computational complexity by Colleen Robichaux (University of Illinois).
Kirillov-Reshetikhin crystals and Cacti representations by Balazs Elek (University of Toronto).