Lie Theory Workshop on Quantum Groups

Abstract. We propose an analog of the Little map for reduced expressions for signed permutations. We show that this map respects the transition equations derived from Chevellay's formula on Schubert classes. We discuss many nice properties of the signed Little map which generalize recent work of Hamaker and Young in type A where they proved Lam's conjecture. As a key step in this word, we define shifted dual equivalence graphs building on work of Assaf and Haiman and prove they can be characterized by axioms. These graphs are closely related to both the signed Little map and to the Coxeter-Knuth relations of type B due to Kraskiewicz.

Abstract. We propose an algebraic approach to categorification of quantum groups at a prime root of unity, with the hope of eventually categorifying Witten-Reshetikhin-Turaev three-manifold invariants. This is joint work with Mikhail Khovanov.

Abstract. This talk is a survey of old and recent results on quantum integrable spin chains with special type of integrable boundary conditions called reflecting boundary conditions. These spin chains are related to algebra co-modules in Hopf algebras.

Abstract. A tower of algebras is a graded algebra such that each graded piece is itself an algebra (with a different multiplication). Classic examples include the towers of group algebras of symmetric groups, Hecke algebras of type A, and nilcoxeter algebras. It is known that the Grothendieck groups of towers of algebras satisfying some natural conditions are Hopf algebras, with the product and coproduct coming from induction and restriction functors. We will discuss how certain induction and restriction functors on the category of modules over a tower of algebras categorify the so-called Heisenberg double of the Hopf algebra associated to that tower. In addition, we prove a Stone-von Neumann type theorem in this general setting. As special cases of our categorification theorem, we recover results of Geissinger, Zelevinsky, and others (for the case of symmetric groups, where the Heisenberg double is the infinite rank Heisenberg algebra) and Khovanov (for the case of nilcoxeter algebras, where the Heisenberg double is the Weyl algebra). For the tower of 0-Hecke algebras, we obtain an algebra that we call the quasi-Heisenberg algebra. As an application of our Stone-von Neumann type theorem in this case, we obtain a new, representation theoretic, proof of the fact that the algebra of quasisymmetric functions is free as a module over the algebra of symmetric functions. (This is joint work with Oded Yacobi.)

Abstract. We introduce crystal operators on a subset of affine tableaux and prove that a large class of 3-point Gromov--Witten invariants of complete flag varieties for $\mathbb C^n$ enumerate the highest weight elements under these operators. This is achieved by defining a sign-reversing involution on the affine tableaux that appear in the product of a Schur times a $k$-Schur function. Special cases of our results include the Schubert structure constants in a product of a Schubert polynomial (and the quantum generalization) times a Schur function $s_\lambda$ where $|\lambda^c|< n$. Another by-product gives the fusion coefficients for the Verlinde algebra when $|\lambda^c|< n$. We also relate the $k$-charge of a $t$-deformation of the $k$-Schur functions to the energy function of Kirillov--Reshetikhin crystals by providing a new description of the Pieri rule for $k$-Schur functions in terms of $k$-bounded partitions. (This is joint work with Jennifer Morse.)

Abstract. Mirkovic-Vilonen (MV) polytopes give nice combinatorial realizations of Kashiwara's crystals for a complex simple Lie algebra. These polytopes were originally defined using the geometry of the affine grassmannian, but they also arise in other contexts, including PBW bases, quiver varieties, and Khovanov-Lauda-Rouquier (KLR) algebras. These ideas make sense for more general Kac-Moody algebras, giving various ways to extend the theory. The story is nicest in affine type, where all the constructions lead to the same (decorated) affine MV polytopes. I will explain what these are, why they are useful, and various places they show up. In particular I will discuss recent work with Dinakar Muthiah showing how to construct affine MV polytopes using affine PBW bases.

Abstract. Take your favorite finite-type Dynkin diagram, equip it with a sink-source orientation, and take its (square) product with a Kronecker quiver: for $\hbox{SU}(N)$ the result should be a quiver that looks something like $N-1$ squares glued together side-by-side, with all the vertical arrows doubled. In the last decade this quiver has turned out to lead a double life in mathematics and physics: physically it encodes part of the BPS spectrum of pure $N=2$ Yang-Mills theory, or the intersection numbers of a certain homology basis of the corresponding Seiberg-Witten curves. Mathematically it describes the cluster coordinates on a special double Bruhat cell of the corresponding Lie group (hence encodes its Poisson and totally positive structure) which in particular is the phase space of an open Toda system.
In either setting, via the formalism of cluster transformations we can read off a distinguished rational automorphism from this quiver, which is on one hand a sort of noncommutative generating function of BPS indices, and on the other hand a symmetry of the Toda system (in fact a discrete recurrence called the $Q$-system, arising in the representation theory of quantum loop algebras). The goal of the talk is to convince you that there's actually a good reason for the same quiver to appear in these seemingly unrelated contexts, with certain irregular Hitchin systems and the spectral networks of Gaiotto, Moore, and Neitzke playing the key intermediate role.