ICTS

In this talk, we introduce crystal bases for the quantum groups associated with general linear and orthosymplectic Lie superalgebras. We introduce a family of highest weight representations which have crystal bases, together with their combinatorial models.

The lectures will focus on the classification, combinatorial construction, characters, and crystals of integrable representations of quantum affine algebras. Following the paper arXiv1907.11796 , the classification fall into three cases: positive level, negative level and level zero, of which the level 0 case is the most “interesting”.

In this talk, we will discuss the branching problems for the pair $(\mathrm{Sp}(2m) \subset \mathrm{Sp}(2n))$. Specifically, we'll explore how an irreducible representation $\pi$ of $\mathrm{Sp}(2n)$ decomposes when restricted to $\mathrm{Sp}(2m)$. In our presentation, we will observe these branching multiplicities as a striking combinatorial matrix.

The aim of the lectures is to give a gentle introduction to the theory of vertex algebras, and present the concept of associated variety. The associated varieties are certain Poisson varieties attached to vertex algebras whose geometry capture important information. They are related to other interesting objects of the theory: the arc spaces and Zhu’s algebras.
 

Let g be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra h. We prove that unique factorization holds for tensor products of parabolic Verma modules. We prove more generally a unique factorization result for products of characters of parabolic Verma modules when restricted to certain subalgebras of h. These include fixed point subalgebras of h under subgroups of diagram automorphisms of g.

We show that the set of all flagged reverse plane partitions of a given skew shape and a flag is a disjoint union of Demazure crystals (up to isomorphism). As a result, the flagged dual stable Grothendieck polynomial is shown to be key positive.

The aim of the lectures is to give a gentle introduction to the theory of vertex algebras, and present the concept of associated variety. The associated varieties are certain Poisson varieties attached to vertex algebras whose geometry capture important information. They are related to other interesting objects of the theory: the arc spaces and Zhu’s algebras.
 

For any n xn Hermitian matrix A, let e(A) = (e_A(1), ..., e_A(n)) be its set of eigenvalues written in descending order. We recall the following classical problem.

Problem 1. (The Hermitian eigenvalue problem) Given two n-tuples of non- increasing real numbers: e' = (e'(1), ..., e'(n)) and e" = (e"(1), ..., e"(n)), determine all possible f = (f(1), ..., f(n)) such that there exist Hermitian matrices A, B, C with e(A) = e'; e(B) = e"; e(C) = f and A + B + C = 0. Even though this problem goes back to the nineteenth century, the first significant result was obtained by H. Weyl in 1912. With contributions from several mathematicians over the century, this problem was finally solved by combining the works of Klyachko (1998), Knutson-Tao (1999), Belkale (2001) and Knutson-Tao-Woodward (2004).

This problem can be generalized for an arbitrary reductive algebraic group as follows: Let G be a connected algebraic group with a maximal compact subgroup K and let g and k be their Lie algebras. Consider the Cartan decomposition g = k+p. Choose a maximal subalgebra (which is necessarily abelian) a of p and let a+ be a dominant chamber in a. Then, any K-orbit in p intersects a+ in a unique point. Then the analogue of the above Hermitian eigenvalue problem is the determination of the following subset C_m (for any m > 2) of (a+)^m: C_m := {(a_1, ..., a_m) in (a+)^m such that there exists (x_1, ..., x_m) in (a+)^m with x_1+ ...+ x_m = 0} and x_i in AdK. a_i}. By works of several mathematicians including Berenstein-Sjamaar (2000), Kapovich-Leeb-Millson (2005), Belkale-Kumar (2006) and Ressayre (2008), C_m has been determined in terms of an irredundant set of inequalities. This series of talks will give a complete solution of the problem. The main tools used are: Geometric Invariant Theory and Topology. We will assume familiarity with basic algebraic geometry and topology. Otherwise, the lectures will be self-contained.

We will also discuss the parallel `saturated' tensor product decomposition problem.

The aim of the lectures is to give a gentle introduction to the theory of vertex algebras, and present the concept of associated variety. The associated varieties are certain Poisson varieties attached to vertex algebras whose geometry capture important information. They are related to other interesting objects of the theory: the arc spaces and Zhu’s algebras.
 

In this talk, we delve into the fascinating intersection of graph theory, Lie superalgebras, and chromatic polynomials. Specifically, we explore the intricate relationship between the Generalized $\bold k$- Chromatic polynomial of a supergraph associated with a BKM Lie superalgebra $\mathfrak L$ and its free roots, where a free root is a root unaffected by Serre relations. By investigating the multi-coloring possibilities of the graph corresponding to these free roots using $q$ colors, we uncover a polynomial expression in $q$ known as the Generalized $\bold k$-Chromatic polynomial.

We establish a connection between this polynomial and the multiplicity of the free root space by employing the denominator identity. This association provides a Lie theoretic interpretation of Chromatic polynomials.

Quantum loop algebras are infinite-dimensional algebras which are naturally involved in the theory of integrable systems as well as in the study of cluster algebras, quiver varieties and Coulomb branches. The category $\mathcal{C}$ of finite-dimensional modules over such an algebra $U_q(\mathfrak{g})$ is an interesting example of a non-semisimple category with generic braidings; that is, isomorphisms of $V \otimes W$ onto $W \otimes V$ for "generic" simple objects $V$ and $W$. Recently, Hernandez constructed similar isomorphisms for a subcategory $\mathcal{O}^-$ of the category of representations of the Borel subalgebra $U_q(\mathfrak{b})$ of $U_q(\mathfrak{g})$. This subcategory $\mathcal{O}^-$, defined by Hernandez--Leclerc in the context of monoidal categorifications of cluster algebras, has the same Grothendieck ring as that of another subcategory $\mathcal{O}^+$ of modules over $U_q(\mathfrak{b})$. This last observation raises the following questions:

Is the category $\mathcal{O}^+$ generically braided? Can we lift the isomorphism between the Grothendieck rings of $\mathcal{O}^+$ and $\mathcal{O}^-$ into an isomorphism of categories?

In this talk, we will answer the above questions positively and give, if time permits, potential applications of our results to generalized quantum affine Schur--Weyl dualities (in the spirit of Kang--Kashiwara--Kim--Oh--Park) and to the representation theory of shifted quantum affine algebras.

The aim of the lectures is to give a gentle introduction to the theory of vertex algebras, and present the concept of associated variety. The associated varieties are certain Poisson varieties attached to vertex algebras whose geometry capture important information. They are related to other interesting objects of the theory: the arc spaces and Zhu’s algebras.
 

Let g = g(A) be a general Borcherds–Kac–Moody (BKM) C-Lie algebra, for Borcherds–Cartan matrix A, and V a highest weight g-module with weight-set wt V . Over Kac–Moody g, we know wt V for:

1) integrable V classically;
2) simple (highest weight) V by Dhillon and Khare, via parabolic Vermas;
3) all highest weight modules V recently by the author and Khare, via holes & higher order Vermas. Over BKM g, even for integrable V , to our knowledge wt V is unknown.

In this talk, we will determine the weight-sets wt V for all highest weight modules V over general BKM g, extending and subsuming at once the formulas in settings 1)–3). Our formula is:
i) explicit, non-recursive and cancellation-free; and
ii) uniform across all types of g (semisimple to Kac– Moody to general BKM) and all V (and all highest weights λ).

We begin with (the combinatorial problem of) describing the comple- ments of the upper-sets in Zn(≥0), thereby determining wt V for all V over g = H ⊕ · · · ⊕ H for H the 3-dimensional Heisenberg Lie algebra and over g = g(A) with A(ii) = 0 ∀i (in connection to finite-dimensional Heisenberg Lie algebras). This reveals a refined notion of holes to study wt V . We construct a novel class (to our knowledge) of “higher order parabolic Vermas” M(λ, H) over BKM g (only singleton holes in H in real directions), subsuming and generalizing i) all integrable V and ii) parabolic Vermas over Kac–Moody g.

We determine wt M(λ, H) over all λ and H, for every BKM g. In particular, we determine wt V for all integrable V – perhaps interestingly in the flavour of Weyl orbit weight-formulas for integrable V over Kac–Moody g (see e.g. Kac’s book). For all V over all BKM g, we recover wt V as the union of wt M(λ, H) with H ⊇ HV . (This talk is based on a recent joint work with Souvik Pal.)

Quantum toroidal algebras Uq(g_tor) occur as the Drinfeld quantum affinizations of quantum affine algebras. In particular, they contain (and are generated by) a horizontal and vertical copy of the affine quantum group. In type A, Miki obtained an automorphism of Uq(g_tor) exchanging these subalgebras, which has since played a crucial role in the investigation of its structure and representation theory.

In this talk, we shall construct an action of the extended double affine braid group B on the quantum toroidal algebra in all untwisted types. In the simply laced cases, using this action and certain involutions of B we obtain automorphisms and anti-automorphisms of Uq(g_tor) which exchange the horizontal and vertical subalgebras, thus generalising the results of Miki.

I will explain how to realize quantum loop groups and shifted quantum loop groups of arbitrary type, possibly non symmetric, using critical convolution algebras. This generalizes Nakajima’s famous construction of symmetric quantum loop groups via quiver varieties. It also provides the first geometrical description of certain families of simple modules such as Kirillov-Reshetikhin or the prefundamental modules (for negative-shift quantum loop groups).

Let \mathfrak{g} be a simple Lie algebra over \mathbb{C}. The KZ connection is a connection on the constant bundle associated with a set of n finite-dimensional irreducible representations of \mathfrak{g} and a
nonzero \kappa \in \mathbb{C}, over the configuration space of n-distinct points on the affine line. Via the work of Schechtman--Varchenko, Looijenga, Belkale--Brosnan--Mukhopadhyay when \kappa \in \mathbb{Q} the associated local systems can be seen to be realizations of naturally defined motivic local systems.

In this talk, we will discuss a basic factorization for the nearby cycles of these motivic local systems as some of the n points coalesce. This leads to the construction of a family (parametrized by \kappa) of deformations over \mathbb{Z}[t] of the representation ring of \mathfrak{g}. We will also discuss how similar constructions for conformal blocks in genus 0 lead to the construction of a family of deformations of the fusion rings. This is a joint work with Prakash Belkale and Najmuddin Fakhruddin.

The semi-infinite flag manifold Q associated with a simple algebraic group G is a "level-zero" variant of the affine flag variety of G. The geometry of Q is quite different from that of the standard affine flag variety; in particular, its Schubert varieties are both infinite-dimensional and infinite-codimensional in Q. Despite this difficulty, a notion of equivariant K-group for Q has been introduced by Kato, Naito, and Sagaki and, through further work of Kato, this K-group has had important applications to the quantum K-theory of (partial) flag varieties G/P. In this talk, I will discuss a nil-DAHA action on the equivariant K-group of Q and explain how it can be used to find algebraic and combinatorial "inverse" Chevalley formulas in this setting (for G of ADE type). This is partly based on joint works with Kouno, Lenart, Naito, and Sagaki.

In this talk, we extend the notion of Weyl modules to toroidal Lie algebras with n variables. Further, using the work of Rao, we identify the level one global Weyl module of toroidal Lie algebra with the suitable submodule of its Fock space representation up to a twist. As an application, we compute the graded character of the level one local Weyl module of toroidal Lie algebra, which generalises the recent work of Kodera.

Given a Coxeter system, we can associate with it its braid monoid. The Coxeter group and the Hecke monoid are natural quotients of it, and both are known to have a lot of applications and interesting representation theory. A particularly important role for geometric applications is played by parabolic elements of the Coxeter group, which form a commutative submonoid of the Hecke monoid. The aim of the present work (joint with Arkady Berenstein and Jianrong. Li) is to classify those homomorphisms of braid monoids which factor through to homomorphisms of both Coxeter groups and Hecke monoids (we call them admissible) and those homomorphisms of Hecke monoids which preserve parabolic elements and their generalizations. As a byproduct, we also obtain a procedure for embedding a non-simply laced Coxeter system into a simply laced one in a way which is compatible with all three structures.

We study general highest weight modules V over an arbitrary complex Kac-Moody algebra \g (one may assume this to be \sl(n) throughout the talk, without sacrificing novelty). We present a "first order" invariant for every V, which yields the convex hull of its weights (for which we classify the faces), but also the Weyl group stabilizer of its character - and for simple (non-integrable) V, the weights of V as well. This connects to earlier results by Satake, Borel-Tits, Cellini-Marietti, Casselman, and Vinberg for finite-dimensional V over semisimple \g (aka Weyl polytopes); and to results of Chari and coauthors.

We then introduce the notions of holes in a module V (over Kac-Moody \g) and of higher order Verma modules. Using these, we present positive formulas (without cancellations) for the weights of arbitrary highest weight modules V. We will end with BGG resolutions and Weyl-Kac character formulas for classes of higher order Verma modules. (Partly joint with Gurbir Dhillon and with G.V.K. Teja.)

Quantum Grothendieck ring in this talk is a deformation of the Grothendieck ring of the monoidal category of finite-dimensional modules over the quantum loop algebra, endowed with a canonical basis consisting of the so-called simple (q,t)-characters. We discuss a collection of isomorphisms among the quantum Grothendieck rings of different Dynkin types respecting the canonical bases, via which the (q,t)-characters of non-simply-laced type inherit several good properties from those of the unfolded simply-laced type. We also discuss their cluster theoretical interpretation, which particularly yields non-trivial birational relations among the (q,t)-characters of different Dynkin types. This is a joint work with David Hernandez, Se-jin Oh, and Hironori Oya.

We propose a geometric approach to the Feigin-Loktev fusion product of modules over the current algebra and compute it in some new cases. Moreover, this establishes a connection between two categories with cluster structure in representation theory: the one of modules over the affine quantum group and the one of perverse coherent sheaves on the affine Grassmannian. Using this connection, we are able to establish the existence of analogs of Q-systems of perverse coherent sheaves in simply-laced types, which conjecturally stand for cluster relations. Based on arXiv:2308.05268.