Schedule

Although the theory applies to all quantum affine algebras we shall focus on some illustrative examples in type GL_n. The goal is to highlight some amazing connections between combinatorics, representations, mathematical physics and probability.

Lecture 1: We’ll study examples of GL_n Macdonald polynomials. The basic tool is the affine Weyl group. This lecture is motivated by the papers of Haglund-Haiman-Loehr 2006 and Lenart 2008. An amazing connection is between tableau formulas for Macdonald polynomials and sequences of elements in the affine Weyl group.

We discuss the known natural bijective correspondence between irreducible (algebraic) self dual representations of the special linear group with those of classical groups, and then discuss how this correspondence relates to tensor product of representations.

The Grothendieck ring of the category of finite dimensional representations over a simple Lie algebra can be described via the character map, as a ring of functions invariant under the action of the Weyl group. This result was generalized to basic Lie superalgebras by A. N. Sergeev and A. P. Veselov with additional invariance conditions. In this talk we will discuss the ring of characters for queer Lie superalgebras. In particular, for the queer Lie supergroup Q(n), we show that the ring is isomorphic to the ring of symmetric Laurent polynomials in x_1,...,x_n such that the evaluation x_1=-x_2=t is independent of t. We shall discuss the representation theoretical meaning of this evaluation.

Generators and relations of graded limits of certain finite dimensional irreducible representations of quantum affine algebras have been determined in recent years. For example, the representations in the Hernandez-Leclerc category corresponding to cluster variables appear to be certain truncations of representations for current algebras and tensor products are related to the notion of fusion products. In this talk we will discuss some known results on this topic and study the characters of prime representations in the HL category.

Let λ, µ and ν be integer partitions with at most n parts each. The Littlewood-Richardson (LR) coefficient c^ν_ λ,µ is the multiplicity of the irreducible representation V (ν) in the decomposition of the tensor product V (λ)⊗V (µ) of irreducible polynomial representations of GL_n. For each permutation w in S_n, the w-refined LR coefficient c^ν_λ,µ(w) is the multiplicity of V(ν) in the decomposition of the so-called Kostant-Kumar submodule K(λ, w, µ) of the tensor product.

The saturation problem asks whether c^ν_λ,µ(w) > 0 given that c^kν_kλ,kµ(w) > 0 for some k ≥ 2. We show that this is true when the permutation w is 312-avoiding or 231-avoiding, by adapting the beautiful combinatorial proof of the LR-saturation conjecture due to Knutson and Tao. This is joint work with K.N. Raghavan and Sankaran Viswanath.

Lecture 2: We’ll construct the standard and simple level 0 modules corresponding to skew shape Young diagrams (for the quantum affine algebra of type GL_n). This lecture is motivated by the papers of Drinfeld 1986, Cherednik 1987 and Nazarov-Tarasov 1998. An amazing result is that the irreducible modules bases indexed by tableaux and that the characters of the standard modules are specialisations of Macdonald polynomials.

TBA

We give an introduction to some questions concerning the restriction of representations of a compact connected group to a closed, connected subgroup.

We introduce global Demazure modules, which are cyclic modules over the current algebras. These modules are endowed with a free action of the polynomial algebras and the spaces of coinvariants are isomorphic to the affine Demazure modules. In particular, in the simply-laced case global Weyl modules are level one global Demazure modules. We describe the algebraic properties of the global modules. We also explain their importance for the study of the arc schemes of the Veronese embeddings of the flag varieties and for the theory of Schubert varieties in the Beilinson-Drinfeld Grassmannians. The talk is based on joint works with Ilya Dumanski and Michael Finkelberg.

Richardson’s theorem implies that the semi-invariants for the adjoint action of a parabolic P on its nilradical m form a polynomial algebra. The generators exactly define hypersurfaces which are orbital varieties. Quite remarkably the number of such invariants depends only the Weyl group representative of the Levi factor L, even though the semi-invariants look very different.

In type A, a Weierstrass section e+V is constructed for this action. It is shown that e belongs to a component of the nilfibre further identified as the closure of some B.u, where B is the Borel and u a subalgebra of stable under the negative part of the L. Sometimes this component is itself an orbital variety but not always. Again it is shown that this component of the nilfibre does not always admit a dense P orbit. Several other components are obtained from this one by permuting indices.

The proofs involve some delicate and fascinating combinatorics which are certainly very deep. Indeed they take to a whole new level of non-triviality a construction of Ringel et al concerning quivers giving a combinatorial proof of Richardson’s theorem in type A.

The talk will consist of a general outline followed by some animation to illustrate the combinatorial constructions.

Lecture 3: We’ll study the tensor power of the level zero representation V = \CC^n[t,t^{-1}] and its relation to R-matrices, transfer matrices, and Macdonald polynomials. This lecture is motivated by the papers of Takhtajan-Fadeev 1979, Kashiwara-Miwa-Stern 1995 and Borodin-Wheeler 2018. An amazing result is that an eigenvector of the transfer matrix is related to the stationary distribution of the ASEP (asymmetric exclusion process) and has coefficients which are specialisations of Macdonald polynomials.

TBA

I'd like to explain a Chevalley type formula for the graded characters of Demazure modules in the level-zero extremal weight modules over the quantum affine algebras. Namely, let $\mu \in P$ be an integral weight for a finite-dimensional simple Lie algebra $\mathfrak{g}$, and $\lambda \in P^{+} \subset P$ a dominant integral weight for $\mathfrak{g}$ which is sufficiently larger than $\mu$. Let $x \in W_{\mathrm{af}}$ be an element of the affine Weyl group $W_{\mathrm{af}}$ of the untwisted affine Lie algebra $\mathfrak{g}_{\mathrm{af}}$ associated to $\mathfrak{g}$. Denote by $V^{-}_{x}(\lambda+\mu)$ the (opposite) Demazure module of highest weight $x(\lambda+\mu)$ in the level-zero extremal weight module $V(\lambda+\mu)$ over the quantum affine algebra $U_{q}(\mathfrak{g}_{\mathrm{af}})$. I'd like to explain a Chevalley type formula for the expansion of the graded character of $V^{-}_{x}(\lambda+\mu)$ as a $(\mathbb{Z}[P])((q^{-1}))$-linear combination of the graded characters of $V^{-}_{y}(\lambda)$, $y \in W_{\mathrm{af}}$. This talk is based on joint works with Kouno, Lenart, and Naito.

In this talk, we introduce HL-modules for quantum affine algebras of type D. These modules are conjecturally the prime representations in certain generalizations of the subcategory C_1 of the category of finite dimensional representations of quantum affine algebras introduced by Hernandez and Leclerc. We prove that their graded limit is isomorphic to some generalized Demazure modules and provide a presentation by generators and relations.

In this talk, we review the fusion product of two evaluation modules of the current algebras, defined by Boris Feigin and Sergej Loktev, in the context of Gröbner degenerations. Using this, we translate a possible answer to Schur positivity for certain differences of products of Schur functions, conjectured by Lam, Postnikov, Pylyavskyy among others, into the search for a Gröbner basis for simple, finite-dimensional modules of the current algebra of type sl_n. This is work in progress and we provide such Gröbner bases for sl_2. This is joint work with Flake and Levandovskyy.