San Francisco State University

08:00   Chun-Ju Lai

08:30   Heather M. Russell

09:00   Julianna Tymoczko

09:30   Jonathan R. Kujawa

10:00   Robert Muth

10:30   Liao Wang

15:00   Mateusz Stroinski

15:30   Joshua Sussan

16:00   You Qi

16:30   Nicolle Gonzáles

17:00   Neil Saunders

17:30   Iva Halacheva

18:00   Fan Zhou

08:00   Louise Sutton

08:30   Christopher D. Bowman-Scargill

09:00   Maud De Visscher

09:30   Monica Vazirani

10:00   Milo James Bechtloff Weising

10:30   Tianyuan Xu

The diagrammatic Hecke category provided the intuition and computational tools necessary to resolve the two most famous conjectures in all of Lie theory: the Lusztig and Kazhdan–Lusztig conjectures. Understanding the modular simple representations of these categories (over a field of prime characteristic p) subsumes the problem of determining the prime divisors of Fibonacci numbers; this is a notoriously difficult problem in number theory, for which a combinatorial solution is highly unlikely.

However, all is not lost. One way to simplify this picture is to let the prime tend to infinity, in this case the situation simplifies and the simple characters can be calculated via a recursive combinatorial algorithm. Another way in which one can simplify this picture is to restrict to certain families with “nice underlying geometries", but to let the prime be arbitrary; this will be the subject of this talk.

We define generalized Schröder polynomials S_λ(q,t,a) for triangular partitions and prove that these polynomials recover the triangular (q,t)-Catalan polynomials at a=0. Moreover, we show that the Poincarè polynomials of the reduced Khovanov-Rozansky homology of Coxeter knots of these partitions are given by S_λ(q,t,a). Finally, combined with recent results of Gorsky-Mazin-Oblomkov, we compute the Poincarè polynomial of the (d, dnm+1)-cable of the (n,m)-torus knot, thus proving a special case of the Oblomkov-Rassmusen-Shende conjecture for generic unibranched planar curves with two Puiseux pairs.

The Bethe subalgebras for gl(n) are a family of maximal commutative subalgebras of the Yangian Y(gl(n)) indexed by points in the Deligne-Mumford compactification of the moduli space M(0,n+2). They can be used to decompose skew representations. In particular, a subalgebra corresponding to an element of the real locus of the parameter space decomposes a fixed skew representation into eigenlines, which overall produces an unramified covering. We will discuss the identification of the fibers with a collection of Gelfand-Tsetlin keystone patterns, which carry a gl(n) crystal structure, as well as the monodromy action realized by a type of cactus group. This is joint work with Anfisa Gurenkova and Leonid Rybnikov.

We show there is a web basis for the Specht module of the Hecke algebra labelled by a two row partition. Using diagrammatic arguments we show that the change of basis matrix between this basis and the standard basis is unitriangular and positive in a strong sense.

The braided Schur algebras (or, Schur algebras of smooth curves) introduced by Maksimau-Minets are analogues of the quiver Schur algebras. These braided Schur algebras admit a diagrammatic presentation, and are intimately related to the representation theory of certain affine KLR algebras, in any characteristic. We’ll talk about a Schur duality for the braided Schur algebras with respect to certain quantum wreath products introduced recently by Lai-Nakano-Xiang.

For a Frobenius superalgebra A with even subalgebra a, we define associated supercategories Web_{A,a} and Web_{A,a}^{aff}, which generalize a number of symmetric web category constructions. These diagrammatic categories are equipped with asymptotically faithful functors to the category of gl_n(A)-modules and its endofunctor category respectively, and serve as ‘thickened’ versions of the wreath and affine wreath algebras over A. For various choices of A/a, these categories have proven connections to RoCK blocks of Hecke algebras, and conjectural connections to blocks of Schur algebras, Sergeev superalgebras, and KLR algebras which I will discuss. This is joint work with Nicholas Davidson, Jonathan Kujawa, and Jieru Zhu.

We propose an approach to categorification of the colored Jones polynomial evaluated at a 2p-th root of unity. This is done by equipping a p-differential discovered by Cautis on the triply graded (colored) Khovanov-Rozansky homology, defined in terms of (singular) Soergel bimodules. This is based on joint work with Louis-Hadrien Robert, Joshua Sussan and Emmanuel Wagner.

A web is a directed, labeled plane graph satisfying certain conditions coming from representation theory. Each web corresponds to a specific invariant vector in a tensor product of fundamental representations of a quantum group. In this talk, we introduce a process called stranding which graphically encodes the monomial terms in a web’s associated vector.

For G connected, reductive algebraic group defined over the complex numbers C, the Springer Correspondence gives a bijection between the irreducible representations of the Weyl group W of G and certain pairs comprising a G-orbit on the nilpotent cone of the Lie algebra of G and an irreducible local system attached to that G-orbit. These irreducible representations can be concretely realised as a W-action on the top degree homology of the fibres of the Springer resolution. These Springer fibres are geometrically very rich and provide interesting Weyl group combinatorics: for example, the irreducible components of these Springer fibres form a basis for the corresponding irreducible representation of W. 

In this talk, I will show how we may better understand the geometry of these Springer fibres in type A and type C via (bi)tableaux combinatorics and how we may describe their irreducible components using cup and cap diagrams in the case of Kato’s Exotic Springer correspondence. One application of this work is a new recursive algorithm between the Weyl group of type C and pairs of standard Young bitableaux, which may have applications for the representation theory for the Hecke algebra with unequal parameters. Work presented here is drawn from separate joint projects with D. Rosso, L. Topley and A. Wilbert.

I will present my recent results on semigroup categories, i.e. monoidal categories without unit objects. Examples of such categories include categories of projective (or injective, or tilting) objects in tensor categories. I will show that even in the absence of a unit object, it is still possible to define dual objects, and that if a semigroup category has "enough duals", then a unit object can be adjoined universally. This procedure builds on a similar construction of Benson-Etingof-Ostrik, which we recast in terms of comonad cohomology. As an application, using ideas from finitary 2-representation theory of Mazorchuk-Miemietz, I will give a characterization of finite tensor categories in terms of their semigroup categories of projective objects. Conversely, from the finitary 2-representation point of view, this provides a characterization of finitary 2-categories whose abelianization is a finite tensor category. Time permitting, I will also present applications to the study of algebra objects in tensor categories.

We construct an action of sl(2) on equivariant Khovanov-Rozansky link homology. We will also give some speculations of an extension of this action to some recently constructed 4-manifold invariants.

This is joint with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner.

The representation theory of cyclotomic KLR algebras is governed by Specht modules, from which irreducible modules arise as certain quotients. One of the major problems in this area is to determine the dimensions of irreducible modules. Classifying irreducible Specht modules would give us families of irreducible modules whose bases and dimensions we recover directly from the theory of Specht modules. In level one, irreducible Specht modules for the Iwahori–Hecke algebras of type A have (almost) been classified. Using this (almost) classification, I will present ongoing work with Matthew Fayers on classifying the irreducible Specht modules for cyclotomic KLR algebras.

A noncrossing matching is a partition into pairs of a finite subset of the number line so that no two pairs alternate endpoints, equivalently so that no two arcs cross when we join the endpoints. We study multicolored noncrossing matchings, namely matchings in which each pair is assigned a color so that the submatching of each color is noncrossing (though arcs of different colors may cross). Multicolored noncrossing matchings arise naturally in certain quantum group representations. A web is a directed, oriented plane graph that encodes an invariant vector within a tensor product of quantum group representations. A stranding of a web is a diagrammatic scheme for identifying the monomial terms in this vector, which can be viewed as a multicolored noncrossing matching within the web graph. We will discuss combinatorial properties of multicolored noncrossing matchings as well as their algebraic implications for strandings.

We study skeins on the 2-torus and 3-torus via the representation theory of the double affine Hecke algebra of type A and its connection to quantum D-modules. As an application we can compute the dimension of the generic SL(N)- and GL(N)-skein module of the 3-torus for arbitrary N. This is joint work with Sam Gunningham and David Jordan.

Parabolic Kazhdan-Lusztig polynomials of type (S_m x S_n, S_{m+n}) have elegant combinatorial descriptions in terms of oriented Temperley-Lieb diagrams or Dyck paths. They control the representation theory of many Lie theoretic objects such as the Khovanov arc algebras and the anti-spherical Hecke categories H_{m,n}. In this talk I will explain how, by delving deeper into the combinatorics of Dyck paths, we can obtain a quiver presentation for the category algebra of H_{m,n}. Moreover, using this presentation, we can show that it is in fact isomorphic to the (extended) Khovanov arc algebra. This talk is based on joint work with C. Bowman, A. Hazi and C. Stroppel.

We define a diagrammatic category which captures the structure of a braided module category generated by a braided vector space and a solution of the reflection equation or boundary Yang-Baxter equation. Our diagrammatic category often behaves like a monoidal category, which is explained via a realization as annular webs. As an example, this category admits a functor to the category of representations of various quantum symmetric pair coideal subalgebras. Quantum symmetric pairs are introduced in the 1990s by Gail Letzter as generalizations of Drinfeld-Jimbo quantum groups. Balagovic–Kolb constructed a universal K matrices which act on representations of the coideal subalgebra and solves the reflection equation. We explicitly compute the K matrices associated to type AIII quantum symmetric pairs and extend the theory to type II representations of quantum gl(n). We expect the annular web category controls this representation category, which we will explore in future works.

In this talk I will introduce a new family of graded representations \widetilde{W}_λ for the positive elliptic Hall algebra E^+ indexed by partitions λ. For λ=∅, \widetilde{W}_∅ recovers the standard E^+ action on symmetric functions. Each of these representations has a homogeneous basis of eigenvectors for the action of the Macdonald element P_{0,1} ∈ E^+ generalizing the symmetric Macdonald functions. The construction of these modules involves the symmetric vector valued Macdonald polynomials of Dunkl-Luque and the partition sequences of the form (n - |λ|, λ) central to Murnaghan’s Theorem. I will also discuss an explicit combinatorial rule for the action of the elements P_{r,0} ∈ E^+ generalizing the Pieri rule for symmetric Macdonald functions.

Let W be a finite simply laced Weyl group of rank n. When W has type E_7, E_8 or D_n for n even, the root system of W contains sets of n mutually orthogonal roots, and we call each product of n orthogonal roots an n-root. The n-roots span an irreducible Macdonald representation M of W. In this talk, we introduce and study simple n-roots, which form a canonical basis of M and play roughly the same role for M as simple roots do for the reflection representation of W. In type D_n for n = 2k, the module M is a natural lift of the Specht module S^{(k,k)} of the symmetric group S_n indexed by the partition (k, k); the n-roots of W are in a natural bijection with pairings of n objects, and the simple n-roots correspond to the web basis of S^{(k,k)} under this bijection. (This is joint work with Richard Green.)

We homologically construct a (functorial) BGG resolution of the finite-dimensional simple module of the nil-Brauer algebra, whose representation theory was shown by Brundan-Wang-Webster to categorify the split ι-quantum group of rank 1, by using infinity-categorical methods following the reconstruction-from-stratification philosophy e.g. appearing in Ayala-Mazel-Gee-Rozenblyum. To do so, we prove a fact of independent interest, that half of the nil-Brauer algebra is Koszul. This BGG resolution categorifies a character formula of Brundan-Wang-Webster, corresponding to a change-of-basis formula in the ι-quantum group. More generally, we have a (functorial) “BGG spectral sequence” which converges to any desired module; this spectral sequence is secretly a resolution when the desired module is finite-dimensional. This spectral sequence also categorifies the character formulae of Brundan-Wang-Webster for any (possibly infinite-dimensional) simple module. We expect the methods used here for producing BGG resolutions to be applicable to other (graded) triangular-based algebras also, especially diagrammatic ones.