San Francisco State University

8:00 - 8:20 am  Anne Schilling,  Uncrowding algorithm for hook-valued tableaux: Part I. 

8:30 - 8:50 am  Jianping Pan,  Uncrowding algorithm for hook-valued tableaux: Part II. 

9:00 - 9:20 am  Suchada Pongprasert,  D_6^{(1)}-geometric crystal at spin node and its ultra-discretization. 

9:30 - 10:00 am  Break and/or informal discussion.

10:00-10:20 am  Rosa Orellana,  A combinatorial model for the decomposition of multivariate polynomial rings as symmetric group modules. 

10:30 - 10:50 am  Tina Kanstrup,  Link homologies and Hilbert schemes via representation theory. 

11:00 - 11:20 am  Julianna Tymoczko,  Comparing different bases for irreducible symmetric group representations. 

2:00 - 2:20 pm  Gail Letzter,  Quantum Weyl Algebras for Homogeneous Spaces. 

2:30 - 2:50 pm  Anna Romanov,  Contravariant duality for Whittaker modules. 

3:00 - 3:20 pm  Jieru Zhu,  Tensor representations for the Drinfeld double of the Taft algebras. 

3:30 - 4:00 pm  Break and/or informal discussion.

4:00 - 4:20 pm  Ana Balibanu,  Steinberg slices in quasi-Poisson varieties. 

4:30 - 4:50 pm  Anne Dranowski,  From q to hbar. 

5:00 - 5:20 pm  Vyjayanthi Chari,  Generalized Demazure Modules and connections with Cluster Algebras. 

8:00 - 8:20 am  Vera Serganova,  Weight modules over Lie algebras of polynomial vector fields. 

8:30 - 8:50 am  Lilit Martirosyan,  Braid rigidity for path algebras. 

9:00 - 9:20 am  Sibylle Schroll,  A geometric complete derived invariant for gentle algebras. 

9:30 - 10:00 am  Break and/or informal discussion.

10:00-10:20 am  Karin Baur,  Structure of Grassmannian cluster categories. 

10:30 - 10:50 am  Rachael Boyd,  Combinatorics of injective words for Temperley-Lieb algebras. 

11:00 - 11:20 am  Gordana Todorov,  Representations of Continuous Quivers of Type A. 

2:00 - 2:20 pm  Hankyung Ko,  Kostant's problem and categorical actions of the Hecke category. 

2:30 - 2:50 pm  Emily Norton,  New decomposition numbers of finite classical groups via categorification. 

3:00 - 3:20 pm  Dwight Williams II,  The diagonal reduction algebra of osp(1|2). 

3:30 - 4:00 pm  Break and/or informal discussion.

4:00 - 4:20 pm  Xin Jin,  Homological mirror symmetry for the universal centralizers. 

4:30 - 4:50 pm  Yue Zhao,  Representations of (degenerate) affine Hecke algebra of type C_n and combinatorics. 

5:00 - 5:20 pm  Tianyuan Xu,  On Kazhdan--Lusztig cells of a-value 2.

We consider a multiplicative analogue of the universal centralizer of a semisimple group G — a family of centralizers parametrized by the regular conjugacy classes of G. This multiplicative analogue has a natural symplectic structure and sits as a transversal in the quasi-Poisson double D(G). We show that D(G) extends to a smooth groupoid over the wonderful compactification of G, and we use this to construct a log-symplectic partial compactification of the multiplicative universal centralizer.

The coordinate ring of the Grassmannian variety of k-dimensional subspaces of an n-dimensional vector space has a cluster algebra structure with Plücker relations giving rise to exchange relations (Scott). An additive categorification of this is given by the category of maximal Cohen Macaulay modules over a quotient of a preprojective algebra (Jensen-King-Su). We study the structure of this category in the infinite types and show links to root combinatorics of an associated Kac Moody Lie algebra. This is joint work with Bogdanic, Garcia Elsener and Li.

This talk will introduce combinatorial properties of the 'complex of planar injective words', a chain complex of modules over the Temperley-Lieb algebra that arose in joint work with Hepworth on homological stability. Despite being a linear rather than a discrete object, this chain complex nevertheless exhibits interesting combinatorial properties. I will introduce the Temperley-Lieb algebras and the complex, before stating our trio of results, inspired by results of Reiner and Webb for the complex of injective words. Our results can be viewed as an interpretation of the n-th Fine number as the 'planar' or 'Dyck path' analogue of the number of derangements of n letters.

We discuss a presentation of certain generalized Demazure modules for simply laced affine Lie algebras. We relate these modules to the classical limit of a family of prime representations of quantum affine algebras which arise from the monoidal categorification of cluster algebras. 

This is based on joint work with Justin Davis, Ryan Moruzzi and Matheus Brito.

We exhibit experimental evidence relating characters of KLR modules and equivariant multiplicities of MV cycles in finite and affine type A. On the categorical side our computations are governed by cluster combinatorics of C_q[N] whereas on the geometric side our computations are governed conjecturally by crystal combinatorics of (affine) MV polytopes.

Abstract. I will present recent work on the homological mirror symmetry about the universal centralizers J_G, for any complex semisimple Lie group G. The A-side is a partially wrapped Fukaya category of J_G, and the B-side is the category of coherent sheaves on the categorical quotient of a dual maximal torus by the Weyl group action (with some modification if G has a nontrivial center).

The aim of this joint work in progress with Roman Bezrukavnikov is to unite different approaches to Khovanov-Rozansky triply graded link homology. The original definition is completely algebraic in terms of Soergel bimodules. It has been conjectured by Gorsky, Negut and Rasmussen that it can also be calculated geometrically in terms of cohomolgy of sheaves on Hilbert schemes. Motivated by string theory Oblomkov and Rozansky constructed a link invariant in terms of matrix factorizations on related spaces and later proved that it coincides with Khovanov-Rozansky homology. In this talk I'll discuss a direct relation between the different constructions and how one might invent these spaces starting directly from definitions.

Abstract. Let g be a semisimple complex Lie algebra and let M be a g-module. Consider A(M), the space of linear endomorphisms on M on which the adjoint action of g is finite. A classical question of Kostant is: for which simple module M is the canonical map from U(g) to A(M) surjective?

I will reformulate this problem using the language of monoidal (or 2-)categories. If M belongs to the BGG category O, then the answer to Kostant’s problem is 'yes' if and only if certain categorical actions of the Hecke category are equivalent, and the latter is determined by decomposing the action of translation functors on M. This leads to a conjectural combinatorial answer to Kostant’s problem in terms of the Kazhdan-Lusztig basis.

This is a joint work with Walter Mazorchuk and Rafael Mrden.

We give a brief overview of quantum Weyl algebras and explain how many of these algebras can be constructed using twisted tensor products defined via R-matrices. Building on this approach, quantum analogs of Weyl algebras are formed for several homogenous spaces using both R-matrices and solutions to reflection equations. We show how these new quantum Weyl algebras interact nicely with related quantized enveloping algebras in terms of module structures and embeddings.

Path algebras are a convenient way of describing decompositions of tensor powers of an object in a tensor category. If the category is braided, one obtains representations of the braid groups B_n for all n ∈ N. We say that such representations are rigid if they are determined by the path algebra and the representations of B_2. We show that besides the known classical cases also the braid representations for the path algebra for the 7-dimensional representation of G_2 satisfies the rigidity condition, provided B_3 generates End(V^{⊗3}). We obtain a complete classification of ribbon tensor categories with the fusion rules of the Lie algebra g(G_2) if this condition is satisfied.

Gerber-Hiss-Jacon conjectured a combinatorial rule for the Harish-Chandra induction and restriction of modular unipotent representations of finite unitary groups. This conjecture was proved by Dudas-Varagnolo-Vasserot by constructing a categorical action of a Kac-Moody algebra on the representation categories in question. Moreover, they showed the categorical action exists for finite odd orthogonal and symplectic groups as well. However, their result had not been used so far to find decomposition numbers (the analog of Kazhdan-Lusztig multiplicities in this setting), about which little is currently known. I will explain some results in this direction from recent and ongoing joint work with Olivier Dudas.

In this talk we showed how multiset-filled tableaux are useful in understanding the decomposition of multivariate polynomial rings when we view them as symmetric group modules. For example, the symmetric group, S_n, acts on the commutative multivariate polynomial ring of m sets of n commuting variables. When we decompose this module, the multiplicities of the irreducible representations is the number of multiset-filled tableaux satisfying column strict conditions.

In this talk, we consider the S_n-module of the polynomial ring with m sets of n commuting variables and m' sets of n anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition λ (a partition of n) is the number of multiset-filled tableaux of shape λ satisfying certain column and row strict conditions.

This is joint work with Mike Zabrocki.

Abstract. Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated to stable canonical Grothendieck polynomials. In this paper, we define a novel uncrowding algorithm for hook-valued tableaux. The algorithm "uncrowds" the entries in the arm of the hooks and yields a set-valued tableau and a column-flagged increasing tableau. We prove that our uncrowding algorithm intertwines with crystal operators. An alternative uncrowding algorithm that “uncrowds” the entries in the leg instead of the arm of the hooks is also given. As an application of uncrowding, we obtain various expansions of the canonical Grothendieck polynomials. This is joint work with Joseph Pappe, Wencin Poh, and Anne Schilling.

Also see Anne Schilling's talk.

Let g be an affine Lie algebra with index set I = {0, 1, 2, ... , n} and g^L be its Langlands dual. It is conjectured that for each Dynkin node k ∈ I \ {0} the affine Lie algebra g has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for g^L . We prove this conjecture for k = 6 and g = D_6^{(1)}.

We introduce a classification of contravariant forms on standard Whittaker modules for a semisimple Lie algebra in terms of Lie algebra homology. This classification can be used to give an algebraic construction of contravariant duality for Whittaker modules. This construction allows us to algebraically define costandard Whittaker modules which align with costandard (twisted) Harish-Chandra sheaves under Beilinson—Bernstein localization. Using this we establish that the category of Whittaker modules has the structure of a highest weight category. This is joint work with Adam Brown (IST Austria).

Abstract. Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated to stable canonical Grothendieck polynomials. In this paper, we define a novel uncrowding algorithm for hook-valued tableaux. The algorithm "uncrowds" the entries in the arm of the hooks and yields a set-valued tableau and a column-flagged increasing tableau. We prove that our uncrowding algorithm intertwines with crystal operators. An alternative uncrowding algorithm that “uncrowds” the entries in the leg instead of the arm of the hooks is also given. As an application of uncrowding, we obtain various expansions of the canonical Grothendieck polynomials. This is joint work with Jianping Pan, Joseph Pappe, and Wencin Poh.

Also see Jianping Pan's talk.

Gentle algebras, a long-studied class of finite dimensional algebras given by a presentation via quiver and relations, have recently played a central role in cluster theory and homological mirror symmetry associated to compact oriented surfaces with boundary. In the latter case, Haiden-Katzarkov-Kontsevich show that the partially wrapped Fukaya category of a surface with stops is triangle equivalent to the derived category of a graded gentle algebra.

In this talk we give an entirely representation theoretic construction of a surface model of the derived category of a (zero-graded) gentle algebra. We show how this surface model enables us to construct a complete invariant which distinguishes gentle algebras, and hence the associated partially wrapped Fukaya categories, up to derived equivalence.

Let W(n) denote the infinite-dimensional Lie algebra of vector fields on the affine space A^n, it has a maximal toral subalgebra T. We present classification of simple W(n)-modules semisimple over T with finite weight multiplicities. In particular, we show that all these modules can be described geometrically as generalized tensor modules (introduced independently by Larsson and Shen). The result is based on previous results of Mathieu and Xue-Lu and the main tool is parabolic induction.

We generalize type A quivers to continuous type A quivers and prove basic results about pointwise finite dimensional representations. We prove analogue of the Bar Code theorem to continuous quivers of type A with alternating orientation.

Starting with this, we define a generalization of the continuous cluster categories. These categories have several new features: continuous clusters and continuous mutations, unlike the original continuous cluster categories.

We describe two different bases for irreducible symmetric group representations: the tableaux basis from combinatorics (and from the geometry of a class of varieties called Springer fibers); and the web basis from knot theory (and from the quantum representations of Lie algebras).  We then describe new results comparing the bases for the case n=3, e.g. showing that the change-of-basis matrix is upper-triangular, and sketch some applications to geometry and representation theory.  This work is joint with H. Russell, as well as with T. Goldwasser and G. Sun.

Reduction algebras associated to a pair of Lie algebras (G, g) have been shown to act irreducibly on the space of primitive vectors of certain G-modules. The role of reduction algebras extends to the super case: Here we consider the diagonal reduction algebra of the pair of Lie superalgebras (osp(1|2) × osp(1|2), osp(1|2)) as a double coset space having an associative diamond product and give generators with relations.

Kazhdan–Lusztig (KL) cells partition Coxeter groups and are important to representation theory. One can compute KL cells of symmetric groups via the Robinson–Schensted correspondence, but for general Coxeter groups combinatorial descriptions of KL cells (or even non-recursive ways to compute them) are largely unknown except for cells of a-value 0 or 1, where a stands for an N-valued function defined on Coxeter groups by Lusztig that is constant on each cell. For example, it is known that every Coxeter group has a unique two-sided KL cell of a-value 1, which consists of all non-identity elements with a unique reduced word. 

We discuss some recent progress on KL cells of a-value 2. In particular, we classify Coxeter groups with finitely many elements of a-value 2, and for such groups we describe all KL cells of a-value 2 via Viennot’s heaps. This is joint work with Richard Green.

Etingof-Freund-Ma functor connects gl_N -modules to representations of degenerate affine Hecke algebra of type C_n and Jordan-Ma functor connects U_q(gl_N)-modules to affine Hecke algebra of type C_n. We consider the images of finite dimensional irreducible gl_N -modules and U_q(gl_N)-modules under these Schur-Weyl type functors. Moreover, we give combinatorial descriptions of these images in terms of standard tableaux.

The Drinfeld double D_n of the Taft algebra, whose ground field contains n-th roots of unity, has a list of 2-dimensional irreducible modules. For each of such module V, we show that there is a well-defined action of the Temperley-Lieb algebra TL_k on the k-fold tensor product of V, and this action commutes with that of D_n. We further establish that when V is self-dual and when k ≤ 2(n − 1), the centralizer algebra End_{D_n}(V^{⊗k}) is isomorphic to TL_k. Our inductive argument uses a rank function on the TL diagrams, which is compatible with the nesting function introduced by Russell-Tymoczko. This is joint work with Georgia Benkart, Rekha Biswal, Ellen Kirkman and Van Nguyen.