(Timetable)
Ben Elias (University of Oregon): Introduction to Soergel bimodules and their diagrammatics
Soergel bimodules are certain bimodules over a polynomial ring, defined by Soergel in the 90s as a simpler and purely algebraic way to access the high-tech categories (from representation theory and geometry) which play a role in the Kazhdan-Lusztig conjectures. In type A, they form a categorification of the group algebra of the symmetric group. In the 2010s, Soergel bimodules were given a diagrammatic description by Elias, Khovanov, and Williamson, making it even easier to study them and do computations. This lecture series will introduce the algebraic and diagrammatic approaches to Soergel bimodules, and will use the opportunity to explain some aspects of diagrammatic philosophy. We focus on type A. In Hogancamp's lecture series, Soergel bimodules will be used to categorify the braid group and study link invariants.
Talk 1: Soergel bimodules, algebraically
We explain what it means to categorify the group algebra of the symmetric group. We introduce Bott-Samelson bimodules and Soergel bimodules and their filtrations by standard bimodules. We state the Soergel categorification theorem.
Talk 2: Soergel bimodules, diagrammatically
We introduce diagrammatics which encode morphisms between Bott-Samelson bimodules. We discuss Frobenius extensions in more depth. We discuss how to find idempotent decompositions diagrammatically. We discuss the quotient of the Soergel category which categorifies the Temperley-Lieb algebra.
Talk 3: Hom forms and bases
We introduce two of the main tools in the diagrammatician's toolkit: hom forms, and double leaves bases. We illustrate how the hom form can be used to guess the relations in the diagrammatic presentation. We demonstrate how the hom form motivates the double leaves basis, and explain how it is constructed.
Matthew Hogancamp (Northeastern University)
TBA
Linhui Shen (Michigan State University): Introduction to Cluster Structures on Braid Varieties
Cluster algebras are a class of commutative algebras introduced by Fomin and Zelevinsky around 2000. An important family of cluster algebras arises from the study of braid varieties and decorated character varieties, with applications in diverse areas including representation theory, contact geometry, and integrable systems. In this lecture series, we present an explicit construction of the cluster structures on braid varieties using the diagrammatic tools of weaves and Lusztig cycles. We prove that the coordinate ring of every braid variety admits a cluster algebra structure, thereby establishing special cases of a conjecture of Bernard Leclerc. As an application, we explain how these cluster structures are connected to the theory of canonical bases in representation theory and to the study of exact Lagrangian fillings of Legendrian knots.
Elise Catania (University of Minnesota - Twin Cities): Dimer Covers, Web Duality, and Grassmannian Cluster Algebras
The coordinate ring of the Grassmannian is one of the most well-studied examples of a cluster algebra. Cluster algebras, introduced by Fomin–Zelevinsky, are commutative rings built from a recursive process, and understanding their generators (cluster variables) is a central problem. Powerful tools for studying these cluster variables come from the combinatorics of dimer covers on plabic graphs as well as SL(k) webs, certain planar graphs that encode tensor invariants. In this talk, we give dimer expansion formulas for images of cluster variables under an important automorphism called the twist map. The coefficients are understood using web duality, where two different web bases are dual under a natural pairing defined by Fraser–Lam–Le. We verify this duality for a large family of SL(3) and SL(4) webs. We also discuss connections to conjectures of Fomin–Pylyavskyy on the classification of cluster variables. This is joint work with Esther Banaian, Christian Gaetz, Miranda Moore, Gregg Musiker, and Kayla Wright.
Donghyun Kim (Ewha Womans University): Symmetric function theory in (m,n)-world
We introduce a combinatorial framework for studying expressions of the form \( f[-MX^{m,n}] \cdot 1 \), where \( f \) is a symmetric function and \( f[-MX^{m,n}] \) denotes the corresponding element in the elliptic Hall algebra. This framework is motivated by Wilson’s conjecture, which suggests a symmetric-function lift of torus link homology. We also discuss a combinatorics of (rational) affine Springer fibers within this framework.
This talk is based on joint work with Jaeseong Oh.
Myungho Kim (Kyung Hee University): I-boxes and weaves
Demazure weaves are combinatorial objects that provide initial seeds for cluster algebra structures on braid varieties. In this talk, we focus on double Bott–Samelson cells, a special class of braid varieties. On the other hand, double Bott–Samelson cells are, up to localization, isomorphic to Grothendieck rings of certain monoidal subcategories of representations of quantum affine algebras. The initial seeds for these categories can be described combinatorially using I-boxes. This naturally leads to a correspondence between double inductive weaves and I-boxes, which we have studied recently. I will explain this correspondence in the talk. This is a joint work with Jisun Huh, Woo-Seok Jung, and Euiyong Park.
Hankyung Ko (Uppsala University): Singular Soergel Calculus
Singular Soergel bimodules encode the translation functors in Lie theoretic representation theory such as that of reductive groups in positive characteristic and category O for semisimple Lie algebras. This talk introduces diagrammatics for the singular Soergel bimodules, based on joint work with Ben Elias, Nicolas Libedinsky, Leonardo Patimo. In particular we discuss a presentation by generators and relations, analogous to Elias-Williamson's Soergel calculus.
Mikhail Mazin (Kensas State University): Recursive computations for Khovanov-Rozansky homology beyond the torus case
The Hogancamp-Mellit recursion, originally used to compute the Khovanov-Rozansky homology of torus knots and links, can also be applied to knots beyond the torus case. Namely, one can compute KR homology of the so called shortcut torus knots, which also happen to be isotopic to monotone knots of Galashin and Lam corresponding to a straight line with not-necessarily integer end points, or Coxeter knots of triangular partitions in the sense of Oblomkov and Rozansky. Furthermore, the recursion allows one to relate the Poincare polynomials of the KR homology of such knots to triangular Catalan and Schroder polynomials, and to the Shuffle theorem under any line of Blasiak, Haiman, Morse, Pun, and Seelinger.
In this talk I will introduce the above-mentioned classes of knots, explain how Hogancamp-Mellit recursion applies to them, and how the answer is related to the Shuffle theorems. The talk is based on a joint paper with Carmen Caprau, Nicolle Gonzales, and Matt Hogancamp.
Nathan Williams (University of Texas at Dallas): Noncrossing Combinatorics, the Full Twist, and Decategorification of Knot Invariants
Much recent work in knot theory has consisted of categorifying, and thereby strengthening, knot invariants. We take the opposite approach: decategorification, more commonly called combinatorics. Using the pure braid group, we introduce a new technique that relates the dual braid group and factorization problems in reflection groups to knot invariants. We prove that the (a,z=0)-Homfly polynomial can be computed as a solution to such a problem. This technique was motivated by and has applications to Coxeter--Catalan combinatorics. For example, in a surprising sort of combinatorial reciprocity, we show that noncrossing partitions arise naturally from our construction applied to positive powers of the full twist, while cluster complexes come from the same construction applied to negative powers. The same method produces noncrossing parking models, and also gives a new proof of EL-shellability and the homotopy type of the noncrossing partition lattice. In crystallographic type, we exploit the conjugacy of all Coxeter elements to give the first reflection subword rational noncrossing Catalan models. This is joint work with Colin Defant.