Flags, Galleries and Reflection Groups

Schubert calculus puzzles from quiver varieties

Abstract: In 2006 Paul Zinn-Justin observed that the puzzle rule for equivariant Schubert calculus on Grassmannians, due to me and Terry Tao in 2003, was based on an "R-matrix", a solution to the Yang-Baxter equation. In 2017 Zinn-Justin and I extended this to discover and create puzzle rules for K-theory of 2- and 3-step flag manifolds. [Maulik and Okounkov '12] trace R-matrices to Nakajima quiver varieties (whose definition I'll recall), and I'll explain how our puzzles can be seen directly from the quiver varieties. Then I'll show some further extensions, some of which were most easily discovered via the quiver variety interpretation.

From puzzles to Lagrangian correspondences: a branching story

Abstract: I will first present a puzzle rule for restricting type A Schubert classes for the Grassmannian to type C in equivariant cohomology, in joint work with Knutson and Zinn-Justin. I will then describe an extension of this rule to cotangent bundles in the context of Lagrangian correspondences between symplectic resolutions, and Maulik-Okounkov classes. Time permitting, I will also discuss a new perspective on the associativity of the product of Schubert classes. This talk is connected to and relies on material from Allen Knutson’s talk.

Dual approach to Coxeter and Artin groups

Abstract: Let $(W,S)$ be a Coxeter system with set of reflections $T = \cup_{w \in W}w(S)$, and let ${\cal A}(W)$ be the related Artin group. About 15 years ago the so called {\em dual approach} to study Coxeter as well as Artin groups has been started. The idea is to systematically replace $S$ by $T$. For Artin groups of spherical type a new presentation could be given, and it has been shown that these groups are so called Garside groups with Garside element a preimage of a Coxeter element in $(W,T)$. In my talk I will explore a generalization of this concept in two different directions. The first direction is to generalize from Coxeter elements to so called quasi Coxeter elements, and the second to move to reflection groups that are extended Coxeter groups and that have applications in representation theory.

The intrinsic noncrossing combinatorics of polynomials with distinct roots

Abstract: The space of all degree n polynomials with distinct roots is naturally equal to the quotient of the complement of the braid arrangement by the symmetric group and it has the n strand braid group as its fundamental group. Another classifying space associated to the n strand braid group is the dual braid complex, a metric simplicial complex built out of orthoschemes that has local links that can be described as a union of a specific set of apartments in a spherical building. In this talk I will describe a concrete connection between the two perspectives, following up on a comment from Daan Krammer and using ideas of Bill Thurston and his collaborators. This is joint work with Michael Dougherty.

An insertion algorithm on multiset partitions with applications to diagram algebras

Abstract: We generalize the Robinson-Schensted-Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length k with entries in [n] and pairs of tableaux of the same shape with one being a standard Young tableau of size n and the other being a standard multiset tableau of content [k]. We also obtain an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson. This is joint work with Laura Colmenarejo, Rosa Orellana, Franco Saliola, and Mike Zabrocki (arXiv:1905.02071).

Coxeter sortable elements and dual braid monoids

Abstract: We explain how to express the simple elements of the dual Garside structure of an Artin group of spherical type in terms of the standard Artin generators, using Reading's Coxeter sortable elements and their connection to noncrossing partitions.

Stabilisers of eigenvectors in complex reflection groups

Abstract: Eigenspaces of elements of finite reflection groups have been studied by Springer, Lehrer-Springer and others, due to their close relationship the polynomial invariants if the group. More recently, in the setting of real reflection groups, Kamgarpour proved an inequality relating the stabiliser of an eigenvector with the corresponding eigenvalue. In this talk, I will present my MPhil research, in which I generalise Kamgarpour's result to the setting of complex reflection groups. I will also discuss connections with braid groups.

Path models in Kazhdan-Lusztig theory

Abstract: The theory of Kazhdan-Lusztig cells plays a fundamental role in the representation theory of Coxeter groups and Hecke algebras. The theory is relatively well understood in the "equal parameter" case, primarily due to the geometric interpretation of Kazhdan-Lusztig theory in this setting and the positivity properties that follow from this interpretation. In contrast much less is known in the unequal parameter case, where these positivity properties no longer hold.

In 2003 Lusztig made 15 conjectures, now known as P1-P15, to capture essential properties of cells for arbitrary parameters. In this talk we outline a proof of P1-P15 for affine types G_2 and C_2 with arbitrary parameters. Our approach is based on two concepts: (a) a balanced system of cell representations for the Hecke algebra, and (b) the asymptotic Plancherel formula. In this talk we will focus on (a), and outline how alcove path combinatorics play a key role in the proof.

This is joint work with Jeremie Guilhot, Universite de Tours.

Coxeter groups and hook formulas

Abstract: We consider "equivariant" cohomology for Coxeter groups in general, and make a hook like formula for them using "equivariant" Schubert calculus.

Twisted quadratic foldings of root systems

Abstract: In this talk I will discuss joint work with Kirill Zainoulline about twisted foldings of root systems. This construction generalises usual involutive foldings corresponding to automorphisms of Dynkin diagrams. Our motivating example is the non-split folding of the root system of type E8 which gives rise to the root system of type H4, appeared already in the Eighties in work of Lusztig. Using moment graph techniques we show that twisted quadratic foldings can be applied to obtain information about equivariant cohomology.

Facets of tensor cone of symmetrizable Kac-Moody Lie algebras

Abstract: This is a joint work with Nicolas Ressayre. In this work, we are interested in the decomposition of the saturated tensor product of two representations of a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$. Let $P_+$ be the set of dominant integral weights. For $\lambda \in P_+,$ $L(\lambda)$ denotes the (irreducible) integrable, highest weight representation of g with highest weight $\lambda$. Let $P_{+,\mathbb{Q}}$ be the rational convex cone generated by $P_+$. Consider the saturated tensor cone

$\Gamma(\mathfrak{g}) :=$ $\{(\lambda_1, \lambda_2, \mu) \in P^3_{+,\mathbb{Q}}$ $\mid \exists N \geq 1$ with $L(N\mu) \subset L(N\lambda_1) \otimes L(N\lambda_2)\}.$

If $\mathfrak{g}$ is finite dimensional, $\Gamma(\mathfrak{g})$ is a convex polyhedral cone described by Belkale-Kumar by an explicit finite list of inequalities, which was proved to be an irredundant set of inequalities by Ressayre. In general, $\Gamma(\mathfrak{g})$ is neither polyhedral, nor closed. Brown-Kumar obtained a list of inequalities that describe $\Gamma(\mathfrak{g})$ conjecturally. In this work, we prove that each of the Brown-Kumar’s inequalities corresponds to a codimension one face of $\Gamma(\mathfrak{g})$. Ressayre has proved that this set of inequalities is sufficient in the case of affine Kac-Moody Lie algebras. The sufficiency question is still open for non-affine $\mathfrak{g}$.

Whittaker modules and parabolic Kazhdan-Lusztig polynomials

Abstract: (Sign) parabolic Kazhdan-Lusztig polynomials describe the relationship between two natural bases of the antispherical module of the Hecke algebra associated to a Coxeter system. They were first introduced by Deodhar, and Kashiwara-Tanisaki showed that in the case of Weyl groups of symmetrisable Kac-Moody Lie algebras, their coefficients give the dimensions of intersection cohomology groups of Schubert varieties in partial flag manifolds. In this talk, I’ll describe how parabolic Kazhdan-Lusztig polynomials are also related to Whittaker modules, which are a class of infinite-dimensional representations of a Lie algebra first studied by Kostant for their relationship to the Whittaker equation in number theory.

Rational Dyck Paths in the Non-Relatively Prime Case

Abstract: The Catalan numbers $1, 2, 5, 14 \ldots$ is one of the most well-known sequences in combinatorics. It enumerates over $100$ families of combinatorial objects. Some of these families include the set of Dyck paths in an $n \times (n+1)$ rectangle, the set of $(+n, +n+1)-$ invariant subsets of $\mathbb{N}$ containing $0$, simultaneous $(n, n+1)$-cores, the ``$(n+1)$-restricted" affine permutations $w \in \widehat S_n / S_n$, the number of cells in a certain affine Springer fibre, a basis of the finite-dimensional representation $e L_{(n+1)/n}$ of the spherical Cherednik algebra $e H_n e$. The above families and the bijections between them all generalize from $(n, n+1)$ to $(n,m)$ when $\gcd(n,m) = 1$. However when $\gcd(dn,dm) = d > 1$, many of these break: in particular some of the sets stay finite while others become infinite.

I will discuss an equivalence relation on the infinite set of $(+dn, +dm)-$invariant subsets of $\mathbb{N}$, such that its equivalence classes are again in bijection with finite set of Dyck paths in a $dn \times dm$ rectangle. Our hope is that this construction will lead to a geometric or representation theoretic interpretation of the dinv statistic from the $d=1$ case.

This is based on joint work with Eugene Gorsky and Mikhail Mazin. (arXiv:1703.02668)

The triumvirate of affine flag varieties

Abstract: This will be a survey talk about the positive level (thin) affine flag variety, the negative level affine flag variety (thick) affine flag variety, and the level 0 (semi-infinite) affine flag variety. The first part will be a discussion of what I know (gallery models for points) and the second part will be a discussion of things I don’t know but would like to know (a universal Borel-Weil-Bott theorem covering all three cases simultaneously and the K-theory of Springer fibres as DAHA-modules).

Coxeter factorizations and the Matrix Tree theorem with generalized Jucys-Murphy weights

Abstract: Questions regarding the enumeration of factorizations in groups go back to at least Hurwitz and Frobenius and the genesis of representation theory. After the 80's and work of Stanley and Jackson, renewed attention brought a flury of researchers in the area. A particularly popular case studies factorizations of the Coxeter element c in complex reflection groups; notably, there is the Chapuy-Stump product formula for the generating series of arbitrary length reflection factorizations of c.

In the type A situation, analogous enumerative results are obtained as corollaries of Kirchhoff's Matric Tree theorem after the determinant calculation of the Laplacian L(K_n). Burman and Zvonkine achieved a beautiful generalization of this result by relating the enumeration of weighted factorizations of the long cycle, where each transpotion (ij) is weighted by its own variable w_ij, with the eigenvalues of the (weighted) Laplacian.

In joint work with Guillaume Chapuy, we consider a (partial) analog of the weighted Laplacian for complex reflection groups. The weights are specified via a flag of parabolic subgroups, generalizing the definition of Jucys-Murphy elements. We prove a product formula for the enumeration of weighted reflection factorizations of Coxeter elements, that subsumes the Chapuy-Stump formula and in part the Burman-Zvonkine formula. Its proof is based on an interesting fact that relates the exterior powers of the reflection representation and the W-characters that are non-zero on the Coxeter class. This fact is uniformly stated but proven in a case-by-case fashion.

Facial weak order in hyperplane arrangements

Abstract: We describe the facial weak order, a poset structure that extends the poset of regions on a central hyperplane arrangement to the set of all faces of the arrangement which was first introduced on the braid arrangements by Krob, Latapy, Novelli, Phan and Schwer. We provide various characterizations of this poset including a local one, a global one, and a geometric one. We then show that the facial weak order is in fact a lattice for simplicial hyperplane arrangements, generalizing a result by Björner, Edelman and Zieglar showing the poset of regions is a lattice for simplicial arrangements.

Flag varieties and representations of p-adic groups

Abstract: Deligne-Lusztig varieties are subvarieties of flag varieties whose cohomology encodes the representations of reductive groups over finite fields. In this talk, we discuss recent progress towards geometric realizations of representations of p-adic groups in three arcs: affine flag varieties, semi-infinite flag varieties, and a finite-ring variant arising from the Moy-Prasad filtration of parahoric subgroups. This is joint work with Alexander Ivanov.

The isomorphism problem for Coxeter groups

Abstract: It is still unknown how to decide if two given finite rank Coxeter groups are isomorphic. I will survey what is known and what is conjectured about this problem.

Coxeter Systems for which the Brink-Howlett automaton is minimal

Abstract: In their celebrated 1993 paper, Brink and Howlett proved that all finitely generated Coxeter groups are automatic. In particular, they constructed a finite state automaton recognising the language of reduced words in a Coxeter group. This automaton is not minimal in general, and recently Christophe Hohlweg, Philippe Nadeau and Nathan Williams stated a conjectural criteria for the minimality. In this talk we will explain these concepts, and prove the conjecture of Hohlweg, Nadeau, and Williams. This work is joint with James Parkinson.

Regularity theorem for totally nonnegative flag varieties

Abstract: The totally nonnegative part TNN(G/P) of a partial flag variety G/P of a split real reductive group G was defined by Lusztig. We show that TNN(G/P) is a regular cell complex, confirming conjectures of Postnikov and Williams. A key ingredient of the proof is an embedding of the projected Richardson stratification of G/P into the affine flag variety of G.

This is joint work with Pasha Galashin and Steven Karp.

Biclosed sets and closure operators on root systems

Abstract: Biclosed subsets of positive roots (closed subsets with closed complement, for a certain closure operator on the positive roots of a Coxeter group) may be viewed as inversion sets of "possibly infinitely long" elements of a conjectural ortholattice completion of weak order on the Coxeter group. Many interesting algebraic, geometric, combinatorial, and representation theoretic objects or structures may (actually or conjecturally) be attached to such biclosed sets, generalizing standard constructions of important objects and structures associated to elements of Coxeter groups. This talk will survey some recent results and conjectures involving various natural closure operators on root systems and related structures, arising out of study of long-standing conjectures in this area.

Retractions from infinity

Abstract: Positively folded galleries arise as images of retractions of buildings onto a fixed apartment and play a role in many areas of math, such as in the study of affine Hecke algebras, Macdonald polynomials, MV-polytopes, and affine Deligne-Lusztig varieties. In this talk, we will define positively folded galleries and then look at a new recursive description of the set of end alcoves of folded galleries which are positive with respect to alcove-induced orientations. These results further allow us to find the images of retractions from certain points at infinity. This talk is based on joint work with Elizabeth Milićević, Petra Schwer and Anne Thomas.

Stratifications of Affine Deligne-Lusztig Varieties

Abstract: Affine Deligne-Lusztig varieties are analogues, in the context of an affine root system, of classical Deligne-Lusztig varieties. Many of their basic properties are closely related to the Bruhat-Tits building of the underlying group. In this talk, I will explain certain stratifications of affine Deligne-Lusztig varieties, their meaning in terms of the building, and how they compare. I also intend to discuss applications to arithmetic geometry, i.e., Rapoport-Zink spaces and Shimura varieties. Part of this is joint work with Xuhua He and Sian Nie, and with Xuhua He and Michael Rapoport.

Reflection subgroups and positivity

Abstract: The first half of this talk will be a brief introduction to some problems in modular representation theory. All these problems relate to the question of determining the dimensions and characters of certain modules for algebraic groups and related finite groups (e.g. symmetric groups). The answers turn out to involve certain "p-canonical bases", which are difficult to calculate directly. I will then discuss a remarkable positivity property of this basis with respect to certain reflection subgroups of affine Weyl groups.