Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Geometry and Representation Theory Seminar

Geometry & Representation - Kaveh Mousavand (UQAM)

Monday, December 3rd, 2018

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Kaveh Mousavand (UQAM)

Title: $\tau$-tilting finiteness of special biserial algebras

Abstract:  $\tau$-tilting theory, recently introduced by Adachi-Iyama-Reiten, is an elegant generalization of the classical tilting theory which fixes the deficiency of the tilting modules with respect to the notion of mutation. In this talk, I view $\tau$-tilting finiteness of algebras as a natural generalization of the representation finiteness property. The natural question then becomes: For which families of algebras does $\tau$-tilting finiteness imply representation finiteness?

First I introduce a reductive method that can be applied to certain families of algebras to reduce this, a priori, intractable problem to a subfamily with nice features. Then, as an interesting class of algebras, I consider the special biserial algebras and for every minimal representation infinite member of this family, I give a full answer to the above question and show. As a corollary, we conclude that a gentle algebra is $\tau$-tilting finite if and only if it is representation finite.

Geometry & Representation - Mike Roth (Queen's University)

Monday, November 26th, 2018

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Mike Roth (Queen's University)

Title: Generating Rays for the Eigencone (after Belkale and Piers)

Abstract:  Let G be a semisimple algebraic group. A fundamental question in the representation theory of G is knowing how to decompose the tensor product of two irreducible representations into its irreducible components, or slightly weaker, which irreducible components appear in a tensor product of two irreducible representations. The irreducible representations of G are parameterized by highest weights, vectors in ℕ^{r}, where r is the rank of G. For a highest weight λ the corresponding irreducible representation is denoted V_{λ} If one takes triples (λ, μ, ν) of highest weights such that V_{ν} appears in V_{λ} ⊗ V_{μ} then these triples generate a polyhedral cone in ℚ^{3r}, known as the eigencone (or sometimes the tensor cone). Trying to find explicit equations for the hyperplanes cutting out the eigencone is a problem with a long history, including fundamental contributions by Weyl, Gelfand, Lidskii, and Wielandt. Finally, twenty years ago, Klyachko found a set of hyperplane inequalities cutting out the eigencone in type A. Progress in the last 20 years has included finding hyperplane inequalities for the eigencones in all types, finding minimal hyperplane inequalities in all types, and finally, also finding descriptions of the linear conditions cutting out higher codimensional faces of the eigencone. Dually to their description by hyperplane inequalities, polyhedral cones may also be described by their generating rays. It is of course natural to then ask for the generating rays of the eigencone. This talk will discuss a recent paper of Belkale and Piers giving a recursive method, valid in all types, of finding generating rays for the eigencone.

Geometry & Representation - Yin Chen (NE Normal University, China)

Monday, November 12th, 2018

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Yin Chen (Northeast Normal University, China)

Title: On vector invariant fields for finite classical groups

Abstract:  In the recent work (https://doi.org/10.1016/j.jpaa.2018.07.015) with David Wehlau, we found a minimal polynomial generating set for the vector and covector invariant field of the general linear group over finite fields. Our method relied on some relations between the generators for the invariant ring of one vector and one covector. The remaining case (without covectors) is more complicated. In this talk, I will present an approach to find polynomial generating sets for the vector invariant fields of the most of finite classical groups. This is a joint work with Zhongming Tang.

Geometry & Representation - Steven Spallone (Indian Institute)

Monday, November 5th, 2018

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Steven Spallone (Indian Institute of Science Education and Research)

Title: Spinoriality of Orthogonal Representations of Reductive Groups

Abstract:  Let G be a connected semisimple complex Lie group and $\pi$ an orthogonal representation of G. We give a simple criterion for whether $\pi$ lifts to the spin group Spin(V), in terms of its highest weights. This is joint work with Rohit Joshi.

Geometry & Representation - Yin Chen (NE Normal University, China)

Monday, October 29th, 2018

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Yin Chen (Northeast Normal University, China)

Title: On vector invariant fields for finite classical groups

Abstract:  In the recent work (https://doi.org/10.1016/j.jpaa.2018.07.015) with David Wehlau, we found a minimal polynomial generating set for the vector and covector invariant field of the general linear group over finite fields. Our method relied on some relations between the generators for the invariant ring of one vector and one covector. The remaining case (without covectors) is more complicated. In this talk, I will present an approach to find polynomial generating sets for the vector invariant fields of the most of finite classical groups. This is a joint work with Zhongming Tang.

Geometry & Representation - Ben Webster (Waterloo)

Monday, October 1st, 2018

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Ben Webster (University of Waterloo/Perimeter Institute)

Title: Representation theory of symplectic singularities

Abstract:  There are a lot of non-commutative algebras out there in the world, so if you want to study some of them, you have to have a theory about which are especially important. One class I find particularly interesting are non-commutative algebras which "almost" commutative and thus can be studied with algebraic geometry, giving a rough dictionary between certain non-commutative algebras and certain interesting spaces. This leads us to a new perspective on some well-known algebras, like universal enveloping algebras, and also to new ones we hadn't previously considered. The representations of the resulting algebras have a lot of interesting structure, and have applications both in combinatorics and in the construction of knot invariants.

Geometry & Representation - Emine Yildrim (Queen's University)

Monday, September 24th, 2018

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Emine Yildrim (Queen's University)

Title: The bounded derived category for cominuscule posets

Abstract:  Cominuscule posets come from root posets and have connections to Lie theory and Schubert calculus. We are interested in whether the bounded derived category of the incidence algebra of a cominuscule poset is fractionally Calabi-Yau. In other words, we ask if some non-zero power of the Serre functor is a shift functor. We answer this question on the level of the Grothendieck groups. On the Grothendieck group this functor becomes an endomorphism called the Coxeter transformation. We show that Coxeter transformation has finite order for two of the three infinite families of cominuscule posets, and for the exceptional cases. Our motivation comes from a conjecture by Chapoton which states that the bounded derived category of incidence algebra of root posets is fractionally Calabi-Yau. Our result can be thought of as a parabolic analogue of Chapoton's conjecture.

Geometry & Representation - Tianyuan Xu (Queen's University)

Monday, September 17th, 2018

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Tianyuan Xu (Queen's University)

Title: Broken lines and the topological ordering of the alternating quiver of type A

Abstract:  The Positivity Conjecture in cluster algebra theory states that the coefficients of the Laurent expansion of any cluster variable in a cluster algebra are always positive integers. In 2014, Gross, Hacking, Keel and Kontsevich constructed a so-called Theta function basis to prove the conjecture for all cluster algebras of geometric type. A key ingredient in the construction of the Theta functions is the broken line model. In this talk, we will discuss the broken lines associated to the alternating quiver of type A, with an emphasis on relating its combinatorial properties to the topological ordering of the quiver, the partial order obtained by taking the transitive and reflexive losure of the relation “v<w if v->w is an edge” on the vertices of the quiver.

The talk is based work in progress with Ba Nguyen, David Wehlau and Imed Zaguia.

Geometry & Representation - Charles Paquette (Queen's/RMC)

Monday, September 10th, 2018

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Charles Paquette (Queen's/RMC)

Title: A quiver construction of some subalgebras of asymptotic Hecke algebras

Abstract:  Lusztig defines an asymptotic Hecke algebra J from a Coxeter system (W,S). This is an algebra that is defined using the Kazhdan-Lusztig (KL) basis of the corresponding Hecke algebra of (W,S). Even though these KL bases are generally hard to understand, there is a two-sided cell C of W that gives rise to a nice subalgebra J_C of J having rich combinatorics and whose algebraic description does not use KL bases. We will see that J_C has a presentation using a quiver with relations, and this allows one to study the representation theory of J_C (and of J) from another perspective. Using quiver representations, we will see that the classification of simple modules, which falls into three categories (finite type, bounded type and unbounded type), can be characterized completely using the shape of the weighted graph G of (W,S).

This is joint work with I. Dimitrov, D. Wehlau and T. Xu.

Geometry & Representation - Mike Zabrocki (York University)

Wednesday, March 28th, 2018

Time: 3:30 p.m.  Place: Jeffery Hall 319

Speaker: Mike Zabrocki (York University)

Title: A Multiset Partition Algebra

Abstract:  Schur-Weyl duality is a statement about the relationship between the actions of the general linear group $Gl_n$ and the symmetric group $S_k$ when these groups act on $V_n^{\otimes k}$ (here $V_n$ is an $n$ dimensional vector space). If we consider the symmetric group $S_n$ as permutation matrices embedded in $Gl_n$, then the partition algebra $P_k(n)$ (introduced by Martin in the 1990's) is the algebra which commutes with the action of $S_n$.

In this talk I will explain how an investigation of characters of the symmetric group leads us to consider analogues of the RSK algorithm involving multiset tableaux. To explain the relationship of the combinatorics to representation theory we were led to the multiset partition algebra as an analogue of the partition algebra and the dimensions of the irreducible representations are the numbers of multiset tableaux.

This is joint work with Rosa Orellana of Dartmouth College.

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