## The University of Leeds Algebra Seminar 2022/23   Tuesdays 15:00 (in MALL or occasionally in other places)

The seminar this semester is organised by Eleonore Faber . The seminar will take place at 15:00 on Tuesdays.

## Schedule:

 DATE and TIME and PLACE SPEAKER TITLE October 11, 15:00  MALL Francesca Fedele   (University of Leeds) Universal localizations of d-homological pairs October 18, 15:00  Baines Wing Miall LT (2.34) Ozgur Esentepe   (University of Leeds) Frobenius structures on the category of maximal Cohen-Macaulay modules October 25, 15:00  MALL Emine Yıldırım   (University of Leeds) Poset quivers November 1, 15:00  MALL Juliet Cooke   (University of Nottingham) Askey-Wilson Algebras as Skein Algebras November 8, 15:00  MALL Amit Shah   (Universitet Aarhus) Defining structure-preserving functors in higher homological algebra November 15, 15:00  hybrid Ezgi Kantarci Oguz   (Galatasaray University) Unimodality of Fences and Oriented posets November 22, 15:00  hybrid Tianyuan Xu   (Haverford College) 2-roots for simply laced Weyl groups November 29, 15:00  MALL Aram Dermenjian   (University of Manchester) Maximal subposets of the $\nu$-Tamari Lattice December 6, 15:00  MALL Ilaria Colazzo   (University of Exeter) Set-theoretic solutions to the Yang-Baxter equation and skew braces December 13, 15:00  MALL Daniel Soskin   (University of Notre Dame) Determinantal inequalities for totally positive matrices

## Abstracts

Francesca Fedele (University of Leeds): Universal localizations of d-homological pairs. Let k be an algebraically closed field and A a finite dimensional k-algebra. The universal localization of A with respect to a set of morphisms between finitely generated projective A-modules always exists. When A is hereditary, Krause and Stovicek proved that the universal localizations of A are in bijection with various natural structures. In this talk I will introduce the higher analogue of universal localizations, that is universal localizations of d-homological pairs with respect to certain wide subcategories, and show a (partial) generalisation of Krause and Stovicek result in the higher setup.

Ozgur Esentepe (University of Leeds): Frobenius structures on the category of maximal Cohen-Macaulay modules. If the words in the title did not mean anything to you, what you will get from this talk is that there is a class of nicely behaved modules over a class of nicely behaved rings. They contain information about singularities of the corresponding geometric object, they enjoy nice homological properties, they appear in many different areas of mathematics including commutative/noncommutative algebra/algebraic geometry, representation theory, mirror symmetry, singularity theory and even conformal field theory. I will gently introduce these modules in the first half of the talk. If you know the words in the title, you may know that the category of maximal Cohen-Macaulay has the (natural) structure of an exact category where the projectives and injectives coincide. I will show you that for any nonzerodivisor you give me in a Gorenstein local ring, I can produce an exact structure on the category of maximal Cohen-Macaulay modules recovering the original when the nonzerodivisor is 1. I will spend the second half of the talk explaining how this works.

Emine Yıldırım (University of Leeds): Poset quivers. Let Q be a quiver. We will talk about a construction of a new quiver which we call a poset quiver Q'=(Q,P) by replacing arrows in Q by linearly ordered posets. We consider the path category of Q' possibly with a weakly admissible ideal. Then we study pointwise finite representations. One of our main interests in this object is to study the decomposition of pointwise finite representations. We further analyse some homological properties in this setting. This is a work in progress with Charles Paquette and Job. D. Rock.

Juliet Cooke (University of Nottingham): Askey-Wilson Algebras as Skein Algebras. In this talk I will give a topological interpretation and diagrammatic calculus for the rank (n ? 2) Askey-Wilson algebra by proving there is an explicit isomorphism with the Kauffman bracket skein algebra of the (n + 1)-punctured sphere.
To do this I will show how the Askey-Wilson algebra can be considered as a subalgebra of the braided tensor product of n copies of either a quantum group or a reflection equation algebra. The Kauffman bracket skein algebra of the (n + 1)-punctured sphere is isomorphic with the Uq(sl2 ) invariants of the Alekseev moduli algebra which completes the correspondence. I will also discuss some corollaries of this result.

Amit Shah (Aarhus Universitet): Defining structure-preserving functors in higher homological algebra. Let d be a positive integer. In an abelian category one can define a d-cluster tilting subcategory. These subcategories are examples of d-abelian categories, which lie at the core of higher homological algebra. The extrinsic structure of a d-abelian is captured by so-called d-exact sequences that are longer analogues of short exact sequences. Just like it can be informative to study functions between sets or group homomorphisms between groups, it can be very helpful to study structure-preserving functors between categories. Between abelian categories these are called exact functors and they send short exact sequences to short exact sequences. The question now is: What would a structure-preserving functor from a d-abelian and to an abelian category be? What’s the right way to send a d-exact sequence to a short exact sequence? Or more than one short exact sequence? In this talk, I’ll report on progress on a project (joint with Raphael Bennett-Tennenhaus, Johanne Haugland and Mads H. Sandøy) trying to answer these questions.

Ezgi Kantarci Oguz (Galatasaray University): Unimodality of Fences and Oriented posets. Fence posets are combinatorial objects that come up in a variety of settings, recently in the work of Morier-Genoud and Ovsienko regarding q-defomed rationals. We prove their conjecture that says that rank polynomials of fence posets are unimodal by introducing a related class of circular fence posets. We further show a piece-by-piece combinatorial method of building fence posets (and others) and calculating their rank polynomials easily via 2x2 matrices. We discuss recent results and further work. (partially based on joint works with Emine Yıldırım, Mohan Ravichandran and Yalım Özel)

Tianyuan Xu (Haverford College): 2-roots for simply laced Weyl groups. We introduce and study 2-roots'', which are symmetrized tensor products of orthogonal roots of Kac--Moody algebras. We concentrate on the case where $W$ is the Weyl group of a simply laced Y-shaped Dynkin diagram with three branches of arbitrary finite lengths $a$, $b$ and $c$; special cases of this include types $D_n$, $E_n$ (for arbitrary $n\geq 6$), and affine $E_6$, $E_7$ and $E_8$.
With motivations from Kazhdan--Lusztig theory, we construct a natural codimension-$1$ submodule $M$ of the symmetric square of the reflection representation of $W$, as well as a canonical basis $\mathcal{B}$ of $M$ that consists of 2-roots. We conjecture that, with respect to $\mathcal{B}$, every element of $W$ is represented by a column sign-coherent matrix in the sense of cluster algebras, and we prove the conjecture in the finite and affine cases. We also prove that if $W$ is not of affine type, then the module $M$ is completely reducible in characteristic zero and each of its nontrivial direct summands is spanned by a $W$-orbit of 2-roots. (This is joint work with Richard Green.)

Aram Dermenjian (University of Manchester): Maximal subposets of the $\nu$-Tamari Lattice. The $\nu$-Tamari lattices were defined by Préville-Ratelle and Viennot. We study the subposets of the $\nu$-Tamari lattice which have either maximal in-degree or maximal out-degree. In particular cases, the maximal out-degree subposet will turn out to be poset isomorphic to smaller $\nu$-Tamari lattices and the maximal in-degree subposet will be poset isomorphic with a greedy order on $\nu$-Dyck paths. No previous knowledge is needed for this talk.

Ilaria Colazzo (University of Exeter): Set-theoretic solutions to the Yang-Baxter equation and skew braces. The Yang-Baxter equation is a fundamental equation in theoretical physics and pure mathematics, with many applications in different fields of mathematics. The importance of this equation led Drinfeld to propose the following problem: studying set-theoretical solutions. A key tool for exploring this family of solutions is the concept of a skew brace, introduced by Rump and Guarnieri and Vendramin. In this talk, we will review the basic theory of this algebraic structure and this family of solutions, describe their connection, discuss some problems, and give some applications.

Daniel Soskin (University of Notre Dame): Determinantal inequalities for totally positive matrices. Totally positive matrices are matrices in which each minor is positive. Lusztig extended the notion to reductive Lie groups. He also proved that specialization of elements of the dual canonical basis in representation theory of quantum groups at q=1 are totally non-negative polynomials. Thus, it is important to investigate classes of functions on matrices that are positive on totally positive matrices. I will discuss two sources of such functions. One has to do with multiplicative determinantal inequalities (joint work with M.Gekhtman). Another deals with majorizing monotonicity of symmetrized Fischer's products which are known for hermitian positive semidefinite case which brings additional motivation to verify if they hold for totally positive matrices as well (joint work with M.Skandera). The main tools we employed are network parametrization, Temperley-Lieb and monomial trace immanants.

For past algebra seminars see: 2021/22 , 2020/21 , 2019/20 , 2018/19 , 2017/18 .