# FY2021 Annual Report

Speyer Unit
Assistant Professor Liron Speyer

## 1. Staff

• Dr Liron Speyer, Assistant Professor
• Dr Chris Chung, Postdoctoral Scholar
• Dr Eoghan McDowell, Postdoctoral Scholar
• Dr Louise Sutton, Postdoctoral Scholar
• Dr Jieru Zhu, Postdoctoral Scholar
• Berta Hudak, PhD Student
• Yukiko Nakagawa, Research Unit Administrator

## 2. Collaborations

• Susumu Ariki, Osaka University
• Nicholas Davidson, Reed College
• Jonathan Kujawa, University of Oklahoma
• Robert Muth, Washington & Jefferson College
• Daniel Tubbenhauer, Bonn University
• Paul Wedrich, Max Planck Institute of Mathematics
• Mark Wildon, Royal Holloway, University of London

## 3. Activities and Findings

### 3.1 Schurian-infinite blocks of type A Hecke algebras

For any algebra $$A$$ over an algebraically closed field $$\mathbb{F}$$, we say that an $$A$$-module $$M$$ is Schurian if $$\mathrm{End}_A(M) \cong \mathbb{F}$$. We say that $$A$$ is Schurian-finite if there are only finitely many isomorphism classes of Schurian $$A$$-modules, and Schurian-infinite otherwise. In this project, we showed that all principal blocks of weight at least 2 of type A Hecke algebras are Schurian-infinite in any characteristic. We also showed that almost all weight 2 and 3 blocks of Hecke algebras are also Schurian-infinite. To prove these results, our paper employs a wide variety of techniques from the (graded) representation theory of Hecke algebras.

### 3.2 The structure of Specht modules labelled by bihooks

The Hecke algebras of type $$B_n$$ have Specht modules indexed by bipartitions of $$n$$, which are an important family of modules for understanding the representation theory of these algebras. Realising the algebras as cyclotomic KLR algebras, via the famous Brundan–Kleshchev isomorphism, these Specht modules can even be graded. The Specht modules are usually indecomposable, but for certain choices of Hecke parameters, many Specht modules are decomposable. For those Specht modules labelled by bihooks – i.e. pairs of hook partitions – we found large families of decomposable Specht modules, which we conjectured to be a complete list of such modules, outside of the degenerate charactersitic 2 case. We derived a formula for the graded composition multiplicities of our decomposable Specht modules in all characteristics, and when the characteristic of the field is not too small, we explicitly determine the structure of all summands.

### 3.3 Small $$\imath$$quantum covering groups

Quantum symmetric pairs are an algebraic quantization of symmetric pairs for universal enveloping algebras. Recent developments by Bao–Wang and others show that many results in the theory for quantum groups admit deep and meaningful generalizations in the setting of quantum symmetric pairs. One such example is the $$\imath$$quantum group at roots of unity by Bao–Sale, a generalization of Lusztig's construction for quantum group at roots of unity, which is closely tied to the modular representation theory of algebraic groups and affine Lie algebras. Quantum covering groups are a generalization of quantum groups that also 'cover' quantum supergroups of anisotropic type. This project investigates the $$\imath$$quantum covering group at roots of unity, generalizing Lusztig’s quantum Frobenius morphism and constructing the small $$\imath$$quantum covering group.

### 3.4 Determination of characters by their values on $$p'$$-classes

Ordinary character values on $$p'$$-classes (that is, classes of order not divisible by $$p$$, a prime) and $$p$$-modular decomposition numbers are connected via the Brauer character table. In particular, two rows of the decomposition matrix are equal if and only if the corresponding pair of ordinary characters agree on $$p'$$-classes. In this project we identify all such pairs for the alternating group and the double covers of the symmetric and alternating groups, for $$p \neq 3$$. To do so, we consider the centre of the group algebra. We show that the $$p'$$-class sums generate most of the centre, and then use some additional character theory to argue that, with specified exceptions, the central characters (and hence the ordinary characters) are uniquely determined by their values on the $$p'$$-classes.

### 3.5 Affinization of web categories for superalgebras

Recent work of Cautis–Kamnitzer–Morrison uses the web category to reformulate a classical result in invariant theory, which describes tensor invariants for $$\operatorname{SL}_2$$ and $$\operatorname{SL}_3$$. This new formulation describes morphisms in a subcategory of $$\operatorname{SL}_n$$ modules using planer graphs. In our work we construct affine webs, i.e. webs with special dot which represents the action of the Casimir tensor element. The corresponding object is obtained by tensoring an arbitrary module with symmetric powers of the natural module. We have computed various relations coming from interactions between the Casimir operators and other generators of these category which are the merge and split maps. These relations are then captured by diagrammatics as pushing dots post trivalent vertices and crossings.

### 3.6 Tilting modules for $$\operatorname{SL}_2$$ in mixed characteristics

In many settings, the Temperley–Lieb algebra is isomorphic to the endomorphism algebra on the tensor representation for  $$\operatorname{SL}_2$$. The Jones–Wenzl projectors give descriptions at the morphisms level for tilting modules in characteristic zero. Their characteristic $$p$$ version is provided by Burrull–Libedinsky–Sentinelli. Tubbenhauer–Wedrich then gave a full list of diagrammatic relations of these $$p$$-JWs in the tilting category. Our project extends this result to the mixed character case, i.e. a field with characteristic $$p$$ containing all $$\ell$$-th roots of unity, and gives fusion rule for tensoring with the natural module at the object and morphism levels. Other interesting facts concern the Müger center and Verlinde quotients of this category.

## 4. Publications

### 4.1 Journals

1. Louise Sutton, Daniel Tubbenhauer, Paul Wedrich, and Jieru Zhu. SL_2 tilting modules in the mixed case, arXiv:2105.07724.
2. Robert Muth, Liron Speyer, and Louise Sutton. Decomposable Specht modules indexed by bihooks II. Algebras and Representation Theory, to appear. doi: 10.1007/s10468-021-10093-3
3. Susumu Ariki and Liron Speyer. Schurian-finiteness of blocks of type A Hecke algebras, arXiv:2112.11148.
4. Yiqiang Li and Jieru Zhu. Quasi-symmetric pairs of $$U(gl_N)$$ and their Schur algebras. Nagoya Mathematical Journal, 245 (2022), 1-27. doi: 10.1017/nmj.2020.16
5. Eoghan McDowell and Mark Wildon. Modular plethystic isomorphisms for two-dimensional linear groups. Journal of Algebra, to appear. arXiv:2105.00538.

### 4.2 Books and other one-time publications

Nothing to report

### 4.3 Oral and Poster Presentations

1. Liron Speyer. Semisimple Specht modules indexed by bihooks. London Algebra Colloquium, Queen Mary University of London, April 2021.
2. Jieru Zhu. Two boundary degenerate affine Hecke-Clifford algebra for gl(n|m) and q(n)​. AMS special session: Interactions between Representation Theory, Poisson Geometry, and Noncommutative Algebra, April 2021.
3. Jieru Zhu. The change-of-coordinate matrix from the polytabloid basis to the web basis​. Reed College math colloquim, April 2021.
4. Jieru Zhu. Two boundary degenerate affine Hecke (Clifford) algebras for Lie super algebras gl(n|m) and q(n)​. AMS special session: Categorical and Combinatorial Methods in Representation Theory, and Related Topics, May 2021.
5. Jieru Zhu. Categorification of Schur algebras of Type B and beyond.AMS special session: Geometric and Categorical Methods in Representation Theory, May 2021.
6. Jieru Zhu. Transitioning between the polytabloid and web bases for the Specht modules. AMS special session: Diagrammatic and Combinatorial Methods in Representation Theory. May 2021.
7. Jieru Zhu. Categorification of the Schur algebras of type B and beyond. Algebraic Lie Theory and Representation theory, June 2021.
8. Jieru Zhu. The SL2 Tilting Category in The Mixed Case. SIAM Conference on Applied Geometry. August 2021.
9. Jieru Zhu. The SL2 Tilting Category in The Mixed Case. University of California at Santa Barbara. Graduate Student Seminar. November 2021.
10. Jieru Zhu. The SL2 Tilting Category in The Mixed Case. University of Colorado Boulder algebra seminar. November 2021.
11. Louise Sutton. Tilting modules for SL2. Conference on Algebraic Representation Theory 2021, November 2021.
12. Liron Speyer. Schurian-infinite blocks of type A Hecke algebras. Physical Algebra and Combinatorics Seminar, OCAMI, Osaka City University, March 2022.

## 5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

## 6. Meetings and Events

### OIST Representation Theory Seminar (on Zoom)

1. April 13, 2021
• Speaker: Stacey Law (University of Cambridge)
• Title: Sylow branching coefficients and a conjecture of Malle and Navarro

2. April 27, 2021

• Speaker: Mark Wildon (Royal Holloway, University of London)
• Title: Plethysms, polynomial representations of linear groups and Hermite reciprocity over an arbitrary field

3. May 28, 2021

• Speaker: Max Gurevich (Technion, Israel)
• Title: New constructions for irreducible representations in monoidal categories of type A

4. June 15, 2021

• Speaker: Sira Gratz (University of Glasgow)
• Title:  Grassmannians, Cluster Algebras and Hypersurface Singularities

5. July 6, 2021

• Speaker: Diego Millan Berdasco (Queen Mary University of London)
• Title: On the computation of decomposition numbers of the symmetric group

6. September 28, 2021

• Speaker: Hankyung Ko (Uppsala University)
• Title: Bruhat orders and Verma modules

7. October 12, 2021

• Speaker: Paul Wedrich (University of Hamburg)
• Title: Knots and quivers, HOMFLYPT and DT

8. October 26, 2021

• Speaker: George Seelinger (University of Michigan)
• Title: Diagonal harmonics and shuffle theorems

9. November 9, 2021

• Speaker: Arik Wilbert (University of South Alabama)
• Title: Real Springer fibers and odd arc algebras

10. November 16, 2021

• Speaker: Samuel Creedon (City, University of London)
• Title: Defining an Affine Partition Algebra

11. November 30, 2021

• Speaker: Tianyuan Xu (University of Colorado at Boulder)
• Title: On Kazhdan–Lusztig cells of a-value 2

12. December 14, 2021

• Speaker: Joanna Meinel
• Title: Decompositions of tensor products: Highest weight vectors from branching

13. February 1, 2022

• Speaker: Daniel Tubbenhauer (University of Sydney)
• Title: On weighted KLRW algebras

14. February 15, 2022

• Speaker: Stephen Doty (Loyola University Chicago)
• Title: Schur-Weyl duality for braid and twin groups via the Burau representation

15. March 1, 2022

• Speaker: Robert Spencer (University of Cambridge)
• Title: (Some) Gram Determinants for $$A_n$$ nets

16. March 22, 2022

• Speaker: John Murray (Maynooth University)
• Title: A Schur-positivity conjecture inspired by the Alperin-Mckay conjecture

## 7. Other

Nothing to report.