# FY2020 Annual Report

Speyer Unit
Assistant Professor Liron Speyer

## 1. Staff

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

## 2. Collaborations

• Mahir Can, Tulane University
• Nicholas Davidson, Reed College
• Yiyang She, Tulane University
• Louise Sutton, University of Manchester
• Jonathan Kujawa, University of Oklahoma
• Robert Muth, Washington & Jefferson College
• Paul Wedrich, Max Planck Institute of Mathematics
• Daniel Tubbenhauer, Bonn University

## 3. Activities and Findings

### 3.1 Strong Gelfand subgroups of $$F\wr S_n$$

A subgroup $$K$$ of a group $$G$$ is called a strong Gelfand subgroup, or a multiplicity-free subgroup, if $$\operatorname{ind}^G_K V$$ is multiplicity-free for every irreducible $$K$$-module V. In characteristic 0, we determined a complete list of all strong Gelfand subgroups, up to conjugacy, of the Coxeter group of type $$B$$, i.e. the hyperoctahedral group $$F\wr S_n$$.

### 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 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.

### 3.4 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. Liron Speyer and Louise Sutton.  Decomposable Specht modules indexed by bihooks.  Pacific J. Math. 304 (2020), no. 2, 655–711. doi: 10.2140/pjm.2020.304.655
2. Mahir Can, Yiyang She and Liron Speyer. Strong Gelfand subgroups of $$F\wr S_n$$. Internat. J. Math., 32 (2021), no. 2, 2150010. doi: 10.1142/S0129167X21500105
3. Jieru Zhu.  Two boundary centralizer algebras for $$\mathfrak{q}(n)$$. Journal of Algebra, Volume 567, Pages 386–458 (2021). doi: 10.1016/j.jalgebra.2020.08.028
4. Mee Seong Im and Jieru Zhu. Transitioning between tableaux and spider bases for Specht modules. Algebras and Representation Theory, to appear. doi: 10.1007/s10468-020-10026-6
5. Robert Muth, Liron Speyer and Louise Sutton.  Decomposable Specht modules indexed by bihooks II, arXiv:2101.11175.
6. Georgia Benkart, Rekha Biswal, Ellen Kirkman, Van Nguyen and Jieru Zhu. McKay matrices for finite-dimensional Hopf algebras. Canadian Journal of Math, Feb 2021, 1-46. doi: 10.4153/S0008414X21000067

### 4.2 Books and other one-time publications

Nothing to report

### 4.3 Oral and Poster Presentations

1. Liron SpeyerSemisimple Specht modules indexed by bihooks. Kyoto Representation Theory Seminar, Kyoto University, July 2020.
2. Chris Chung. $$\imath$$Quantum Covering Groups: Serre presentation and canonical basis. OIST Representation Theory Seminar, January 2021.
3. Liron SpeyerSemisimple Specht modules indexed by bihooks. Algebraic Lie Theory Seminar, University of Colorado Boulder, March 2021.
4. Jieru ZhuTransitioning between the tableaux bases and the web bases for Specht modules. University of Georgia Algebra Seminar, March 2021.
5. Jieru ZhuTransitioning between the tableaux bases and the web bases for Specht modules. AMS Spring Southeastern Sectional Meeting, Special session on Superalgebras, quantum groups and related topics, March 2021.

## 5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

## 6. Meetings and Events

### OIST Representation Theory Seminar (on Zoom)

1. September 15, 2020
• Speaker: Chris Bowman (University of Kent)
• Title: Tautological p-Kazhdan–Lusztig Theory for cyclotomic Hecke algebras

2. September 29, 2020

• Speaker: Mahir Can (Tulane University)
• Title: Spherical Varieties and Combinatorics

3. October 13, 2020

• Speaker: Eoghan McDowell (Royal Holloway, University of London)
• Title: The image of the Specht module under the inverse Schur functor

4. October 27, 2020

• Speaker: Rob Muth (Washington and Jefferson College)
• Title:  Specht modules and cuspidal ribbon tableaux

5. November 10, 2020

• Speaker: Jieru Zhu (Hausdorff Institute of Mathematics)
• Title: Double centralizer properties for the Drinfeld double of the Taft algebras

6. November 17, 2020

• Speaker: Qi Wang (Osaka University)
• Title: On $$\tau$$-tilting finiteness of Schur algebras

7. December 8, 2020

• Speaker: Nicolas Jacon (University of Reims Champagne-Ardenne)
• Title: Cores of Ariki-Koike algebras

8. January 12, 2021

• Speaker: Mathew Fayers (Queen Mary University of London)
• Title: The Mullineux map

9. January 26, 2021

• Speaker: Chris Chung (OIST)
• Title: $$\imath$$Quantum Covering Groups: Serre presentation and canonical basis

10. February 2, 2021

• Speaker: Alistair Savage (University of Ottawa)
• Title: Affinization of monoidal categories

11. February 16, 2021

• Speaker: Nick Davidson (Reed College)
• Title: Type P Webs and Howe Duality

12. March 2, 2021

• Speaker: Aaron Yi Rui Low (National University of Singapore)

13. March 16, 2021

• Speaker: Catharina Stroppel (University of Bonn)
• Title: Verlinde rings and DAHA actions

14. March 30, 2021

• Speaker: Alexander Kleshchev (University of Oregon)
• Title: Irreducible restrictions from symmetric groups to subgroups

## 7. Other

Nothing to report.