# FY2022 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
- Martín Forsberg Conde, PhD Student
- Berta Hudak, PhD Student
- Yukiko Nakagawa, Research Unit Administrator

## 2. Collaborations

- Susumu Ariki, Osaka University
- Chris Bowman, University of York
- Nicholas Davidson, Reed College
- Matt Fayers, Queen Mary University of London
- Jonathan Kujawa, University of Oklahoma
- Sinéad Lyle, University of East Anglia
- Andrew Mathas, University of Sydney
- Robert Muth, Washington & Jefferson College
- 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 blocks of weight at least 2 of type A Hecke algebras are Schurian-infinite in any characteristic. To prove these results, our paper employs a wide variety of techniques from the (graded) representation theory of Hecke algebras.

### 3.2 Graded decomposition matrices for type C KLR algebras

Graded decomposition numbers are well-studied for type A cyclotomic Khovanov–Lauda–Rouquier algebras, in large part thanks to the Brundan–Kleshchev isomorphism that links them to cyclotomic Hecke algebras. In both affine and finite type C, a Specht module theory for cyclotomic KLR algebras has been constructed much more recently, and has been recently shown to arise from a cellular structure on the algebra. This development allows us to study graded decomposition numbers for these algebras. Here, we determine all graded decomposition matrices in level 1, for \(n \leqslant 12\). We in fact determine a two-parameter analogue of these matrices, where the extra parameter records layers in a certain Jantzen filtration of the Specht modules. Along the way, we prove that the defect is always non-negative for these type C algebras, and determine the submodule structures of Specht modules for \(n \leqslant 10\). We also give the first level 1 example of a characteristic 0 graded decomposition number not matching the corresponding dual canonical basis coefficient, and the first example of a negative degree graded decomposition number in characteristic 0. Neither of these phenomena occur in type A.

### 3.3 Representation type for level 1 cyclotomic KLR algebras in type \(C\)

Classically, the representation type of block algebras of the Iwahori–Hecke algebra of the symmetric group was described by Erdmann and Nakano. Beyond type \(A\), a general representation type classification in the style of Erdmann–Nakano for the block algebras of cyclotomic KLR algebras is a subject of active research. In type \(C\), this was done by Ariki–Park for \(\Lambda = \Lambda_0\). We determine the representation type for block algebras of the quiver Hecke algebras \(R^{\Lambda}(\beta)\) of type \(C^{(1)}_\ell\) for level one i.e. \(\Lambda = \Lambda_k\), generalising the results of Ariki–Park.

### 3.4 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.

### 3.5 Identities connecting partition combinatorics and quiver representations

Recently Rimányi–Weigandt–Yong established a weight-preserving bijection between families of partitions, where the families were indexed by combinatorial objects (lacing diagrams) which also index the simple modules for a type A quiver algebra. This establishes a symmetric polynomial identity, and furthermore they show that the powers which arise are the codimensions of the corresponding simple modules. This project aims to investigate this phenomeon for larger classes of quiver algebras. This involves identifying the correct families of partitions to correspond to the simple modules, finding suitable bijections, and demonstrating that the resulting identity carries the relevant representation-theoretic information.

### 3.6 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.

### 3.7 Large \(p\)-core \(p'\)-partitions

A partition which has no parts divisible by \(p\) is called a \(p'\)-partition, and arise as labels of \(p'\)-classes of the symmetric group. A partition which has no hook lengths divisible by \(p\) is called \(p\)-core, and arise as labels of the \(p\)-blocks of the symmetric group. Partitions which satisfy both properties simultaneously can be used, together with the Murnaghan–Nakayama rule, to identify many zeros in the character table of the symmetric group. This project investigates the largest \(p\)-core \(p'\)-partition for given \(p\), including giving a lower bounds for its size. In particular this demonstrates that the upper bound on the size given by McSpirit–Ono is of optimal degree.

### 3.8 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.

### 3.9 Irreducible Specht modules for type B Hecke algebras

The representation theory of the cyclotomic Hecke algebras is governed by a special family of modules, called Specht modules, for which dimensions, bases and presentations are known. Irreducible modules arise as heads of certain Specht modules, however, the dimensions of irreducible modules are not known in general. One can begin the study of irreducible modules by studying irreducible Specht modules. A classification of irreducible Specht modules of the Iwahori–Hecke algebras of type A when \(q\neq -1\) is now known thanks to several authors: James–Mathas, Fayers, Lyle, James–Lyle–Mathas. Simultaneously using this classification together with the theory developed above on skew cyclotomic KLR algebras, we are working towards proving our conjectural classification of irreducible Specht modules for the Iwahori–Hecke algebra of type B.

### 3.10 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.

### 3.11 Carter–Payne homomorphisms for type C KLR algebras

In this project we aim to prove a Carter–Payne-style theorem which guarantees homomorphisms between Specht modules in cyclotomic KLR algebras of type C. So far we have investigated the case where each component has only one row, and have proved that the conjectured theorem holds in several cases under this restriction.

### 3.12 Spechts homomorphisms in quantum characteristic 2

The Iwahori–Hecke algebra of type A, denoted \(H_n\), has a very similar representation theory to that of the symmetric group and in particular, it is cellular. For every partition \(\lambda\) of n, there exists a corresponding Specht module \(S^\lambda\). These modules arise as the cell modules for \(H_n\) and therefore are the simple \(H_n\)-modules in the semisimple case. In his paper, using the Brundan–Kleshchev isomorphism, Loubert completely determined the homomorphisms between Specht modules labelled by arbitrary shapes and those labelled by hooks in the case when the quantum characteristic is not equal to 2. This project, using a similar strategy to Loubert, extends his results for quantum characteristic 2. In doing so, we also give a complete description of the generators on the basis elements of \(S^\lambda\) where \(\lambda\) is a hook. Furthermore, we generalise James's well-known result for the trivial module \(S^{(n)}\) and describe all shapes \(\mu\) such that \(\dim\hom(S^\mu,S^{(n)})\neq0\).

## 4. Publications

### 4.1 Journals

- Mee Seong Im and
**Jieru Zhu**.*Transitioning between tableaux and spider bases for Specht modules*. Algebras and Representation Theory**25**(2022), 387–399. doi: 10.1007/s10468-020-10026-6 - Georgia Benkart, Rekha Biswal, Ellen Kirkman, Van Nguyen, and
**Jieru Zhu**.*McKay matrices for finite-dimensional Hopf algebras*. Canadian Journal of Math,**74**(2022) no. 3, 686–731. doi: 10.4153/S0008414X21000067 **Eoghan McDowell**and Mark Wildon.*Modular plethystic isomorphisms for two-dimensional linear groups*. Journal of Algebra,**602**(2022), 441–483. doi: 10.1016/j.jalgebra.2022.02.025- Georgia Benkart, Rekha Biswal, Ellen Kirkman, Van Nguyen, and
**Jieru Zhu**.*Tensor representations for the Drinfeld double of the Taft algebra*. Journal of Algebra,**606**(2022), 764–796. doi: 10.1016/j.jalgebra.2022.04.041 - Robert Muth,
**Liron Speyer**, and**Louise Sutton**.*Decomposable Specht modules indexed by bihooks II*. Algebras and Representation Theory,**26**(2023), no. 1, 241–280, doi: 10.1007/s10468-021-10093-3 **Louise Sutton**, Daniel Tubbenhauer, Paul Wedrich, and**Jieru Zhu**.*SL_2 tilting modules in the mixed case*, Selecta Mathematica,**29**(2023), Paper No. 39. doi: 10.1007/s00029-023-00835-0.- Susumu Ariki, Sinéad Lyle, and
**Liron Speyer**.*Schurian-finiteness of blocks of type A Hecke algebras*, arXiv:2112.11148 **Eoghan McDowell**.*Characters and projective characters of alternating and symmetric groups determined by values on l'-classes*, arXiv:2205.06505**Eoghan McDowell**.*Large p-core p′-partitions and walks on the additive residue graph.*Annals of Combinatorics, to appear. doi: 10.1007/s00026-022-00622-2- Sinéad Lyle and
**Liron Speyer**.*Schurian-finiteness of blocks of type A Hecke algebras II*, arXiv:2208.05711 **Eoghan McDowell**.*Flagged Schur polynomial duality via a lattice path bijection*. Electronic Journal of Combinatorics,**30**(2023) no. 1, Paper No. 1.5. doi: 10.37236/11200- Nicholas Davidson, Jonathan Kujawa, Robert Muth, and
**Jieru Zhu**.*Superalgebra deformations of web categories: finite webs*, arXiv:2302.04073

### 4.2 Books and other one-time publications

Nothing to report

### 4.3 Oral and Poster Presentations

**Eoghan McDowell**.*Determination of characters by their values on p'-classes.*Algebraic Lie Theory and Representation Theory (ALTReT) 2022, Chiba University, May 2022.**Liron Speyer**.*Schurian-infinite blocks of type A Hecke algebras.*York Algebra Seminar, University of York, September 2022.**Liron Speyer**.*Schurian-infinite blocks of type A Hecke algebras.*Mathematical Society of Japan Autumn Meeting 2022, Hokkaido University, September 2022.**Louise Sutton**.*Irreducible modules of the cyclotomic KLR algebras.*Recent Developments in Combinatorial Representation Theory, RIMS, University of Kyoto, November 2022.**Martín Forsberg Conde**.*The representation theory of type A Iwahori–Hecke algebras I.*Silver workshop V: Complex Geometry and related topics, OIST, November 2022.**Liron Speyer**.*The representation theory of type A Iwahori–Hecke algebras II.*Silver workshop V: Complex Geometry and related topics, OIST, November 2022.**Liron Speyer**.*Schurian-infinite blocks of type A Hecke algebras.*Conference in Algebraic Representation Theory (CART) 2022, University of Tsukuba, November 2022.**Liron Speyer**.*Graded decomposition matrices for type C KLR algebras.*Representation Theory, Combinatorics and Geometry, National University of Singapore, December 2022.**Eoghan McDowell**.*Determination of characters by their values on \(p'\)-classes.*Representation Theory, Combinatorics and Geometry, National University of Singapore, December 2022.**Louise Sutton**.*Irreducible (Specht) modules for cyclotomic KLR algebras.*Representation Theory, Combinatorics and Geometry, National University of Singapore, December 2022.**Berta Hudak**.*Representation type of level 1 KLR algebras \(R^{\Lambda_k}(\beta)\) in type \(C^{(1)}_\ell\).*Representation Theory, Combinatorics and Geometry, National University of Singapore, December 2022.**Louise Sutton**.*Irreducible Specht modules for cyclotomic KLR algebras.*Categorification in representation theory, University of Sydney, February 2023.**Louise Sutton**.*Irreducible Specht modules for symmetric groups and beyond.*Women at the Intersection of Mathematics and Theoretical Physics Meet in Okinawa, OIST, March 2023.**Berta Hudak**.*(Super)symmetric polynomials and the Pieri rule.*Women at the Intersection of Mathematics and Theoretical Physics Meet in Okinawa, OIST, March 2023.

## 5. Intellectual Property Rights and Other Specific Achievements

**Grants and Fellowships**

**Louise Sutton**. Simons-Centre de Recherches Mathématiques Scholar-in-Residence Program, University of Montreal, offered summer 2022, unable to go due to COVID restrictions.**Liron Speyer**. Awarded JSPS Kakenhi Grant-in-aid for Scientific Research (C) – FY2023–2025.**Louise Sutton**. Awarded JSPS Kakenhi Grant-in-aid for Early-Career Scientists – FY2023–2024.**Eoghan McDowell**and Stacey Law. Awarded Heilbronn Small Grant to support research visit in May 2023.

## 6. Meetings and Events

### OIST Representation Theory Seminar (on Zoom)

- April 5, 2022

- Speaker: Dean Yates (Queen Mary University of London)
- Title: Spin representations of the symmetric group

2. April 20, 2022

- Speaker: Kay Jin Lim (Nanyang Technological University)
- Title: Descent Algebra of Type A

3. June 14, 2022

- Speaker: Shunsuke Tsuchioka (Tokyo Institute of Technology)
- Title: An example of A2 Rogers-Ramanujan bipartition identities of level 3

4. July 5, 2022

- Speaker: Rob Muth (Duquesne University)
- Title: Superalgebra deformations of web categories

5. July 19, 2022

- Speaker: Alice Dell'Arciprete (University of East Anglia)
- Title: Scopes equivalence for blocks of Ariki-Koike algebras

6. July 21, 2022

- Speaker: Sinead Lyle (University of East Anglia)
- Title:Rouquier blocks for Ariki-Koike algebras

7. August 10, 2022

- Speaker: Andrew Mathas (University of Sydney)
- Title: Content systems and KLR algebras

8. September 20, 2022

- Speaker: Kai Meng Tan (National University of Singapore)
- Title: Young’s seminormal basis vectors and their denominators

9. October 11, 2022

- Speaker:Haralampos Genanios (University of York)
- Title: Young’s seminormal basis vectors and their denominators

10. October 25, 2022

- Speaker: Chun-Ju Lai (Academia Sinica)
- Title: Quasi-hereditary covers, Hecke subalgebras and quantum wreath product

11. November 1, 2022

- Speaker: Loic Poulain d'Andecy (University of Reims Champagne-Ardenne)
- Title: KLR-type presentation of affine Hecke algebras of type B

12. November 15, 2022

- Speaker: Giada Volpato (University of Florence)
- Title: On the restriction of a character of \(\mathfrak{S}_n\) to a Sylow p-subgroup

13. November 29, 2022

- Speaker: Nicolle Gonzalez
- Title: Higher Rank Rational (q,t)-Catalan Polynomials and a Finite Shuffle Theorem

14. January 10, 2023

- Speaker: Rowena Paget (University of Kent)
- Title: Plethysm and the Partition Algebra

15. January 24, 2023

- Speaker: Pavel Turek (Royal Holloway, University of London)
- Title: On stable modular plethysms of the natural module of SL2(Fp)in characteristic p

16. March 14, 2023

- Speaker: Soichi Okada (Nagoya University)
- Title: Intermediate symplectic characters and enumeration of shifted plane partitions

## 7. Other

Nothing to report.