# FY2023 Annual Report

**Speyer Unit
Assistant Professor Liron Speyer**

## 1. Staff

- Dr Liron Speyer, Assistant Professor
- Dr Louise Sutton, Staff Scientist
- Dr Chris Chung, Postdoctoral Scholar
- Dr Eoghan McDowell, Postdoctoral Scholar
- Dr Kaveh Mousavand, Postdoctoral Scholar
- Dr Duc-Khanh Nguyen, 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
- Matt Fayers, Queen Mary University of London
- Stacey Law, University of Cambridge
- Sinéad Lyle, University of East Anglia
- Andrew Mathas, University of Sydney
- Robert Muth, Duquesne University
- Charles Paquette, Royal Military College of Canada
- Linliang Song, Tongji University
- Qi Wang, Tsinghua University

## 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 RoCK blocks and skew cyclotomic KLR algebras

We consider cuspidal ribbon tilings of multicores and RoCK multipartitions, showing that core blocks and RoCK blocks may be distinguished via these tilings. We define and study skew cyclotomic KLR algebras, showing that for any convex preorder, the simple imaginary semicuspidal modules for KLR algebras in affine type A arise as heads of skew Specht modules associated to level 1 RoCK blocks. Using this machinery, we describe the simple modules for type A KLR algebras as heads of certain explicitly described skew Specht modules.

### 3.4 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.

### 3.5 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.

### 3.6 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 Spin representations which reduce modulo 2 to a multiple of a Specht module

For a finite group, it is interesting to determine when two ordinary irreducible representations have the same \(p\)-modular reduction; that is, when two rows of the decomposition matrix in characteristic \(p\) are equal, or equivalently when the corresponding \(p\)-modular Brauer characters are the same. We complete this task for the double covers of the symmetric group when \(p=2\), by determining when the \(2\)-modular reduction of an irreducible spin representation coincides with a \(2\)-modular Specht module. In fact, we obtain a more general result: we determine when an irreducible spin representation has \(2\)-modular Brauer character proportional to that of a Specht module.

### 3.9 Interactions between bricks and some modern notions of rigidity

Conceptual interactions between different sets of indecomposable modules often provide valuable insights into the representation theory of algebras and related areas. For a finite dimensional algebra A, an A-module is called a brick provided that every nonzero endomorphism is invertible. Moreover, those A-modules with no nontrivial self-extensions are called rigid. As shown in the seminal work of Adachi-Iyama-Reiten on tau-tilting theory, an important subset of indecomposable rigid modules is specified by the modern notion of tau-rigidity. Furthermore, Demonet-Iyama-Jasso have established an injective map from the set of indecomposable tau-rigid modules into the set of bricks. Building upon this new direction of research, in our recent work we consider those algebras over which every brick is tau-rigid and show that this property gives a new characterization of an important classical family of algebras, known as locally representation-directed. We further generalize this classical setting from the geometric viewpoint and introduce and study a novel geometric counterpart of the locally representation-directed algebras. Our results on the new family of algebras provide important evidence for the correctness of some open brick-Brauer-Thrall conjectures that we posed in our earlier work.

### 3.10 Applications of the (K-theoretic) Peterson isomorphism

The K-k-Schur functions \(g_{\lambda}^{(k)}\) introduced and characterized via the Pieri rule by Lam, Shimozono and Morse, are the polynomial representatives of Schubert classes of the K-homology \(K_*(Gr_{SL_{k+1}})\) of the affine Grassmannian. The quantum Grothendieck polynomials \(G_{w}^{Q}\) studied by Lenart, Maeno represent the Schubert classes in quantum K-theory \(QK^*(Fl_{k+1})\). A quantum K-Pieri rule for \(G_w^{Q}\) was conjectured by Lenart, Maeno and proved by Naito, Sagaki. Ikeda, Iwao, Maeno gave an explicit K-theoretic Peterson isomorphism \(\Phi_{k+1}\) between \(K_*(Gr_{SL_{k+1}})\) and \(QK^*(Fl_{k+1})\) after appropriate localization. We are interested in the following question. It extends the work of Lam and Shimozono for the Peterson isomorphism between homology of affine Grassmannians and quantum cohomology of flag varieties: Apply the K-theoretic isomorphism to the Pieri rule in \(QK^*(Fl_{k+1})\) to obtain a formula in \(K_*(Gr_{SL_{k+1}})\). We hoped that the result should be a Pieri rule in \(K_*(Gr_{SL_{k+1}})\). However, the result we have obtained recently is a more general formula than the Pieri formula in \(K_*(Gr_{SL_{k+1}})\). In particular, we obtain a new Monk formula in \(K_*(Gr_{SL_{k+1}})\). This Monk formula seems to be the formula conjectured in another work of Dalal and Morse, but in a different language. We are now trying to prove that they are the same. This is joint work with Naito.

### 3.11 Generalised Carter–Payne homomorphisms for type A KLR algebras

The main open problem in the representation theory of the symmetric group concerns the structure of the Specht modules in terms of the irreducible modules in the case of positive characteristic. One of the ways to probe the structure of the Specht modules is to investigate the space of homomorphisms between Specht modules. Classically, the main result in this direction was given by Carter and Payne in 1980. In this project, we make use of new methods developed using the formalism of KhovanovーLaudaーRouquier (KLR) algebras in order to achieve: - In KLR algebras of type A, the unification of the results of Carter and Payne with more modern results by Lyle and Mathas, and Witty. - For the symmetric group, the first generalization of the classical result by Carter and Payne since it was stated in 1980. The classical CarterーPayne theorem concerns homomorphisms between Specht modules indexed by partitions whose Young diagrams differ by a single row. Our new result concerns homomorphisms between Specht modules indexed by partitions whose Young diagrams differ by partition-shaped subsets of nodes.

### 3.12 Graded Young tableau combinatorics for infinite quantum characteristic

Young tableaux play a pivotal role in the representation theory of cyclotomic Hecke algebras. Brundan and Kleshchev showed that the cyclotomic Hecke algebras are in fact graded algebras, and further work by Brundan, Kleshchev and Wang showed that the Specht modules have a homogeneous basis, providing the set of Young tableaux with a natural grading. Despite its theoretical importance, very few combinatorial results are known about graded tableaux. In this project we study graded tableaux in the case of infinite quantum characteristic, and apply the results to the graded representation theory of cyclotomic Hecke algebras.

### 3.13 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 \(\dim\hom(S^\mu,S^{(n)})\neq0\).

### 3.14 Representation type for higher level cyclotomic KLR algebras in type \(C\)

Representation type serves as a fundamental tool in the representation theory of finite dimensional algebras, especially, over an algebraically closed field. For Iwahori–Hecke algebras of type A, representation type was described by Erdmann and Nakano. Beyond type A, a similar representation type classification is a subject of active research. In this project, we consider the representation type of cyclotomic quiver Hecke algebras whose Lie type are affine type C when the level is at least two. Following the strategy developed earlier by Ariki—Song—Wang, we first explain the construction of a connected quiver for the elements of the set of all dominant weights. Then, using this quiver, we completely determine the representation type of these algebras. We find that unlike in level one, the defect no longer governs the representation type. Furthermore, for tame cases when the characteristic of the underlying field is not two, we explicitly describe their basic algebras.

## 4. Publications

### 4.1 Journals

- Susumu Ariki, Sinéad Lyle, and
**Liron Speyer**.*Schurian-finiteness of blocks of type A Hecke algebras*. Journal of the London Mathematical Society**108**(2023), no. 6, 2333–2376. doi: 10.1112/jlms.12808 **Eoghan McDowell**.*Characters and spin characters of alternating and symmetric groups determined by values on l'-classes*. Arkiv för Matematik, to appear. arXiv:2205.06505**Eoghan McDowell**.*Large p-core p′-partitions and walks on the additive residue graph*. Annals of Combinatorics**27**, (2023) 857–871. doi: 10.1007/s00026-022-00622-2**Christopher Chung**and**Berta Hudak**.*Representation type of level 1 KLR algebras \(R^\Lambda(\beta)\) in type C*. Osaka Journal of Mathematics, to appear. arXiv:2304.10184**Kaveh Mousavand**and Charles Paquette.*Geometric interactions between bricks and τ-rigidity*, arXiv:2311.14863**Berta Hudak**.*Homomorphisms into Specht modules labelled by hooks in quantum characteristic two*, arXiv:2402.08942- Susumu Ariki,
**Berta Hudak**, Linliang Song, and Qi Wang.*Representation type of higher level cyclotomic quiver Hecke algebras in affine type C*, arXiv:2402.09940 - Matthew Fayers and
**Eoghan McDowell**.*Spin characters of the symmetric group which are proportional to linear characters in characteristic 2*, arXiv:2403.08243

### 4.2 Books and other one-time publications

Nothing to report

### 4.3 Oral and Poster Presentations

**Berta Hudak**.*Homomorphisms into Specht modules labelled by hooks when e=2.*Algebraic Lie Theory and Representation Theory (ALTReT) 2023, Tokyo Institute of Technology, May 2023.**Chris Chung**.*Structure and representation type for type C cyclotomic KLR algebras.*Algebraic Lie Theory and Representation Theory (ALTReT) 2023, Tokyo Institute of Technology, May 2023.**Eoghan McDowell**.*Determination of characters of the symmetric and alternating groups.*Algebra and Representation Theory Seminar, University of Cambridge, May 2023.**Berta Hudak**.*Homomorphisms into Specht modules labelled by hooks when e=2.*Representation Theory of Hecke Algebras and Categorification, OIST, June 2023.**Eoghan McDowell**.*Determination of characters of the symmetric and alternating groups.*Representation Theory of Hecke Algebras and Categorification, OIST, June 2023.**Louise Sutton**.*Idempotent truncations of KLR algebras.*Hecke algebras and applications, Spetses, July 2023.**Duc-Khanh Nguyen**.*A generalization of the Murnaghan-Nakayama rule for K-k-Schur and k-Schur functions.*International Conference on Enumerative Combinatorics and Applications, University of Haifa (online), September 2023.**Kaveh Mousavand**.*Generic brick-Brauer-Thrall Conjecture**.*tau-research School, Cologne, Germany, September 2023.**Liron Speyer**.*Graded decomposition matrices for type C KLR algebras.*LMS Regional workshop on Lie theory, University of York, September 2023.**Kaveh Mousavand**.*The 2nd brick-Brauer-Thrall conjecture and a torus action on representation varieties.*Silting in Representation Theory, Singularities, and Noncommutative Geometry, Oaxaca, Mexico, September 2023.**Duc-Khanh Nguyen**.*A generalization of the Murnaghan-Nakayama rule for K-k-Schur and k-Schur functions.*Combinatorial aspects of representation theory and related topics, Waseda University, October 2023.**Kaveh Mousavand**.*Distribution of bricks-- algebraic and geometric viewpoints*, Representation Theory Seminar, Bielefeld, Germany, October 2023.**Martín Forsberg Conde**.*Specht Module の Carter–Payne 準同型の一般化. (A generalization of Carter–Payne homomorphisms between Specht Modules.)*Combinatorial aspects of representation theory and related topics, Waseda University, October 2023.**Kaveh Mousavand**.*Locally representation-directed algebras-- realized and generalized via generically tau-reduced components*, Current trends in representation theory of algebras, cluster algebras and geometry, Luminy, France, November 2023.**Kaveh Mousavand***. Distribution of bricks and one-parameter families of stable modules**.*McKay correspondence, Tilting theory and related topics, Kavli IPMU, Japan, December 2023.**Kaveh Mousavand***. Rigidity of bricks and brick-Brauer-Thrall conjectures,*Tokyo-Nagoya Algebra Seminar, Nagoya, December 2023.**Liron Speyer**.*Schurian-infinite blocks of type A Hecke algebras.*Algebra seminar, University of Sydney, February 2024.**Kaveh Mousavand***. A \(\tau\)-tilting counterpart of cluster algebras of minimal infinite type*. Advances in Cluster Algebras 2024, Nagoya, March 2024.**Martín Forsberg Conde**.*New homomorphisms between Specht modules of the symmetric group.*Mathematical Society of Japan Spring Meeting 2024, Osaka Metropolitan University, March 2024.**Liron Speyer**.*Graded decomposition matrices for type C KLR algebras.*Mathematical Society of Japan Spring Meeting 2024, Osaka Metropolitan University, March 2024.

## 5. Intellectual Property Rights and Other Specific Achievements

**Grants and Fellowships**

**Kaveh Mousavand**. JSPS Kakenhi Grant-in-aid for Early-Career Scientists – FY2024–2025.**Liron Speyer**. JSPS Kakenhi Grant-in-aid for Scientific Research (C) – FY2023–2025.**Louise Sutton**. 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)

- July 28, 2023

- Speaker: Ziqing Xiang (Southern University of Science and Technology)
- Title: Quantum wreath product

2. August 29, 2023

- Speaker:Nicholas Williams (Lancaster University)
- Title: Higher-dimensional cluster combinatorics and representation theory

3. October 10, 2023

- Speaker: Benjamin Sambale (Leibniz Universität Hannover)
- Title: Groups of p-central type

4. October 24, 2023

- Speaker:Emily Norton (University of Kent)
- Title: Groups of p-central type

5. November 7, 2023

- Speaker: Ulrich Thiel (University of Kaiserslautern-Landau)
- Title: The rank one property for free Frobenius extensions

6. November 28, 2023

- Speaker: Alexander Yong (University of Illinois at Urbana-Champaign)
- Title: Newell-Littlewood numbers

7. December 5, 2023

- Speaker: Eric Marberg (The Hong Kong University of Science and Technology (HKUST))
- Title: From Klyachko models to perfect models

8. December 12, 2023

- Speaker: Kaveh Mousavand (OIST)
- Title: Some applications of bricks in classical and modern problems in representation theory

9. January 19, 2024

- Speaker: Travis Scrimshaw (Hokkaido University)
- Title: An Overview of Kirillov-Reshtikhin Modules and Crystals

10. January 31, 2024

- Speaker: Peigen Cao (Nagoya University)
- Title: Bongartz co-completions in cluster algebras and its applications

11. February 28, 2024

- Speaker: Yuta Kimura (Osaka Metropolitan University)
- Title: Classifying torsion classes of Noetherian algebras

## 7. Other

Nothing to report.