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# OIST Representation Theory Seminar

#### Recordings of talks will be available on this page and here if the speaker agrees to have their talk recorded.

### Tuesday 27th October 2020, 9:00–10:00am JST (UTC+9), online on Zoom

### Rob Muth, Washington and Jefferson College

### Title: Specht modules and cuspidal ribbon tableaux

**Abstract:** Representation theory of Khovanov-Lauda-Rouquier (KLR) algebras in affine type A can be studied through the lens of Specht modules, associated with the cellular structure of cyclotomic KLR algebras, or through the lens of cuspidal modules, associated with categorified PBW bases for the quantum group of affine type A. Cuspidal ribbons provide a sort of combinatorial bridge between these approaches. I will describe some recent results on cuspidal ribbon tableaux, and some implications in the world of KLR representation theory, such as bounds on labels of simple factors of Specht modules, and the presentation of cuspidal modules. Portions of this talk are joint work with Dina Abbasian, Lena Difulvio, Gabrielle Pasternak, Isabella Sholtes, and Frances Sinclair.

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#### Meeting ID: 98121636946 - password announced 24 hours before the talk.

### Tuesday 10th November 2020, 4:30–5:30pm JST (UTC+9), online on Zoom

### Jieru Zhu, Hausdorff Institute of Mathematics

### Title: Double centralizer properties for the Drinfeld double of the Taft algebras

**Abstract:** The Drinfeld double of the taft algebra, \(D_n\), whose ground field contains \(n\)-th roots of unity, has a known list of 2-dimensional irreducible modules. For each of such module \(V\), we show that there is a well-defined action of the Temperley-Lieb algebra \(TL_k\) on the \(k\)-fold tensor product of \(V\), and this action commutes with that of \(D_n\). When \(V\) is self-dual and when \(k \leq 2(n-1)\), we further establish a isomorphism between the centralizer algebra of \(D_n\) on \(V^{\otimes k}\), and \(TL_k\). Our inductive argument uses a rank function on the TL diagrams, which is compatible with the nesting function introduced by Russell-Tymoczko. This is joint work with Georgia Benkart, Rekha Biswal, Ellen Kirkman and Van Nguyen.

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#### Meeting ID: 98046374253 - password announced 24 hours before the talk.

### Tuesday 17th November 2020, 4:30–5:30pm JST (UTC+9), online on Zoom

### Qi Wang, Osaka University

### Title: On \(\tau\)-tilting finiteness of Schur algebras

**Abstract:** Support \(\tau\)-tilting modules are introduced by Adachi, Iyama and Reiten in 2012 as a generalization of classical tilting modules. One of the importance of these modules is that they are bijectively corresponding to many other objects, such as two-term silting complexes and left finite semibricks. Let \(V\) be an \(n\)-dimensional vector space over an algebraically closed field \(\mathbb{F}\) of characteristic \(p\). Then, the Schur algebra \(S(n,r)\) is defined as the endomorphism ring \(\mathsf{End}_{\mathbb{F}G_r}\left ( V^{\otimes r} \right )\) over the group algebra \(\mathbb{F}G_r\) of the symmetric group \(G_r\). In this talk, we discuss when the Schur algebra \(S(n,r)\) has only finitely many pairwise non-isomorphic basic support \(\tau\)-tilting modules.

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#### Meeting ID: 98985731067 - password announced 24 hours before the talk.

### Tuesday 8th December 2020, 4:30–5:30pm JST (UTC+9), online on Zoom

### Nicolas Jacon, University of Reims Champagne-Ardenne

### Title: Cores of Ariki-Koike algebras

**Abstract:** We study a natural generalization of the notion of cores for l-partitions : the (e, s)-cores. We relate this notion with the notion of weight as defined by Fayers and use it to describe the blocks of Ariki-Koike algebras.

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#### Meeting ID: 96582466553 - password announced 24 hours before the talk.

## Past talks

### Tuesday 13th October 2020, 4:30–5:30pm JST (UTC+9), online on Zoom

### Eoghan McDowell, Royal Holloway, University of London

### Title: The image of the Specht module under the inverse Schur functor

**Abstract:** The Schur functor and its inverses give an important connection between the representation theories of the symmetric group and the general linear group. Kleshchev and Nakano proved in 2001 that when the characteristic of the field is at least 5, the image of the Specht module under the inverse Schur functor is isomorphic to the dual Weyl module. In this talk I will address what happens in characteristics 2 and 3: in characteristic 3, the isomorphism holds, and I will give an elementary proof of this fact which covers also all characteristics other than 2; in characteristic 2, the isomorphism does not hold for all Specht modules, and I will classify those for which it does. Our approach is with Young tableaux, tabloids and Garnir relations.

### Tuesday 29th September 2020, 9:00–10:00am JST (UTC+9), online on Zoom

### Mahir Can, Tulane University

### Title: Spherical Varieties and Combinatorics

**Abstract:** Let G be a reductive complex algebraic group with a Borel subgroup B. A spherical G-variety is an irreducible normal G-variety X where B has an open orbit. If X is affine, or if it is projective but endowed with a G-linearized ample line bundle, then the group action criteria for the sphericality is in fact equivalent to the representation theoretic statement that a certain space of functions (related to X) is multiplicity-free as a G-module. In this talk, we will discuss the following question about a class of spherical varieties: if X is a Schubert variety for G, then when do we know that X is a spherical L-variety, where L is the stabilizer of X in G.

### Tuesday 15th September 2020, 4:30–5:30pm JST (UTC+9), online on Zoom

### Title: Tautological p-Kazhdan–Lusztig Theory for cyclotomic Hecke algebras

**Abstract:** We discuss a new explicit isomorphism between (truncations of) quiver Hecke algebras and Elias–Williamson’s diagrammatic endomorphism algebras of Bott–Samelson bimodules. This allows us to deduce that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are tautologically equal to the associated p-Kazhdan–Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. This allows us to give an elementary and explicit proof of the main theorem of Riche–Williamson’s recent monograph and extend their categorical equivalence to cyclotomic Hecke algebras, thus solving Libedinsky–Plaza’s categorical blob conjecture.