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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 26th January 2021, 4:30–5:30pm JST (UTC+9), L4E48, and online on Zoom

### Title: \(\imath\)Quantum Covering Groups: Serre presentation and canonical basis

**Abstract:** In 2016, Bao and Wang developed a general theory of canonical basis for quantum symmetric pairs \((\mathbf{U}, \mathbf{U}^\imath)\), generalizing the canonical basis of Lusztig and Kashiwara for quantum groups and earning them the 2020 Chevalley Prize in Lie Theory. The \(\imath\)divided powers are polynomials in a single generator that generalize Lusztig's divided powers, which are monomials. They can be similarly perceived as canonical basis in rank one, and have closed form expansion formulas, established by Berman and Wang, that were used by Chen, Lu and Wang to give a Serre presentation for coideal subalgebras \(\mathbf{U}^\imath\), featuring novel \(\imath\)Serre relations when \(\tau(i) = i\). Quantum covering groups, developed by Clark, Hill and Wang, are a generalization that `covers' both the Lusztig quantum group and quantum supergroups of anisotropic type. In this talk, I will talk about how the results for \(\imath\)-divided powers and the Serre presentation can be extended to the quantum covering algebra setting, and subsequently applications to canonical basis for \(\mathbf{U}^\imath_\pi\), the quantum covering analogue of \(\mathbf{U}^\imath\), and quantum covering groups at roots of 1.

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#### Meeting ID: 93326482515, the password will be announced 24 hours before the seminar.

### Tuesday 2nd February 2021, 9:30–10:30am JST (UTC+9), online on Zoom

### Title: Affinization of monoidal categories

**Abstract:** We define the affinization of an arbitrary monoidal category, corresponding to the category of string diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to the category. The affinization formalizes and unifies many constructions appearing in the literature. We describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants.

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#### Meeting ID: 92259155615, the password will be announced 24 hours before the seminar.

### Tuesday 16th February 2021, 9:30–10:30am JST (UTC+9), online on Zoom

### Title: Type P Webs and Howe Duality

**Abstract:** Webs are combinatorially defined diagrams which encode homomorphisms between tensor products of certain representations of Lie (super)algebras. I will describe some recent work with Jon Kujawa and Rob Muth which defines webs for the type P Lie superalgebra, and then uses these webs to deduce an analog of Howe duality for this Lie superalgebra.

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#### Meeting ID: TBA, the password will be announced 24 hours before the seminar.

### Tuesday 2nd March 2021, 4:30–5:30pm JST (UTC+9), online on Zoom

### Aaron Yi Rui Low, National University of Singapore

### Title: TBA

**Abstract:** TBA

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#### Meeting ID: TBA, the password will be announced 24 hours before the seminar.

### Tuesday 16th March 2021, 4:30–5:30pm JST (UTC+9), online on Zoom

### TBA

### Title: TBA

**Abstract:** TBA

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#### Meeting ID: TBA, the password will be announced 24 hours before the seminar.

### Tuesday 30th March 2021, 9:30–10:30pm JST (UTC+9), online on Zoom

### Title: TBA

**Abstract:** TBA

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#### Meeting ID: TBA, the password will be announced 24 hours before the seminar.

## Past talks

### Tuesday 12th January 2021, 4:30–5:30pm JST (UTC+9), online on Zoom

### Matthew Fayers, Queen Mary University of London

### Title: The Mullineux map

**Abstract:** In characteristic p, the simple modules for the symmetric group \(S_n\) are the James modules \(D^\lambda\), labelled by p-regular partitions of n. If we let \(sgn\) denote the 1-dimensional sign module, then for any p-regular \(\lambda\), the module \(D^\lambda\otimes sgn\) is also a simple module. So there is an involutory bijection \(m_p\) on the set of p-regular partitions such that \(D^\lambda\otimes sgn=D^{m_p(\lambda)}\). The map \(m_p\) is called the Mullineux map, and an important problem is to describe \(m_p\) combinatorially. There are now several known solutions to this problem. I will describe the history of this problem and explain the known combinatorial solutions, and then give a new solution based on crystals and regularisation.

### 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.

### 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.

### 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.

### 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.

### 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.