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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 April 2021, 4:30–5:30pm JST (UTC+9), online on Zoom

### Mark Wildon, Royal Holloway, University of London

### Title: Plethysms, polynomial representations of linear groups and Hermite reciprocity over an arbitrary field

**Abstract:** Let \(E\) be a \(2\)-dimensional vector space. Over the complex numbers the irreducible polynomial representations of the special linear group \(SL(E)\) are the symmetric powers \(Sym^r E\). Composing polynomial representations, for example to form \(Sym^4 Sym^2 E\), corresponds to the plethysm product on symmetric functions. Expressing such a plethysm as a linear combination of Schur functions has been identified by Richard Stanley as one of the fundamental open problems in algebraic combinatorics. In my talk I will use symmetric functions to prove some classical isomorphisms, such as Hermite reciprocity \(Sym^m Sym^r E \cong Sym^r Sym^m E\), and some others discovered only recently in joint work with Rowena Paget. I will then give an overview of new results showing that, provided suitable dualities are introduced, Hermite reciprocity holds over arbitrary fields; certain other isomorphisms (we can prove) have no modular generalization. The final part is joint work with my Ph.D student Eoghan McDowell.

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#### Meeting ID: 91705645780, the password will be announced 24 hours before the seminar.

### Friday 28th May 2021, 4:30–5:30pm JST (UTC+9), online on Zoom, note the unusual day

### Title: New constructions for irreducible representations in monoidal categories of type A

**Abstract:** One ever-recurring goal of Lie theory is the quest for effective and elegant descriptions of collections of simple objects in categories of interest. A cornerstone feat achieved by Zelevinsky in that regard, was the combinatorial explication of the Langlands classification for smooth irreducible representations of p-adic GL_n. It was a forerunner for an exploration of similar classifications for various categories of similar nature, such as modules over affine Hecke algebras or quantum affine algebras, to name a few. A next step - reaching an effective understanding of all reducible finite-length representations remains largely a difficult task throughout these settings.

#### Recently, joint with Erez Lapid, we have revisited the original Zelevinsky setting by suggesting a refined construction of all irreducible representations, with the hope of shedding light on standing decomposition problems. This construction applies the Robinson-Schensted-Knuth transform, while categorifying the determinantal Doubilet-Rota-Stein basis for matrix polynomial rings appearing in invariant theory. In this talk, I would like to introduce the new construction into the setting of modules over quiver Hecke (KLR) algebras. In type A, this category may be viewed as a quantization/gradation of the category of representations of p-adic groups. I will explain how adopting that point of view and exploiting recent developments in the subject (such as the normal sequence notion of Kashiwara-Kim) brings some conjectural properties of the RSK construction (back in the p-adic setting) into resolution. Time permits, I will discuss the relevance of the RSK construction to the representation theory of cyclotomic Hecke algebras.

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#### Meeting ID: 97679648498, the password will be announced 24 hours before the seminar.

## Past talks

### Tuesday 13th April 2021, 4:30–5:30pm JST (UTC+9), online on Zoom

### Stacey Law, University of Cambridge

### Title: Sylow branching coefficients and a conjecture of Malle and Navarro

**Abstract:** The relationship between the representation theory of a finite group and that of its Sylow subgroups is a key area of interest. For example, recent results of Malle–Navarro and Navarro–Tiep–Vallejo have shown that important structural properties of a finite group \(G\) are controlled by the permutation character \(\mathbb{1}_P\big\uparrow^G\), where \(P\) is a Sylow subgroup of \(G\) and \(\mathbb{1}_P\) denotes the trivial character of \(P\). We introduce so-called Sylow branching coefficients for symmetric groups to describe multiplicities associated with these induced characters, and as an application confirm a prediction of Malle and Navarro from 2012, in joint work with E. Giannelli, J. Long and C. Vallejo.

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

### Title: Irreducible restrictions from symmetric groups to subgroups

**Abstract:** We motivate, discuss history of, and present a solution to the following problem: describe pairs (G,V) where V is an irreducible representation of the symmetric group S_n of dimension >1 and G is a subgroup of S_n such that the restriction of V to G is irreducible. We do the same with the alternating group A_n in place of S_n. The latest results on the problem are joint with Pham Huu Tiep and Lucia Morotti.

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

### Title: Verlinde rings and DAHA actions

**Abstract:** In this talk we will briefly recall how quantum groups at roots give rise Verlinde algebras which can be realised as Grothendieck rings of certain monoidal categories. The ring structure is quite interesting and was very much studied in type A. I will try to explain how one gets a natural action of certain double affine Hecke algebras and show how known properties of these rings can be deduced from this action and in which sense modularity of the tensor category is encoded.

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

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

### Title: Adjustment matrices

**Abstract:** James's Conjecture predicts that the adjustment matrix for weight \(w\) blocks of the Iwahori-Hecke algebras \(\mathcal{H}_{n}\) and the \(q\)-Schur algebras \(\mathcal{S}_{n}\) is the identity matrix when \(w<\mathrm{char}(\mathbb{F})\). Fayers has proved James's Conjecture for blocks of \(\mathcal{H}_{n}\) of weights 3 and 4. We shall discuss some results on adjustment matrices that have been used to prove James's Conjecture for blocks of \(\mathcal{S}_{n}\) of weights 3 and 4 in an upcoming paper. If time permits, we will look at a proof of the weight 3 case.

### 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.

### 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.

### 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.

### 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.