Title: Deformations of Frobenius algebras and noncommutative 2d topological quantum field theories
Abstract: In this talk I will present an attempt to define noncommutative 2d topological quantum field theories using deformations of Frobenius algebras. First, we will overview the importance and uses of 2d topological quantum field theories, as well as their equivalence to commutative Frobenius algebras. Then, we will consider the deformations given by cotwisted tensor products, characterize when these are Frobenius algebras, and explain their deficiencies for our goal. Afterwards, I will introduce the notion of warped tensor products of Frobenius algebras and characterize when these are Frobenius algebras. This notion enables us to construct a family of bifunctors that could potentially yield nonsymmetric monoidal structures on the category of Frobenius algebras, which would then deserve to be called noncommutative 2d topological quantum field theories. This is work in progress with Rohan Das and Julia Plavnik.
Tuesday 12th November 2024, 3:00–4:00pm JST (UTC+9), L4F01 and online on Zoom
Title: Toward a Pieri rule for double quantum Grothendieck polynomials
Abstract: In a joint work with Satoshi Naito (arXiv:2211.01578), we proved a Pieri rule (conjectured by Lenart and Maeno) for quantum Grothendieck polynomials, which describes the product of the quantum Grothendieck polynomial associated to a cyclic permutation and an arbitrary quantum Grothendieck polynomial as a \(\mathbb{Z}[Q_1,Q_2,\dots]\)-linear combination of quantum Grothendieck polynomials. Recently, in a joint work with Satoshi Naito and Duc-Khanh Nguyen, we are trying to extend this result to the case of double quantum Grothendieck polynomials. In this talk, I'd like to report on the progress of the joint work.
Tuesday 29th October 2024, 9:30–10:30am JST (UTC+9), online on Zoom
Abstract: Many mathematical and scientific problems concern systems of linear operators \((A_1, ..., A_n)\). Spectral theory is expected to provide a mechanism for studying their properties, just like the case for an individual operator. However, defining a spectrum for non-commuting operator systems has been a difficult task. The challenge stems from an inherent problem in finite dimension: is there an analogue of eigenvalues in several variables? Or equivalently, is there a suitable notion of joint characteristic polynomial for multiple matrices \(A_1, ..., A_n\)? A positive answer to this question seems to have emerged in recent years.
Definition. Given square matrices \(A_1, ..., A_n\) of equal size, their characteristic polynomial is defined as \[Q_A(z):=\det(z_0I+z_1A_1+\cdots+z_nA_n), z=(z_0, ..., z_n)\in \mathbb{C}^{n+1}.\] Hence, a multivariable analogue of the set of eigenvalues is the eigensurface (or eigenvariety) \(Z(Q_A):=\{z\in \mathbb{C}^{n+1}\mid Q_A(z)=0\}\). This talk will review some applications of this idea to problems involving projection matrices and finite dimensional complex algebras. The talk is self-contained and friendly to graduate students.
Tuesday 15th October 2024, 3:00–4:00pm JST (UTC+9), L4F01 and online on Zoom
Title: Cores and core blocks of Ariki-Koike algebras
Abstract: This talk will consist of two parts. In the first part, we will see how certain results (such as the Nakayama 'Conjecture') for the symmetric groups and Iwahori-Hecke algebras of type A can be generalised to Ariki-Koike algebras using the map from the set of multipartitions to that of (single) partitions first defined by Uglov. In the second part, we look at Fayers's core blocks, and see how these blocks may be classified using the notation of moving vectors first introduced by Yanbo Li and Xiangyu Qi. If time allows, we will discuss Scopes equivalences between these blocks arising as a consequence of this classification.
Abstract: Algebras defined over fields of characteristic zero and positive characteristic usually do not behave the same way. In the recent preprint with David J. Benson, we initiate the study by focusing on the integral basic algebras. That is, we consider a p-modular system \((K,\mathcal{O},k)\) and an \(\mathcal{O}\)-algebra \(A\) where both the algebras \(K\otimes_\mathcal{O} A\) and \(k\otimes_\mathcal{O}A\) are basic. When the algebra satisfies the right hypotheses, we have equalities of the dimensions of their cohomology groups between simple modules and equalities of graded Cartan numbers. As a case study, we focus on the descent algebras of Coxeter groups. They have been extensively studied since the introduction by Louis Solomon in 1976. We investigate their invariants as mentioned previously, their Ext quivers and representation type. The classification of the representation type in the \(p=0\) case has previously achieved by Manfred Schocker. In a recent preprint, together with Karin Erdmann, we complete the classification in the \(p>0\) case.
Title: Application of (K-theoretic) Peterson isomorphism
Abstract: The theory of symmetric polynomials plays a key role in Representation Theory, Schubert Calculus, and Algebraic Combinatorics. Fundamental rules like the Pieri, Murnaghan-Nakayama, and Littlewood-Richardson rules describe the decomposition of products of Schubert classes into Schubert classes. We focus on the decomposition of polynomial representatives of Schubert classes in homology and K-homology of the affine Grassmannian of SL_n, as well as quantum Schubert classes in quantum cohomology and K-cohomology of the full flag manifold of type A. Specifically, we explore how to use the Peterson isomorphism to connect formulas between homology and quantum cohomology, and between K-homology and quantum K-cohomology, extending techniques from the work of Lam-Shimozono on Schubert classes.
Tuesday 16th April 2024, 3:00–4:00pm JST (UTC+9), L4E48 and online on Zoom
Title: Spin representations of the symmetric group which reduce modulo 2 to Specht modules
Abstract: When do two ordinary irreducible representations of a group have the same p-modular reduction? In this talk I will address this question for the double cover of the symmetric group, and more generally give a necessary and sufficient condition for a spin representation of the symmetric group to reduce modulo 2 to a multiple of a Specht module (in the sense of Brauer characters or in the Grothendieck group). I will explain some of the techniques used in the proof, including describing a function which swaps adjacent runners in an abacus display for the labelling partition of a character. This is joint work with Matthew Fayers.
Wednesday 28th February 2024, 1:30–2:30pm JST (UTC+9), L4E48 and online on Zoom
Title: Classifying torsion classes of Noetherian algebras
Abstract: Let R be a commutative Noetherian ring and A a Noetherian R-algebra. In this talk, we study classification of torsion classes, torsion free classes and Serre subcategories of modA. In the case where A=R, such subcategories were classified by Gabriel, Takahashi and Stanley-Wang by using prime ideals of R. If R is a field, then A is a finite dimensional algebra, and there are many studies of such subcategories relating with tilting theory. For a Noetherian algebra case, localization of A at a prime ideal of R plays an important role. We see that classification can be reduced to finite dimensional algebras. If A is commutative, our results cover cases of commutative rings. This is joint work with Osamu Iyama.
Wednesday 31st January 2024, 1:30–2:30pm JST (UTC+9), L4F01 and online on Zoom
Peigen Cao, Nagoya University
Title: Bongartz co-completions in cluster algebras and its applications
Abstract: A cluster algebra is a Z-subalgebra of a rational function field generated by a special set of generators called cluster variables, which are grouped into overlapping subsets of fixed size, called clusters. One can travel from one cluster to the others by a recursive process called mutation. In this talk I will introduce Bongartz co-completions in cluster algebras and give its applications to Fomin-Zelevinsky’s conjectures on denominator vectors and exchange graphs of cluster algebras.
Friday 19th January 2024, 2:00–3:00pm JST (UTC+9), L4F01 and online on Zoom
Title: An Overview of Kirillov-Reshtikhin Modules and Crystals
Abstract: Kirillov-Reshetikhin (KR) modules are an important class of finite dimensional representations associated to an affine Lie algebra and the associated Yangian and quantum group. KR modules are known to appear in many integrable systems and govern the dynamics. In this talk, we will give an overview of the role KR modules play in the category of finite dimensional representations, R-matrices and the fusion construction, their (conjectural) crystal bases, and how they relate to Demazure modules. In particular, we will focus on how to construct their crystal bases combinatorially and the different types of character theories. As time permits, we will discuss some of the relations with (quantum) integrable systems.
Tuesday 12th December 2023, 4:30–5:30pm JST (UTC+9), L4E48 and online on Zoom
Title: Some applications of bricks in classical and modern problems in representation theory
Abstract: Bricks (also known as Schur representations) form a special subfamily of indecomposable modules, and they are used in the algebraic and geometric study of representation theory of algebras. We start by looking at some classical results on bricks, including a characterization of locally representation-directed algebras (due to Dräxler). Then, we consider some new directions of research in which bricks have played crucial roles. More specifically, we briefly recall an elegant correspondence between bricks and indecomposable \(\tau\)-rigid-modules (due to Demonet-Iyama-Jasso), which has many applications in \(\tau\)-tilting theory. We use the notion of \(\tau\)-rigidity to give a new characterization of locally representation-directed algebras, and to further generalize this family. If time permits, we also report on some new results on an open conjecture (so-called the 2nd brick-Brauer-Thrall conjecture) which I posed in 2019. Part of this talk is based on my recent joint work with Charles Paquette.
Tuesday 5th December 2023, 16:30–17:30am JST (UTC+9), L4E48 and online on Zoom
Eric Marberg, The Hong Kong University of Science and Technology (HKUST)
Title: From Klyachko models to perfect models
Abstract: In this talk a "model" of a finite group or semisimple algebra means a representation containing a unique irreducible subrepresentation from each isomorphism class. In the 1980s Klyachko identified an elegant model for the general linear group over a finite field with \(q\) elements. There is an informal sense in which taking the \(q \to 1\) limit of Klyachko's construction gives a model for the symmetric group, which can be extended to its Iwahori-Hecke algebra. The resulting Hecke algebra representation is a special case of a "perfect model", which is a more flexible construction that can be considered for any finite Coxeter group. In this talk, I will classify exactly which Coxeter groups have perfect models, and discuss some notable features of this classification. For example, each perfect model gives rise to a pair of related W-graphs, which are dual in types B and D but not in type A. Various interesting questions about these W-graphs remain open. This is joint work with Yifeng Zhang.
Tuesday 28th November 2023, 9:30–10:30am JST (UTC+9), online on Zoom
Alexander Yong, University of Illinois at Urbana-Champaign
Title: Newell-Littlewood numbers
Abstract: The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients from algebraic combinatorics. Both appear in representation theory as tensor product multiplicities for a classical Lie group. This talk concerns the question: Which multiplicities are nonzero? In 1998, Klyachko established common linear inequalities defining both the eigencone for sums of Hermitian matrices and the saturated Littlewood-Richardson cone. We prove some analogues of Klyachko's nonvanishing results for the Newell-Littlewood numbers. This is joint work with Shiliang Gao, Gidon Orelowitz, and Nicolas Ressayre. The presentation is based on arXiv:2005.09012, arXiv:2009.09904, and arXiv:2107.03152.
Title: The rank one property for free Frobenius extensions
Abstract: The Cartan matrix of a finite-dimensional algebra is the matrix of multiplicities of simple modules in indecomposable projective modules. This is crucial information about the representation theory of the algebra. In my talk I will present a general setting including several important examples from Lie theory, such as restricted quantized enveloping algebras at roots of unity, in which we could prove that the Cartan matrix has the remarkable property of being blockwise of rank one. This is joint work with Gwyn Bellamy.
Tuesday 24th October 2023, 4:30–5:30pm JST (UTC+9), online on Zoom
Title: Decomposition numbers for unipotent blocks with small \(sl_2\)-weight in finite classical groups
Abstract: There are many familiar module categories admitting a categorical action of a Lie algebra. The combinatorial shadow of such an action often yields answers to module-theoretic questions, for instance via crystals. In proving a conjecture of Gerber, Hiss, and Jacon, it was shown by Dudas, Varagnolo, and Vasserot that the category of unipotent representations of a finite classical group has such a categorical action. In this talk I will explain how we can use the categorical action to deduce closed formulas for certain families of decomposition numbers of these groups. This is joint work in progress with Olivier Dudas.
Abstract: A finite group \(G\) with center \(Z\) is of central type if there exists an irreducible character \(\chi\) such that \(\chi(1)^2=|G:Z|\). Howlett–Isaacs have shown that such groups are solvable. A corresponding theorem for \(p\)-Brauer characters was proved by Navarro–Späth–Tiep under the assumption that \(p\ne 5\). I have shown that there are no exceptions for \(p=5\). Moreover, I give some applications to \(p\)-blocks with a unique Brauer character.
Title: Higher-dimensional cluster combinatorics and representation theory
Abstract: Perhaps the most prominent example of a cluster algebra is the type A cluster algebra, where clusters are in bijection with triangulations of a convex polygon, as observed by Fomin and Zelevinsky. A categorical version of this relationship is that triangulations of a convex polygon are in bijection with cluster-tilting objects in the cluster category of the path algebra of the type A quiver. In each case, mutating the cluster or cluster-tilting object corresponds to flipping a diagonal inside a quadrilateral. It is natural to wonder whether any similar relationship exists for triangulations of higher-dimensional polytopes. Indeed, in a beautiful paper Oppermann and Thomas show that triangulations of even-dimensional cyclic polytopes are in bijection with cluster-tilting objects in the cluster categories of the higher Auslander algebras of type A, which were introduced by Iyama. Mutating the cluster-tilting objects corresponds to bistellar flips of triangulations, which are the higher-dimensional analogues of flipping a diagonal inside a quadrilateral. In this talk, we will outline the work of Oppermann and Thomas, and explain the odd-dimensional half of the picture too. Indeed, the speaker has shown that triangulations of odd-dimensional cyclic polytopes are in bijection with equivalence classes of maximal green sequences for the higher Auslander algebras of type A, where maximal green sequences are maximal chains of cluster-tilting objects.
Friday 28th July 2023, 10:30–11:30am JST (UTC+9), L4E48 and online on Zoom, note the unusual day and time
Ziqing Xiang, Southern University of Science and Technology
Title: Quantum wreath product
Abstract: The classical wreath product \(G \wr \Sigma_d\) is a semidirect product \(G^d \rtimes \Sigma_d\) with \(\Sigma_d\) acting on \(G^d\) by permutation. We deform this classical wreath product by deforming \(G\) into an associative algebra \(B\), deforming \(\Sigma_d\) into a Hecke algebra, and deforming the action. The result is called a quantum wreath product \(B \wr H(d)\). Many variants of Hecke algebras can be viewed as quantum wreath products, hence could be treated in a unified manner.
In this talk, we will discuss necessary and sufficient conditions for quantum wreath products to have a basis of suitable size. We will also discuss some other structural results, the Schur algebras of these quantum wreath products, and their representations.
Tuesday 14th March 2023, 10:30–11:30am JST (UTC+9), online on Zoom, note the unusual time
Title: Intermediate symplectic characters and enumeration of shifted plane partitions
Abstract: The intermediate symplectic characters, introduced by R. Proctor, interpolate between Schur functions and symplectic characters. They arise as the characters of indecomposable representations of the intermediate symplectic group, which is defined as the group of linear transformations fixing a (not necessarily non-degenerate) skew-symmetric bilinear form. In this talk, we present Jacobi-Trudi-type determinant formulas and bialternant formulas for intermediate symplectic characters. By using the bialternant formula, we can derive factorization formulas for sums of intermediate symplectic characters, which allow us to give a proof and variations of Hopkins' conjecture on the number of shifted plane partitions of double-staircase shape with bounded entries.
Title: On stable modular plethysms of the natural module of \(\textrm{SL}_2(\mathbb{F}_p)\) in characteristic \(p\)
Abstract: To study polynomial representations of general and special linear groups in characteristic zero one can use formal characters to work with symmetric functions instead. The situation gets more complicated when working over a field \(k\) of non-zero characteristic. However, by describing the representation ring of \(k\textrm{SL}_2(\mathbb{F}_p)\) modulo projective modules appropriately we are able to use symmetric functions with a suitable specialisation to study a family of polynomial representations of \(k\textrm{SL}_2(\mathbb{F}_p)\) in the stable category. In this talk we describe how this introduction of symmetric functions works and how to compute various modular plethysms of the natural \(k\textrm{SL}_2(\mathbb{F}_p)\)-module in the stable category. As an application we classify which of these modular plethysms are projective and which are `close' to being projective. If time permits, we describe how to generalise these classifications using a rule for exchanging Schur functors and tensoring with an endotrivial module.
Abstract: The symmetric group \(S_{mn}\) acts naturally on the collection of set partitions of a set of size mn into n sets each of size m. The irreducible constituents of the associated ordinary character are largely unknown; in particular, they are the subject of the longstanding Foulkes Conjecture. There are equivalent reformulations using polynomial representations of infinite general linear groups or using plethysms of symmetric functions. I will review plethysm from these three perspectives before presenting a new approach to studying plethysm: using the Schur-Weyl duality between the symmetric group and the partition algebra. This method allows us to study stability properties of certain plethysm coefficients. This is joint work with Chris Bowman. If time permits, I will also discuss some new results with Chris Bowman and Mark Wildon.
Title: Higher Rank Rational (q,t)-Catalan Polynomials and a Finite Shuffle Theorem
Abstract: The classical shuffle theorem states that the Frobenius character of the space of diagonal harmonics is given by a certain combinatorial sum indexed by parking functions on square lattice paths. The rational shuffle theorem, conjectured by Gorsky-Negut and proven by Mellit, states that the geometric action on symmetric functions (described by Schiffmmann-Vasserot) of certain elliptic Hall algebra elements \(P_{(m,n)}\) yield the bigraded Frobenius character of a certain Sn representation. This character is known as the Hikita polynomial. In this talk I will introduce the higher rank rational (q,t)-Catalan polynomials and show these are equal to finite truncations of the Hikita polynomial. By generalizing results of Gorsky-Mazin-Vazirani and constructing an explicit bijection between rational semistandard parking functions and affine compositions, I will derive a finite analog of the rational shuffle theorem in the context of spherical double affine Hecke algebras.
Tuesday 15th November 2022, 4:30–5:30pm JST (UTC+9), online on Zoom
Giada Volpato, University of Florence
Title: On the restriction of a character of \(\mathfrak{S}_n\) to a Sylow \(p\)-subgroup
Abstract: The relevance of the McKay Conjecture in the representation theory of finite groups has led to investigate how irreducible characters decompose when restricted to Sylow \(p\)-subgroups. In this talk we will focus on the symmetric groups. Since the linear constituents of the restriction to a Sylow \(p\)-subgroup has been studied a lot by E. Giannelli and S. Law, we will concentrate on constituents of higher degree. In particular, we will describe the set of the irreducible characters which allow a constituent of a fixed degree, separating the cases of \(p\) being odd and \(p=2\). This is a joint work with Eugenio Giannelli.
Title: KLR-type presentation of affine Hecke algebras of type B
Abstract: KLR algebras of type A have been a revolution in the representation theory of Hecke algebras of a type A flavour, thanks to the the Brundan-Kleshchev-Rouquier isomorphism relating them explicitly to the affine Hecke algebra of type A. KLR algebras of other types exist but are not related to affine Hecke algebras of other types. In this talk I will present a generalisation of the KLR presentation for the affine Hecke algebra of type B and I will discuss some applications. This talk is based on joint works with Salim Rostam and Ruari Walker.
Title: Quasi-hereditary covers, Hecke subalgebras and quantum wreath product
Abstract: The Hecke algebra is in general not quasi-hereditary, meaning that its module category is not a highest weight category; while it admits a quasi-hereditary cover via category O for certain rational Cherednik algebras due to Ginzburg-Guay-Opdam-Rouquier. It was proved in type A that this category O can be realized using q-Schur algebra, but this realization problem remains open beyond types A/B/C. An essential step for type D is to study Hu's Hecke subalgebra, which deforms from a wreath product that is not a Coxeter group. In this talk, I'll talk about a new theory allowing us to take the ``quantum wreath product'' of an algebra by a Hecke algebra. Our wreath product produces the Ariki-Koike algebra as a special case, as well as new ``Hecke algebras'' of wreath products between symmetric groups. We expect them to play a role in answering the realization problem for complex reflection groups. This is a joint work with Dan Nakano and Ziqing Xiang.
Tuesday 11th October 2022, 4:30–5:30pm JST (UTC+9), online on Zoom
Title: On self-extensions of irreducible modules for symmetric groups
Abstract: We work in the context of the modular representation theory of the symmetric groups. A long-standing conjecture, from the late 80s, suggests that there are no (non-trivial) self-extensions of irreducible modules over fields of odd characteristic. In this talk we will highlight several new positive results on this conjecture. This is a joint work with S. Kleshchev and L. Morotti.
Title: Young’s seminormal basis vectors and their denominators
Abstract: The dual Specht module of the symmetric group algebra over \(\mathbb{Q}\) has two distinguished bases, namely the standard basis and Young’s seminormal basis. We study how the Young’s seminormal basis vectors are expressed in terms of the standard basis, as well as the denominators of the coefficients in these expressions. We obtain closed formula for some Young’s seminormal basis vectors, as well as partial results for the denominators in general. This is a joint work with Ming Fang (Chinese Academy of Sciences) and Kay Jin Lim (Nanyang Technological University).
Abstract: In 1901 Young gave an explicit construction of the ordinary irreducible representations of the symmetric groups. In doing this, he introduced content functions for partitions, which are now a key statistic in the semisimple representation theory of the symmetric groups. In this talk I will describe a generalisation of Young's ideas to the cyclotomic KLR algebras of affine types A and C. This is quite surprising because Young's seminormal forms are creatures from the semisimple world whereas the cyclotomic KLR algebras are rarely semisimple. As an application, we show that these algebras are cellular and construct their irreducible representations. A special case of these results gives new information about the symmetric groups in characteristic p>0. If time permits, I will describe how these results lead to an explicit categorification of the corresponding integrable highest weight modules. This is joint work with Anton Evseev.
Abstract: The Rouquier blocks, also known as the RoCK blocks, are important blocks of the symmetric groups algebras and the Hecke algebras of type \(A\), with the partitions labelling the Specht modules that belong to these blocks having a particular abacus configuration. We generalise the definition of Rouquier blocks to the Ariki-Koike algebras, where the Specht modules are indexed by multipartitions, and explore the properties of these blocks.
Title: Scopes equivalence for blocks of Ariki-Koike algebras
Abstract: We consider representations of the Ariki-Koike algebra, a \(q\)-deformation of the group algebra of the complex reflection group \(C_r \wr S_n\). The representations of this algebra are naturally indexed by multipartitions of \(n\). We examine blocks of the Ariki-Koike algebra, in an attempt to generalise the combinatorial representation theory of the Iwahori-Hecke algebra. In particular, we prove a sufficient condition such that restriction of modules leads to a natural correspondence between the multipartitions of \(n\) whose Specht modules belong to a block \(B\) and those of \(n-\delta_i(B)\) whose Specht modules belong to the block \(B'\), obtained from \(B\) applying a Scopes' equivalence.
Tuesday 5th July 2022, 4:30–5:30pm JST (UTC+9), L4E48, and online on Zoom
Title: Superalgebra deformations of web categories
Abstract: For a superalgebra A, and even subalgebra a, one may define an associated diagrammatic monoidal supercategory Web(A,a), which generalizes a number of symmetric web category constructions. In this talk, I will define and discuss Web(A,a), focusing on two interesting applications: Firstly, Web(A,a) is equipped with an asymptotically faithful functor to the category of gl_n(A)-modules generated by symmetric powers of the natural module, and may be used to establish Howe dualities between gl_n(A) and gl_m(A) in some cases. Secondly, Web(A,a) yields a diagrammatic presentation for the ‘Schurification' T^A_a(n,d). For various choices of A/a, these Schurifications have proven connections to RoCK blocks of Hecke algebras, and conjectural connections to RoCK blocks of Schur algebras and Sergeev superalgebras. This is joint work with Nicholas Davidson, Jonathan Kujawa, and Jieru Zhu.
Title: An example of A2 Rogers-Ramanujan bipartition identities of level 3
Abstract: In the 1970s, Lepowsky-Milne discovered a similarity between the infinite products of the Rogers-Ramanujan identities (RR identities, for short) and the principal characters of the level 3 standard modules of the affine Lie algebra of type \(A^{(1)}_{1}\). Subsequently, Lepowsky-Wilson gave a Lie-theoretic interpretation and a proof of the RR identities with the vertex operators. In this talk, I will present some results (arXiv:2205.04811) for the level 3 case of type \(A^{(1)}_{2}\).
Abstract: For a finite Coxeter group W, L. Solomon defined certain subalgebra of the group algebra kW which is now commonly known as the Solomon’s descent algebra. As usual, the type A and B cases have special interest for both the algebraists and combinatorists. In this talk, I will be particularly focusing on the type A and modular case. It is closely related to the representation theory of the symmetric group and the (higher) Lie representations.
Title: Spin representations of the symmetric group
Abstract: Spin representations of the symmetric group S_n can be thought of equivalently as either projective representations of S_n, or as linear representations of a double cover S_n+ of S_n. Whilst the linear representation theory of S_n is dictated by removing ‘rim-hooks’ from (the Young diagrams of) partitions of n, the projective representation theory of S_n is controlled by removing ‘bars’ from bar partitions of n (i.e. partitions of n into distinct parts). We will look at some combinatorial results on bar partitions from a recent paper of the author before discussing methods for determining the modular decomposition of spin representations over fields of positive characteristic.
Title: A Schur-positivity conjecture inspired by the Alperin-Mckay conjecture
Abstract: The McKay conjecture asserts that a finite group has the same number of odd degree irreducible characters as the normalizer of a Sylow 2-subgroup. The Alperin-McKay (A-M) conjecture generalizes this to the height-zero characters in 2-blocks.
In his original paper, McKay already showed that his conjecture holds for the finite symmetric groups S_n. In 2016, Giannelli, Tent and the speaker established a canonical bijection realising A-M for S_n; the height-zero irreducible characters in a 2-block are naturally parametrized by tuples of hooks whose lengths are certain powers of 2, and this parametrization is compatible with restriction to an appropriate 2-local subgroup.
Now corresponding to a 2-block of the symmetric group S_n, there is a 2-block of a maximal Young subgroup of S_n of the same weight. An obvious question is whether our canonical bijection is compatible with restriction of height-zero characters between these blocks.
Attempting to prove this compatibility lead me to formulate a conjecture asserting the Schur-positivity of certain differences of skew-Schur functions. The corresponding skew-shapes have triangular inner-shape, but otherwise do not refer to the 2-modular theory. I will describe my conjecture and give positive evidence in its favour.
Abstract: The nets giving a diagrammatic description of the category of (tensor products of) fundamental representations of \(sl_n\) form a cellular category. We can then ask about the natural inner form on certain cell modules. In this talk, we will calculate the determinant of some of these forms in terms of certain traces of clasps or magic weave elements (for which there is a conjectured formula due to Elias). The method appears moderately general and gives a result which is hopefully illuminating and applicable to other monoidal, cellular categories.
Title: Schur-Weyl duality for braid and twin groups via the Burau representation
Abstract: The natural permutation representation of the symmetric group admits a q-analogue known as the Burau representation. The symmetric group admits two natural covering groups: the braid group of Artin and the twin group of Khovanov, obtained respectively by forgetting the cubic and quadratic relations in the Coxeter presentation of the symmetric group. By computing centralizers of tensor powers of the Burau representation, we obtain new instances of Schur-Weyl duality for braid groups and twin groups, in terms of the partial permutation and partial Brauer algebras. The method produces many representations of each group that can be understood combinatorially. (This is joint work with Tony Giaquinto.)
Abstract: Weighted KLRW algebras are diagram algebras that depend on continuous parameters. Varying these parameters gives a way to interpolate between various algebras that appear in (categorical) representation theory such as semisimple algebras, KLR algebras, quiver Schur algebras and diagrammatic Cherednik algebras. This talk is a friendly (and diagrammatic!) introduction explaining these algebras, with no prior knowledge about any of these assumed. Based on joint work A. Mathas.
Title: Decompositions of tensor products: Highest weight vectors from branching
Abstract: We consider tensor powers of the natural sl_n-representation, and we look for descriptions of highest weight vectors therein: We discuss explicit formulas for n=2, a recursion for n=3, and for bigger n we demonstrate how Jucys-Murphy elements allow us to compute highest weight vectors (both in theory and in practice using sage). This is joint work with Pablo Zadunaisky.
Tuesday 30th November 2021, 9:30–10:30am JST (UTC+9), online on Zoom
Abstract: The Kazhdan–Lusztig (KL) cells of a Coxeter group are subsets of the group defined using the KL basis of the associated Iwahori–Hecke algebra. The cells of symmetric groups can be computed via the Robinson–Schensted correspondence, but for general Coxeter groups combinatorial descriptions of KL cells are largely unknown except for cells of a-value 0 or 1, where a refers to an N-valued function defined by Lusztig that is constant on each cell. In this talk, we will report some recent progress on KL cells of a-value 2. In particular, we classify Coxeter groups with finitely many elements of a-value 2, and for such groups we characterize and count all cells of a-value 2 via certain posets called heaps. We will also mention some applications of these results for cell modules. This is joint work with Richard Green.
Tuesday 16th November 2021, 4:30–5:30pm JST (UTC+9), online on Zoom
Samuel Creedon, City, University of London
Title: Defining an Affine Partition Algebra
Abstract: In this talk we motivate the construction of a new algebra called the affine partition algebra. We summarise some of its basic properties and describe an action which extends the Schur-Weyl duality between the symmetric group and partition algebra. We establish connections to the affine partition category defined recently by Brundan and Vargas and show that such a category is a full subcategory of the Heisenberg category.
Abstract: Arc algebras were introduced by Khovanov in a successful attempt to lift the quantum sl2 Reshetikhin-Turaev invariant for tangles to a homological invariant. When restricted to knots and links, Khovanov’s homology theory categorifies the Jones polynomial. Ozsváth-Rasmussen-Szabó discovered a different categorification of the Jones polynomial called odd Khovanov homology. Recently, Naisse-Putyra were able to extend odd Khovanov homology to tangles using so-called odd arc algebras which were originally constructed by Naisse-Vaz. The goal of this talk is to discuss a geometric approach to understanding odd arc algebras and odd Khovanov homology using Springer fibers over the real numbers. This is joint work with J. N. Eberhardt and G. Naisse.
Tuesday 26th October 2021, 9:30–10:30am JST (UTC+9), online on Zoom
Abstract: The Shuffle Theorem, conjectured by Haglund, Haiman, Loehr, Remmel and Ulyanov, and proved by Carlsson and Mellit, describes the characteristic of the \(S_n\)-module of diagonal harmonics as a weight generating function over labeled Dyck paths under a line with slope −1. The Shuffle Theorem has been generalized in many different directions, producing a number of theorems and conjectures. We provide a generalized shuffle theorem for paths under any line with negative slope using different methods from previous proofs of the Shuffle Theorem. In particular, our proof relies on showing a "stable" shuffle theorem in the ring of virtual GL_l-characters. Furthermore, we use our techniques to prove the Extended Delta Conjecture, yet another generalization of the original Shuffle Conjecture.
Abstract: I will describe a surprising connection between the colored HOMFLY-PT polynomials of knots and the motivic Donaldson-Thomas invariants of certain symmetric quivers, which was conjectured by Kucharski-Reineke-Stosic-Sulkowski. I will outline a proof of this correspondence for arborescent links via quivers associated with 4-ended tangles. Finally, I will speculate about how much of the HOMFLY-PT skein theory might carry over to the realm of DT quiver invariants and what kind of geometric information about knots might be encoded in these quivers. This is joint work with Marko Stosic.
Abstract: The Bruhat order on a Weyl group has a representation theoretic interpretation in terms of Verma modules. The talk concerns resulting interactions between combinatorics and homological algebra. I will present several questions around the above realization of the Bruhat order and answer them based on a series of recent works, partly joint with Volodymyr Mazorchuk and Rafael Mrden.
Title: On the computation of decomposition numbers of the symmetric group
Abstract: The most important open problem in the modular representation theory of the symmetric group is finding the multiplicity of the simple modules as composition factors of the Specht modules. In characteristic 0 the Specht modules are just the simple modules of the symmetric group algebra, but in positive characteristic they may no longer be simple. We will survey the rich interplay between representation theory and combinatorics of integer partitions, review a number of results in the literature which allow us to compute composition series for certain infinite families of Specht modules from a finite subset of them, and discuss the extension of these techniques to other Specht modules.
Title: Grassmannians, Cluster Algebras and Hypersurface Singularities
Abstract: Grassmannians are objects of great combinatorial and geometric beauty, which arise in myriad contexts. Their coordinate rings serve as a classical example of cluster algebras, as introduced by Fomin and Zelevinsky at the start of the millennium, and their combinatorics is intimately related to algebraic and geometric concepts such as to representations of algebras and hypersurface singularities. At the core lies the idea of generating an object from a so-called “cluster” via the concept of “mutation”. In this talk, we offer an overview of Grassmannian combinatorics in a cluster theoretic framework, and ultimately take them to the limit to explore the a priori simple question: What happens if we allow infinite clusters? We introduce the notion of a cluster algebra of infinite rank (based on joint work with Grabowski), and of a Grassmannian category of infinite rank (based on joint work with August, Cheung, Faber and Schroll).
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
