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Nicolle González: Higher Rank Rational (q,t)-Catalan Polynomials and a Finite Shuffle Theorem

29th November, 2022

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 analogue of the rational shuffle theorem in the context of spherical double affine Hecke algebras.

Leonardo Maltoni: Towards a Bernstein presentation of the affine Hecke category

28th November, 2022

The affine Hecke algebra has a remarkable commutative subalgebra corresponding to the coroot lattice in the affine Weyl group. Its nature is encoded in the Bernstein presentation and reveals important representation-theoretic properties of the algebra. If one considers categorifications of the Hecke algebra, for instance the diagrammatic category, the above subalgebra corresponds to a class of complexes in the homotopy category called Wakimoto sheaves, which can be seen as Rouquier complexes. In this talk I will introduce the affine Hecke algebra, the diagrammatic category and the objects mentioned above. Then I will describe some reduced representarives for Rouquier complexes and present some results about the extension groups between Wakimoto sheaves in affine type A1.

Loïc Poulain d’Andecy: KLR-type presentation of affine Hecke algebras of type B

1st November, 2022

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.

Chun-Ju Lai: Quasi-hereditary covers, Hecke subalgebras and quantum wreath product

25th October, 2022

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.

Alice Dell’Arciprete: Scopes equivalence for blocks of Ariki-Koike algebras

19th July, 2022

We consider representations of the Ariki-Koike algebra, a q-deformation of the group algebra of the complex reflection group Cr ≀ Sn. 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-δi(B) whose Specht modules belong to the block B', obtained from B applying a Scopes equivalence.

Rob Muth: Superalgebra deformations of web categories

5th July, 2022

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 𝔤𝔩n(A)-modules generated by symmetric powers of the natural module, and may be used to establish Howe dualities between 𝔤𝔩n(A) and 𝔤𝔩m(A) in some cases. Secondly, Web(A,a) yields a diagrammatic presentation for the ‘Schurification' TAa(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.

CJ Lai: Wreath products, Schur dualities, and quasi-hereditary covers

28th February, 2022

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 by the category 𝒪 of a certain rational Cherednik algebra due to Ginzburg-Guay-Opdam-Rouquier. It was later shown in type A that this category 𝒪 can be realized concretely as the module category of Dipper-James's q-Schur algebra, but this realization problem remains open beyond types A and B. An essential step for type D, i.e., the complex reflection group G(2,2,n), is to study Hu's Hecke subalgebra, which deforms a wreath product that is not a Coxeter group. In this talk, I'll introduce a new theory allowing us to take the wreath product of an algebra by a Hecke algebra. Before our work, wreath products related to Hecke algebras were worked out at the degenerate level by Wan-Wang. Our wreath product produces the Ariki-Koike algebras as special cases as well as new 'Hecke algebras' of wreath products between symmetric groups. These are the first steps towards answering the realization problem for complex reflection groups.

Tianyuan Xu: On Kazhdan–Lusztig cells of a-value 2

30th November, 2021

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.

Tamanna Chatterjee: Parity Sheaves Arising from Graded Lie Algebras II

13th September, 2021

Let G be a complex, connected, reductive, algebraic group, and χ : ℂ× → G be a fixed cocharacter that defines a grading on 𝔤, the Lie algebra of G. In my first talk I have talked about the grading, derived category of equivariant perverse sheaves, bijection between the simple objects and some pairs that we are familiar with. In positive characteristic parity sheaves will play an important role. In this talk I will define parabolic induction and restriction both on nilpotent cone and graded setting. We will dive into the results of Lusztig in characteristic 0 in the graded setting. Under some assumptions on the field k and the group G we will recover some results of Lusztig in characteristic 0. These assumptions together with Mautner's cleanness conjecture will play a vital role. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair. Lusztig's work on ℤ-graded Lie algebras is related to representations of affine Hecke algebras, so a long term goal of this work will be to interpret parity sheaves in the context of affine Hecke algebras.

Alistair Savage: Affine Hecke algebras and the elliptic Hall algebra

13th May, 2021

The elliptic Hall algebra has appeared in many different contexts in representation theory and geometry under different names. We will explain how this algebra is categorified by the quantum Heisenberg category, which is a diagrammatic category modelled on affine Hecke algebras. This categorification can be used to construct large families of representations for the elliptic Hall algebra.