Seminars in Schur Algebras

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Karin Erdmann: The Hemmer-Nakano Theorem and relative dominant dimension

30th May, 2024

Let ℋq(d) be the Iwahori-Hecke algebra of the symmetric group where q is a primitive ℓ-th root of unity, and let A = Sq(n,d) be the q-Schur algebra. Hemmer and Nakano proved amongst others that for ℓ ≥ 4, the Schur functor gives an equivalence between the category of A-modules with Weyl filtration, and the category of ℋq(d)-modules with dual Specht filtration, and that certain extension groups get identified. This has been a surprise and has inspired further research. In this talk we discuss some extensions of this result.

Jie Du: The quantum queer supergroup via their q-Schur superalgebras

28th May, 2024

Using a geometric setting of q-Schur algebras, Beilinson-Lusztig-MacPherson discovered a new basis for quantum 𝔤𝔩n (i.e., the quantum enveloping algebra Uq(𝔤𝔩n) of the Lie algebra 𝔤𝔩n) and its associated matrix representation of the regular module of Uq(𝔤𝔩n). This beautiful work has been generalized (either geometrically or algebraically) to quantum affine 𝔤𝔩n, quantum super 𝔤𝔩m|n, and recently, to some i-quantum groups of type AIII.

In this talk, I will report on a completion of the work for a new construction of the quantum queer supergroup using their q-Schur superalgebras. This work was initiated 10 years ago, and almost failed immediately after a few months’ effort, due to the complication in computing the multiplication formulas by odd generators. Then, we moved on testing special cases or other methods for some years and regained confidence to continue. Thus, it resulted in a preliminary version which was posted on arXiv in August 2022.

The main unsatisfaction in the preliminary version was the order relation used in a triangular relation and the absence of a normalized standard basis. It took almost two more years for us to tune the preliminary version up to a satisfactory version, where the so-called SDP condition, involving further combinatorics related to symmetric groups and Clifford generators, and an extra exponent involving the odd part of a labelling matrix play decisive roles to fix the problems.

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.

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.

Aaron Yi Rui Low: Adjustment matrices

2nd March, 2021

James's Conjecture predicts that the adjustment matrix for weight w blocks of the Iwahori-Hecke algebras ℋn and the q-Schur algebras 𝒮n is the identity matrix when w < char(F). Fayers has proved James's Conjecture for blocks of ℋn 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 𝒮n of weights 3 and 4 in an upcoming paper. If time permits, we will look at a proof of the weight 3 case.

Qi Wang: On τ-tilting finiteness of Schur algebras

17th November, 2020

Support τ-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 𝔽 of characteristic p. Then, the Schur algebra S(n,r) is defined as the endomorphism ring End𝔽Gr(V⊗r) over the group algebra 𝔽Gr of the symmetric group Gr. In this talk, we discuss when the Schur algebra S(n,r) has only finitely many pairwise non-isomorphic basic support τ-tilting modules.