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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.

Haralampos Geranios: On self-extensions of irreducible modules for symmetric groups

11th October, 2022

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.

Kai Meng Tan: Young’s seminormal basis vectors and their denominators

20th September, 2022

The dual Specht module of the symmetric group algebra over 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.

Andrew Mathas: Content systems and KLR algebras

10th August, 2022

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 generalization 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.

Sinéad Lyle: Rouquier blocks for Ariki-Koike algebras

21st July, 2022

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 generalize 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.

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.

Irina Sviridova: Hook theorem for identities and its generalizations

15th July, 2022

The hook theorem is one of the key result of the classical theory of polynomial identities of algebras in the case of a field of characteristic zero. This well known result is fundamental for applications of the technique of the classic representation theory of the symmetric group to study identities. It has essential connections with many important facts of PI-theory, and implies many important and interesting consequences. In particular, it is one of the basic results for Kemer's positive solution of the Specht problem. Also it is the base to construct the growth theory for varieties of associative algebras over a field of of characteristic zero.

In the last years, one of the most popular directions of the theory of polynomial identities is to consider algebras with some additional structures (such as gradings, involutions, actions by automorphisms, etc.), and to study identities of such algebras with the additional signature.

We will discuss the versions of the hook theorem for various types of such identities with complementary structures. In particular, we will represent some version of the hook theorem for identities with some types of actions. This result generalizes the analogous results known before, for example, for graded identities or identities with involution. We also will discuss some possible consequences and applications of this theorem.

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.

Kay Jin Lim: Descent Algebra of Type A

20th April, 2022

For a finite Coxeter group W, L. Solomon defined certain subalgebra of the group algebra kW which is now commonly known as 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.

Dean Yates: Spin representations of the symmetric group

5th April, 2022

Spin representations of the symmetric group Sn can be thought of equivalently as either projective representations of Sn, or as linear representations of a double cover Sn+ of Sn. Whilst the linear representation theory of Sn is dictated by removing 'rim-hooks' from (the Young diagrams of) partitions of n, the projective representation theory of Sn 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.