Tag - Representation theory

Karlee Westrem: A new symmetric function identity with an application to symmetric group character values

7th October, 2024

Symmetric functions show up in several areas of mathematics including enumerative combinatorics and representation theory. Tewodros Amdeberhan conjectures equalities of Σn character sums over a new set called Ev(λ). When investigating the alternating sum of characters for Ev(λ) written in terms of the inner product of Schur functions and power sum symmetric functions, we found an equality between the alternating sum of power sum symmetric polynomials and a product of monomial symmetric polynomials. As a consequence, a special case of an alternating sum of Σn characters over the set Ev(λ) equals 0.

Kay Jin Lim: Integral basic algebras

1st October, 2024

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,𝒪,k) and an 𝒪-algebra A where both the algebras K⊗𝒪A and k⊗𝒪A 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.

Duc-Khanh Nguyen: Application of (K-theoretic) Peterson isomorphism

17th September, 2024

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 SLn, 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.

Karthik Ganapathy: GL-algebras and the Noetherianity problem

16th September, 2024

Draisma recently proved that finite length polynomial representations of the infinite general linear group GL are topologically GL-noetherian, i.e., the descending chain condition holds for GL-stable closed subsets. The scheme-theoretic variant of this theorem is a major open problem in the area. I will briefly outline the rich history of this problem and provide a negative answer in characteristic 2.

Sondre Kvamme: Higher torsion classes and silting complexes

20th June, 2024

Higher Auslander-Reiten theory was introduced by Iyama in 2007 as a generalization of classical Auslander-Reiten theory. The main objects of study in the theory are d-cluster tilting subcategories of module categories. It turns out that many notions in algebra and representation theory have generalizations to higher Auslander-Reiten theory. In particular, in 2016 Jørgensen introduced a generalization of torsion classes, called higher torsion classes.

In this talk, I will recall the definition of higher torsion classes. I will then explain how functorially finite d-torsion classes give rise to (d+1)-term silting complexes, and hence to derived equivalences. The construction is analogous to the construction of 2-term silting complexes due to Adachi-Iyama-Reiten in 2014. I will illustrate the constructions and results on higher Nakayama algebras of type An.

Kent Vashaw: A Chinese remainder theorem and Carlson theorem for monoidal triangulated categories

6th June, 2024

Carlson's connectedness theorem for cohomological support varieties is a fundamental result which states that the support variety for an indecomposable module of a finite group is connected. In this talk, we will discuss a generalization, where it is proved that the Balmer support for an arbitrary monoidal triangulated category satisfies the analogous property. This is shown by proving a version of the Chinese remainder theorem in this context, that is, giving a decomposition for a Verdier quotient of a monoidal triangulated category by an intersection of coprime thick tensor ideals.

Cornelius Pillen: Proving and disproving conjectures by Humphreys, Verma and Donkin

31st May, 2024

Let G be a simple, simply connected algebraic group defined over a field of positive characteristic p, Gr be its rth Frobenius kernel, and, for q = pr, G(q) denotes the group of rational points over a field with q elements.

Motivated by work of Curtis and Steinberg, who showed that the simple Gr- and the simple G(q)-modules can be lifted to G, Humphreys and Verma conjectured that the projective covers of the simple Gr-modules also afford a G-module structure. Donkin later refined this conjecture by suggesting that these Gr-projectives lift uniquely to G in the form of tilting modules.

Ballard and Jantzen verified Donkin’s Tilting Module Conjecture for primes that are roughly twice the Coxeter number of the underlying root system or larger. But it was shown by Nakano and his collaborators that the conjecture fails in general. Counterexamples exist for all root systems with the exception of types B2, where the conjecture holds, and type A, where the conjecture remains completely open.

In this talk we delve into the rich history of these and closely related conjectures and report on their current status.

Emilie Wiesner: Representations of the Virasoro algebra

31st May, 2024

The Virasoro algebra is the central extension of derivations on Laurent polynomials. It plays an important role in mathematical physics and is itself a nice case study of an infinite-dimensional Lie algebra with triangular decomposition. I’ll give an overview of several families of representations of the Virasoro algebra and some connections between them.

Paul Sobaje: Combinatorics for Special Tilting Modules

31st May, 2024

The quest to find a character formula for the simple modules of a reductive algebraic group in positive characteristic took an unexpected turn roughly a decade ago when Williamson found a large number of counterexamples to the Lusztig Conjecture. Since then, the path to the simple characters has gone through the characters of the indecomposable tilting modules, thanks to the work of Riche and Williamson. However, the combinatorics required for determining all tilting characters are quite complicated, and the vast majority of these characters are not necessary to determine the simple characters. This talk is based on our pursuit of a more simplistic model in terms of what we’ve called the 'Steinberg quotient' of special tilting characters.

Aslak Bakke Buan: From exceptional to τ-exceptional sequences in module categories

30th May, 2024

Exceptional sequences and their mutations were first considered in triangulated categories by the Moscow school of algebraic geometers. In the early nineties, Crawley-Boevey and Ringel studied exceptional sequences for module categories of hereditary algebras. We first recall their definitions and their main results, and then proceed to discuss a natural generalization to all (not necessarily hereditary) finite-dimensional algebras. This is the theory of τ-exceptional sequences, which was developed in joint work with Marsh, motivated by τ-tilting theory, by Adachi-Iyama-Reiten, by Jasso's reduction techniques for such modules and corresponding torsion pairs, and by the introduction of signed exceptional sequences by Igusa-Todorov.

The interplay between theories for τ-rigid modules, torsion pairs, and wide subcategories is central to our discussions.