Algebra

NOTES: extra seminar this week on an unusual day (Friday); no seminar next week.
Unrefinable partitions, arising quite unexpectedly in a combinatorial problem in group theory, represent a special subset of integer partitions into distinct parts, constrained by an additional additive relationship between the parts. Despite being a natural combinatorial object, they remain relatively unexplored in the literature, with only a few known properties and results.
In this talk, we explore the foundational aspects of unrefinable partitions, showing some of their initial properties. We will present an algorithm designed to efficiently test for unrefinability in a given partition. By establishing a bound on the largest part in such partitions, we introduce the concept of maximal unrefinable partitions, a subclass with its own distinctive structure. We will show how to count such maximal unrefinable partitions using explicit bijections, providing a clearer understanding of their combinatorial structure.

Brauer graph algeras are finite dimensional algebras constructed from the combinatorial data of a graph called a Brauer graph. Kauer proved that derived equivalences of Brauer graph algebras can be obtained from the move of one edge in the corresponding Brauer graph. Moreover, this derived equivalence is entirely described thanks to a tilting object which can be interpreted in terms of silting mutation. In this talk, I will be interested in skew Brauer graph algebras which generalize the class of Brauer graph algebras. These algebras are constructed from the combinatorial data of a Brauer graph where some edges might be "degenerate". I will explain how Kauer's results can be generalized for the move of multiple edges and to the case of skew Brauer graph algebras.

The positive integers that come up in the solutions of the Diophantine equation $x^2+y^2+z^2=3xyz$ are called Markov numbers. They play an important role in the theory of rational expansions and come with a 100+ year old open problem called the Uniqueness conjecture. Recently discovered connections to cluster algebras have revitalized the theory, leading to generalizations and deformations motivated by the cluster model. In this talk, we will use oriented posets to construct a combinatorial model for $q$-deformed Markov numbers. We will also discuss future directions and further avenues of research.

This is based on the equally named paper which is on arxiv and to appear in Journal of Algebra. I will discuss the connection with the representations of reductive groups and Lie algebras: the hyper algebra, the restricted enveloping algebra and their centres. Then I will move to divided power algebras, truncated polynomial algebras and their invariants. I will discuss some results ($GL_n$ only) from my paper, a conjecture concerning the so-called “restriction property” and the connection with “special symmetrization map” from Okounkov-Olshanskii. If time allows, I will mention extensions of the above results to several matrices and vectors and covectors.

Nichols algebras appear in several areas of mathematics, from Hopf algebras and quantum groups to Schubert calculus and conformal field theories. In this talk, I will review the main problems related to Nichols algebras and discuss some recent classification theorems.

There is an equivalence between the category of simplicial abelian groups and the category of differential graded abelian groups called the Dold-Kan equivalence. There is also a class of curious objects called Crossed-Simplicial Groups defined by a distributive law between a collection of groups indexed by the natural numbers and the simplicial category $\Delta$. There have been attempts to extend the Dold-Kan to crossed-simplicial setting by explicitly constructing extensions of the differential graded side. But these are few and far between. In this talk, I'll start by a new but a weakened analogue of the Dold-Kan by using certain induction and restriction functors, and by passing to the homotopy categories on both sides. Then show that this homotopical version of the equivalence extends to the crossed-simplicial setting.