Köln Algebra and Representation Theory Seminar

Abstract 2:  In cluster theory, establishing explicit formulae for cluster variables is an important problem. In the setting of surface cluster algebras this may be given combinatorially via snake graphs and homologically by the CC-map. Recently a combinatorial formula was introduced by Musiker--Ovenhouse--Zhang in an attempt to introduce super cluster algebras of type A (which computes super lambda-lengths in Penner-Zeitlin's super-Teichmüller spaces). This formula is given in terms of double dimer covers on snake graphs. Motivated by this construction, we propose a representation theoretic interpretation of super lambda-lengths and introduce a super-CC formula which recovers the combinatorial model. This is joint work in progress with Fedele, Garcia Elsener and Serhiyenko.

Abstract: One of the topics of *the asymptotic representation theory *is to study concrete examples of reducible representations of some classical groups with the following question in mind: *what can we say about a random (or "typical") irreducible component?* These questions become interesting in the asymptotic setup when the groups which we study become bigger and bigger, and the dimension of the representation tends to infinity. In this talk I will concentrate on the specific example of the representation of a product of two general linear groups, known as *"skew Howe duality"*. This is a joint work in progress with Greta Panova, Stephen Moore, and Jacinta Torres. 

Abstract: Let A be an arbitrary hereditary abelian category which may not have enough projective objects. Lu and Peng defined the modified Ringel-Hall algebra SDH(A) of A and proved that SDH(A) has a natural basis and is isomorphic to the Drinfeld Double Ringel-Hall algebra of A. In this talk, we introduce a coproduct formula on SDH(A) with respect to the basis of SDH(A) and prove that this coproduct is compatible with the product of SDH(A), thereby the modified Ringel-Hall algebra of A is endowed with a bialgebra structure which is identified with the bialgebra structure of the Drinfeld Double Hall algebra of A.

Abstract: Degenerate flag varieties are certain flat degenerations of flag manifolds introduced by Feigin in 2010, that proved useful in the representation theory of the special linear and symplectic Lie algebra. It has been shown by Cerulli Irelli, Lanini and Littelmann that these degenerate flag varieties in types A and C are actually Schubert varieties for larger classical groups of same type. We propose a framework (called Dynkin abelianizations) that generalizes their ideas to other types and to "partially" degenerate flag varieties, realizing those partial degenerations as Schubert varieties as well. This is joint work (in progress) with Shreepranav Enugandla, Xin Fang and Ghislain Fourier.

Abstract: Geometric models associated to triangulations of Riemann surfaces arose in the context of cluster algebras and have since been used as an important tool to study representation theory of algebras and provide connections with algebraic geometry and symplectic geometry.

Significant applications of geometric models include a description of extensions and a classification of support tau-tilting modules over gentle algebras. Gentle algebras are a particular subclass of string algebras, which are of tame representation type, meaning it is often possible to get a global understanding of their representation theory.

In this talk I will describe the module category of a gentle algebra via partial triangulations of unpunctured surfaces, explain how to extend this model to a geometric model of the module category of any string algebra and use this model to obtain a classification of support tau-tilting modules. This is based on joint work in progress with Karin Baur.

Abstract: Studying higher simplex equations (Zamolodchikov equations), in 1989 Manin and Schechtman introduced the notion of a higher Bruhat order on the $d$-element subsets of a set $[n]=\{1,2,\ldots,n\}$. Among other results in this field, they proved that the set of such orders for $n,d$ fixed, endowed with natural local transformations, constitutes a poset with one minimal and one maximal elements. I will report on a wider model, involving the so-called convex order on certain path systems in an acyclic directed graph. We introduce local transformations, or flips, on such orders and prove that the resulting structure gives a poset with one minimal and one maximal elements as well, and get relations between convex orders in neighboring dimensions, yielding a generalization of the above-mentioned classical results. This is a joint work with V.Danilov and A.Karzanov. 

Abstract 1: Gentle algebras constitute a class of finite dimensional algebras which naturally arise in rather diverse contexts and whose representation theory is non-trivial but accessible. In this talk, I will explain a classification of semiorthogonal decompositions of their bounded derived categories obtained in joint work with Jakub Kopriva (arXiv:2209.14496). It is given by certain cuts of the surface attached to a gentle algebra in the work of Baur and Coelho Simoes (arXiv:1803.05802, published in IMRN, 2021), and Opper, Plamondon and Schroll (arXiv:1801.09659).