Köln Algebra and Representation Theory Seminar

Abstract: A Jacobi matrix is a matrix $A=(a_{i,j})$ of infinite size indexed over $\Z$ verifying the next conditions: there exists $N>0$ depending on $A$ such that $a_{i,j}=0$ for $\vert i-j \vert>N$. The Lie algebra of such matrices plays a key role in soliton theory and it is important to determine its central extension. In this talk, we will explain some key steps of the computation of its homology, and its variants. Some interesting open questions will be also given. 

Abstract: Howe duality gives a simple formula for decomposing the symmetric and exterior algebra of $\mathbb{C}^n \otimes \mathbb{C}^m$ as a natural $\mathfrak{gl}_n \times \mathfrak{gl}_m$-module. Taking the corresponding characters yields the well-known Cauchy identities, which can alternatively be deduced from the Robinson-Schensted-Knuth (RSK) correspondence. In this talk, I will first explain how both Howe duality and the RSK correspondence can be recovered from the combinatorics of crystals and tableaux. Then, I will describe a generalisation of these constructions in the case of the symplectic Lie algebras $\mathfrak{sp}_{2n}$ and $\mathfrak{sp}_{2m}$. 

Abstract: Thanks to recent important work on gentle algebras by a number of people (among others, Amiot, Baur, Coelhos, Opper, Padrol, Palu, Pilaud, Plamondon, and Schroll) representations of these algebras correspond to curves drawn on a corresponding surface. In general the combinatorics of these curves on surfaces can be complicated. I will explain how, for a particular class of algebras, we can give a very simple combinatorial description of the band modules which are bricks (i.e., have one-dimensional endomorphism ring). Surprisingly, a subset of the combinatorics that arises turns out to have been recently studied in the combinatorics on words community, under the name of "perfectly clustering words". I will explain what perfectly clustering words are, and how our representation-theoretic approach allowed us to resolve a conjecture on perfectly clustering words. This is joint work with Benjamin Dequêne, Mélodie Lapointe, Yann Palu, Pierre-Guy Plamondon, and Christophe Reutenauer.

Abstract: The mutation theory of quivers with potential and their representations developed around 15 years ago by Derksen-Weyman-Zelevinsky allows to express cluster variables in skew-symmetric cluster algebras as Caldero-Chapoton functions of quiver representations. In ongoing joint work, Bea Schumann and I show that under certain "optimization" hypothesis, the radicals of indecomposable projectives behave well under Derksen-Weyman-Zelevinsky mutations of representations. This provides a new expression of certain "optimized" frozen cluster variables as F-polynomials of radicals of indecomposable projectives. 

Abstract: TBC

Abstract: We present a construction for cluster charts in simple Lie groups compatible with Poisson structures in the Belavin-Drinfeld classification. The key ingredient is a birational Poisson map from the group to itself that transform a Poisson bracket associated with a nontrivial Belavin-Drinfeld data into the standard one.  It allows us to obtain a cluster chart as a pull-back of the Berenstein-Fomin-Zelevinsky cluster coordinates on the open double Bruhat cell. This is a joint work with M. Shapiro and A. Vainshtein.

Abstract: This is a report on an ongoing project with Daniel Labardini-Fragoso and Jon Wilson.  We show that for a punctured surface with non-empty boundary and tagged triangulation T there is a bijection between the MSW-Laminations of the surfaces and the generically τ-reduced, decorated  components of the representation varieties for the Jacobian algebra P(Q_T, W_T), which intertwines shear coordinates with generic g-vectors and respects compatibility.  A key-step is to understand the above situation for the case of signature 0 triangulations, which correspond to skewed-gentle Jacobian algebras.

Abstract: I will talk about our joint work with A.Belavin and V.Belavin on the multiple Calaby-Yau (CY) mirror phenomenon which appears in Berglund-H\"ubsch-Krawitz (BHK) mirror symmetry. We show that, for any pair of Calabi–Yau orbifolds that are BHK mirrors of a loop–chain type pair of Calabi–Yau manifolds in the same weighted projective space, the periods of the holomorphic nonvanishing form coincide. 

Abstract: I will explain a general procedure to construct Newton-Okounkov bodies for a certain class of (partial) compactifications of cluster varieties. This class consists of the (partial) minimal models of cluster varieties with enough theta functions. This construction applies for example to Grassmannians and Flag varieties, among others. Our construction depends on a choice of torus in the atlas of the cluster variety and the associated Newton-Okounkov body lives inside a real vector space. Time permitting, I will explain how to compare the Newton-Okounkov bodies associated with different tori and elaborate on the "intrinsic Newton-Okounkov body", which is an object that does not depend on the choice of torus and lives inside the real tropicalization of the mirror cluster variety. This is based on upcoming work with Lara Bossinger, Man-Wai Cheung and Timothy Magee. 

Abstract: For algebras of finite representation type, AR-theory offers a beautiful and comprehensive set of techniques to understand their module categories. For a general finite dimensional algebra A, this framework is harder to apply. Instead of working with all modules, we may look at certain subsets of mod A. In this talk, we will consider such a subset, in particular the set of tau-rigid modules, i.e modules M such that \Hom_A(M,\tau M) = 0 (\tau is the Auslander-Reiten translate in the module category). These have interesting combinatorial properties, and seem to capture at least “some” of the interesting representation theory of A.

Abstract: TBC