Convegno
“QUIVER REPRESENTATIONS, QUIVER VARIETIES AND COMBINATORICS”

We will first work through the construction of moduli spaces of quiver representations as GIT quotients, and collect basic geometric properties. We will then work out classes of examples where these spaces can be described explicitly. We will describe geometric techniques for studying moduli spaces, for example coordinates, vector bundles, torus localization, Hilbert scheme.

In these lectures I will present the calculation of the title in the case of the star-shaped quivers related to character varieties based on my joint work with E. Letellier and T. Hausel. The starting point will be a formula of Hua for a general quiver. The basic tool used is the combinatorics of symmetric functions and generating functions, which I will discuss from scratch.

Semi-invariants of quivers and their applications.

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In 2021 Fang and Reineke described the support of linear degenerations of flag varieties in terms of Motzkin paths, by using Knight-Zelevinsky multi-segment duality. In a joint project with Esposito and Marietti (IMRN 2023, arXiv 2206.10281) we give a new characterization of supports in representation-theoretic terms by what we call excessive multi-segments. To do so we consider an algebraic structure on the set of Motzkin paths that we call Motzkin monoid. By using a universal property of the Motzkin monoid, we show that excessive multi segments are parametrized in a natural way by Motzkin paths. Moreover, we show that this parametrization coincides exactly with the Fang-Reineke parametrization. As a byproduct we have an elementary combinatorial criterion to decide if a multisegment is a support. We have an inductive procedure to describe the inverse of the Fang-Reineke map. In this term there is a very beautiful (as yet conjectural) formula for the coefficients.

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Symmetric quivers and symmetric varieties In this talk I will report on ongoing joint work with Ryan Kinser and Jenna Rajchgot on varieties of symmetric quiver representations. These varieties are acted upon by a reductive group via change of basis, and it is natural to ask for a parametrisation of the orbits, for the closure inclusion relation among them, for information about the singularities arising in orbit closures. Since the Eigthies, same (and further) questions about representation varieties for type A quivers have been attached by relating such varieties to Schubert varieties in type A flag varieties (Zelevinsky, Bobinski-Zwara, ...). I will explain that in the symmetric setting it is possible to interpret the above questions in terms of certain symmetric varieties. More precisely, we show that singularities of an orbit closure of a symmetric quiver representation variety are smoothly equivalent to singularities of an appropriate Borel orbit closure in a symmetric variety.

This talk will be an introduction to the field of topological data analysis, emphasizing the role played in it by quiver representation theory. Specifically, I will describe how the supports of the indecomposables can be used as descriptors for data, with stability guarantees under suitable choices of metrics on the representation categories of the quivers under consideration. I will also explain how one proceeds when those quivers are of wild representation type.

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