Seminari del Dipartimento di Matematica

Seminario del 2022

Symmetric quivers and symmetric varieties

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

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. For example, 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 on a symmetric variety. As a consequence, we obtain an infinite class of symmetric quiver loci that are normal and Cohen-Macaulay.