Tag - Quiver representations

Martin Frankland: Toda brackets in n-angulated categories

17th October, 2024

Geiss, Keller, and Oppermann introduced n-angulated categories to capture the structure found in certain cluster tilting subcategories in quiver representation theory. Jasso and Muro investigated Toda brackets and Massey products in such cluster tilting subcategories by using the ambient triangulated category. In joint work with Sebastian Martensen and Marius Thaule, we introduce Toda brackets in n-angulated categories, generalizing Toda brackets in triangulated categories (the case n = 3). We will look at different constructions of the brackets, their properties, some examples, and some applications.

Ryan Kinser: K-polynomials of type A quiver orbit closures and lacing diagrams

16th August, 2016

Orbit closures of type A quiver representations are algebraic varieties that arise naturally in several areas of mathematics: for example, in Lusztig's geometric realization of Ringel's work on quantum groups; as generalizations of determinantal varieties in commutative algebra; and in the theory of degeneracy loci of maps of vector bundles.

For equioriented type A quivers, a formula due to Knutson-Miller-Shimozono expresses the equivariant cohomology class of each orbit closure as a sum, over certain 'lacing diagrams', of products of Schubert polynomials. Lacing diagrams were introduced by Abeasis and del Fra in 1982 to visualize direct sum decompositions of type A quiver representations.

In joint work with Allen Knutson and Jenna Rajchgot, we proved a 2004 conjecture of Buch and Rimnyi that generalizes this formula in two ways: to arbitrarily oriented type A quivers, and to equivariant K-classes (a.k.a. K-polynomials), from which equivariant cohomology can be recovered.

The aim of this talk is to explain the combinatorics of (K-theoretic) lacing diagrams and carefully state the formula. Time permitting, I will give some idea of the Gröbner degeneration technique used in the proof.