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Algebra and Combinatorics Seminar: Raymond Maresca, Brandeis University, Combinatorics of exceptional collections in type A-tilde
November 15, 2023 | 12:45 pm - 1:45 pm EST
We will define quivers of type A-tilde, their representations, and exceptional collections of these representations. We will then introduce a combinatorial model of these representations, based on the one constructed by Garver, Igusa, Matherne, and Ostroff for type A, by drawing strands on a copy of the integers. We will see that collections of strands called strand diagrams are in bijection with exceptional collections in type A-tilde. After, we will introduce a different combinatorial model consisting of arcs on an annulus that is also in bijection with exceptional collections in type A-tilde. Using the arc diagrams, we will place exceptional collections into finitely many infinite families, and then using a labeling of the strand diagrams, we will show that for a certain orientation of the quiver, the number of families is counted by a generalization of the Catalan numbers known as the Rothe numbers, or the Rothe-Hagan coefficients of the first type.