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# Dr. Daiping Weng, UC Davis – Cluster Structures on Double Bott-Samelson Cells

## November 18, 2022 @ 4:00 pm - 5:00 pm

Mode: In-person

Title: **Cluster Structures on Double Bott-Samelson Cells**

Abstract: * By the Bruhat decomposition theorem, relative positions between flags in G/B are encoded by the corresponding Weyl group W. Given a pair of positive braids (b,d) associated with the Weyl group W, we define the double Bott-Samelson cell Conf^b_d(G) to be the moduli space of flags satisfying relative position conditions imposed by positive braid words of b and d. We construct cluster structures on double Bott-Samelson cells Conf^b_d(G) and describe their Donaldson-Thomas transformation as a cyclic rotation on the circle of flags. As a consequence, this gives a new geometric proof of the Zamolodchikov periodicity conjecture. We also construct Deodhar stratifications on double Bott-Samelson cells and develop a formula for their F_q point counts. Moreover, in the case where G is of type A, the F_q point count of Conf^b_d(G) give a link invariant for rainbow closures of positive braids. This is based on joint work with L. Shen (1904.07992).
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