Braid varieties for SLn are smooth affine varieties associated to any positive braid. Their cohomology contains information about the Khovanov-Rozansky homology of a related link. One can analogously define braid varieties for any simple algebraic group. Special cases of braid varieties include Richardson varieties, double Bruhat cells, and double Bott-Samelson cells. Cluster algebras, introduced by Fomin and Zelevinsky, are a class of commutative rings which are completely determined by a combinatorial input called a seed. I’ll discuss joint work with P. Galashin, T. Lam and D. Speyer in which we show the coordinate rings of braid varieties are cluster algebras. In the SLn case, seeds for these cluster algebras come from ‘3D plabic graphs’, which are bicolored graphs embedded in a 3-dimensional ball and generalize Postnikov’s plabic graphs for positroid varieties.
This video is part of the University of Georgia‘s Algebra seminar.
