School of Mathematics
206 Church St SE
Minneapolis, MN 55403
Email: cfraser at umn dot edu
Office: 204 Vincent
Phone: 612 626 0843
I am an RTG Postdoc at the University of Minnesota. I received my Ph.D from the University of Michigan under the supervision of Sergey Fomin. I am on the academic job market during Fall 2020.
Combinatorial representation theory, especially cluster algebras and total positivity. My CV is available here.
Papers and preprints:
This arXiv search produces each of my arXiv preprints. Individual arXiv links, a brief abstract, and some slides and videos, can be found below.
We study the points in Gr(k,n) fixed by a given power of the cyclic shift map, giving a cell decomposition of the TNN points and studying the existence of (generalized) cluster structures on these spaces. This extends Karp's description of the points fixed by the cyclic shift itself. This has relations with Chekhov and Shapiro's work on decorated Teichmuller theory of orbifolds, Gekhtman-Shapiro-Vainshtein's work on periodic band matrices, and Gleitz's work on quantum affine algebras at roots of unity.
We describe clusters in open positroid variety indexed by plabic graphs with appropriately permuted boundary labels., and conjecture that all such clusters are related by quasi-cluster transformations. This construction induces isomorphisms between open positroid varieties via twistlike maps.
We study the relationship between modules over quantum affine algebras and Grassmannian cluster algebras. We give (subject to a mild conjecture) an explicit basis for the coordinate ring of the Grassmannian, containing the cluster monomials, whose elements are labeled by semistandard Young tableaux of rectangular shape. We discuss the resulting tableau-theoretic rules for Grassmannian cluster combinatorics.
We describe a precise relationship between the r-fold dimer model on bipartite graphs in the disk and SL_r webs. We deduce a complete set of skein relations for webs from Postnikov's moves on plabic graphs.
We describe an action of the the affine extended braid group on d strands on the open positroid variety in Gr(k,n), where d = gcd(k,n). This action permutes the cluster monomials in the Grassmannian cluster algebra.