Cellular resolutions of cycles, associahedra, and standard Young tableauAlgebra & Discrete Mathematics
|Speaker:||Anton Dochtermann, Stanford|
|Start time:||Fri, Feb 4 2011, 2:10PM|
Let I_n denote the Stanley-Reisner ring of an n-cycle, i.e., the monomial ideal in k[x_1,...x_n] generated by all diagonals of an n-gon. We construct a minimal cellular resolution of I_n using algebraic discrete Morse theory applied to a natural monomial labeling of the associahedron A_n (the polytopal sphere encoding dissections of a polygon). The matching that gives rise to the discrete Morse function is constructed via an encoding of the facial structure of A_n with Young tableau, and in particular we see that the Betti numbers of I_n are given by the number of standard Young tableau corresponding to certain collection of partitions of n. We conjecture that these complexes are in fact polytopal and self-dual, although we can only show this for (very) small dimensions.