Algorithms for Counting Paths of Fixed Faces

Abstract

There is a Hasse graph associated with each symmetry of every n-dimensional polytope, and there is an algebra associated with each Hasse graph. Each level of the graph represents the number of k-dimensional faces that remain fixed under a given automorphism (or symmetry) of the polytope. For each symmetry, we determine a polynomial f(t) where the power of t represents the length of each path in the graph. The coefficient of t0 is the number of points, the coefficient of t1 is the number of paths of length 1, . . . , and the coefficient of ti is the number of unique paths of length i in the Hasse graph. Our goal is to determine the structure of all the algebras associated with finite Coxeter groups (consisting of 4 families and 6 exceptional groups) by determining all Hasse graph polynomials f(t). Duffy and past student research groups have accomplished finding the Hasse graph polynomials for the algebras associated with the An; Bn; Dn; I2(p) families and H3. We are working on the 600-Cell (H4).