Date: Friday, February 20, 2015
Location: 3866 East Hall (4:10 PM to 5:00 PM)
Title: Symmetry and Geometric Structure for the Worpitzky identity
Abstract: The classical Worpitzky theorem in combinatorics expresses powers in terms of Eulerian numbers. It corresponds to the volume decomposition of a simplex into hypersimplices having volumes given by the Eulerian polynomials A_{n1}(q).
We replace the simplex and hypersimplex volumes with S_nmodules coming from simplicial decompositions. Let \sigma in S_n have cycle lengths \lambda_1,..., \lambda_k. Characters have number theoretic properties.
Theorem: The character of \sigma on the module of the simplex \Delta_r^n is \delta_{g,1}r^{k1}, where g=gcd(\lambda_1,..., \lambda_k,r).
Theorem: The character of \sigma on the module of the hypersimplex B_{a,b} is the coefficient of q^a in \delta_{g,1}qA_{k1}(q).[\lambda_1]...[\lambda_k],$ with A_{k1}(q) the Eulerian polynomial and g=gcd(\lambda_1,..., \lambda_k,a,na).
The two characters are connected by an equivariant Worpitzky identity
\chi_{\Delta_r^n}\simeq\chi_\text{poly}\otimes \chi_{B_{a,b}}.
We construct the character values geometrically as volumes and as linear module dimensions of new generalized hypersimplices.
This is joint work with Adrian Ocneanu.
Files:
Speaker: Nick Early
Institution: Pennsylvania State
Event Organizer:
