Equivariant Euler characteristics on permutohedral varieties
- Vincenzo Galgano
Abstract
In the literature cohomology has been shown to be a useful tool for detecting invariants in combinatorics. Moreover, equivariant cohomology provides even finer invariants, and it becomes particularly practical for toric varieties. Binomial coefficients can be interpreted as a solution to an intersection problem on a permutohedral variety X. Via Hirzebruch-Riemann-Roch, this is equivalent to computing Euler characteristic of a specific element of K-theory of X which admits a natural lifting to equivariant K-theory. Thus the Euler characteristic (and hence binomial coefficients) may be upgraded to a Laurent polynomial. We provide and implement three different approaches, in particular a recursive one, to computing these polynomials. This is a joint work with Hanieh Keneshlou and Mateusz Michałek.