Equivariant Euler characteristics on permutohedral varieties

TL;DR

This paper explores equivariant Euler characteristics on permutohedral varieties, providing new computational methods including a recursive approach to calculate Laurent polynomial invariants linked to binomial coefficients.

Contribution

It introduces three novel approaches, notably a recursive method, for computing equivariant Euler characteristics as Laurent polynomials on permutohedral varieties.

Findings

Developed three methods for computing Laurent polynomial Euler characteristics.

Connected intersection theory on permutohedral varieties to combinatorial binomial coefficients.

Implemented recursive algorithms for efficient calculations.

Abstract

By the work of J.Huh, one can interpret binomial coefficients as a solution to an intersection problem on a permutohedral variety $X_E$. Applying Hirzebruch-Riemann-Roch, this intersection problem is equivalent to computing Euler characteristic of a specific element of $K$-theory of $X_E$. This element has a natural lifting to equivariant $K$-theory and thus the Euler characteristic may be upgraded to a Laurent polynomial. We provide and implement three different approaches, in particular a recursive one, to computing these polynomials.