Equivariant log-concavity and equivariant K\"ahler packages

TL;DR

This paper demonstrates that certain cohomology rings related to tori and their classifying spaces are equivariantly log-concave and satisfy the Kähler package, extending classical geometric properties to an equivariant algebraic setting.

Contribution

It establishes equivariant log-concavity and the Kähler package for specific cohomology rings, providing explicit representation maps and proving the hard Lefschetz theorem in this context.

Findings

Cohomology of tori and classifying spaces are equivariantly log-concave.

The Kähler package holds in an equivariant algebraic setting.

Explicit S_n-representation maps satisfy the hard Lefschetz theorem.

Abstract

We show that the exterior algebra $\Lambda_{R}\left[\alpha_{1}, \cdots, \alpha_{n}\right]$, which is the cohomology of the torus $T=(S^{1})^{n}$, and the polynomial ring $\mathbb{R}\left[t_{1}, \ldots, t_{n}\right]$, which is the cohomology of the classifying space $B (S^{1})^{n}=\left(\mathbb{C} \mathbb{P}^{\infty}\right)^{n}$, are $S_{n}$-equivariantly log-concave. We do so by explicitly giving the $S_{n}$-representation maps on the appropriate sequences of tensor products of polynomials or exterior powers and proving that these maps satisfy the hard Lefschetz theorem. Furthermore, we prove that the whole K\"ahler package, including algebraic analogies of the Poincar\'e duality, hard Lefschetz, and Hodge-Riemann bilinear relations, holds on the corresponding sequences in an equivariant setting.