On Frobenius-Perron Dimension

TL;DR

This paper introduces a new Frobenius-Perron dimension concept for infinite rank modules, computes it for certain representation modules, and establishes bounds for quantum cohomology classes in Grassmannians.

Contribution

It defines Frobenius-Perron dimension for infinite rank modules and computes it for specific representation modules, also providing bounds in quantum cohomology.

Findings

Frobenius-Perron dimension for certain infinite modules is defined and computed.

Explicit Frobenius-Perron dimensions are obtained for complex representations of unitary groups.

A lower bound for Frobenius-Perron dimensions of Schubert classes in quantum cohomology is established.

Abstract

We propose a notion of Frobenius-Perron dimension for certain free $\mathbb{Z}$-modules of infinite rank and compute it for the $\mathbb{Z}$-modules of finite dimensional complex representations of unitary groups with nonnegative dominant weights. We also provide a lower bound for the Frobenius-Perron dimension of Schubert classes in the quantum cohomology of complex Grassmannians.