Galkin's lower bound conjecture holds for the Grassmannian

TL;DR

This paper proves Galkin's lower bound conjecture for the eigenvalues of quantum multiplication by the first Chern class on Grassmannians, establishing a key spectral property related to their dimension.

Contribution

It confirms Galkin's conjecture for Grassmannians, showing the largest eigenvalue of the quantum multiplication operator meets the conjectured lower bound.

Findings

Largest eigenvalue of c_1 meets the lower bound dim(Gr(k,n))+1.

Equality holds only for projective spaces.

Supports Galkin's conjecture in the case of Grassmannians.

Abstract

Let Gr$(k,n)$ be the Grassmannian. The quantum multiplication by the first Chern class $c_1({\rm Gr}(k,n))$ induces an endomorphism $\hat c_1$ of the finite-dimensional vector space $\mathrm{QH}^*({\rm Gr}(k,n))_{|q=1}$ specialized at $q=1$. Our main result is a case that a conjecture by Galkin holds. It states that the largest real eigenvalue of $\hat{c}_1$ is greater than or equal to $\dim {\rm Gr}(k,n)$+1 with equality if and only if Gr$(k,n)=\mathbb{P}^{n-1}$.