Galkin's lower bound conjecture holds for the Grassmannian

Abstract

Let Gr$(k,n)$ be the Grassmannian. The quantum multiplication by the first Chern class $c_1(\mathrm{Gr}(k,n))$ induces an endomorphism ${\hat{c}}_1$ of the finite-dimensional vector space ${\mathrm{QH}}^{\ast }(\mathrm{Gr}(k,n){)}_{|q=1}$ 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 $\dim \mathrm{Gr}(k,n)$+1 with equality if and only if Gr$(k,n)={\mathbb{P}}^{n-1}$.