  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$(k,n)=\mathbb{P}^{n-1}$.
