Grassmannian quantum cohomology in the infinite limit and total positivity

Abstract

The theory of total positivity was shown by Lusztig to be intrinsically linked to the canonical basis with its positivity properties. When we restrict ourselves to studying total positivity just for the set of lower-triangular unipotent Toeplitz matrices, say in type $A$, then there is a similar link with the quantum cohomology rings of flag varieties and the Schubert bases and their positivity properties. Namely, this builds on a theory of Dale Peterson that gives a uniform Lie-theoretic description of all of the quantum cohomology rings $qH^*(G/P)$. In a precursor to this paper, the Schubert basis and quantum parameters in $qH^*(SL_n/B)$, which restrict to positive-valued functions on totally positive Toeplitz matrices, were analysed with respect to their limiting behaviour as $n\to\infty$, uncovering a novel connection with the classical Edrei theorem on parametrising the infinite totally positive Toeplitz matrices. In this paper we study the Grassmannian case, using the conventions from the $SL_{n}/B$ setting as a guide, and we determine the quantum parameter and Schubert class asymptotics in different scenarios. Along the way, we obtain a new interpretation of the strange duality involution on the localised quantum cohomolgy ring of the Grassmannian. Finally, we prove an asymptotic formula for quantum parameters in a partial flag setting, and we furthermore formulate some conjectures concerning partial flag varieties and related quantum cohomology asymptotics.