On the quantum parameter in the quantum cohomology of a family of odd symplectic partial flag varieties
Connor Bean, Caleb Shank, Ryan M. Shifler
TL;DR
The paper analyzes the quantum cohomology of the non-homogeneous family of odd symplectic partial flag varieties $IF$. It uses curve neighborhood methods on the associated moment graph and leverages the divisor axiom to compute contributions to the quantum product $\tau_{Div_i} \star \tau_{id}$. The main finding is that the monomial $q_1 q_2 \cdots q_m$ arises with multiplicity $m$ in the Schubert expansion, tied to $m$ irreducible curve neighborhood components with coefficients determined by evaluation maps. This provides a concrete multiplicity pattern for quantum parameters in a non-homogeneous setting, extending known phenomena from homogeneous cases.
Abstract
We will consider a particular family of odd symplectic partial flag varieties denoted by $\mbox{IF}$. In the quantum cohomology ring $\mbox{QH}^*(\mbox{IF})$, we will show that $q_1q_2\cdots q_m$ appears $m$ times in the quantum product $τ_{Div_i} \star τ_{id}$ when expressed as a sum in terms of the Schubert basis.