Equality of the right-quiver matrix with the U^∨ factor in Z_P

Prove that, for the partial flag variety G/P, the matrix \tilde{u}_R constructed from the decorated right-quiver Q_{P,R} (as defined in Section 8.2) is equal to the unique matrix u_R \in U^{\vee} that satisfies b_P = u_L \kappa_P \dot{w}_P \bar{w}_0 u_R \in B_-^{\vee}, where u_L is the left-quiver matrix from Section 7.3 and \kappa_P is the highest-weight map from Section 7.2. This establishes that the right-quiver construction recovers precisely the U^{\vee} factor required for b_P to lie in B_-^{\vee}.

Background

Section 8 develops a second quiver construction, Q_{P,R}, intended to describe the right-hand factor u_R in the product b_P = u_L \kappa_P \dot{w}_P \bar{w}_0 u_R \in Z_P. The matrix g_R and its U^{\vee} factor \tilde{u}_R are defined analogously to g_L and u_L from the left-quiver Q_P.

Conjecture 8.3 asserts that \tilde{u}_R coincides with u_R, thereby giving a fully explicit quiver interpretation of both factors u_L and u_R in Z_P. The authors note that this equality holds for Grassmannians (by Marsh and Rietsch) and expect similar techniques to apply, but they have been unable to complete the proof for general partial flag varieties.

The conjecture is restated equivalently as the identity b_P = g_L \kappa_P \bar{w}_0 \tilde{g}_R, using relationships between Weyl group representatives and the quiver constructions; the authors provide supporting evidence and reductions but indicate that a full proof remains elusive.