The Littlewood–Richardson rule for $\Lambda$-cominuscule elements

Let $G$ be a Kac–Moody group, let $P$ be a parabolic subgroup, and let $X=G/P$. Let $W$ be the Weyl group, and let $\Lambda$ be the dominant weight associated with $P$. Let $c_{u,v}^w$ be the Schubert structure constants and let $t_{u,v}^w$ be the jeu de taquin integer for $u,v\leq w$. For a $\Lambda$-cominuscule element $w=s_{\alpha_1}\cdots s_{\alpha_l}$, define $m(w)$ by

$$m(w):=\prod_{\substack{i\in[1,l],\ \alpha\in S(\Lambda),\\(\alpha,\alpha)>(\alpha_i,\alpha_i),\ i\geq(\alpha,1)}}\frac{(\alpha,\alpha)}{(\alpha_i,\alpha_i)},$$

where $S(\Lambda)=\{\alpha:\langle\Lambda,\alpha^\vee\rangle>0\}$, $(\cdot,\cdot)$ is a $W$-invariant scalar product, and $(\alpha,1)$ is the minimal element of the heap colored by $\alpha$. Set $m_{u,v}^w=m(w)/(m(u)m(v))$. The $\Lambda$-cominuscule Littlewood–Richardson conjecture. For $w$ a $\Lambda$-cominuscule element and $u$ and $v$ in $W$ smaller than $w$, we have

$$c_{u,v}^w=m_{u,v}^w t_{u,v}^w.$$

This extends the proposed jeu de taquin rule from $\Lambda$-minuscule to $\Lambda$-cominuscule elements by incorporating the root-length correction factor $m_{u,v}^w$. The source provides no resolution status.