ScalingStacks

Corollary 4.11 (Khovanov). Given $b,b^{\prime}\in B_{n}$ and $r\in\{1,\ldots,n-1\}$, we have

$$t_{1}^{-1/2}t_{2}^{1/2}Y_{b\sigma_{r}^{-1}b^{\prime}}+t_{1}^{1/2}t_{2}^{-1/2}Y_{b\sigma_{r}b^{\prime}}=\sqrt{-1}(t_{1}^{-1/2}-t_{1}^{1/2})Y_{bb^{\prime}}.$$