# On $\Psi$-generating paths in the Bruhat order of a Weyl group

Let $$W$$ be a Weyl group with roots $$R$$ and positive roots $$R^+$$. Let $$v\in W$$ of length $$r$$. We call $$\mathbb{m}=(\alpha_1,\ldots,\alpha_r)\in(R^+)^r$$ a Bruhat path from $$1$$ to $$v$$ if $$1\lessdot s_{\alpha_r}\lessdot s_{\alpha_{r-1}}s_{\alpha_r}\lessdot\cdots\lessdot s_{\alpha_1}\cdots s_{\alpha_r}=v$$ where $$\lessdot$$ means the covering relation in the Bruhat order. Let $$\mathbb{m}=(\alpha_1,\ldots,\alpha_r)$$ be a Bruhat path from $$1$$ to $$v$$, then we write $$x_{\mathbb{m}}=x_{\alpha_1}\otimes\cdots\otimes x_{\alpha_r}\in V_W^{\otimes r}$$ where $$V_W$$ is the Yetter-Drinfeld $$W$$-module defined by generators $$x_\alpha$$ where $$\alpha\in R$$ and relations $$x_{-\alpha}=-x_\alpha$$ for all $$\alpha\in R$$. Let $$\mathbb{B}_r$$ be the braid group with generators $$\sigma_1,\ldots,\sigma_{r-1}$$. This group acts on $$V_W^{\otimes r}$$ by $$\sigma_i x=\Psi_{i,i+1}x$$ where $$1\leq i\leq r-1$$, $$x\in V_W^{\otimes r}$$, and where $$\Psi_{i,i+1}$$ is the braiding of the $$i$$th and the $$(i+1)$$th tensor factor. Let $$t\colon\mathbb{S}_{r}\to\mathbb{B}_r$$ be the Matsumoto section, then Bazlov conjectures that $$\left(\sum_{\sigma\in\mathbb{S}_r}t(\sigma)\right)x_{\mathbf{m}_0}=\sum_{\text{all Bruhat paths \mathbf{m} from 1 to v}}x_{\mathbf{m}}$$ for some distinguished (not necessarily unique) Bruhat path $$\mathbf{m}_0$$ from $$1$$ to $$v$$. A Bruhat path $$\mathbb{m}_0$$ from $$1$$ to $$v$$ which satisfies the above identity is then called a $$\Psi$$-generating path from $$1$$ to $$v$$.

While I believe that this conjecture is not really true (although I would appreciate a proof as well), I would like to ask a simpler

Question. Let $$\mathbf{m}_0$$ be a $$\Psi$$-generating path from $$1$$ to $$v$$. Is then necessarily $$\mathbf{m}_0=(\beta_1,\ldots,\beta_r)$$ for some reduced expression $$v=s_{\beta_1}\cdots s_{\beta_r}$$?

Second question. Do you know any further literature except that is related to this problem?