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?


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