Relation of different definition of twists

Let $ G=GL_n$ and $ W$ the Weyl group of $ G$ . Let $ B$ be a Borel subgroup of $ G$ and $ U$ the unipotent radical of $ B$ . Let $ B_-$ be the Borel subgroup of $ G$ such that $ B_- \cap B = T$ . In the paper and the paper, Berenstein-Fomin-Zelevinsky twist on $ U^w = U \cap B_-wB_-$ is a map $ \eta: U^w \to U^w$ which sends $ g \in U^w$ to the unique element in $ U \cap B_- w g^T$ .

Let $ u, v \in W$ , $ G^{u,v} = BuB \cap B_-vB_-$ . For $ a \in G$ which has a LDU decomposition, denote $ a = [a]_- [a]_0 [a]_+$ be the LDU decomposition of $ a$ . In type $ A$ , denote $ \overline{s}_i = \varphi_i\left( \begin{matrix} 0 & -1 \ 1 & 0 \end{matrix} \right)$ and $ \overline{\overline{s}}_i = \varphi_i\left( \begin{matrix} 0 & 1 \ -1 & 0 \end{matrix} \right)$ , where $ \varphi_i: SL_2 \to G$ is the group homomorphism given by $ \varphi_i\left( \begin{matrix} 1 & t \ 0 & 1 \end{matrix} \right) = x_i(t)$ and $ \varphi_i\left( \begin{matrix} 1 & 0 \ t & 1 \end{matrix} \right) = x_\overline{i}(t)$ , $ x_i(t) = \exp(te_i)$ , $ x_{\overline{i}}(t) = \exp(tf_i)$ , $ e_i, f_i, h_i$ are standard generators of the Lie algebra $ \mathfrak{g}$ of $ G$ .

On the other hand, in the paper, a twist map on $ \zeta_{u,v}: G^{u,v} \to G^{u^{-1}, v^{-1}}$ is defined by \begin{align} \zeta_{u,v}(g) = ( [\overline{\overline{u^{-1}}} x]_-^{-1} \overline{\overline{u^{-1}}} x \overline{v^{-1}} [x \overline{v^{-1}}]_+^{-1} )^{\theta}, \end{align} where in type $ A$ , $ \theta: SL_n \to SL_n$ is the map such that $ \theta(x)_{ij}$ is the determinant of the submatrix of $ x$ obtained from $ x$ by deleting the $ i$ th row and $ j$ th column.

Let $ u=e$ and $ v=w$ . Then $ G^{u,v}=B \cap B_-wB_- = T U^w$ and $ \zeta_{e,w}: TU^w \to TU^{w^{-1}}$ .

I think that $ \eta: U^w \to U^w$ is different from $ \zeta_{e,w}:TU^w \to TU^{w^{-1}}$ . What are the relation between these two maps? Thank you very much.