# 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.