Let ${G}$ be a simple algebraic group over $\mathbb{C}$ with maximal torus $T$ and set of simple roots $\{\alpha_i\}_{i\in \Delta}$. We then have a Borel supgroup $B=TU$ with unipotent radical $U$. Let $w_0\in W$ be the long element of the Weyl group.

For $w\in W$, we say the string of integers $(i_1,\ldots,i_l)$ is a reduced word decomposition for $w$ if $w=s_{i_1}\cdots s_{i_l}$ and $l(w) = l$. To each reduced word decomposition $\underline{i}=(i_1,i_2,\ldots,i_m)$ of $w_0$, one obtains two rational maps $\mathbb{C}^m\longrightarrow U$:

The first map is the obvious one: $$ (t_1,t_2,\ldots,t_m)\mapsto x_{i_1}(t_1)x_{i_2}(t_2)\cdots x_{i_m}(t_m),$$

where $x_{i}:\mathbb{G}_a\longrightarrow U$ is the simple root group associated to $\alpha_i$. This map is not surjective, and is related to the study of positive subvariety $U_{>0}\subset U$. For example see [1].

The second parametrization uses the fact that a reduced word decomposition of $w_0$ induces an enumeration of the positive roots: for $1\leq k\leq m$, set $$\beta_k^{\underline{i}}=s_{i_1}\cdots s_{i_{k-1}}\alpha_{i_k}.$$ With this enumeration, we obtain the map \begin{equation} (s_1,\ldots s_m)\mapsto x_{\beta_1}(s_1)x_{\beta_2}(s_2)\cdots x_{\beta_m}(s_m). \end{equation} This gives an honest parametrization of $U$, inducing an isomorphism of affine varieties $U\cong \mathbb{A}^m$. I am curious about the change of coordinates $$ x_{i_1}(t_1)x_{i_2}(t_2)\cdots x_{i_m}(t_m) = x_{\beta_1}(s_1)x_{\beta_2}(s_2)\cdots x_{\beta_m}(s_m).$$ Obviously, you could try to describe it in vague terms using the Steinberg commutation relations, but that is not very useful for applications.

For a fixed reduced word decomposition $\underline{i}$ of $w_0$, is anything known in general about the transition map $$ (t_1,\ldots, t_m) \mapsto (s_1,\ldots, s_m).$$ Can it be described in representation theoretic or combinatorial terms? I would be happy with a reference.

The reason for my interest is that certain constructions which apply for any $u\in U$ seem to have more useful interpretations when understood on $U_{>0}$ in terms of the $t_i$-coordinates.

A quick glance at SGA3 Expose XXIII did not return any results. I have worked out a few examples (mostly type A or rank 2), and $s_j$ will be a polynomial in the $t_i$ of degree $ht(\beta_j)$, where $ht(\sum_ic_i\alpha_i)=\sum_ic_i$, and the coefficients will be coefficients which arise in the commutation relations, but I am not sure what else to say in general. One might hope that there is a nicer statement arising from the connection to $\underline{i}$.

[1] Berenstein, Arkady; Zelevinsky, Andrei, Tensor product multiplicities, canonical and totally positive varieties, Invent. Math. 143, No. 1, 77-128 (2001). ZBL1061.17006.


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