# How does the Frobenius map permute the roots

How can a Frobenius map permute the roots of an algebraic group?

According to Carter (in Finite groups of Lie type), a root subgroup $X_{\alpha}$ is the 1-dimensional unipotenet subgroup giving rise to the root $\alpha$. The permutation $\rho$ on the roots, which is given by a Frobenius map $F: G \rightarrow G$ is defined to be $F(X_{\alpha}) = X_{\rho(\alpha)}$.

But I don't understand the permutation. For example, according to the figure on page 37, there is certain Frobenius map for a group of type $A_l$, $l \geq 2$, such that two of the simple roots are permuted.

But I don't know how can this take place.

Now I am considering a special case. Let $K = \bar{\mathbb F_2}$, the algebraic closure of the field of two elements, and $G = SL(3,K)$, the group of $3 \times 3$ special linear group over $K$, which is of type $A_2$. If $F: G \rightarrow G$, $(a_{ij}) \mapsto (a_{ij}^2)$, and for two simple roots $\alpha, \beta$ of $G$, $X_{\alpha} =\left\{ \begin{pmatrix} 1 & a & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} | a \in K \right\}$, $X_{\beta} =\left\{ \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & b\\ 0 & 0 & 1 \end{pmatrix} | b \in K \right\}$. Then both $F(X_{\alpha})$ and $F(X_{\beta})$ are all themselves, with no permutation done. From this, I can see no hint as to the case when the permutation can be caused by a Frobenius map.

Is there any concerete example of this permutation? Any type of algebraic groups is OK. Thanks very much.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 20, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9217805862426758, "perplexity_flag": "head"}