Arun Ram: The group SL3

The group $S L_{3}$

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 25 June 2007. This page is the result of joint work with James Parkinson and Christoph Schwer.

The Chevalley group $G (𝔽) = S L_{3} (𝔽)$ is generated by the elements

x_{α_{1}} (f) = (\begin{matrix} 1 & f & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}), x_{α_{2}} (f) = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & f \\ 0 & 0 & 1 \end{matrix}), x_{ϕ} (f) = (\begin{matrix} 1 & 0 & f \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}),

and

x_{- α_{1}} (f) = x_{α_{1}} (f)^{t}

x_{- α_{2}} (f) = x_{α_{2}} (f)^{t}

, and

x_{- ϕ} (f) = x_{ϕ} (f)^{t}

. Then

n_{α} (g) = x_{α} (g) x_{- α} (- g^{- 1}) x_{α} (g) = h_{α^{\lor}} (g) n_{α}

with

h_{α_{1}^{\lor}} (g) = (\begin{matrix} g & 0 & 0 \\ 0 & g^{- 1} & 0 \\ 0 & 0 & 0 \end{matrix}), h_{α_{2}^{\lor}} (g) = (\begin{matrix} 0 & 0 & 0 \\ 0 & g & 0 \\ 0 & 0 & g^{- 1} \end{matrix}), h_{ϕ^{\lor}} (g) = (\begin{matrix} g & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & g^{- 1} \end{matrix}),

and

n_{α_{1}} = (\begin{matrix} 0 & 1 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}), n_{α_{2}} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{matrix}), n_{ϕ} = (\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ - 1 & 0 & 0 \end{matrix}),

Then

x_{α_{0}} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ c t^{- 1} & 0 & 1 \end{matrix}) so that n_{α_{0}} (c) = (\begin{matrix} 0 & 0 & - c^{- 1} t \\ 0 & 1 & 0 \\ c t^{- 1} & 0 & 0 \end{matrix}) = (\begin{matrix} c^{- 1} t & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & c t^{- 1} \end{matrix}) (\begin{matrix} 0 & 0 & - 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix})