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The longest word in type $A_3$ Weyl group written as a matrix is \begin{align} w_0=s_1s_2s_1=\left(\begin{array}{cccc} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{array}\right). \end{align}

What is the longest word in type $C_2$ Weyl group written in terms of a matrix? In this case, $w_0 = s_1 s_2 s_1 s_2$. How to write $s_1, s_2$ as matrices? Thank you very much.

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  • $\begingroup$ Do you mean "what is the matrix, in terms of a basis of simple roots, of the transformation of the root system induced by …"? In this case, the matrices of simple reflections very nearly come from the Cartan matrix. Although you must usually specify your numbering of simple roots, of course here it doesn't matter. If $\alpha_1$ is the short root and $\alpha_2$ the long, then $s_1 = \begin{pmatrix} -1 & 2 \\ 0 & 1 \end{pmatrix}$ and $s_2 = \begin{pmatrix} 1 & 0 \\ 1 & -1 \end{pmatrix}$. $\endgroup$
    – LSpice
    Commented Jul 7, 2018 at 19:48
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    $\begingroup$ It's scalar multiplication by $-1$. $\endgroup$
    – Ben Webster ♦
    Commented Jul 7, 2018 at 20:22

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To answer these questions we have great computer algebra systems such as Sage

W = WeylGroup("C2")
W.long_element()

You can run short snippets of code like these on SageCell

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The Chevalley basis of $sp_4$ is generated by \begin{align} & e_1=\left(\begin{array}{cccc} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 \end{array}\right), \ e_2 = \left(\begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right), \\ & f_1 = \left(\begin{array}{cccc} 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 \end{array}\right), f_2 = \left(\begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \end{array}\right). \end{align} We have \begin{align} \overline{s}_i = \exp(-e_i)\exp(f_i)\exp(-e_i). \end{align} Therefore \begin{align} & \overline{s}_1 = \left(\begin{array}{cccc} 0 & 1 & 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & -1 & 0 \end{array}\right), \quad \overline{s}_2 = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & -1 & 0 & 0 \end{array}\right), \\ & \overline{s}_1 \overline{s_2} \overline{s}_1 \overline{s}_2 = \left(\begin{array}{cccc} 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \end{array}\right), \quad s_1 s_2 s_1 s_2 = \left(\begin{array}{cccc} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \end{array}\right). \end{align}

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