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Calculate the Weyl groups of the classical root systems.

The Weyl group of a root system is the group generated by the reflections across the hyperplanes defined by the roots. Indeed, the roots come in pairs, positive and negative, and each such pair defines a hyperplane.

Furthermore, the root system is typically described in terms of an underlying alphabet {$\left \{ 1,2,...,n \right \}$} with possibly an orientation {$\left \{+,-\right \}$} which are affected by this reflection as follows:

 Hyperplane = {$\pm$} Root Transposition {$\pm(e_2-e_1)$} {$e_1 \Leftrightarrow e_2, -e_1 \Leftrightarrow -e_2$} {$\pm(e_2--e_1)$} {$e_1 \Leftrightarrow -e_2, -e_1 \Leftrightarrow e_2$} {$\pm(e_1)$} {$e_1 \Leftrightarrow -e_1$}

The classical root systems are:

• {$A_n$}: {$\pm (e_i-e_j)$}
• {$B_n$}: {$\pm (e_i-e_j), \pm (e_i--e_j), \pm (e_i-0)$}
• {$C_n$}: {$\pm (e_i-e_j), \pm (e_i--e_j), \pm (e_i--e_i)$}
• {$D_n$}: {$\pm (e_i-e_j), \pm (e_i--e_j)$} where {$i\neq j$}

Thus {$A_n$} offers the transpositions {$e_1 \Leftrightarrow e_2$}, which generate the symmetric group {$S_n$}.

The root system {$B_n$} offers the transpositions {$e_i \Leftrightarrow e_j$} as well as the transpositions {$e_i \Leftrightarrow -e_i$}, generating the hyperoctahedral group. The root system {$C_n$} likewise offers these transpositions and generates the hyperoctahedral group.

However, the root system {$D_n$} does not provide the transpositions {$e_i \Leftrightarrow -e_i$}. Instead, it provides transpositions which simultaneously reflect along a pair of axes: {$e_1 \Leftrightarrow -e_2, -e_1 \Leftrightarrow e_2$}. Thus the Weyl group is a subgroup of the hyperoctahedral group which includes only even numbers of reflections along the axes.

Hyperoctahedral group is shuffling cards that can also be flipped or not (front or back).

#### RootSystemsWeylGroups

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