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Andrius Kulikauskas

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See: Classical Lie groups

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 = {$\pm$} RootTransposition
{$\pm(e_2-e_1)$}{$e_1 \Leftrightarrow e_2, -e_1 \Leftrightarrow -e_2$}
{$\pm(e_2--e_1)$}{$e_1 \Leftrightarrow -e_2, -e_1 \Leftrightarrow e_2$}
{$\pm(e_1)$}{$e_1 \Leftrightarrow -e_1$}

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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