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Book: RootSystemsWeylGroups


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:

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).

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Puslapis paskutinį kartą pakeistas 2018 lapkričio 11 d., 17:20