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## Book.RootSystemsWeylGroups istorija

2018 lapkričio 11 d., 17:20 atliko AndriusKulikauskas -
Pridėtos 31-33 eilutės:

Hyperoctahedral group is shuffling cards that can also be flipped or not (front or back).
2018 lapkričio 02 d., 20:18 atliko AndriusKulikauskas -
2018 lapkričio 02 d., 19:56 atliko AndriusKulikauskas -
Pridėta 30 eilutė:
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.
2018 lapkričio 02 d., 19:52 atliko AndriusKulikauskas -
Pridėtos 25-28 eilutės:

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.
2018 lapkričio 02 d., 19:47 atliko AndriusKulikauskas -
Pakeista 12 eilutė iš:
Furthermore, the root system is typically described in terms of an underlying alphabet {${1,2,...,n}$} with possibly an orientation {$+,-$} which are affected by this reflection as follows:
į:
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:
2018 lapkričio 02 d., 19:46 atliko AndriusKulikauskas -
Pakeistos 12-13 eilutės iš
Furthermore, the root system is typically described in terms of an underlying alphabet ${{which is affected by this reflection as follows: į: Furthermore, the root system is typically described in terms of an underlying alphabet {${1,2,...,n}$} with possibly an orientation {$+,-$} which are affected by this reflection as follows: Pakeistos 24-25 eilutės iš * {$D_n$}: {$\pm (e_i-e_j), \pm (e_i--e_j)$} where {$i\neq j$} į: * {$D_n$}: {$\pm (e_i-e_j), \pm (e_i--e_j)$} where {$i\neq j$} 2018 lapkričio 02 d., 19:42 atliko AndriusKulikauskas - Pakeistos 12-13 eilutės iš Furthermore, the root system is typically described in terms of an underlying alphabet which is affected by this reflection as follows: į: Furthermore, the root system is typically described in terms of an underlying alphabet${{which is affected by this reflection as follows:
Pakeistos 21-24 eilutės iš
* {$A_n$}: {$\pm (x_i-x_j)$}
* {$B_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j), \pm (x_i-0)$}
* {$C_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j), \pm (x_i--x_i)$}
* {$D_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j)$} where {$i\neq j$}
į:
* {$A_n$}: {$\pm (e_i-e_j)$}
* {$B_n$}: {$\pm (e_i-e_j), \pm (e_i--e_j), \pm (e_i-0)$}
* {$C_n$}: {$\pm (e_i-e_j), \pm (e_i--e_j), \pm (e_i--e_i)$}
* {$D_n$}: {$\pm (e_i-e_j), \pm (e_i--e_j)$} where {$i\neq j$}
2018 lapkričio 02 d., 19:39 atliko AndriusKulikauskas -
Pakeista 22 eilutė iš:
* {$B_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j), \pm x_i$}
į:
* {$B_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j), \pm (x_i-0)$}
Pakeista 24 eilutė iš:
* {$D_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j) where i\neq j$}
į:
* {$D_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j)$} where {$i\neq j$}
2018 lapkričio 02 d., 19:39 atliko AndriusKulikauskas -
Pakeista 24 eilutė iš:
* {$D_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j) where i\neq j$}
į:
* {$D_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j) where i\neq j$}
2018 lapkričio 02 d., 19:38 atliko AndriusKulikauskas -
Pakeista 24 eilutė iš:
* {$D_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j) i\neq j$}
į:
* {$D_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j) where i\neq j$}
2018 lapkričio 02 d., 19:38 atliko AndriusKulikauskas -
Pakeistos 24-26 eilutės iš
* {$D_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j) i\neq j$}

||'''roots beyond '''||||{$\pm \pm x_i$}||{$\pm (x_i+x_j), \pm 2x_i$}||{$\pm (x_i+x_j), i\neq j$}||
į:
* {$D_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j) i\neq j$}
2018 lapkričio 02 d., 19:37 atliko AndriusKulikauskas -
Pakeistos 17-26 eilutės iš
||{$\pm(e_1)$}||{$e_1 \Leftrightarrow -e_1$}||
į:
||{$\pm(e_1)$}||{$e_1 \Leftrightarrow -e_1$}||

The classical root systems are:

* {$A_n$}: {$\pm (x_i-x_j)$}
* {$B_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j), \pm x_i$}
* {$C_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j), \pm (x_i--x_i)$}
* {$D_n$}: {$\pm (x_i-x_j), \pm (x_i--x_j) i\neq j$}

||'''roots beyond '''||||{$\pm \pm x_i$}||{$\pm (x_i+x_j), \pm 2x_i$}||{$\pm (x_i+x_j), i\neq j$}||
2018 lapkričio 02 d., 19:29 atliko AndriusKulikauskas -
Pakeista 16 eilutė iš:
||{$\pm(e_2+e_1)$}||{$e_1 \Leftrightarrow -e_2, -e_1 \Leftrightarrow e_2$}||
į:
||{$\pm(e_2--e_1)$}||{$e_1 \Leftrightarrow -e_2, -e_1 \Leftrightarrow e_2$}||
2018 lapkričio 02 d., 19:29 atliko AndriusKulikauskas -
Pakeistos 8-17 eilutės iš
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į:
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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 which is affected by this reflection as follows:

||'''Hyperplane =''' {$\pm$} '''Root'''||'''Transposition'''||
||{$\pm(e_2-e_1)$}||{$e_1 \Leftrightarrow e_2, -e_1 \Leftrightarrow -e_2$}||
||{$\pm(e_2+e_1)$}||{$e_1 \Leftrightarrow -e_2, -e_1 \Leftrightarrow e_2$}||
||{$\pm(e_1)$}||{$e_1 \Leftrightarrow -e_1$}||
2018 lapkričio 02 d., 19:13 atliko AndriusKulikauskas -
Pridėtos 1-8 eilutės:
>>bgcolor=#E9F5FC<<
-------------
See: [[Classical Lie groups]]

'''Calculate the Weyl groups of the classical root systems.'''

-------------
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#### RootSystemsWeylGroups

Naujausi pakeitimai

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