表示论¶
普通特征标 (Ordinary characters)¶
如何在 Sage 中计算有限群的特征标表？可以使用 Sage-GAP 接口来计算特征标表。
你可以使用 PermutationGroup 类的 character_table 方法，
或通过 GAP 命令 CharacterTable 的接口，将置换群 \(G\) 的特征标值表构建为 Sage 矩阵。
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: G.order()
8
sage: G.character_table() # random
[ 1 1 1 1 1]
[ 1 -1 -1 1 1]
[ 1 -1 1 -1 1]
[ 1 1 -1 -1 1]
[ 2 0 0 0 -2]
sage: CT = libgap(G).CharacterTable()
sage: CT.Display() # random
CT1
2 3 2 2 2 3
1a 2a 2b 4a 2c
2P 1a 1a 1a 2c 1a
3P 1a 2a 2b 4a 2c
X.1 1 1 1 1 1
X.2 1 -1 -1 1 1
X.3 1 -1 1 -1 1
X.4 1 1 -1 -1 1
X.5 2 . . . -2
>>> from sage.all import *
>>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4))]])
>>> G.order()
8
>>> G.character_table() # random
[ 1 1 1 1 1]
[ 1 -1 -1 1 1]
[ 1 -1 1 -1 1]
[ 1 1 -1 -1 1]
[ 2 0 0 0 -2]
>>> CT = libgap(G).CharacterTable()
>>> CT.Display() # random
CT1
<BLANKLINE>
2 3 2 2 2 3
<BLANKLINE>
1a 2a 2b 4a 2c
2P 1a 1a 1a 2c 1a
3P 1a 2a 2b 4a 2c
<BLANKLINE>
X.1 1 1 1 1 1
X.2 1 -1 -1 1 1
X.3 1 -1 1 -1 1
X.4 1 1 -1 -1 1
X.5 2 . . . -2
下面是另一个示例:
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
sage: G.character_table() # random
[ 1 1 1 1]
[ 1 -zeta3 - 1 zeta3 1]
[ 1 zeta3 -zeta3 - 1 1]
[ 3 0 0 -1]
sage: G = libgap.eval("Group((1,2)(3,4),(1,2,3))"); G
Group([ (1,2)(3,4), (1,2,3) ])
sage: T = G.CharacterTable()
sage: T.Display() # random
CT2
2 2 . . 2
3 1 1 1 .
1a 3a 3b 2a
2P 1a 3b 3a 1a
3P 1a 1a 1a 2a
X.1 1 1 1 1
X.2 1 A /A 1
X.3 1 /A A 1
X.4 3 . . -1
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
>>> from sage.all import *
>>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3))]])
>>> G.character_table() # random
[ 1 1 1 1]
[ 1 -zeta3 - 1 zeta3 1]
[ 1 zeta3 -zeta3 - 1 1]
[ 3 0 0 -1]
>>> G = libgap.eval("Group((1,2)(3,4),(1,2,3))"); G
Group([ (1,2)(3,4), (1,2,3) ])
>>> T = G.CharacterTable()
>>> T.Display() # random
CT2
<BLANKLINE>
2 2 . . 2
3 1 1 1 .
<BLANKLINE>
1a 3a 3b 2a
2P 1a 3b 3a 1a
3P 1a 1a 1a 2a
<BLANKLINE>
X.1 1 1 1 1
X.2 1 A /A 1
X.3 1 /A A 1
X.4 3 . . -1
<BLANKLINE>
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
其中 \(E(3)\) 表示单位立方根，\(ER(-3)\) 表示 \(-3\) 的平方根，即 \(i\sqrt{3}\),
而 \(b3 = \frac{1}{2}(-1+i \sqrt{3})\)。
请注意添加的 print Python 命令。这会令输出更美观。
sage: irr = G.Irr(); sorted(irr)
[Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, E(3)^2, E(3), 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, E(3), E(3)^2, 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 3, 0, 0, -1 ] )]
sage: irr.Display() # random
[ [ 1, 1, 1, 1 ],
[ 1, E(3)^2, E(3), 1 ],
[ 1, E(3), E(3)^2, 1 ],
[ 3, 0, 0, -1 ] ]
sage: CG = G.ConjugacyClasses(); CG
[ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ]
sage: gamma = CG[2]; gamma
(2,4,3)^G
sage: g = gamma.Representative(); g
(2,4,3)
sage: chi = irr[1]; chi # random
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, E(3)^2, E(3), 1 ] )
sage: g^chi # random
E(3)
>>> from sage.all import *
>>> irr = G.Irr(); sorted(irr)
[Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, E(3)^2, E(3), 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, E(3), E(3)^2, 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 3, 0, 0, -1 ] )]
>>> irr.Display() # random
[ [ 1, 1, 1, 1 ],
[ 1, E(3)^2, E(3), 1 ],
[ 1, E(3), E(3)^2, 1 ],
[ 3, 0, 0, -1 ] ]
>>> CG = G.ConjugacyClasses(); CG
[ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ]
>>> gamma = CG[Integer(2)]; gamma
(2,4,3)^G
>>> g = gamma.Representative(); g
(2,4,3)
>>> chi = irr[Integer(1)]; chi # random
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, E(3)^2, E(3), 1 ] )
>>> g**chi # random
E(3)
最后一个量是特征 chi 在群元素 g 处的值。
或者，如果你关闭 IPython 的“美观打印”，那么表格将打印得更好。
sage: %Pprint
Pretty printing has been turned OFF
sage: G = libgap.eval("Group((1,2)(3,4),(1,2,3))"); G
Group([ (1,2)(3,4), (1,2,3) ])
sage: T = G.CharacterTable(); T
CharacterTable( Alt( [ 1 .. 4 ] ) )
sage: T.Display()
CT3
2 2 2 . .
3 1 . 1 1
1a 2a 3a 3b
2P 1a 1a 3b 3a
3P 1a 2a 1a 1a
X.1 1 1 1 1
X.2 1 1 A /A
X.3 1 1 /A A
X.4 3 -1 . .
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
sage: irr = G.Irr(); irr
[ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3)^2, E(3) ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3), E(3)^2 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 3, -1, 0, 0 ] ) ]
sage: irr.Display()
[ [ 1, 1, 1, 1 ],
[ 1, 1, E(3)^2, E(3) ],
[ 1, 1, E(3), E(3)^2 ],
[ 3, -1, 0, 0 ] ]
sage: %Pprint
Pretty printing has been turned ON
>>> from sage.all import *
>>> %Pprint
Pretty printing has been turned OFF
>>> G = libgap.eval("Group((1,2)(3,4),(1,2,3))"); G
Group([ (1,2)(3,4), (1,2,3) ])
>>> T = G.CharacterTable(); T
CharacterTable( Alt( [ 1 .. 4 ] ) )
>>> T.Display()
CT3
<BLANKLINE>
2 2 2 . .
3 1 . 1 1
<BLANKLINE>
1a 2a 3a 3b
2P 1a 1a 3b 3a
3P 1a 2a 1a 1a
<BLANKLINE>
X.1 1 1 1 1
X.2 1 1 A /A
X.3 1 1 /A A
X.4 3 -1 . .
<BLANKLINE>
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
>>> irr = G.Irr(); irr
[ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3)^2, E(3) ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3), E(3)^2 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 3, -1, 0, 0 ] ) ]
>>> irr.Display()
[ [ 1, 1, 1, 1 ],
[ 1, 1, E(3)^2, E(3) ],
[ 1, 1, E(3), E(3)^2 ],
[ 3, -1, 0, 0 ] ]
>>> %Pprint
Pretty printing has been turned ON
布劳尔特征标 (Brauer characters)¶
GAP 中的布劳尔特征标表尚未具有“原生”接口。
要访问它们，你可以使用 libgap.eval 命令直接与 GAP 交互。
下面的示例通过使用 GAP 接口来说明语法。
sage: G = libgap.eval("Group((1,2)(3,4),(1,2,3))"); G
Group([ (1,2)(3,4), (1,2,3) ])
sage: irr = G.IrreducibleRepresentations(GF(7)); irr # random arch. dependent output
[ [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^4 ] ] ],
[ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^2 ] ] ],
[ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^0 ] ] ],
[ (1,2)(3,4), (1,2,3) ] ->
[ [ [ Z(7)^2, Z(7)^5, Z(7) ], [ Z(7)^3, Z(7)^2, Z(7)^3 ],
[ Z(7), Z(7)^5, Z(7)^2 ] ],
[ [ 0*Z(7), Z(7)^0, 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^0 ],
[ Z(7)^0, 0*Z(7), 0*Z(7) ] ] ] ]
sage: brvals = [[chi.Image(c.Representative()).BrauerCharacterValue()
....: for c in G.ConjugacyClasses()] for chi in irr]
sage: brvals # random architecture dependent output
[ [ 1, 1, E(3)^2, E(3) ],
[ 1, 1, E(3), E(3)^2 ],
[ 1, 1, 1, 1 ],
[ 3, -1, 0, 0 ] ]
sage: T = G.CharacterTable()
sage: T.Display() # random
CT3
2 2 . . 2
3 1 1 1 .
1a 3a 3b 2a
2P 1a 3b 3a 1a
3P 1a 1a 1a 2a
X.1 1 1 1 1
X.2 1 A /A 1
X.3 1 /A A 1
X.4 3 . . -1
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
>>> from sage.all import *
>>> G = libgap.eval("Group((1,2)(3,4),(1,2,3))"); G
Group([ (1,2)(3,4), (1,2,3) ])
>>> irr = G.IrreducibleRepresentations(GF(Integer(7))); irr # random arch. dependent output
[ [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^4 ] ] ],
[ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^2 ] ] ],
[ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^0 ] ] ],
[ (1,2)(3,4), (1,2,3) ] ->
[ [ [ Z(7)^2, Z(7)^5, Z(7) ], [ Z(7)^3, Z(7)^2, Z(7)^3 ],
[ Z(7), Z(7)^5, Z(7)^2 ] ],
[ [ 0*Z(7), Z(7)^0, 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^0 ],
[ Z(7)^0, 0*Z(7), 0*Z(7) ] ] ] ]
>>> brvals = [[chi.Image(c.Representative()).BrauerCharacterValue()
... for c in G.ConjugacyClasses()] for chi in irr]
>>> brvals # random architecture dependent output
[ [ 1, 1, E(3)^2, E(3) ],
[ 1, 1, E(3), E(3)^2 ],
[ 1, 1, 1, 1 ],
[ 3, -1, 0, 0 ] ]
>>> T = G.CharacterTable()
>>> T.Display() # random
CT3
<BLANKLINE>
2 2 . . 2
3 1 1 1 .
<BLANKLINE>
1a 3a 3b 2a
2P 1a 3b 3a 1a
3P 1a 1a 1a 2a
<BLANKLINE>
X.1 1 1 1 1
X.2 1 A /A 1
X.3 1 /A A 1
X.4 3 . . -1
<BLANKLINE>
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3