Linear representation theory of alternating group:A10: Difference between revisions

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group = alternating group:A10|
group = alternating group:A10|
connective = of}}
connective = of}}
==Summary==
<section begin="summary"/>
{| class="sortable" border="1"
! Item !! Value
|-
| [[degrees of irreducible representations]] over a [[splitting field]] (such as <math>\overline{\mathbb{Q}}</math> or <math>\mathbb{C}</math>) || 1, 9, 35, 36, 42, 75, 84, 90, 126, 160, 210, 224, 224, 225, 252, 288, 300, 315, 350, 384, 384, 450, 525, 567<br>grouped form (by default each occurs once): 1, 9, 35, 36, 42, 75, 84, 90, 126, 160, 210, 224 (2 times) 225, 252, 288, 300, 315, 350, 384 (2 times), 450, 525, 567<br>[[maximum degree of irreducible representation|maximum]] 567, [[number of irreducible representations equals number of conjugacy classes|number]]: 24, [[sum of squares of degrees of irreducible representations equals order of group|sum of squares]]: 1814400
|}
<section end="summary"/>
==GAP implementation==
===Degrees of irreducible representations===
These can be computed using the [[GAP:CharacterDegrees|CharacterDegrees]] function:
<pre>gap> CharacterDegrees(CharacterTable(AlternatingGroup(10)));
[ [ 1, 1 ], [ 9, 1 ], [ 35, 1 ], [ 36, 1 ], [ 42, 1 ], [ 75, 1 ], [ 84, 1 ], [ 90, 1 ], [ 126, 1 ], [ 160, 1 ], [ 210, 1 ], [ 224, 2 ], [ 225, 1 ],
  [ 252, 1 ], [ 288, 1 ], [ 300, 1 ], [ 315, 1 ], [ 350, 1 ], [ 384, 2 ], [ 450, 1 ], [ 525, 1 ], [ 567, 1 ] ]</pre>
===Character table===
These can be computed using the [[GAP:Irr|Irr]] and [[GAP:CharacterTable|CharacterTable]] functions:
<pre>gap> Irr(CharacterTable(AlternatingGroup(10)));
[ Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 9, 5, 1, 6, 2, 3, -1, 0, 3, -1, 0, 1, 4, 0, 1, -1, 1, -1, 2, -1, -1, -1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 35, 11, 3, 14, 2, 2, 2, -1, 3, 3, 0, -1, 5, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, -1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 36, 8, -4, 15, -1, 3, -1, 0, 2, -2, -1, 0, 6, -2, 0, 1, -1, 1, 1, 1, 1, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 42, 6, 2, 0, 0, 3, 3, -3, 0, -4, 0, 2, -3, 1, 0, 2, -1, -1, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 75, 15, 3, 15, 3, 0, 0, 3, 1, -3, 1, -1, 0, 0, 0, 0, 0, 0, -2, 1, 1, -1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 84, 0, -4, 21, -3, 3, 3, 3, -2, 2, 1, 0, 4, 0, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 90, 14, 2, 6, 2, 3, -1, 0, 0, 4, 0, 2, -5, -1, 1, 0, -1, 1, -1, -1, -1, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 126, -14, 6, 21, 1, 6, -2, 0, -4, 0, -1, -2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 160, 16, 0, 34, -2, -2, -2, -2, 0, 0, 0, 0, 5, 1, -1, 0, 0, 0, -1, -1, -1, 0, 1, 1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 210, 6, -6, -21, 3, 0, 0, 3, -4, 0, -1, 2, 5, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 224, -16, 0, 14, 2, 2, 2, -1, 0, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, -1, 2 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 224, -16, 0, 14, 2, 2, 2, -1, 0, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 2, -1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 225, 5, 9, 15, -1, -6, 2, 0, -1, 3, -1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, -1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 252, 8, 4, -21, -1, 3, -1, 0, -2, 2, 1, 0, 2, -2, -1, 2, 1, -1, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 288, 16, 0, -6, -2, 6, -2, 0, 0, 0, 0, 0, -7, 1, -1, -2, 0, 0, 1, 1, 1, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 300, 0, 4, -15, -3, 3, 3, 3, 2, -2, -1, 0, 0, 0, 0, 0, 1, 1, -1, -1, -1, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 315, 19, -5, 21, 1, -3, 1, 0, -1, -1, -1, -1, -5, -1, 1, 0, 1, -1, 0, 0, 0, 1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 350, -10, -2, 35, -1, -1, -1, -1, -2, -2, 1, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 384, 0, 0, -24, 0, 0, 0, -3, 0, 0, 0, 0, 4, 0, 1, -1, 0, 0, -1,
      E(21)^2+E(21)^8+E(21)^10+E(21)^11+E(21)^13+E(21)^19, E(21)+E(21)^4+E(21)^5+E(21)^16+E(21)^17+E(21)^20, 0, 0, 0 ] ), Character( CharacterTable( Alt(
    [ 1 .. 10 ] ) ), [ 384, 0, 0, -24, 0, 0, 0, -3, 0, 0, 0, 0, 4, 0, 1, -1, 0, 0, -1, E(21)+E(21)^4+E(21)^5+E(21)^16+E(21)^17+E(21)^20,
      E(21)^2+E(21)^8+E(21)^10+E(21)^11+E(21)^13+E(21)^19, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 450, 10, 2, -15, 1, -3, 1, 0,
      -2, -2, 1, -2, 0, 0, 0, 0, -1, 1, 2, -1, -1, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 525, -15, 5, 0, 0, -3, -3, 3, 3, -1,
      0, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 567, -9, -9, 0, 0, 0, 0, 0, 3, 3, 0, -1, -3, 1,
      0, 2, 0, 0, 0, 0, 0, -1, 0, 0 ] ) ]</pre>

Latest revision as of 01:13, 19 April 2012

This article gives specific information, namely, linear representation theory, about a particular group, namely: alternating group:A10.
View linear representation theory of particular groups | View other specific information about alternating group:A10

Summary

Item Value
degrees of irreducible representations over a splitting field (such as or ) 1, 9, 35, 36, 42, 75, 84, 90, 126, 160, 210, 224, 224, 225, 252, 288, 300, 315, 350, 384, 384, 450, 525, 567
grouped form (by default each occurs once): 1, 9, 35, 36, 42, 75, 84, 90, 126, 160, 210, 224 (2 times) 225, 252, 288, 300, 315, 350, 384 (2 times), 450, 525, 567
maximum 567, number: 24, sum of squares: 1814400

GAP implementation

Degrees of irreducible representations

These can be computed using the CharacterDegrees function: 

gap> CharacterDegrees(CharacterTable(AlternatingGroup(10)));
[ [ 1, 1 ], [ 9, 1 ], [ 35, 1 ], [ 36, 1 ], [ 42, 1 ], [ 75, 1 ], [ 84, 1 ], [ 90, 1 ], [ 126, 1 ], [ 160, 1 ], [ 210, 1 ], [ 224, 2 ], [ 225, 1 ],
  [ 252, 1 ], [ 288, 1 ], [ 300, 1 ], [ 315, 1 ], [ 350, 1 ], [ 384, 2 ], [ 450, 1 ], [ 525, 1 ], [ 567, 1 ] ]

Character table

These can be computed using the Irr and CharacterTable functions: 

gap> Irr(CharacterTable(AlternatingGroup(10)));
[ Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 9, 5, 1, 6, 2, 3, -1, 0, 3, -1, 0, 1, 4, 0, 1, -1, 1, -1, 2, -1, -1, -1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 35, 11, 3, 14, 2, 2, 2, -1, 3, 3, 0, -1, 5, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, -1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 36, 8, -4, 15, -1, 3, -1, 0, 2, -2, -1, 0, 6, -2, 0, 1, -1, 1, 1, 1, 1, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 42, 6, 2, 0, 0, 3, 3, -3, 0, -4, 0, 2, -3, 1, 0, 2, -1, -1, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 75, 15, 3, 15, 3, 0, 0, 3, 1, -3, 1, -1, 0, 0, 0, 0, 0, 0, -2, 1, 1, -1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 84, 0, -4, 21, -3, 3, 3, 3, -2, 2, 1, 0, 4, 0, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 90, 14, 2, 6, 2, 3, -1, 0, 0, 4, 0, 2, -5, -1, 1, 0, -1, 1, -1, -1, -1, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 126, -14, 6, 21, 1, 6, -2, 0, -4, 0, -1, -2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 160, 16, 0, 34, -2, -2, -2, -2, 0, 0, 0, 0, 5, 1, -1, 0, 0, 0, -1, -1, -1, 0, 1, 1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 210, 6, -6, -21, 3, 0, 0, 3, -4, 0, -1, 2, 5, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 224, -16, 0, 14, 2, 2, 2, -1, 0, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, -1, 2 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 224, -16, 0, 14, 2, 2, 2, -1, 0, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 2, -1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 225, 5, 9, 15, -1, -6, 2, 0, -1, 3, -1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, -1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 252, 8, 4, -21, -1, 3, -1, 0, -2, 2, 1, 0, 2, -2, -1, 2, 1, -1, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 288, 16, 0, -6, -2, 6, -2, 0, 0, 0, 0, 0, -7, 1, -1, -2, 0, 0, 1, 1, 1, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 300, 0, 4, -15, -3, 3, 3, 3, 2, -2, -1, 0, 0, 0, 0, 0, 1, 1, -1, -1, -1, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 315, 19, -5, 21, 1, -3, 1, 0, -1, -1, -1, -1, -5, -1, 1, 0, 1, -1, 0, 0, 0, 1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 350, -10, -2, 35, -1, -1, -1, -1, -2, -2, 1, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 384, 0, 0, -24, 0, 0, 0, -3, 0, 0, 0, 0, 4, 0, 1, -1, 0, 0, -1,
      E(21)^2+E(21)^8+E(21)^10+E(21)^11+E(21)^13+E(21)^19, E(21)+E(21)^4+E(21)^5+E(21)^16+E(21)^17+E(21)^20, 0, 0, 0 ] ), Character( CharacterTable( Alt(
    [ 1 .. 10 ] ) ), [ 384, 0, 0, -24, 0, 0, 0, -3, 0, 0, 0, 0, 4, 0, 1, -1, 0, 0, -1, E(21)+E(21)^4+E(21)^5+E(21)^16+E(21)^17+E(21)^20,
      E(21)^2+E(21)^8+E(21)^10+E(21)^11+E(21)^13+E(21)^19, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 450, 10, 2, -15, 1, -3, 1, 0,
      -2, -2, 1, -2, 0, 0, 0, 0, -1, 1, 2, -1, -1, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 525, -15, 5, 0, 0, -3, -3, 3, 3, -1,
      0, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 567, -9, -9, 0, 0, 0, 0, 0, 3, 3, 0, -1, -3, 1,
      0, 2, 0, 0, 0, 0, 0, -1, 0, 0 ] ) ]
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