Linear representation theory of alternating group:A9: Difference between revisions
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| [[degrees of irreducible representations]] over a [[splitting field]] (such as <math>\overline{\mathbb{Q}}</math> or <math>\mathbb{C}</math>) || 1, 8, 21, 21, 27, 28, 35, 35, 42, 48, 56, 84, 105, 120, 162, 168, 189, 216<br>grouped form (each occurs once by default): 1, 8, 21 (2 times), 27, 28, 35 (2 times), 42, 48, 56, 84, 105, 120, 162, 168, 189, 216<br>[[maximum degree of irreducible representation|maximum]]: 216, [[number of irreducible representations equals number of conjugacy classes|number]]: 18, [[sum of squares of degrees of irreducible representations equals order of group|sum of squares]]: 181440 | | [[degrees of irreducible representations]] over a [[splitting field]] (such as <math>\overline{\mathbb{Q}}</math> or <math>\mathbb{C}</math>) || 1, 8, 21, 21, 27, 28, 35, 35, 42, 48, 56, 84, 105, 120, 162, 168, 189, 216<br>grouped form (each occurs once by default): 1, 8, 21 (2 times), 27, 28, 35 (2 times), 42, 48, 56, 84, 105, 120, 162, 168, 189, 216<br>[[maximum degree of irreducible representation|maximum]]: 216, [[number of irreducible representations equals number of conjugacy classes|number]]: 18, [[sum of squares of degrees of irreducible representations equals order of group|sum of squares]]: 181440 | ||
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| [[minimal splitting field]], i.e., smallest field of realization of all irreducible representations (characteristic zero) || <math>\mathbb{Q}(\zeta + \zeta^2 + \zeta^4 + \zeta^8)</math> where <math>\zeta</matH> is a primitive fifteenth root of unity<br>Same as <math>\mathbb{Q}(\sqrt{-15})</math><br>Same as [[field generated by character values]] | |||
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| condition for a field of characteristic not 2,3,5,7 to be a splitting field || -15 should be a square in that field | |||
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Revision as of 01:08, 19 April 2012
This article gives specific information, namely, linear representation theory, about a particular group, namely: alternating group:A9.
View linear representation theory of particular groups | View other specific information about alternating group:A9
Summary
| Item | Value |
|---|---|
| degrees of irreducible representations over a splitting field (such as or ) | 1, 8, 21, 21, 27, 28, 35, 35, 42, 48, 56, 84, 105, 120, 162, 168, 189, 216 grouped form (each occurs once by default): 1, 8, 21 (2 times), 27, 28, 35 (2 times), 42, 48, 56, 84, 105, 120, 162, 168, 189, 216 maximum: 216, number: 18, sum of squares: 181440 |
| minimal splitting field, i.e., smallest field of realization of all irreducible representations (characteristic zero) | where is a primitive fifteenth root of unity Same as Same as field generated by character values |
| condition for a field of characteristic not 2,3,5,7 to be a splitting field | -15 should be a square in that field |
GAP implementation
Degrees of irreducible representations
These can be computed using the CharacterDegrees function:
gap> CharacterDegrees(CharacterTable(AlternatingGroup(9))); [ [ 1, 1 ], [ 8, 1 ], [ 21, 2 ], [ 27, 1 ], [ 28, 1 ], [ 35, 2 ], [ 42, 1 ], [ 48, 1 ], [ 56, 1 ], [ 84, 1 ], [ 105, 1 ], [ 120, 1 ], [ 162, 1 ], [ 168, 1 ], [ 189, 1 ], [ 216, 1 ] ]