Linear representation theory of double cover of alternating group:A7
From Groupprops
This article gives specific information, namely, linear representation theory, about a particular group, namely: double cover of alternating group:A7.
View linear representation theory of particular groups | View other specific information about double cover of alternating group:A7
Summary
Item | Value |
---|---|
degrees of irreducible representations over a splitting field (such as ![]() ![]() |
in grouped form: 1 (1 time), 4 (2 times), 6 (1 time), 10 (2 times), 14 (4 times), 15 (1 time), 20 (2 times), 21 (1 time), 35 (1 time), 36 (1 time) number: 16, maximum: 36, lcm: 1260, sum of squares: 5040 |
Family contexts
Family | Parameter values | General discussion of linear representation theory of family |
---|---|---|
double cover of alternating group ![]() |
![]() ![]() |
linear representation theory of double cover of alternating group |
GAP implementation
Although the group can be constructed and manipulated explicitly as PerfectGroup(5040,1), this is a time-consuming process. Instead, we can use the stored information about the group using the "2.A7" symbol for it.
Degrees of irreducible representations
These can be accessed using the CharacterDegrees, and CharacterTable functions as follows:
gap> CharacterDegrees(CharacterTable("2.A7")); [ [ 1, 1 ], [ 4, 2 ], [ 6, 1 ], [ 10, 2 ], [ 14, 4 ], [ 15, 1 ], [ 20, 2 ], [ 21, 1 ], [ 35, 1 ], [ 36, 1 ] ]
Character table
This can be computed as follows:
gap> Irr(CharacterTable("2.A7")); [ Character( CharacterTable( "2.A7" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "2.A7" ), [ 6, 6, 2, 3, 3, 0, 0, 0, 0, 1, 1, -1, -1, -1, -1, -1 ] ), Character( CharacterTable( "2.A7" ), [ 10, 10, -2, 1, 1, 1, 1, 0, 0, 0, 0, 1, E(7)+E(7)^2+E(7)^4, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6, E(7)^3+E(7)^5+E(7)^6 ] ), Character( CharacterTable( "2.A7" ), [ 10, 10, -2, 1, 1, 1, 1, 0, 0, 0, 0, 1, E(7)^3+E(7)^5+E(7)^6, E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4, E(7)+E(7)^2+E(7)^4 ] ), Character( CharacterTable( "2.A7" ), [ 14, 14, 2, 2, 2, -1, -1, 0, 0, -1, -1, 2, 0, 0, 0, 0 ] ), Character( CharacterTable( "2.A7" ), [ 14, 14, 2, -1, -1, 2, 2, 0, 0, -1, -1, -1, 0, 0, 0, 0 ] ), Character( CharacterTable( "2.A7" ), [ 15, 15, -1, 3, 3, 0, 0, -1, -1, 0, 0, -1, 1, 1, 1, 1 ] ), Character( CharacterTable( "2.A7" ), [ 21, 21, 1, -3, -3, 0, 0, -1, -1, 1, 1, 1, 0, 0, 0, 0 ] ), Character( CharacterTable( "2.A7" ), [ 35, 35, -1, -1, -1, -1, -1, 1, 1, 0, 0, -1, 0, 0, 0, 0 ] ), Character( CharacterTable( "2.A7" ), [ 4, -4, 0, -2, 2, 1, -1, 0, 0, -1, 1, 0, -E(7)-E(7)^2-E(7)^4, E(7)+E(7)^2+E(7)^4, -E(7)^3-E(7)^5-E(7)^6, E(7)^3+E(7)^5+E(7)^6 ] ), Character( CharacterTable( "2.A7" ), [ 4, -4, 0, -2, 2, 1, -1, 0, 0, -1, 1, 0, -E(7)^3-E(7)^5-E(7)^6, E(7)^3+E(7)^5+E(7)^6, -E(7)-E(7)^2-E(7)^4, E(7)+E(7)^2+E(7)^4 ] ), Character( CharacterTable( "2.A7" ), [ 14, -14, 0, 2, -2, -1, 1, E(8)-E(8)^3, -E(8)+E(8)^3, -1, 1, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "2.A7" ), [ 14, -14, 0, 2, -2, -1, 1, -E(8)+E(8)^3, E(8)-E(8)^3, -1, 1, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "2.A7" ), [ 20, -20, 0, -4, 4, -1, 1, 0, 0, 0, 0, 0, -1, 1, -1, 1 ] ), Character( CharacterTable( "2.A7" ), [ 20, -20, 0, 2, -2, 2, -2, 0, 0, 0, 0, 0, -1, 1, -1, 1 ] ), Character( CharacterTable( "2.A7" ), [ 36, -36, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 1, -1 ] ) ]
Here is a command to display the character table in a more user-friendly fashion:
gap> Display(CharacterTable("2.A7")); 2.A7 2 4 4 3 3 3 1 1 3 3 1 1 2 1 1 1 1 3 2 2 1 2 2 2 2 . . . . 1 . . . . 5 1 1 . . . . . . . 1 1 . . . . . 7 1 1 . . . . . . . . . . 1 1 1 1 1a 2a 4a 3a 6a 3b 6b 8a 8b 5a 10a 12a 7a 14a 7b 14b 2P 1a 1a 2a 3a 3a 3b 3b 4a 4a 5a 5a 6a 7a 7a 7b 7b 3P 1a 2a 4a 1a 2a 1a 2a 8b 8a 5a 10a 4a 7b 14b 7a 14a 5P 1a 2a 4a 3a 6a 3b 6b 8b 8a 1a 2a 12a 7b 14b 7a 14a 7P 1a 2a 4a 3a 6a 3b 6b 8a 8b 5a 10a 12a 1a 2a 1a 2a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 6 6 2 3 3 . . . . 1 1 -1 -1 -1 -1 -1 X.3 10 10 -2 1 1 1 1 . . . . 1 B B /B /B X.4 10 10 -2 1 1 1 1 . . . . 1 /B /B B B X.5 14 14 2 2 2 -1 -1 . . -1 -1 2 . . . . X.6 14 14 2 -1 -1 2 2 . . -1 -1 -1 . . . . X.7 15 15 -1 3 3 . . -1 -1 . . -1 1 1 1 1 X.8 21 21 1 -3 -3 . . -1 -1 1 1 1 . . . . X.9 35 35 -1 -1 -1 -1 -1 1 1 . . -1 . . . . X.10 4 -4 . -2 2 1 -1 . . -1 1 . -B B -/B /B X.11 4 -4 . -2 2 1 -1 . . -1 1 . -/B /B -B B X.12 14 -14 . 2 -2 -1 1 A -A -1 1 . . . . . X.13 14 -14 . 2 -2 -1 1 -A A -1 1 . . . . . X.14 20 -20 . -4 4 -1 1 . . . . . -1 1 -1 1 X.15 20 -20 . 2 -2 2 -2 . . . . . -1 1 -1 1 X.16 36 -36 . . . . . . . 1 -1 . 1 -1 1 -1 A = E(8)-E(8)^3 = ER(2) = r2 B = E(7)+E(7)^2+E(7)^4 = (-1+ER(-7))/2 = b7