Linear representation theory of double cover of alternating group:A7

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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 \overline{\mathbb{Q}} or \mathbb{C}) 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 2 \cdot A_n n= 7, so the group is 2 \cdot A_7 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
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