Linear representation theory of projective special linear group:PSL(2,13)

This article gives specific information, namely, linear representation theory, about a particular group, namely: projective special linear group:PSL(2,13).
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Summary

Item	Value
degrees of irreducible representations over a splitting field (such as ${\overline {\mathbb {Q} }}$ or $\mathbb {C}$)	1,7,7,12,12,12,13,14,14 in grouped form: 1 (1 time), 7 (2 times), 12 (3 times), 13 (1 time), 14 (2 times) number: 9, sum of squares: 1092, maximum: 14, quasirandom degree: 7, lcm: 1092
number of irreducible representations (equals number of conjugacy classes)	9 As $PSL(2,q),q=13$ ($q$ odd): $(q+5)/2=(13+5)/2=9$

Family contexts

Family name	Parameter values	General discussion of linear representation theory of family
projective special linear group of degree two $PSL(2,q)$ over a finite field of size $q$	$q=13$, i.e., field:F13, so the group is $PSL(2,13)$	linear representation theory of projective special linear group of degree two over a finite field

GAP implementation

Degrees of irreducible representations

These can be computed using the CharacterDegrees, GAP:CharacterTable, and PSL functions:

gap> CharacterDegrees(CharacterTable(PSL(2,13)));
[ [ 1, 1 ], [ 7, 2 ], [ 12, 3 ], [ 13, 1 ], [ 14, 2 ] ]

Character table

This can be computed using the Irr, CharacterTable, and PSL functions:

gap> Irr(CharacterTable(PSL(2,13)));
[ Character( CharacterTable( Group([ (3,13,11,9,7,5)(4,14,12,10,8,6),
      (1,2,9)(3,8,10)(4,5,12)(6,13,14) ]) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( Group([ (3,13,11,9,7,5)(4,14,12,10,8,6),
      (1,2,9)(3,8,10)(4,5,12)(6,13,14) ]) ),
    [ 7, 0, 0, 0, 1, -1, -1, -E(13)-E(13)^3-E(13)^4-E(13)^9-E(13)^10-E(13)^12,
      -E(13)^2-E(13)^5-E(13)^6-E(13)^7-E(13)^8-E(13)^11 ] ),
  Character( CharacterTable( Group([ (3,13,11,9,7,5)(4,14,12,10,8,6),
      (1,2,9)(3,8,10)(4,5,12)(6,13,14) ]) ),
    [ 7, 0, 0, 0, 1, -1, -1, -E(13)^2-E(13)^5-E(13)^6-E(13)^7-E(13)^8-E(13)^11
        , -E(13)-E(13)^3-E(13)^4-E(13)^9-E(13)^10-E(13)^12 ] ),
  Character( CharacterTable( Group([ (3,13,11,9,7,5)(4,14,12,10,8,6),
      (1,2,9)(3,8,10)(4,5,12)(6,13,14) ]) ),
    [ 12, -E(7)^3-E(7)^4, -E(7)-E(7)^6, -E(7)^2-E(7)^5, 0, 0, 0, -1, -1 ] ),
  Character( CharacterTable( Group([ (3,13,11,9,7,5)(4,14,12,10,8,6),
      (1,2,9)(3,8,10)(4,5,12)(6,13,14) ]) ),
    [ 12, -E(7)^2-E(7)^5, -E(7)^3-E(7)^4, -E(7)-E(7)^6, 0, 0, 0, -1, -1 ] ),
  Character( CharacterTable( Group([ (3,13,11,9,7,5)(4,14,12,10,8,6),
      (1,2,9)(3,8,10)(4,5,12)(6,13,14) ]) ),
    [ 12, -E(7)-E(7)^6, -E(7)^2-E(7)^5, -E(7)^3-E(7)^4, 0, 0, 0, -1, -1 ] ),
  Character( CharacterTable( Group([ (3,13,11,9,7,5)(4,14,12,10,8,6),
      (1,2,9)(3,8,10)(4,5,12)(6,13,14) ]) ), [ 13, -1, -1, -1, 1, 1, 1, 0, 0
     ] ), Character( CharacterTable( Group([ (3,13,11,9,7,5)(4,14,12,10,8,6),
      (1,2,9)(3,8,10)(4,5,12)(6,13,14) ]) ), [ 14, 0, 0, 0, -1, 2, -1, 1, 1
     ] ), Character( CharacterTable( Group([ (3,13,11,9,7,5)(4,14,12,10,8,6),
      (1,2,9)(3,8,10)(4,5,12)(6,13,14) ]) ), [ 14, 0, 0, 0, -1, -2, 1, 1, 1
     ] ) ]