# Linear representation theory of projective special linear group:PSL(2,11)

## Contents

This article gives specific information, namely, linear representation theory, about a particular group, namely: projective special linear group:PSL(2,11).
View linear representation theory of particular groups | View other specific information about projective special linear group:PSL(2,11)

## Summary

Item Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 1,5,5,10,10,11,12,12
in grouped form: 1 (1 time), 5 (2 times), 10 (2 times), 11 (1 time), 12 (2 times)
number: 8, sum of squares: 660, maximum: 12, quasirandom degree: 5, lcm: 660
number of irreducible representations (equals number of conjugacy classes) 8
As $PSL(2,q), q = 11$ ($q$ odd): $(q+ 5)/2 = (11 + 5)/2 = 8$

## 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 $q$ $q = 11$, i.e., field:F11, so the group is $PSL(2,11)$ 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(2,11)));
[ [ 1, 1 ], [ 5, 2 ], [ 10, 2 ], [ 11, 1 ], [ 12, 2 ] ]

### Character table

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

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