# Linear representation theory of projective special linear group of degree two over a finite field

## Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: projective special linear group of degree two.
View linear representation theory of group families | View other specific information about projective special linear group of degree two

This article describes the linear representation theory of the projective special linear group of degree two over a finite field. The order (size) of the field is $q$, and the characteristic prime is $p$. $q$ is a power of $p$. We denote the group as $PSL(2,q)$ or $PSL_2(q)$.

See also the linear representation theories of: general linear group of degree two, projective general linear group of degree two, and special linear group of degree two.

## Summary

Item Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) Case $q$ congruent to 1 mod 4 (e.g., $q=5,9,13,17,25,29$): 1 (1 time), $(q + 1)/2$ (2 times), $q - 1$ ($(q - 1)/4$ times), $q$ (1 time), $q + 1$ ($(q - 5)/4$ times)
Case $q$ congruent to 3 mod 4 (e.g., $q = 3,7,11,19,23,27$): 1 (1 time), $(q - 1)/2$ (2 times), $q - 1$ ($(q - 3)/4$ times), $q$ (1 time), $q + 1$ ($(q - 3)/4$ times)
Case $q$ even (e.g., $q=2,4,8,16,32$): 1 (1 time), $q - 1$ ($q/2$ times), $q$ (1 time), $q + 1$ ($(q - 2)/2$ times)
number of irreducible representations Case $q$ odd: $(q + 5)/2$; Case $q$ even: $q + 1$.
See number of irreducible representations equals number of conjugacy classes, element structure of projective special linear group of degree two over a finite field#Conjugacy class structure
quasirandom degree (minimum possible degree of nontrivial irreducible representation) Case $q$ congruent to 1 mod 4 (e.g., $q=5,9,13,17,25,29$): $(q + 1)/2$
Case $q$ congruent to 3 mod 4 (e.g., $q = 3,7,11,19,23,27$): $(q - 1)/2$
Case $q$ even: $q - 1$
maximum degree of irreducible representation over a splitting field $q + 1$
lcm of degrees of irreducible representations over a splitting field Case $q$ odd: $q(q+1)(q-1)/2 = (q^3-q)/2$, Case $q$ even: $q(q+1)(q-1) = q^3 - q$
sum of squares of degrees of irreducible representations over a splitting field Case $q$ odd: $(q^3 - q)/2$, case $q$ even: $q^3 - q$
equal to the group order. See sum of squares of degrees of irreducible representations equals group order

## Particular cases

$q$ (field size) $p$ (underlying prime, field characteristic) Case for $q$ Group $PSL(2,q)$ Order of the group (= $q^3 - q$ if $q$ even, $(q^3 - q)/2$ if $q$ odd) Degrees of irreducible representations (ascending order) Number of irreducible representations (= $q + 1$ if $q$ even, $(q + 5)/2$ if $q$ odd) Linear representation theory page
2 2 even symmetric group:S3 6 1,1,2 3 linear representation theory of symmetric group:S3
3 3 3 mod 4 alternating group:A4 12 1,1,1,3 4 linear representation theory of alternating group:A4
4 2 even alternating group:A5 60 1,3,3,4,5 5 linear representation theory of alternating group:A5
5 5 1 mod 4 alternating group:A5 60 1,3,3,4,5 5 linear representation theory of alternating group:A5
7 7 3 mod 4 projective special linear group:PSL(3,2) 168 1,3,3,6,7,8 6 linear representation theory of projective special linear group:PSL(3,2)
8 2 even projective special linear group:PSL(2,8) 504 1,7,7,7,7,8,9,9,9 9 linear representation theory of projective special linear group:PSL(2,8)
9 3 1 mod 4 alternating group:A6 360 1,5,5,8,8,9,10 7 linear representation theory of alternating group:A6
11 11 3 mod 4 projective special linear group:PSL(2,11) 660 1,5,5,10,10,11,12,12 8 linear representation theory of projective special linear group:PSL(2,11)
13 13 1 mod 4 projective special linear group:PSL(2,13) 1092 1,7,7,12,12,12,13,14,14 9 linear representation theory of projective special linear group:PSL(2,13)
16 2 even projective special linear group:PSL(2,16) 4080 1,15 (8 times),16,17 (7 times) 17 linear representation theory of projective special linear group:PSL(2,16)
17 17 1 mod 4 projective special linear group:PSL(2,17) 2448 1,9,9,16,16,16,16,17,18,18,18 11 linear representation theory of projective special linear group:PSL(2,17)
19 19 3 mod 4 projective special linear group:PSL(2,19) 3420 1,9,9,18,18,18,18,19,20,20,20,20 12 linear representation theory of projective special linear group:PSL(2,19)