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

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