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

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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)