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
, and the characteristic prime is
.
is a power of
. We denote the group as
or
.
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 or ) |
Case congruent to 1 mod 4 (e.g., ): 1 (1 time), (2 times), ( times), (1 time), ( times) Case congruent to 3 mod 4 (e.g., ): 1 (1 time), (2 times), ( times), (1 time), ( times) Case even (e.g., ): 1 (1 time), ( times), (1 time), ( times)
|
number of irreducible representations |
Case odd: ; Case even: . 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 congruent to 1 mod 4 (e.g., ):  Case congruent to 3 mod 4 (e.g., ):  Case even:
|
maximum degree of irreducible representation over a splitting field |
|
lcm of degrees of irreducible representations over a splitting field |
Case odd: , Case even:
|
sum of squares of degrees of irreducible representations over a splitting field |
Case odd: , case even:  equal to the group order. See sum of squares of degrees of irreducible representations equals group order
|
Particular cases
(field size) |
(underlying prime, field characteristic) |
Case for |
Group |
Order of the group (= if even, if odd) |
Degrees of irreducible representations (ascending order) |
Number of irreducible representations (= if even, if 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)
|