Linear representation theory of special linear group:SL(2,3)

This article gives specific information, namely, linear representation theory, about a particular group, namely: special linear group:SL(2,3).
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Summary

Item	Value
degrees of irreducible representations over a splitting field	1,1,1,2,2,2,3 maximum: 3, lcm: 6, number: 7, sum of squares: 24
Schur index values of irreducible representations over a splitting field	1,1,1,?,?,?,1

Irreducible representations over the complex numbers

Trivial one-dimensional representation

There is a trivial one-dimensional representation, namely, the representation sending all elements to $(1)$.

Nontrivial one-dimensional representations

There are two nontrivial one-dimensional representations. For both of them, the kernel is the quaternion group, which is the commutator subgroup. These correspond to the two nontrivial one-dimensional representations of the quotient group, i.e., the abelianization, which is a cyclic group of order three.

Both representations map the two non-identity cosets of the commutator subgroup to primitive cuberoots of unity. The two representations differ in which cuberoot of unity is assigned to which coset.

Irreducible three-dimensional representation with center as kernel

There is an irreducible faithful three-dimensional representation whose kernel is the center of the group, which has order two. This corresponds to the three-dimensional irreducible representation of the alternating group of degree four, which is the quotient by the center. This representation comes by restricting to the alternating group the standard representation of the symmetric group of degree four.

Three irreducible two-dimensional representations

There are three irreducible two-dimensional representations. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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Orthogonality relations and numerical checks

• The degrees of the irreducible representations are $1,1,1,2,2,2,3$. We have $1^{2}+1^{2}+1^{2}+2^{2}+2^{2}+2^{2}+3^{2}=24$. This confirms the fact that sum of squares of degrees of irreducible representations equals order of group.
• The center is the unique largest abelian normal subgroup. It has order $2$ and index $12$. The degrees of irreducible representations all divide $12$, and their squares are all less than $12$. This confirms that: degree of irreducible representation divides index of abelian normal subgroup and order of inner automorphism group bounds square of degree of irreducible representation.