# irreducible representations of the special linear group over $\mathbb{F}_{p}$

Let $p\geq 5$ be a prime and let $G=\operatorname{SL}(2,\mathbb{F}_{p})$ be the special linear group over the field with $p$ elements.

###### Lemma.

The group $G=\operatorname{SL}(2,\mathbb{F}_{p})$ has no non-trivial $1$ dimensional irreducible representations over $\mathbb{F}_{p}$.

###### Proof.

See an attached entry (http://planetmath.org/MathitSL2mathbbF_pHasNo1DimensionalIrreducibleRepresentationsOverMathbbF_p) for the proof. ∎

Next, we construct several irreducible representations for $G=\operatorname{SL}(2,\mathbb{F}_{p})$. For $0\leq m\leq p-1$, let $V_{m}$ be the $\mathbb{F}_{p}$ vector space of homogeneous polynomials of degree $m$ in the independent variables $x$ and $y$ (of course, for $m=0$, the representation is trivial). We give $V_{m}$ a structure of $\mathbb{F}_{p}[G]$-module as follows. Let $p=p((x,y))\in V_{m}$ and $A\in\operatorname{SL}(2,\mathbb{F}_{p})$. We define:

 $A\cdot p:=p(A\cdot(x,y)^{t})=p(a_{11}x+a_{12}y,a_{21}x+a_{22}y)$

where $t$ denotes transpose. The representations $V_{m}$ are, in a sense, all the irreducible representations of $G$.

###### Theorem.

For $0\leq m\leq p-1$, the representations $V_{m}$ are irreducible representations of dimension $m+1$ over $\mathbb{F}_{p}$. Furthermore, up to isomorphism, there are no other irreducible representations of $G$ over $\mathbb{F}_{p}$.

## References

• 1 Charles B. Thomas, Representations of Finite and Lie Groups, Imperial College Press, London.
Title irreducible representations of the special linear group over $\mathbb{F}_{p}$ IrreducibleRepresentationsOfTheSpecialLinearGroupOvermathbbFp 2013-03-22 15:09:53 2013-03-22 15:09:53 alozano (2414) alozano (2414) 5 alozano (2414) Theorem msc 20G15 GroupRepresentation SpinNetworksAndSpinFoams