irreducible representations of the special linear group over ${𝔽}_p$

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

Lemma.

The group $G\mathrm{=}\mathrm{SL}\mathit{}\mathrm{(}\mathrm{2}\mathrm{,}{\mathrm{F}}_p\mathrm{)}$ has no non-trivial $\mathrm{1}$ dimensional irreducible representations over ${\mathrm{F}}_p$.

Proof.

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

Next, we construct several irreducible representations for $G=\mathrm{SL}(2,{𝔽}_p)$. For $0\le m\le p-1$, let $V_m$ be the ${𝔽}_p$ 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 ${𝔽}_p[G]$-module as follows. Let $p=p((x,y))\in V_m$ and $A\in \mathrm{SL}(2,{𝔽}_p)$. 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 $\mathrm{0}\mathrm{\le }m\mathrm{\le }p\mathrm{-}\mathrm{1}$, the representations $V_m$ are irreducible representations of dimension $m\mathrm{+}\mathrm{1}$ over ${\mathrm{F}}_p$. Furthermore, up to isomorphism, there are no other irreducible representations of $G$ over ${\mathrm{F}}_p$.