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

Let $p\ge 5$ be a prime and let $G=\mathrm{SL}(2,{\mathbb{F}}_{p})$ 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 $G=\mathrm{SL}(2,{\mathbb{F}}_{p})$. For $0\le m\le p-1$, 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 $A\in \mathrm{SL}(2,{\mathbb{F}}_{p})$. 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 $\mathrm{0}\mathrm{\le}m\mathrm{\le}p\mathrm{-}\mathrm{1}$, 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 ${\mathrm{F}}_{p}$.

## References

- 1 Charles B. Thomas, Representations of Finite and Lie Groups, Imperial College Press, London.

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

Canonical name | IrreducibleRepresentationsOfTheSpecialLinearGroupOvermathbbFp |

Date of creation | 2013-03-22 15:09:53 |

Last modified on | 2013-03-22 15:09:53 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 5 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 20G15 |

Related topic | GroupRepresentation |

Related topic | SpinNetworksAndSpinFoams |