irreducible representations of the special linear group over 𝔽p

Let p5 be a prime and let G=SL(2,𝔽p) be the special linear groupMathworldPlanetmath over the field with p elements.


The group G=SL(2,Fp) has no non-trivial 1 dimensional irreducible representations over Fp.


See an attached entry ( for the proof. ∎

Next, we construct several irreducible representations for G=SL(2,𝔽p). For 0mp-1, let Vm be the 𝔽p vector space of homogeneous polynomialsMathworldPlanetmath of degree m in the independent variables x and y (of course, for m=0, the representation is trivial). We give Vm a structure of 𝔽p[G]-module as follows. Let p=p((x,y))Vm and ASL(2,𝔽p). We define:


where t denotes transpose. The representations Vm are, in a sense, all the irreducible representations of G.


For 0mp-1, the representations Vm are irreducible representations of dimensionMathworldPlanetmath m+1 over Fp. Furthermore, up to isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, there are no other irreducible representations of G over Fp.


  • 1 Charles B. Thomas, Representations of Finite and Lie Groups, Imperial College Press, London.
Title irreducible representations of the special linear group over 𝔽p
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