Modular Representation Theory - Some Orthogonality Relations For Brauer Characters

Some Orthogonality Relations For Brauer Characters

When a projective module is lifted, the associated character vanishes on all elements of order divisible by p, and (with consistent choice of roots of unity), agrees with the Brauer character of the original characteristic p module on p-regular elements. The (usual character-ring) inner product of the Brauer character of a projective indecomposable with any other Brauer character can thus be defined: this is 0 if the second Brauer character is that of the socle of a non-isomorphic projective indecomposable, and 1 if the second Brauer character is that of its own socle. The multiplicity of an ordinary irreducible character in the character of the lift of a projective indecomposable is equal to the number of occurrences of the Brauer character of the socle of the projective indecomposable when the restriction of the ordinary character to p-regular elements is expressed as a sum of irreducible Brauer characters.