Duality of simple and projective Brauer characters

If

S_{i}

are the simple modules of a split group algebra and

P_{i}

are their projective covers, then

⟨ χ_{P_{i}}, χ_{S_{j}} ⟩_{p^{'}} = δ_{ij} .

Table of contents
- Column orthogonality for Brauer characters Duality of simple and projective Brauer characters

Column orthogonality for Brauer characters

For p-regular elements

g, h

,

\sum_{S} χ_{S} (g^{- 1}) χ_{P_{S}} (h) = {∣ C_{G} (g) ∣ 0 g and h are conjugate, otherwise,

where

S

ranges over the simple modules and

P_{S}

is the projective cover of

S

.

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 2 / c / Solution

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