Brauer character

Let $M$ be a representation over a splitting field $k$ of characteristic $p$. For a p-regular element $g$, the eigenvalues of $g$ on $M$ have order prime to $p$. Lift them to characteristic-zero roots of unity by the Teichmuller lift; their sum is the Brauer character value ${\chi }_M(g)$.

The Brauer character of a finite-dimensional modular representation determines its semisimplification. More strongly, the irreducible Brauer characters are linearly independent as complex-valued functions on the p-regular conjugacy classes.

The Brauer character of a projective module over a group algebra is called a projective character. It is the restriction to p-regular elements of the ordinary character of a lifted projective lattice; that ordinary character vanishes on p-singular elements.

For class functions on the p-regular elements of a finite group,Equivalently, summing over p-regular conjugacy-class representatives $x$ gives weights $1/|C_G(x)|$.

If $S_i$ are the simple modules of a split group algebra and $P_i$ are their projective covers, then

For p-regular elements $g,h$,where $S$ ranges over the simple modules and $P_S$ is the projective cover of $S$.