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.

Over a splitting field for finite group representations of characteristic $p$, the irreducible Brauer characters form a complex basis of the class functions on the p-regular elements. Independence is the character form of the Brauer–Nesbitt theorem. To obtain spanning, extend such a class function by zero on the p-singular classes. Ordinary irreducible characters form a basis of all class functions by character orthogonality. Restricting them to p-regular elements yields Brauer characters of reductions of an integral form of a group representation in a compatible splitting p-modular system; if needed, first extend scalars, which does not change the simple-module list under the splitting hypothesis. Each restriction is a nonnegative integral sum of simple Brauer characters by exact-sequence additivity and the Jordan–Hölder theorem. Thus these restrictions span, proving the assertion. Consequently the number of simple modules equals the number of p-regular conjugacy classes, and evaluation identifies the complexified modular representation ring with the product of one copy of $\mathbb{C}$ for each such class.

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 $S$.