Brauer Character

A Brauer character of a finite-dimensional representation of a finite group over an algebraically closed field of characteristic is the characteristic-zero trace function defined on the elements whose orders are prime to . For each such element, its eigenvalues are lifted to the corresponding prime-to- roots of unity in characteristic zero and then added. The choice of compatible lifting is fixed throughout. The irreducible Brauer characters correspond to the simple modules, and their number in a block is denoted by . For a finite -group there is only one irreducible Brauer character, with value 1 on the identity. The blockwise Alperin weight conjecture compares with a count of weights.