  Formulas for the primitive idempotents of the trivial source algebra, in
characteristic zero, have been given by Boltje and Bouc--Th\'{e}venaz. We shall
give another formula for those idempotents, expressing them as linear
combinations of the elements of a canonical basis for the integral ring. The
formula is an inversion formula analogous to the Gluck--Yoshida formula for the
primitive idempotents of the Burnside algebra. It involves all the irreducible
characters of all the normalizers of $p$-subgroups. As a corollary, we shall
show that the linearization map from the monomial Burnside ring has a matrix
whose entries can be expressed in terms of the above Brauer characters and some
reduced Euler characteristics of posets.
