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Formulas for the primitive idempotents of the trivial source algebra, incharacteristic zero, have been given by Boltje and Bouc--Th\'{e}venaz. We shallgive another formula for those idempotents, expressing them as linearcombinations of the elements of a canonical basis for the integral ring. Theformula is an inversion formula analogous to the Gluck--Yoshida formula for theprimitive idempotents of the Burnside algebra. It involves all the irreduciblecharacters of all the normalizers of $p$-subgroups. As a corollary, we shallshow that the linearization map from the monomial Burnside ring has a matrixwhose entries can be expressed in terms of the above Brauer characters and somereduced Euler characteristics of posets.

An inversion formula for the primitive idempotents of the trivial source algebra