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arXiv · 1809.10984

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

Laurence Barker
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Abstract

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.

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Laurence Barker. 2018-09-28. An inversion formula for the primitive idempotents of the trivial source algebra. https://arxiv.org/abs/1809.10984

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