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

TL;DR

This paper presents a new inversion formula for primitive idempotents in the trivial source algebra, expressing them as linear combinations of a canonical basis, involving all irreducible characters of normalizers of p-subgroups.

Contribution

It introduces an alternative inversion formula for primitive idempotents in the trivial source algebra, extending previous formulas and connecting to the Burnside algebra and Brauer characters.

Findings

New inversion formula for primitive idempotents

Expression of linearization map matrix entries in terms of Brauer characters

Connection to reduced Euler characteristics of posets

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.