Formulas for primitive idempotents in Frobenius algebras and an application to decomposition maps

In the first part of this paper we present explicit formulas for primitive idempotents in arbitrary Frobenius algebras using the entries of representing matrices coming from projective indecomposable modules with respect to a certain choice of basis. The proofs use a generalisation of the well-known Frobenius-Schur relations for semisimple algebras.

The second part of this paper considers $\mathcal{O}$-free $\mathcal{O}$-algebras of finite $\mathcal{O}$-rank over a discrete valuation ring $\mathcal{O}$ and their decomposition maps under modular reduction modulo the maximal ideal of $\mathcal{O}$, thereby studying the modular representation theory of such algebras.

Using the formulas from the first part we derive general criteria for such a decomposition map to be an isomorphism that preserves the classes of simple modules involving explicitly known matrix representations on projective indecomposable modules.

Finally, we show how this approach could eventually be used to attack a conjecture by Gordon James in the formulation of Meinolf Geck for Iwahori-Hecke algebras, provided the necessary matrix representations on projective indecomposable modules could be constructed explicitly.