On trace functions
Let \(A\) be a finite-dimensional algebra over a field \(K\) of characteristic \(p\neq 0\). A trace function on \(A\) is a \(K\)-linear map \(\tau:A\to K\) satisfying \(\tau(ab)=\tau(ba)\) and \(\tau(a^p)=\tau(a)^p\) for all \(a,b\in A\). The trace functions on \(A\) form a vector space \(T_K(A)\) over the prime subfield \(K_1\) of \(K\). In case \(A/J(A)\) is separable the dimension of \(T_K(A)\) coincides with the number of isomorphism classes of simple \(A\)-modules. The authors use \(T_K(A)\) in order to give elementary proofs of some of the basic results on finite group representations in characteristic \(p\): \(KG/J(KG)\) is separable for a finite group \(G\), the characters of the simple \(KG\)-modules form a basis for \(T_K(KG)\), every simple \(KG\)- module can be realized over the subfield containing all character values. There are also modular versions of Brauer's induction theorem and of Brauer's characterization of characters: A class function \(\chi:G\to K\) is a character if and only if its restriction to each cyclic subgroup is a character.
scientific article; zbMATH DE number 734557