PROJECTIVE ESSENTIAL IDEMPOTENTS

This paper introduces the concept of projective essential idempotents. These are primitive central idempotents in a twisted group algebra. The first main result provides conditions for the existence of them. In the second main result, we prove that every $q$-ary simplex code can be seen as an ideal of a twisted group algebra generated by a projective essential idempotent. Conversely, we show that every projective essential idempotent in a twisted group algebra of a cyclic group of order $n=(q^k-1)/(q-1)$ over $\F$ generates a $q$-ary simplex code.