On symmetric quotients of symmetric algebras

Abstract

We investigate symmetric quotient algebras of symmetric algebras, with an emphasis on finite group algebras over a complete discrete valuation ring ${\mathcal O}$. Using elementary methods, we show that if an ordinary irreducible character $\chi$ of a finite group $G$ gives rise to a symmetric quotient over ${\mathcal O}$ which is not a matrix algebra, then the decomposition numbers of the row labelled by $\chi$ are all divisible by the characteristic $p$ of the residue field of ${\mathcal O}$.