On symmetric quotients of symmetric algebras

We investigate symmetric quotient algebras of symmetric algebras,

with an emphasis on finite group algebras over a complete discrete

valuation ring R with residue field of positive characteristic p. Using elementary methods, we show that if an

ordinary irreducible character of a finite group gives

rise to a symmetric quotient over R which is not a matrix algebra,

then the decomposition numbers of the row labelled by the character are

all divisible by p. In a different direction, we show that if is P is a finite

p-group with a cyclic normal subgroup of index p, then every ordinary irreducible character of P gives rise to a

symmetric quotient of RP. This is joint work with Shigeo Koshitani and Markus Linckelmann.