All-involution table algebras and finite projective spaces

摘要

Table algebras all of whose nonidentity basis elements are involutions (in the sense of Zieschang), which serve as a counterpoint to the generic Hecke algebras parametrized by Coxeter groups, are classified. If two-generated, they are the family H (n) (for all n a parts per thousand yen 3), which for suitable n arise from schemes defined by affine planes of order n - 1. Otherwise, the basis involutions correspond to the points of a finite projective space whose incidence geometry determines the algebra multiplication. This generalizes to table algebras a previous result of van Dam for association schemes. An algebraic characterization is also given.