The integral structure of Hecke algebras for finite generalized polygons

Abstract

Suppose (P, B, F) are the points, blocks and flags of generalized m-gon and H(F) the associated rank 2 Iwahori-Hecke algebra. H(F) acts naturally on the integral standard module ZF based on F. This work gives arithmetic conditions on subring R, where R contains the integers and is contained in the rationals, that insure the associated R-ary Iwahori-Hecke algebra is completely reducible on RF. The constituent multiplicities are related to the R-normal form of the incidence matrix of (P, B, F).