Irreducible maps and bilinear forms

Bautista showed in 1982 that the possible multiplicities of indecomposable summands of the domains and ranges of previous termirreduciblenext termprevious termmapsnext term between modules over artin algebras are given by numerical invariants of certain previous termbilinearnext termprevious termformsnext term associated with the algebra. We obtain further information about these multiplicities by relating the previous termformsnext term to those studied elsewhere in algebra and geometry. One spectacular result is that the allowable multiplicities for some algebras over the field of real numbers depend on J.F. Adams’ determination of the number of linearly independent vector fields on a sphere.