Bilinear forms on G-algebras
Given a G-algebra A (in the sense of \textit{J. A. Green} [Math. Z. 107, 133-150 (1968; Zbl 0164.340)]), with a G-invariant linear form the authors construct bilinear forms on the ideal \(Tr^ G_ P(A^ P)\) of \(A^ G\), for each p-subgroup P of G. The rank of this form is computed, and special cases then yield information about the number of blocks with given defect group and the multiplicities of Scott modules as direct summands of modules of the form \(M\otimes M^*\) (cf., \textit{D. J. Benson} and \textit{J. F. Carlson} [J. Algebra 104, 329-350 (1986; Zbl 0612.20005)]). An extension of the multiplicity formula to the context of pointed groups is presented; it was communicated to the authors by L. Puig. A second part of the paper, nearly independent of the first part, discusses the connections between Hecke algebras and group algebras, with applications to fusion and the structure of permutation modules. At the end, the results are linked with those in the first part to obtain a new proof of the main result of \textit{G. R. Robinson}'s earlier paper [J. Algebra 84, 493-502 (1983; Zbl 0519.20011)] on the number of blocks with given defect group.
scientific article; zbMATH DE number 3991756
scientific article