The Green ring and modular representations of finite groups of Lie type
For G a finite group with split BN-pair and \(U=O_ p(B)\) a Sylow p- subgroup of G, the permutation module \(Y=ind_{G,U}(k)\) where k is a sufficiently large field of characteristic p has first been studied by \textit{C. W. Curtis} and \textit{F. Richen} [Lect. Notes Math. 131, B1-B39 (1970; Zbl 0233.20005)]. They proved that the indecomposable summands \(Y_ i\) of Y are in one to one correspondence with the simple kG-modules and have simple head and socle as well. Later on \textit{H. Sawada} [Math. Z. 155, 29-41 (1977; Zbl 0345.20009)] and \textit{J. Green} [J. Lond. Math. Soc., II. Ser. 18, 247-252 (1978; Zbl 0403.16009)] obtained the results by use of the Hecke algebra \(End_{KG}(Y)\). \textit{N. B. Tinberg} then computed the dimensions and vertices of the \(Y_ i's\) [J. Algebra 61, 508-526 (1979; Zbl 0434.20002)] and Cabanes the Green correspondents which give rise to verify Alperin's weight conjecture for those groups. In this paper the author starts with the significant results of Curtis and Richen. Interpreting these in the Green ring and using ``Möbius inversion'' he obtains a formula for the \(Y_ i's\) from which the results of Tinberg and Cabanes follow in a quite elegant way. Moreover he discovers a natural duality on the set \(\{Y_ i\}\) which turns out to be an isometry on the subgroup of the Green ring which is spanned by the \(Y_ i's\).
scientific article; zbMATH DE number 4118578