On certain quotients of the Green rings of dihedral 2-groups.
Let \(k\) be an algebraically closed field of characteristic 2, let \(G\) be a dihedral 2-group such that \(|G|\geq 8\), and let \(A(kG)\) be the Green ring of the group algebra \(kG\). By a theorem of Benson and Carlson, the isomorphism classes of the even-dimensional indecomposable \(kG\)-modules span an ideal \(A(kG;2)\) of \(A(kG)\), and 0 is the only nilpotent element in \(\overline A(kG):=A(kG)/A(kG;2)\). The author shows that the images of the isomorphism classes of the odd-dimensional indecomposable \(kG\)-modules in \(\overline A(kG)\) form an Abelian group \(\Gamma\) which is torsion-free and infinitely generated. The precise nature of \(\Gamma\) remains open. The author shows, however, that the multiplication in \(\Gamma\) induces graph automorphisms between components of the Auslander-Reiten quiver of \(kG\).
scientific article; zbMATH DE number 5285326
scientific article