Lambda and psi operations on Green rings
A(G) denotes the Green ring formed from (\({\mathbb{C}}\)-combinations of) finite-dimensional (right) kG-modules, where k is a field of characteristic p and G is a finite group. A species is a \({\mathbb{C}}\)- algebra epimorphism \(s:A(G)\to {\mathbb{C}},\quad x\mapsto(s,x).\) For instance if g is a p'-element of G, the evaluation map \(b_ g\) of the Brauer character value \((b_ g,M)\) for each module M on the element g is a species. For each \(n\in {\mathbb{N}}\), a \({\mathbb{C}}\)-algebra endomorphism \(\psi^ n\) of A(G) is constructed; these \(\psi^ n\) are analogous to Adams operators defined in other contexts; they satisfy \((b_{g^ n},x)=(b_ g,\psi^ n(x))\) for Brauer species \(b_ g\) and \(b_{g^ n}.\) More generally the power \(s^ n\) (again a species) of s is defined as \((s^ n,n)= (s,\psi^ n(x)). \lambda^ n(x)\) is also defined in terms of \(\psi\) and for \(n<p\), \(\lambda^ n(x)\) is the exterior nth power of x. This makes A(G) into a special \(\lambda\)-ring (coefficients of A(G) taken in \({\mathbb{Z}}[1/p])\). An examination of the vertices and origins of powers of a species is given.
scientific article; zbMATH DE number 3857358
scientific article