Characters of permutation summands
A permutation lattice for a finite group \(G\) over the ring \(A\) of integers in a number field \(K\) is a free \(A\)-module with a finite \(A\)-basis which is permuted by \(G\). Direct summands of these, as \(AG\)-modules, are called permutation summands for \(G\) over \(A\). Characters of permutation summands of \(G\) over \(A\) are always \(\mathbb{Q}\)-valued. The Grothendieck ring of the category of all permutation summands for \(G\) over \(A\) is denoted by \(\Omega_A(G)\). The author studies the image of the map \(\phi\colon\Omega_A(G)\to\overline R_\mathbb{Q}(G)\) which sends a lattice \(L\) to the \(K\)-character \(\phi_L\) of \(K\otimes_AL\) in the ring of \(\mathbb{Q}\)-valued characters of \(G\). The image of \(\phi\) always has finite index in \(\overline R_\mathbb{Q}(G)\) and grows with \(A\). Let \(\mathbb{Q}_p^{\text{nr}}\) be the maximal unramified extension of the \(p\)-adic complete field \(\mathbb{Q}_p\). A character of \(G\) is \(p'\)-linear if it is one-dimensional with values roots of unity of order prime to \(p\). Let \(R_{\mathbb{Q}_2^{\text{nr}}}(G)\) be the ring of characters of \(\mathbb{Q}_2^{\text{nr}}\)-representations of \(G\), \({\mathcal R}(G)\) be the quotient of \(R_{\mathbb{Q}_2^{\text{nr}}}(G)\) by the subgroup generated by induced characters \(\text{ind}_H^G\phi\) of \(2'\)-linear characters \(\phi\). The author shows that the image of \(\phi\colon\Omega_A(G)\to\overline R_\mathbb{Q}(G)\) is always contained in the kernel of the composite map \(\overline R_\mathbb{Q}(G)\to R_{\mathbb{Q}_2^{\text{nr}}}(G)\to{\mathcal R}(G)\), and if \(A\) is big enough this containment is an equality.
scientific article; zbMATH DE number 1308226
scientific article