The Cartan ring of a finite group
Let \(G\) be a finite group, and \(F\) be an algebraically closed field of prime characteristic \(p>0\). Denote by \(G_{0}(FG)\) the Grothendieck ring of the group algebra \(FG\) and by \(K_{0}(FG)\) the ideal of \(G_{0}(FG)\) generated by finitely projective modules. The Cartan ring of \(G\), \(Cart(FG)\), is defined as the quotient \(G_{0}(FG)/K_{0}(FG)\). As an abelian group, it is known to be a \(p\)-group [Theorem 3.6.32 in \textit{H. Nagao} and \textit{Y. Tsushima}, ``Representations of finite groups'', Academic Press, Boston (1989; Zbl 0673.20002)]. The authors describe in this paper the {ring} structure of \(Cart(FG)\). This goes as follows: let \(\overline{\mathbb{Q}}_{p}\) be an algebraic closure of the \(p\)-adic rationals, and \(\mathbb{Z}_{p}\) denote the \(p\)-adic integers. Moreover, denote by \(\xi\) a \(| G| _{p'}\)-primitive root of unity in \(\overline{\mathbb{Q}}_{p}\), where \(| G| _{p'}\) denotes the \(p'\)-part of the order of \(G\), and \(G_{p'}\) denotes the set of \(p'\)-elements of \(G\). Finally, let \(IBr(G)\) be the set of irreducible Brauer characters \(G_{p'}\to \mathbb{Z}_{p}\) and \(RIBr(G)\) stand for the set of \(R\)-linear combinations of \(IBr(G)\) for any subring \(R\subset\overline{\mathbb{Q}}_{p}\). Now, let \(\Gamma\) be the Galois group of the Galois extension \(\mathbb{Q}_{p}[\xi]| \mathbb{Q}_{p}\), the authors observe that there is a natural action of \(G\times \Gamma\) on \(G_{p'}\), let \(G_{p'}/\!\!\sim_{G\times\Gamma}\) be the orbit space under this action, and \(m_s=| \Gamma :\Gamma_{s}| \) for each \(s\in G_{p'}\). The main theorem is the following: {For \(s\in G_{s}\), let \(R_{s}\) be the ring of integers in the unique unramified extension \(K_{s}\) of \(\mathbb{Q}_{p}\) of degree \(m_{s}\). The Cartan ring of \(G\) is isomorphic as a ring to \[ \prod_{s\in G_{p'}/\sim_{G\times\Gamma}}R_{s}/| C_{G}(s)| R_{s}, \] where \[ R_{s}=\mathbb{Z}_{p}[\phi(s):\phi\in IBr(G)]. \] } As a consequence of the above, the authors give a more precise description of the Cartan ring of \(G\): Corollary 3. Let \(s_{1},\dots s_{r}\in G_{p'}\) be representatives of the \(G\times\Gamma\)-orbits of elements in \(G_{p'}\) which lie in a conjugacy class of positive defect, let \(d_{1},\dots ,d_{r}\) be these corresponding defects, and \(m_i=| \Gamma :\Gamma_{s_{i}}| \). Then, (1) The Cartan ring of \(G\) is the direct product of uniserial rings \(A_{1},\dots ,A_{r}\) each of Loewy length \(d_{i}\). (2) The Jacobson radical of the Cartan ring of \(G\) is \(pCart(G)\) and the quotient is the direct product of \(r\) finite fields of \(p^{m_{i}}\) elements. (3) The Cartan ring of \(G\) has precisely \((d_{1}+1)\cdots (d_{r}+1)\) ideals.
scientific article; zbMATH DE number 2153363