On the lifting problem of representations of a metacyclic group (Q6612184)

Let \(p\) be a prime and let \(Q\) be a cyclic group of order \(q := p^h\) for some positive integer \(h\). Further, let \(C\) be a cyclic group of order \(m\) acting on \(Q\), where \(m\) is coprime to \(p\).\N\NThe authors investigate the problem of lifting representations of the metacyclic group \(G = Q \rtimes C\) from characteristic \(p\) to characteristic \(0\). More precisely, let~\(k\) be an algebraically closed field of characteristic \(p\) and let \(R\) be a local integral domain of characteristic \(0\) with residue class field \(k\), and which contains a \(q\)th root of unity. Given a finite dimensional \(k[G]\)-module \(M\), does there exist an \(R[G]\)-module~\(V\), free as \(R\)-module, such that \(k \otimes_R V \cong M\) as \(k[G]\)-modules?\N\NThe main theorem describes the isomorphism types of the \(k[G]\)-modules \(M\) which allow such a lift. The simple \(k[G]\)-modules are \(1\)-dimensional and are the inflations of the simple \(k[C]\)-modules \(\epsilon^0, \epsilon, \ldots , \epsilon^{m-1}\) for some generator \(\epsilon\) of the \(k\)-character group of \(C\). Every finite dimensional \(k[G]\)-module \(M\) is of the form \(M = M_1 \oplus \cdots \oplus M_t\) for indecomposable \(k[G]\)-modules \(M_1, \ldots , M_t\). There are \(qm\) such indecomposable \(k[G]\)-modules up to isomorphism, and they are uniserial; their isomorphism type is determined by their socle and their dimension, which varies between \(1\) and \(q\). Let us write \(V( \ell, \kappa )\) for the indecomposable \(k[G]\)-module with socle \(\epsilon^{\ell}\) and dimension \(\kappa\).\N\NWe can now present the main theorem of the paper. Let \(m'\) denote the order of the kernel of the action of \(C\) on \(Q\). Let \(M\) be an indecomposable \(k[G]\)-module. Then \(M\) lifts to an indecomposable \(R[G]\)-module, which is free as \(R\)-module, if and only if the following three conditions are satisfied:\N\N\begin{itemize}\N\item[a.] We have \(\dim(M) \leq q\);\N\item[b.] We have \(\dim(M) \equiv a\,(\mbox{mod}\,m/m')\) with \(a \in \{ 0, 1 \}\);\N\item[c.] There is a decomposition \(M = V( \ell_1, \kappa_1 ) \oplus \cdots \oplus V( \ell_t, \kappa_t )\) such that \(\ell_{j+1}\) depends in a specific way on \(\ell_{j}\) and \(\kappa_j\) for \(1 \leq j < t\).\N\end{itemize}\NNotice the typographical error in condition b of Theorem 1, where the congruence is taken modulo \(m\). Also, in view of Definition 9 and Proposition 38, the condition c of Theorem 1 cannot be accurate as stated. Presumably the correct condition is \N\[\N\ell_{j+1} = (\ell_{j}+m'{\kappa_j})\!\mod\,m\N\]\Nfor \(1 \leq j < t\). The reviewer feels that a corrigendum would be appropriate.\N\NThe proofs are elementary. The authors start with the presentation \[G = \langle \tau, \sigma \mid \tau^q, \sigma^m, \sigma \tau \sigma^{-1} = \tau^{\alpha} \rangle\] for some integer \(\alpha\) coprime to~\(p\) with \(1 \leq \alpha \leq q - 1\). They then use liner algebra and an explicit description of the actions of~\(\tau\) and~\(\sigma\) on the modules \(V( \ell, \kappa )\) to construct two matrices \(T\) and \(S\) over \(R\) satisfying the above defining relations.\N\NThe authors proceed in the utmost generality, where the quantities studied depend on the parameter \(\alpha\), and where \(m\) is arbitrary. In the opinion of the reviewer, the proofs could be simplified by considering the \(p\)-blocks of \(k[G]\), as the lifting problem can be studied block-wise. Then, starting with the principal block, one could assume that \(m' = 1\), i.e.\ that the action of~\(C\) on~\(Q\) is faithful. The passage to arbitrary~\(m\) and other blocks could then be achieved by translating the results for the principal block through the multiplication with a linear \(R\)-character of~\(G\).