Quasi-permutation representations of the groups \(\text{SL}(3,q)\) and \(\text{PSL}(3,q)\)
Let \(G\) be a finite group and let \(\chi\) be a complex character of \(G\). We say that \(\chi\) is a quasi-permutation character if \(\chi(g)\) is a non-negative rational integer for each element \(g\) in \(G\). Let \(c(G)\) denote the minimal degree of a faithful quasi-permutation character of \(G\). Let \(q(G)\) denote the minimal degree of a faithful quasi-permutation character \(\chi\) of \(G\), where \(\chi\) is the character of a representation of \(G\) defined over the field of rational numbers. Let \(p(G)\) denote the minimal degree of a faithful permutation representation of \(G\). Finally, let \(r(G)\) denote the minimal degree of a faithful rational-valued character of \(G\). The inequalities \(r(G)\leq c(G)\leq q(G)\leq p(G)\) hold. The authors determine the values of these invariants in the case that \(G\) is either the group \(\text{PSL}(3,q)\) or the group \(\text{SL}(3,q)\), where \(q\) is a power of a prime. Suppose that \(G\) is \(\text{PSL}(3,q)\). Then \(G\) is a finite simple group and it is straightforward to use the character table of \(G\) to prove that \(r(G)=q(q+1)\), \(c(G)=p(G)=q(G)=q^2+q+1\). Suppose now that \(G\) is \(\text{SL}(3,q)\). Only the case that \(q\equiv 1\bmod 3\) is of interest here, since otherwise \(G\) is isomorphic to \(\text{PSL}(3,q)\). We will not describe the authors' findings, as they are more complicated to state and depend upon the power of 3 that divides \(q-1\). We note here an error in part (c) of the main theorem. In the case that \(G\) is \(\text{SL}(3,4)\), it is stated that \(c(G)=q(G)=45\). The correct value is \(42\). The same error occurs on p.~149 of the paper. The value for \(p(G)\) is \(63\). We also note that in part (b) of the main theorem, in the second case of the enunciation of the value of \(r(G)\), it should say \(q\not\equiv 1\bmod 9\).
scientific article; zbMATH DE number 1808545