The Schur indices of the irreducible characters of the special linear groups
The author provides the definitive solution to the problem of computing the Schur index over the field of rational numbers of any irreducible complex character of the finite special linear group \(\text{SL}_n(q)\), where \(q\) is a power of the prime \(p\). The reviewer has worked on this problem and obtained partial information, with complete solutions in some cases. Turull's solution shows that the outstanding cases not considered by the reviewer are substantially more complicated than might have been anticipated. A brief history of the problem is as follows. Let \(G\) denote the general linear group \(\text{GL}_n(q)\) and let \(U\) denote a Sylow \(p\)-subgroup of \(G\). Let \(H\) denote \(\text{SL}_n(q)\). \(H\) is a normal subgroup of \(G\) and the quotient group \(G/H\) is cyclic of order \(q-1\). Let \(\chi\) be an irreducible complex character of \(G\). \textit{A.~V.~Zelevinsky} showed [in Representations of finite classical groups, Lect. Notes Math. 869, Springer, Berlin (1981; Zbl 0465.20009)] that the restriction of \(\chi\) to \(U\) contains some complex linear character of \(U\) (depending on \(\chi\)) with multiplicity 1. Given the structure of the normalizer of \(U\) in \(G\), this is sufficient to prove that \(\chi\) has Schur index 1 over the rationals. While Zelevinsky's theorem also applies to the irreducible characters of \(H\) (note that \(H\) contains the subgroup \(U\)), this does not imply that all characters of \(H\) have Schur index 1, since the normalizer of \(U\) in \(H\) does not necessarily have the required structure. Nevertheless, one can prove without too much extra work that the Schur index of an irreducible complex character of \(H\) is 1 if either \(q\) is a power of 2, or \(n\) is odd, or if the power of 2 dividing \(n\) is greater than the power of 2 dividing \(p-1\). On the other hand, while the Schur index is always at most 2, examples exist where it is exactly 2. In previous papers [in J. Algebra 227, No. 1, 124-132, 133-158 (2000; Zbl 0976.20009, Zbl 0959.20010)], Turull has developed generalizations of the theory of the Brauer group to exploit both the cyclic quotient group structure and Zelevinsky's multiplicity one theorem in the context of the special linear group. He uses this background work to good effect in the current paper. Rather than attempting to give an exhaustive account of all his main results, we will describe some of the highlights. In Theorem 5.9, he gives a method to decide whether the Schur index of a given character of \(H\) (described in terms of a parametrization) is 1 or 2. He shows as a consequence that, as we vary \(n\) and \(q\), there is no upper bound for the number of finite primes at which the local Schur index of an irreducible character of \(H\) is 2. This even holds good for those characters of \(H\) which are trivial on the center of \(H\) and may thus be identified with characters of the quotient group \(\text{PSL}_n(q)\). Previously, it was not even known whether all Schur indices for this group were equal to 1.
scientific article; zbMATH DE number 1563183
scientific article