The Schur index of projective characters of symmetric and alternating groups.

*(English)*Zbl 0792.20015As the title suggests, the author gives an explicit rule for determining the Schur index of every irreducible projective character of the symmetric and alternating groups, or equivalently, for every irreducible character of the representation groups of these groups. The results for the ‘ordinary’ characters are well known; in that case, the Schur index is always one. This paper solves the problem for the two-valued or ‘spin’ characters where, as usual, the problem is far more complex and interesting. In an earlier paper [J. Lond. Math. Soc., II. Ser. 35, 421- 432 (1987; Zbl 0613.20006)], the author has already shown that the Schur index is always one or two and also has dealt with the exceptional cases, \(A_ 6\) and \(A_ 7\) where he shows that it is one for all characters. As usual, it is a difficult problem to determine whether the answer is one or two. Thus, it is an achievement in this case to provide a complete answer and to calculate the Schur index of each irreducible spin character through a precise and simple combinatorial algorithm.

The Schur index is the rank of a division algebra associated with each character; here, a rule is given to calculate this algebra from the strict partition associated with the character. A natural group structure is attached to the set of classes of partitions yielding the same algebras. It is interesting that this group structure is closely related to the Brauer group and the Brauer-Wall group.

The Schur index is the rank of a division algebra associated with each character; here, a rule is given to calculate this algebra from the strict partition associated with the character. A natural group structure is attached to the set of classes of partitions yielding the same algebras. It is interesting that this group structure is closely related to the Brauer group and the Brauer-Wall group.

Reviewer: A.O.Morris (Aberystwyth)

##### MSC:

20C30 | Representations of finite symmetric groups |

20C25 | Projective representations and multipliers |