Generators for the centre of the group algebra of a symmetric group.
It has been known since the early work of Frobenius that the structure of the centre of the integral group algebra \(\mathbb{Z} G\) of a finite group \(G\) provides delicate information about \(G\) and its representation theory. The first main manifestation of the importance of this commutative algebra is its role in the proof of the integrality of the central characters on the class sums, a key theorem in all applications of group character theory. Brauer's theory of \(p\)-blocks sought to investigate the centre of the group algebra over a field of characteristic \(p\) and thereby deduce results about the structure of \(G\) and its representations in characteristic 0. This theory also exploits the integrality of the central characters. In the paper under review, the author takes \(G\) to be the symmetric group \(S_n\) of degree \(n\) and investigates generators for the centre of both \(\mathbb{Z} S_n\) and \(\mathbb{F} S_n\), where \(\mathbb{F}\) is a field of characteristic 2. Using results of Farahat and Higman on the centre of \(\mathbb{Z} S_n\), the author shows that the centre of \(\mathbb{F} S_n\) is generated by class sums of elements of 2-power order. He shows that this has as a consequence the fact that, given a 2-subgroup \(D\) of \(S_n\), there is at most one 2-block of \(\mathbb{F} S_n\) with \(D\) as defect group. While this is a known consequence of the Nakayama conjecture, the proof given is a pleasant consequence of the author's investigation of how the centre of the modular group algebra can be generated in unexpected ways, and is especially interesting in that there can be no analogue of this result for odd primes. This is by no means the main theorem of this wide-ranging paper, but we will not attempt to give a detailed description of its content. We note the appearance of the Jucys-Murphy elements, which have achieved prominence in the representation theory of \(S_n\) in the past twenty years, and also of the Catalan numbers \(C_n\), which the author proves to be odd if and only if \(n+1\) is a power of 2 (a result used elsewhere, for instance in the geometry of the Grassmannian variety).
scientific article; zbMATH DE number 2037109
scientific article