Cores and quotients of partitions through the Frobenius symbol

Abstract

The Frobenius symbol was first introduced in 1900 by Frobenius as a way to encode an integer partition. In 1941, motivated by the modular representation theory of the symmetric group, Nakayama introduced the idea of a p-core partition, for p prime, using hook removals. In the following decade, Robinson, Littlewood, Staal and Farahat codified the $p$-quotient of such a partition, using variations of the star diagram. Since the 1970s, the convention has been to build up the theory of both core and quotients with the abacus construction, first introduced by G. James. In this paper we return to the earlier point of view. First we show that, for any positive integer t, the t-core and t-quotient of an integer partition can be directly obtained from its Frobenius symbol. The argument also works in the opposite direction: that is, given the Frobenius symbol of a t-core and a $t$-tuple of Frobenius symbols, one can recover the Frobenius symbol of the corresponding partition. One immediate application is the calculation of the Durfee number of the associated partition from the Frobenius symbols of the core and quotient. In 1991, J. Scopes gathered together the $p$-core partitions into families to prove that Donovan's conjecture holds for the symmetric groups at the prime p. We describe, using our methods, the action of the affine Weyl group W_p of type $A$ on Frobenius symbols, and use this to parametrize and compute the explicit number of Scopes families. In particular we enumerate both the infinite and finite Scopes families. Core partitions have also attracted recent interest in number theory. By constructing explicit and combinatorial bijections, we revisit some well-known identities originally obtained using sophisticated methods. We end with a close study of the relationship between certain hooks in the quotient and certain hooks in the associated partition.