Some combinatorial results involving Young diagrams

Abstract

1. Introduction. In the first half of this paper we introduce a new method of examining the g-hook structure of a Young diagram, and use it to prove most of the standard results about <?-cores and ^-quotients. In particular, we give a quick new proof of Chung's Conjecture (2), which determines the number of diagrams with a given g-weight and says how many of them are ^-regular. In the case where q is prime, this tells us how many ordinary and g-modular irreducible representations of the symmetric group <& n there are in a given g-block. None of the results of section 2 is original. In the next section we give a new definition, the ^>-power diagram, which is closely connected with the ^-quotient. This concept is interesting because when p is prime a condition involving the p-power diagram appears to be a necessary and sufficient criterion for the diagram to be ^-regular and the corresponding ordinary irreducible representation of @ n to remain irreducible modulo p. In this paper we derive combinatorial results involving the ^>-power diagram, and in a later article we investigate the relevant representation theory.

Cubist algebras

“…We describe a method due to Gordon James [16] for representing partitions which is useful in this context. We consider an abacus with p vertical half-infinite runners, with positions labelled 0, 1, .…”

Rouquier blocks

“…. ) be a partition of n. It is useful to represent λ on an abacus with e runners; see [14]. Choose an integer b greater than the number of non-zero parts of λ and for 1 ≤ i ≤ b, let β i = λ i + b − i.…”

“…Take b = 14. The beta numbers for λ are (23,17,16,14,11,10,8,7,5,4, 3, 2, 1, 0) and the beta numbers for µ are (23,19,17,14,11,10,8,7,5,4, 3, 2, 1, 0). The abacus configurations for λ and µ are given below, as well as the abacus configuration for the 3-core ρ which is the 3-core of both λ and µ.…”

Symmetric group blocks of small defect

“…By r-restricting D(µ) k times, we obtain an S n−k -module which contains D(µ) as a submodule. If λ has fewer than k removable r-nodes then by r-restricting S(λ) k times we obtain the zero module, so We now recall the notion of an abacus from [6]. A p-abacus has p runners, which we label as runner 1 to runner p, reading from left to right.…”