Bar weights of bar partitions and spin character zeros

Abstract

The main combinatorial result in this article is a classification of bar partitions of n which are of maximal p-bar weight for all odd primes p ≤ n. As a consequence, we show that apart from very few exceptions any irreducible spin character of the double covers of the symmetric and alternating groups vanishes on some element of odd prime order.

Separating Characters by Blocks

“…Again, we note that the exceptions in Corollary 2.7 have already turned up in [2]. In the language here, we had classified in particular those irreducible characters of A n which cannot be separated from the trivial character by any prime p.…”

“…(ii) The exceptions in Corollary 2.3(2) have already turned up in [2] where we have classified partitions which are of maximal p-weight for all primes p. In the language here, this means that in particular we had determined all irreducible characters which cannot be separated from the trivial character.…”

“…Again, this also implies a corresponding result for the spin characters ofà n (see [19]). In [1] also the 2-block distribution of spin characters was determined; this needs the notion of the doubling of a bar partition: for a bar partition λ its doubling dbl(λ) is the partition obtained from λ by replacing each odd part 2t + 1 by t + 1, t and each even part 2t by t, t. Then by [1, Theorem 4.1] (see also [19,Theorem 13.20]), two irreducible spin characters ofS n labelled by λ and µ belong to the same 2-block if and only if dbl(λ) (2) = dbl(µ) (2) . Thus, checking the exceptions in Corollary 3.3, we obtain the following immediately.…”

“…For G of split orthogonal type D n , n 4, it was shown in [14, Section 3G] that two Zsigmondy primes l(2n − 2, q) and l(n, q) separate characters when n is odd, while for n even, l(2n − 2, q) and l(n − 1, q) only leave two unipotent characters of l(2n − 4, q)-defect zero, or G is the Atlas group G = O + 8 (2). This completes the proof of the theorem.…”