A refinement of a partition identity and blocks of some modular characters
type: Item
instance of: scholarly article
title: A refinement of a partition identity and blocks of some modular characters
zbMATH Open document ID: 0864.20005
DOI: 10.1007/BF01273340
author: George E. Andrews
author: Christine Bessenrodt
author: Jørn B. Olsson
published in: Archiv der Mathematik
publication date: 1996-11-22T00:00:00Z
review text: In an earlier paper [Trans. Am. Math. Soc. 344, No. 2, 597-615 (1994; Zbl 0806.05065)], the authors have collaborated in a remarkable way to prove a partition theoretic result which simultaneously settles a recent conjecture by the last two authors and the reviewer [J. Algebra 164, No. 1, 146-172 (1994; Zbl 0835.20018)] concerning labels of irreducible modular characters of the covering groups \(\widehat S_n\) of the symmetric group \(S_n\) and an older conjecture by the first author in 1974 in the context of extensions of the Rogers-Ramanujan identities. The purpose of this paper is two-fold. On the other hand, a purely representation theoretic proof of a conjecture in the earlier paper that the number of labels of 5-modular spin characters in a given block equals the number of (associate classes) of such characters. This is done by proving that the number of bar partitions of \(n\) into parts prime to 5 and in the set \({\mathcal D}_5(n)\), the set of 5-modular spin character labels (too technical to describe here) with the same 5-bar core is the same. This part also contains a lucid account of this subject area. The other part proves a refinement of the principal result of the earlier paper -- the proof of this part follows the same lines of that result. This new result is a refined partition theorem; it proves that two sets of partitions, both sets having definitions which are too technical to be given here, are equally numerate. The conjecture proved in the first part is a consequence of this result.
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 11P81
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 937054
zbMATH Keywords: labels
zbMATH Keywords: irreducible modular characters
zbMATH Keywords: covering groups
zbMATH Keywords: symmetric groups
zbMATH Keywords: Rogers-Ramanujan identities
zbMATH Keywords: 5-modular spin characters
zbMATH Keywords: blocks
zbMATH Keywords: number of bar partitions
zbMATH Keywords: partition theorems
MaRDI profile type: Publication
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cites work: Some combinatorial results involving Young diagrams
cites work: Q3952291
cites work: On p-quotients for spin characters
cites work: Decomposition matrices for spin characters of symmetric groups
cites work: The Number of Modular Characters in Certain p -Blocks
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description: scientific article; zbMATH DE number 937054