A refinement of a partition identity and blocks of some modular characters
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.
scientific article; zbMATH DE number 937054
scientific article