A proof of the Mullineux conjecture
type: Item
instance of: scholarly article
title: A proof of the Mullineux conjecture
zbMATH Open document ID: 0958.20018
DOI: 10.1007/PL00004340
author: Ben Ford
author: Alexander S. Kleshchev
published in: Mathematische Zeitschrift
publication date: 1997-12-10T00:00:00Z
review text: Let \(\lambda=(\nu^{\alpha_1}_1,\dots,\nu^{\alpha_k}_k)\), \(\nu_1>\cdots>\nu_k>0\), \(\alpha_i>0\), be a partition of \(n\in\mathbb{N}\). \(\lambda\) is said to be \(p\)-regular for a prime \(p\) if \(\alpha_1,\dots,\alpha_k<p\). Let \(F\) be a field of characteristic \(p>0\). All irreducible representations of the symmetric group \(S_n\) are naturally parametrized by \(p\)-regular partitions. We denote by \(D^\lambda\) the irreducible module corresponding to \(\lambda\). Let \(\text{sgn}_n(g)=\text{sgn}(g)\) be the sign of a permutation \(g\). Then, given an irreducible \(D^\lambda\), the tensor product \(D^\lambda\otimes\text{sgn}_n\) is also irreducible. Put \(D^\lambda\otimes\text{sgn}_n=D^{b_n(\lambda)}\). In this way a bijection \(b_n\) on the set \(P_n\) of \(p\)-regular partitions of \(n\) is defined for each \(n\in\mathbb{N}\). Mullineux proposed an algorithm which defines a bijection \(m_n\colon P_n\to P_n\) and conjectured that \(b_n=m_n\). The authors present a proof of this conjecture.
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 1096019
zbMATH Keywords: Mullineux conjecture
zbMATH Keywords: irreducible representations
zbMATH Keywords: symmetric groups
zbMATH Keywords: \(p\)-regular partitions
zbMATH Keywords: irreducible modules
zbMATH Keywords: tensor products
Wikidata QID: Q123007188
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: pl00004340
OpenAlex ID: W2033422411
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description: scientific article; zbMATH DE number 1096019