# Block characters of the symmetric groups

Authors
• 1 Queen Mary University of London, School of Mathematical Sciences, Mile End Road, London, E1 4NS, UK , London (United Kingdom)
• 2 Institute for Information Transmission Problems, Bolshoy Karetny 19, Moscow, 127994, Russia , Moscow (Russia)
• 3 Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA , Cambridge (United States)
Type
Published Article
Journal
Journal of Algebraic Combinatorics
Publisher
Springer US
Publication Date
Aug 29, 2012
Volume
38
Issue
1
Pages
79–101
Identifiers
DOI: 10.1007/s10801-012-0394-9
Source
Springer Nature
Keywords
A block character of a finite symmetric group is a positive definite function which depends only on the number of cycles in a permutation. We describe the cone of block characters by identifying its extreme rays, and find relations of the characters to descent representations and the coinvariant algebra of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathfrak{S}}_{n}$\end{document}. The decomposition of extreme block characters into the sum of characters of irreducible representations gives rise to certain limit shape theorems for random Young diagrams. We also study counterparts of the block characters for the infinite symmetric group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathfrak{S}}_{\infty}$\end{document}, along with their connection to the Thoma characters of the infinite linear group GL∞(q) over a Galois field.