Lie Elements and Knuth Relations

2002/09/24 by Manfred Schocker
Mathematics · #math.RA #math.CO #msc:17B01 #msc:05E10 #msc:20C30 #msc:16W30

paper · pdf

published as Canadian J. Math. 56 (2004), No. 4, 871-882 · 15 pages

arxiv created 2002/09/24 · arxiv updated 2009/11/30

Abstract

A coplactic class in the symmetric group S_n consists of all permutations in S_n with a given Schensted Q-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of S_n which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the Solomon descent algebra.

Related

• Cyclic descents and P-partitions
• Modular Idempotents for the Descent Algebras of Type A and Higher Lie Powers and Modules
• Generating series formulas for the structure constants of Solomon's descent algebra
• Representation Theory of Symmetric Groups and Related Hecke Algebras
• Crossed morphisms, (integration of) post-Lie algebras and the post-Lie Magnus expansion