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# Lie Elements and Knuth Relations

Published:2004-08-01
Printed: Aug 2004
• Manfred Schocker
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## Abstract

A coplactic class in the symmetric group $\Sym_n$ consists of all permutations in $\Sym_n$ 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 $\Sym_n$ 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 direct sum of all Solomon descent algebras.
 Keywords: symmetric group, descent set, coplactic relation, Hopf algebra, convolution product
 MSC Classifications: 17B01 - Identities, free Lie (super)algebras 05E10 - Combinatorial aspects of representation theory [See also 20C30] 20C30 - Representations of finite symmetric groups 16W30 - Coalgebras, bialgebras, Hopf algebras (See also 16S40, 57T05); rings, modules, etc. on which these act