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Lie Elements and Knuth Relations

Published online by Cambridge University Press:  20 November 2018

Manfred Schocker
Manfred Schocker*
Affiliation:
Mathematical Institute, 24–29 St. Giles’, Oxford OX1 3LB, U.K. e-mail: schocker@maths.ox.ac.uk
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Abstract

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A coplactic class in the symmetric group ${{\mathcal{S}}_{n}}$ consists of all permutations in ${{\mathcal{S}}_{n}}$ 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 ${{\mathcal{S}}_{n}}$ 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 direct sum of all Solomon descent algebras.

Keywords

17B0105E1020C3016W30symmetric groupdescent setcoplactic relationHopf algebraconvolution product
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 4 , 01 August 2004 , pp. 871 - 882
Copyright
Copyright © Canadian Mathematical Society 2004

Footnotes

The author was supported by the Research Chairs of Canada.

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