The Algebra of Quasi-Symmetric Functions Is Free over the Integers

Author: Hazewinkel M.

Source: Advances in Mathematics, Volume 164, Number 2, December 2001 , pp. 283-300(18)

Publisher: Academic Press

Abstract:

Let denote the Leibniz–Hopf algebra, which also turns up as the Solomon descent algebra and the algebra of noncommutative symmetric functions. As an algebra =ZZ₁, Z₂,…, the free associative algebra over the integers in countably many indeterminates. The coalgebra structure is given by (Z_n)=ⁿ_i=0 Z_iZ_n-i, Z₀=1. Let be the graded dual of . This is the algebra of quasi-symmetric functions. The Ditters conjecture says that this algebra is a free commutative algebra over the integers. In this paper the Ditters conjecture is proved. © 2001 Elsevier Science.

Keywords: Leibniz–; Hopf algebra; quasi-symmetric functions; Ditters conjecture; Lie–; Hopf algebra; Solomon descent algebra; shuffle algebra; overlapping shuffle algebra; noncommutative symmetric functions; divided power sequences; coalgebra; Hopf algebra; free coalgebra; formal group; Lyndon words; symmetric group; symmetric functions; Hecke algebra

Language: English

Document Type: Research article

Affiliations: CWI, Amsterdam, The Netherlands

Publication date: 2001-12-01

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