Difference between revisions of "Leibniz-Hopf algebra and quasi-symmetric functions"
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| − | Let | + | Let $\mathcal{M}$ be the graded dual of the [[Leibniz–Hopf algebra]] over the integers. The strong Ditters conjecture states that $\mathcal{M}$ is a free commutative algebra with as generators the concatenation powers of elementary [[Lyndon word]]s. This conjecture is still open (as of 2001); the initial proof contains mistakes (so the assertion of its proof in [[Leibniz–Hopf algebra]] is incorrect), and so does a later version [[#References|[a1]]] of it. Meanwhile, the weak Ditters conjecture, which states that $\mathcal{M}$ is free over the integers without giving a concrete set of generators, has been proved; see [[Quasi-symmetric function]] and [[#References|[a2]]]. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.J. Ditters, A.C.J. Scholtens, "Free polynomial generators for the Hopf algebra | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> E.J. Ditters, A.C.J. Scholtens, "Free polynomial generators for the Hopf algebra $\mathit{Qsym}$ of quasi-symmetric functions" ''J. Pure Appl. Algebra'' , '''144''' (1999) pp. 213–227</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Hazewinkel, "Quasi-symmetric functions" D. Krob (ed.) A.A. Mikhalev (ed.) A.V. Mikhalev (ed.) , ''Formal Power Series and Algebraic Combinatorics (Moscow 2000)'' , Springer (2000) pp. 30–44</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
Latest revision as of 19:16, 17 June 2016
Let $\mathcal{M}$ be the graded dual of the Leibniz–Hopf algebra over the integers. The strong Ditters conjecture states that $\mathcal{M}$ is a free commutative algebra with as generators the concatenation powers of elementary Lyndon words. This conjecture is still open (as of 2001); the initial proof contains mistakes (so the assertion of its proof in Leibniz–Hopf algebra is incorrect), and so does a later version [a1] of it. Meanwhile, the weak Ditters conjecture, which states that $\mathcal{M}$ is free over the integers without giving a concrete set of generators, has been proved; see Quasi-symmetric function and [a2].
References
| [a1] | E.J. Ditters, A.C.J. Scholtens, "Free polynomial generators for the Hopf algebra $\mathit{Qsym}$ of quasi-symmetric functions" J. Pure Appl. Algebra , 144 (1999) pp. 213–227 |
| [a2] | M. Hazewinkel, "Quasi-symmetric functions" D. Krob (ed.) A.A. Mikhalev (ed.) A.V. Mikhalev (ed.) , Formal Power Series and Algebraic Combinatorics (Moscow 2000) , Springer (2000) pp. 30–44 |
How to Cite This Entry:
Leibniz-Hopf algebra and quasi-symmetric functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leibniz-Hopf_algebra_and_quasi-symmetric_functions&oldid=15349
Leibniz-Hopf algebra and quasi-symmetric functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leibniz-Hopf_algebra_and_quasi-symmetric_functions&oldid=15349
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article