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Difference between revisions of "Leibniz-Hopf algebra and quasi-symmetric functions"

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130030/l1300301.png" /> be the graded dual of the [[Leibniz–Hopf algebra|Leibniz–Hopf algebra]] over the integers. The strong Ditters conjecture states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130030/l1300302.png" /> 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|Leibniz–Hopf algebra]] is incorrect), and so does a later version [[#References|[a1]]] of it. Meanwhile, the weak Ditters conjecture, which states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130030/l1300303.png" /> is free over the integers without giving a concrete set of generators, has been proved; see [[Quasi-symmetric function|Quasi-symmetric function]] and [[#References|[a2]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130030/l1300304.png" /> 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>
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<table>
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<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>
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<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>
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</table>
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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
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article