RIGIDITY FOR MPR, THE MALVENUTO-POIRIER-REUTENAUER HOPF ALGEBRA OF PERMUTATIONS

Hazewinkel, Michiel

• Accepted : 2006.08.05
• Published : 2007.12.25
• 30 2

Abstract

In this note it is proved that MPR is rigid as a Hopf algebra with distinguished basis. I.e. there are no nontrivial automorphisms that preserve the multiplication and comultiplication and take the distinguished basis of all permutations into itself (as a graded set).

Keywords

MPR Hopf algebra;rigidity;Hopf algebra of permutations

References

1. Marcelo Aguiar, Frank Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, Preprint, Dept. Math. Texas A&M University, Dept. Math., University of Massachusetts at Amherst., 2002.
2. Michiel Hazewinkel, Hopf algebras of endomorphisms of Hopf algebras, Preprint, CWI, 2004.
3. Dirk Kreimer, Knots and Feynman diagrams, Cambridge Univ. Press, 2000.
4. A. Liulevicius, Arrows, symmetries and representation rings, J. Pure and Appl. Algebra 19(1980), 259-273. https://doi.org/10.1016/0022-4049(80)90103-6
5. Jean-Louis Loday, Maria O. Ronco, Hopf algebra of planary binary trees, Adv. in Math. 139:2(1998), 293-309. https://doi.org/10.1006/aima.1998.1759
6. Jean-Louis Loday, Maria O. Ronco, Order structure on the algebra of permutations of planar binary trees, Preprint, Inst. Math. Louis Pasteur, Strassbourg, 2001.
7. C. Malvenuto, Chr, Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. of Algebra 177(1994), 967-982.
8. Stephane Poirier, Christophe Reutenauer, Algebres de Hopf de tableaux, Ann. Sci. Math. Quebec 19:1(1995), 79-90.
9. C. Schensted, Longest increasing and decreasing subsequences, Canadian J. Math. 13(1961), 179-191. https://doi.org/10.4153/CJM-1961-015-3