# zbMATH — the first resource for mathematics

Hopf algebra of the planar binary trees. (English) Zbl 0926.16032
Let $$k[S_n]$$ be the group algebra of the symmetric group $$S_n$$ over the field $$k$$. The Solomon descent algebra $$Sol_n$$ is a subalgebra of dimension $$2^{n-1}$$ of $$k[S_n]$$, and has a basis $$Q_n$$ such that the linear dual of the composite map $$k[Q_n]\simeq Sol_n\to k[S_n]$$ is induced by a set-theoretic map $$S_n\to Q_n$$. It is known that a graded Hopf algebra structure on $$k[S_\infty]=\bigoplus_{n\geq 0}k[S_n]$$ can be constructed such that $$Sol_\infty=\bigoplus_{n\geq 0}Sol_n$$ is a Hopf subalgebra. There is a set $$Y_n$$, made of planar binary trees with $$n$$ vertices, such that the projection from $$S_n$$ to $$Q_n$$ is the composite of two maps $$\phi\colon Y_n\to Q_n$$ and $$\psi\colon S_n\to Y_n$$. The main result of the paper is to show that the graded vector space $$k[Y_\infty]=\bigoplus_{n\geq 0}k[Y_n]$$ may be equipped with a structure of a graded Hopf algebra such that $$\phi$$ and $$\psi$$ induce Hopf algebra morphisms.

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 20C30 Representations of finite symmetric groups 16W50 Graded rings and modules (associative rings and algebras) 20C05 Group rings of finite groups and their modules (group-theoretic aspects)
Full Text:
##### References:
 [1] Bergeron, F.; Bergeron, N.; Howlett, R.B.; Taylor, D.E., A decomposition of the descent algebra of a finite Coxeter group, J. algebraic combin., 1, 23-44, (1992) · Zbl 0798.20031 [2] Gelfand, I.M.; Krob, D.; Lascoux, A.; Leclerc, B.; Retakh, V.S.; Thibon, J.-Y., Noncommutative symmetric functions, Adv. math., 112, 218-348, (1995) · Zbl 0831.05063 [3] Hanlon, Phil, The fixed-point partition lattices, Pacific J. math., 96, 319-341, (1981) · Zbl 0474.06012 [4] Loday, Jean-Louis, Série de Hausdorff, idempotents eulériens et algèbres de Hopf, Exposition. math., 12, 165-178, (1994) · Zbl 0807.17003 [5] Jean-Louis, Loday, Dialgebras, IRMA, Strasbourg, 1998 [6] Malvenuto, C.; Reutenauer, C., Duality between quasi-symmetric functions and the Solomon descent algebra, J. algebra, 177, 967-982, (1995) · Zbl 0838.05100 [7] Milgram, R.James, Iterated loop spaces, Ann. math. ser. 2, 84, 386-403, (1966) · Zbl 0145.19901 [8] Poirier, S.; Reutenauer, C., Algèbres de Hopf de tableaux, Ann. sci. math. Québec, 19, 79-90, (1995) · Zbl 0835.16035 [9] Reutenauer, C., Free Lie algebras, London mathematical society monographs, new series, 7, (1993), Oxford Science/Clarendon Press/Oxford Univ. Press New York [10] Maria, O. Ronco, The Solomon algebra of the symmetric group and other Coxeter groups, 1996 [11] Solomon, Louis, A MacKey formula in the group ring of a Coxeter group, J. algebra, 41, 255-264, (1976) · Zbl 0355.20007 [12] Stasheff, James Dillon, Homotopy associativity of H-spaces, I, Trans. amer. math. soc., 108, 275-292, (1963) · Zbl 0114.39402 [13] Andy, Tonks, Relating the associahedron and the permutohedron, Operads: Proceedings of Renaissance Conferences, Contemp. Math. 202, 33, 36, Hartford, CT/Luminy, 1995, Amer. Math. Soc. Providence, RI, 1997 · Zbl 0873.51016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.