Structure of The Malvenuto-Reutenauer Hopf Algebra of Permutations (Extended Abstract)

By Marcelo Aguiar, Frank Sottile. 2002

159 citations recorded by the source corpus.

We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. We also describe the structure constants of the multiplication as a certain number of facets of the permutahedron. Our results reveal a close relationship between the structure of this Hopf algebra and the weak order on the symmetric groups.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4ec33b71da6a627ba01acfa53c4853f058817dbd

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Citations in the corpus

Hopf Algebras in Combinatorics (2014, background): “Available at http://www.math.cornell.edu/~maguiar/a.pdf [7] M. Aguiar and F. Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations.”
Sets with two associative operations (2003, background): “Jean-Louis Loday informed me that the integer un is the number of permutations with no global descent [1].”
Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables (2005, background): “This is a surprising fact since these numbers also happen to be the number of free generators/primitives of the Malvenuto-Reutenauer Hopf algebra of permutations (see [1], [5]).”
Noncrossing partitions and the shard intersection order (2009, unknown): “The latter play a role in the Malvenuto-Reutenauer Hopf algebra of permutations [1].”
MATHEMATICAL DEVELOPMENTS FROM THE ANALYSIS OP RIFFLE SHUFFLING (2003, background): “Aguiar and Sottile (2002), Billera, Hsiao and Van Willigenburg (2001) and Garsia, Wallach (2002) are relevant, significant studies.”
Cocommutative Hopf Algebras of Permutations and Trees (2004, methodology): “deﬁnition of φ(or its inverse) makes it clear that φ(u\v) = φ(u)\φ(v) for any permutations uand v. In particular, \-irreducible heap-ordered trees correspond to \-irreducible permutations under φ. In [2], we introduced a linear basis Mw of SSym, indexed by permutations w, which is obtained from the ”
The peak algebra and the descent algebras of types B and D (2003, background): “ coproduct of w∈ Sn is ∆(w) := Xn p=0 w(p)⊗w ′ (p) where w(p) and w ′ (p) are as in (8.1). This is the Hopf algebra of Malvenuto and Reutenauer [24]. A detailed study of this Hopf algebra is given in [3] (the object denoted by SSym in this reference is the dual Hopf algebra to QS). Similar construct”
Bidendriform bialgebras, trees, and free quasi-symmetric functions (2005, background): “4.1 Recalls (See [ 3 , 5, 18]). The algebra FQSym is the vector space generated by the elements (Fu)u∈S, where S is the disjoint union of the symmetric groups Sn (n ∈ N). Its product and its coproduct are given in the following way: for all u ∈ Sn, v ∈ Sm, by putting u = (u1...un),”
Combinatorial Hopf algebras and generalized Dehn–Sommerville relations (2003, unknown): “The proof of part (a) is complete.”
Structure of the Loday–Ronco Hopf algebra of trees (2004, background): “For any permutations ρ, σ, τ , (ρ ∨ σ)\τ = ρ ∨ (σ\τ) ,(2) ρ/(σ∨τ) = (ρ/σ)∨τ .”
Combinatorial Hopf algebras and K-homology of Grassmanians (2007, unknown): “…= x; and (2) all v ∈ SMm+n+p−1 such that v|[m] = w, st(v|[m,m+n−1]) = u and st(v|[m+n,m+n+p−1]) = x; and (3) all v ∈ SMm+n+p−1 such that v|[m] = w, st(v|[m+1,m+n]) = u and st(v|[m+n,m+n+p−1]) = x; and (4) all v ∈ SMm+n+p−2 such that v|[m] = w, st(v|[m,m+n−1]) = u and st(v|[m+n−1,m+n+p−2]) = x.”