FPSAC 202, Nagoya, Japan DMTCS proc AR, 202, 93 204 A polynomial realization of the Hopf algebra of uniform block permutations Rémi Maurice Institut Gaspard Monge, Université Paris-Est Marne-la-Vallée, 5 Boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France Abstract We investigate the combinatorial Hopf algebra based on uniform block permutations and we realize this algebra in terms of noncommutative polynomials in infinitely many bi-letters Résumé Nous étudions l algèbre de Hopf combinatoire dont les bases sont indexées par les permutations de blocs uniformes et nous réalisons cette algèbre en termes de polynômes non-commutatifs en une infinité de bi-lettres Keywords: combinatorial Hopf algebras, uniform block permutations Introduction For several years, Hopf algebras based on combinatorial objects have been thoroughly investigated Examples include the Malvenuto-Reutenauer Hopf algebra FQSym whose bases are indexed by permutations [7, 4, 2], the Loday-Ronco Hopf algebra PBT whose bases are indexed by planar binary trees [6], the Hopf algebra of free symmetric functions FSym whose bases are indexed by standard Young tableaux [] The product and the coproduct, which define the structure of a combinatorial Hopf algebra, are, in general, rules of composition and decomposition given by combinatorial algorithms such as the shuffle and the deconcatenation of permutations, as in FQSym [4], of packed words, as in WQSym [9] or of parking functions, as in PQSym [0] These can also be given by the disjoint union and the admissible cuts of rooted trees, as in the Connes-Kreimer Hopf algebra [3], or by concatenation and subgraphs/contractions of graphs [8] Computing with these structures can be difficult, and polynomial realization can bring up important simplifications We can see a polynomial realization as the image by an injective Hopf algebra morphism from a combinatorial Hopf algebra to an algebra of polynomials commutative or not The idea is to encode the combinatorial objects by polynomials, in such a way that the product of the combinatorial Hopf algebra become the ordinary product of polynomials, and that the coproduct be given by the disjoint union of alphabets, endowed with some extra structure such as an order relation This can be done, eg, for FQSym [4], WQSym [9], PQSym [0] and Connes-Kreimer [5] In this paper, we realize the Hopf algebra UBP of uniform block permutations introduced by Aguiar and Orellana [] We begin by recalling the preliminary notions of uniform block permutations in Section 365 8050 c 202 Discrete Mathematics and Theoretical Computer Science DMTCS, Nancy, France
94 Rémi Maurice 2 We recall, in Section 3, the Hopf algebra structure of uniform block permutations, we translate this structure via the decomposition and we describe the dual structure Using polynomial realizations of Hopf algebras based on permutations and set partitions, we realize this algebra in Section 4 We finish by finding the Hopf algebra WQSym as a quotient of UBP and the dual as a subalgebra of UBP Acknowledgements The author would like to thank Jean-Yves Thibon for his advice and reading of this paper 2 Definitions and decompositions Definition A uniform block permutation of size n is a bijection between two set partitions of {,, n} that maps each block to a block of the same cardinality Notation Let A {A, A 2,, A k } and A {A, A 2,, A k } be two set partitions of the same size We denote the uniform block permutation f : A A for which the image of A i is A i by A A 2 A k A A 2 A k Example Here are all uniform block permutations of size and 2: n 2 2 2 n 2,, 2 2 2 In the case where the set partitions consist of singletons ie, A { {},, {n} }, we find the permutations of size n Definition 2 Let f and g be two uniform block permutations The concatenation of f and g, denoted by f g, is the uniform block permutation obtained by adding the size of f to all entries of g and concatenating the result to f Example 2 3 2 23 3 2 46 5 46 2 5 3 3 2 46 5 79 8 23 79 5 48 6 Definition 3 Let f : A A and g : B B be two uniform block permutations of the same size n The composition g f : C C of f and g is defined by the following process The blocks C of the set partition C are the subsets of {,, n} which are minimal for the two properties C is a union of blocks of A 2 fc is a union of blocks B i of B The image of C is the union of images gb i Note that if f is a permutation, then the set partition C is B, and if g is a permutation, then the set partition C is A
Uniform block permutations 95 Example 3 25 346 236 45 5 2 3 4 6 46 2 5 3 2 3 4 5 6 5 4 2 6 3 4 2 36 5 23 45 6 36 245 236 45 4 236 5 2 456 3 The data of a permutation and a set partition is enough to describe a uniform block permutation Given a uniform block permutation f : A A of size n, there is a permutation σ in S n such that A { } {σx} H A x H Conversely, a set partition of the set {,, n} and a permutation of S n determine a uniform block permutation of size n Any uniform block permutation can thus decompose non-uniquely as, f σ Id A Id A σ The first equality expresses the fact that one can group the images σ i whose indices are in the same block in A, the second expresses the fact that one can group the images σ i whose values are in the same block in A 3 Algebraic structures In [], Aguiar and Orellana introduced a Hopf algebra whose basis is the set of uniform block permutations It will be denoted by UBP, standing for Uniform Block Permutations Let us recall the convolution of two permutations Let τ and τ be two permutations: τ τ σ σu v stduτ stdvτ where the standardization is a process that associates a permutation with a word The standardized of w is defined as the permutation having the same inversions as w For example: stda 2 a 3 a 2 a a 6 a a 4 354726, 2 2 243 342 432 234 243 342 The product and the coproduct are defined as follows Definition 4 Let f and g be two uniform block permutations of respective sizes m and n The convolution, denoted by f g, is the sum of all uniform block permutations of the form 2 m n f g σ σ 2 σ mn where ie, σ appears in 2 m 2 n stdσ σ m 2 m and stdσ m σ mn 2 n
96 Rémi Maurice Example 4 2 2 [ 2 3 2 3 23 23 2 3 2 3 23 2 3 2 3 3 2 23 3 2 ] 23 23 We denote by stde the standardization of a collection of disjoint sets of integers obtained by numbering each integer a in E by one plus the number of integers in E smaller than a Example 5 We set C {,,i} std { {, 6}, {9, 3}, {3, 5, 2} } { {, 4}, {5, 7}, {2, 3, 6} } and let IC be the set of integers i such that for any set partition C { H C; H {, 2,, i} }, C {i,,n} std C C {,,i} {, 2,, n} C {,,i} C {i,,n} Example 6 Let C { {, 3}, {2}, {4, 6, 7}, {5} } be a set partition: C {,,2} { {2} } C {,,3} { {, 3}, {2} } C {4,,7} { {, 3, 4}, {2} } IC {0, 3, 7} Definition 5 Let f : A A be a uniform block permutation of size n By Lemma 22 in [], i is in IA if and only if there exists a unique permutation σ in S n appearing in the convolution product 2 i 2 n i and unique uniform block permutations f i of size i and f n i of size n i such that f f i f n i 2 n σ σ 2 σ n The coproduct is then Example 7 3 256 4 since 3 256 4 f : 3 256 4 i IA f i f n i 3 245 5 234 3 256 4 [ ] 3 256 4 2 3 4 5 6 2 3 4 5 6 [ 3 245 ] 2 3 4 5 6 5 234 [ 3 256 4 ] 2 3 6 4 5 2 3 4 5 6 2 3 4 5 6
Uniform block permutations 97 One can rewrite this product and coproduct using the decomposition of a uniform block permutation Given f : A A and g : B B two uniform block permutations, f σ Id A Id A σ and g τ Id B Id B τ, we have where f g σ τ Id A B 2 f f {,,i} f {i,,n} 3 i IA Example 8 f {,,i} Id A {,,i} σ {,,i} f {i,,n} Id A {i,,n} stdσ {i,,n} 2 2 Id { 2 Id { } {}} {,2} 2 Id { {},{2,3}} 23 23 32 Id { {},{2,3}} 23 23 23 23 2 3 3 2 Example 9 We set A { {, 5}, {2, 3, 4}, {6} } 3 256 4 Id A 25634 ε Id A 25634 Id { 2534 Id { {,5},{2,3,4}} 3 256 4 3 245 5 234 {} } IdA 25634 ε 3 256 4 It is proved in [] that UBP is a self-dual Hopf algebra We can describe the dual structure Let us first define another product and coproduct of uniform block permutations Let f : A A and g : B B be two uniform block permutations, f σ Id A Id A σ and g τ Id B Id B τ: where f g Id A B σ τ 4 f i IA f f i f i f n 5 f f i stdσ σ i Id A {,,i} f i f n stdσ i σ n Id A {i,,n}
98 Rémi Maurice Example 0 2 2 Id {{},{2,3}} 2 Id {{},{2,3}} 23 23 23 23 2 3 3 2 23 23 23 3 256 4 25634 Id { {,3},{2,5,6},{4}} ε 25634 Id { 25634 Id { {,3},{2,5,6},{4}} 3 256 4 3 256 4 The two structures of Hopf algebra are dual to each other for the scalar product In other words, we have: 4 Polynomial realization {,3},{2,5,6},{4} } ε f, g δ f,g 6 f, g h f, g h 7 f, g h f, g h 8 Given two alphabets B {b i ; 0 i} and C {c i ; 0 i}, with C totally ordered, we consider an { b } alphabet A i ; 0 i, j of bi-letters endowed with two relations: c j for all i and k, for all j and l, bi c j bi c j bk c l bk c l if and only if c j c l, if and only if b i b k The product of two biwords is the concatenation u u2 u u 2 v v 2 v v 2 u We extend the definition of the standardization to biwords Let w be a biword The standard- v ized of w is the permutation stdv
Uniform block permutations 99 Example [ ] b b std b 3 b b 2 2435 c 2 c 3 c 2 c c 6 Let f : A A be a uniform block permutation Let ξ f be the minimum permutation for the lexicographic order such that f ξ f Id A We say that a biword w is f compatible if and We set: - stdw ξ f - if i and j are in the same block in A, then the bi-letters w i and w j satisfy w i w j G f A w;w is f compatible w 9 3 256 4 Example 2 Let f be a uniform block permutation The permutation σ 25634 is the minimum permutation associated with f and we have bi b G f A j b i b k b j b j c c 2 c 3 c 4 c 5 c 6 i,j,k stdc c 2 c 6σ Theorem The polynomials G f A provide a realization of the Hopf algebra of uniform block permutations That is to say G f A G g A G h A h f g Let A be an alphabet isomorphic to the alphabet A such that, for any bi-letter w of A and for any bi-letter w of A, we have w w Define A A as the disjoint union of A and A endowed with the relations and If we allow the bi-letters of A and A to commute and identify P AQA with P Q, we have: G f A A G f 0 Example 3 G AG A b B c C 2 b c b,b 2 B b,b 2 B c,c 2 C;c c 2 G 2 2 c,c 2 C b b 2 c c 2 b b 2 c c 2 A G 2 2 b,b 2 B c,c 2 C;c >c 2 A b b 2 c c 2
200 Rémi Maurice G 2 AG A bb b v c b B b B 2 stdv2 c C bi b i b i2 v v 2 i,i 2 stdv 2 Example 4 We set τ 243 G 3 24 3 24 stdv 2 i,i 2 c c 2 c 3 G 2 3 2 3 A A bi b i b i2 c c 2 c 3 i,i 2 c c 3<c 2 G 2 3 3 2 i,j c c 2 c 3 c 4 G 3 24 3 24 bi b j b i b j c c 2 c 3 c 4 A G bi b i b i2 c c 2 c 3 G 2 3 23 i,j c c 2 c 3 c 4 3 24 3 24 i,i 2 c 3<c c 2 b i b j b i b j A c c 2c 3c 4 bi b i b i2 c c 2 c 3 G 3 2 4 23 4 5 Packed words A A i,j,k stdc c 2c 3c 4τ i,j,k stdc c 2 c3c4τ G 3 2 4 23 4 G 2 3 23 bi b j b i b k c c 2 c 3 c 4 bi b j b ib k c c 2c 3 c 4 A G AG i,j,k stdc c 2 c 3 c4τ i,j,k stdc c 2 c 3 c 4 τ AG 3 2 2 3 b i b j b i b k c c 2c 3c 4 b i b j b i b k c c 2c 3c 4 A G 3 2 4 23 4 We recall the definition of a packed word and the structure of Hopf algebra given in [9] The packed word packw associated with a word w A is the word u obtained by the following process If b < b 2 < < b r are the letters occuring in w, u is the image of w by the homomorphism b i a i A word w is said to be packed if packw is w A A
Uniform block permutations 20 5 WQSym The Hopf algebra WQSym is the K vector space generated by packed words endowed with the shifted shuffle product and the coproduct given by, using notations in [5]: u v u v[max u], w u vw packu packv 2 Let Ψ be the map that associates the word w of the length n with a permutation σ of size n and a set partition A of {, 2,, n} constructed by the following process The ith letter u i of u is the minimum of the letters σ j whose values σ j are in the same block in A as σ i The word w is obtained by packing The map Ψ induces a linear map from the vector space generated by uniform block permutations to that generated by packed words We define it by writing uniform block permutations f in the form Id A σ and applying Ψ to σ and A We also call this map Ψ Example 5 [ ] 2 Ψ 2 [ 2 3 Ψ 23 ] [ 3 24 22 Ψ 2 34 ] 22 If we endow the vector spaces with the shifted shuffle product, the map Ψ is an algebra morphism Proposition Let f and g be two uniform block permutations Example 6 Ψ [ 2 3 3 2 2 2 Ψf g Ψf Ψg 3 ] 234 324 342 324 342 342 324 342 342 342 2 2 [ 2 3 Ψ 3 2 ] Ψ [ 2 2 The map Ψ cannot be a coalgebra morphism as shown in the example [ 2 3 2 3 Ψ Ψ Ψ Ψ 23 23 [ ] 2 3 Ψ Ψ 23 ε 22 22 ε 2 2 ε 22 2 22 ε 22 ] ]
202 Rémi Maurice The idea is that the coproduct on packed words deconcatenates the word in all suffixes and prefixes, whereas the coproduct on uniform block permutations preserves the blocks This suggests to define an another coproduct: w packu packv u vw alphu alphv Endowed with the shifted shuffle product and with the coproduct, the vector space generated by packed words is a combinatorial Hopf algebra denoted by WQSym Proposition 2 The Hopf algebras WQSym and WQSym are isomorphic The morphism of Hopf algebras is given by the following process We define the following relation on packed words of the same length Let u and v be two packed words, the cover relation is u v if and only if stdu stdv { vk if v there exists a i such that u k k i v k otherwise 2 22 22 2 32 2 22 32 23 23 32 The map Γ defined as WQSym WQSym w u is an isomorphism of Hopf algebras For example, the image of a permutation is a permutation, the image of the word of length n consisting only of is the sum of all the non-decreasing packed words of length n u w Example 7 Γ Γ2 2 2 2 23 2 32 3 32 2 Γ Γ Γ Γ 2 2 ε 32 ε 2 2 ε 2 ε 32 2 32 Γ2 Proposition 3 The map Ψ is a surjective Hopf algebra morphism from UBP to WQSym
Uniform block permutations 203 52 WQSym We denote by WQSym the K vector space generated by the packed words endowed with the convolution product and the coproduct given by u W v w, 4 u wu v P W ;packu u,packv v 0 k max u u {,,k} packu {k,,max u} 5 The dual WQSym of WQSym is the Hopf algebra endowed with the product u W v w 6 for any packed word u and v, and the coproduct wu v P W packu u packv v alphu alphv Proposition 4 The Hopf algebra WQSym and WQSym are isomorphic In fact, the map Γ induces a dual map This is Γ : WQSym WQSym w u w u For example: Γ 22334 22334 22223 22333 22222 This map is an isomorphism of Hopf algebras Example 8 Γ 22 W Γ 2233 3322 323 2233 2222 3322 22 323 22 W Γ 22 W Γ Γ Γ 3234 Γ Γ ε 3234 223 2 2 323 3234 ε ε 3234 3233 223 222 2 2 323 3234 3233 ε 3234 3233 Γ 3234
204 Rémi Maurice Proposition 5 The map is an injective morphism of Hopf algebras References ϕ : WQSym UBP w f f;ψfw [] M Aguiar and RC Orellana The Hopf algebra of uniform block permutations J Alg Combin, 28 :5 38, 2008 [2] M Aguiar and F Sottile Structure of the Malvenuto-Reutenauer Hopf algebra of permutations Adv Math, 92:225 275, 2005 [3] A Connes and D Kreimer Hopf algebras, renormalization and noncommutative geometry Comm Math Phys, pages 203 242, 998 [4] G Duchamp, F Hivert, and J-Y Thibon Noncommutative symmetric functions VI: Free quasisymmetric functions and related algebras Internat J Alg Comput, 2:67 77, 2002 [5] L Foissy, J-C Novelli, and J-Y Thibon Polynomial realizations of some combinatorial Hopf algebras arxiv:0092067v [mathco], 200 [6] J-L Loday and M-O Ronco Hopf algebra of the planar binary trees Adv Math, 39:293 309, 998 [7] C Malvenuto and C Reutenauer Duality between quasi-symmetric functions and the Solomon descent algebra J alg, 77:967 982, 995 [8] D Manchon On bialgebras and Hopf algebras of oriented graphs arxiv:03032v3 [mathco], 20 [9] J-C Novelli and J-Y Thibon Polynomial realizations of some trialgebras FPSAC San Diego, 2006 [0] J-C Novelli and J-Y Thibon Hopf algebras and dendriform structures arising from parking functions Fundamenta Mathematicae, 93:89 24, 2007 [] S Poirier and C Reutenauer Algèbres de Hopf de tableaux Ann Sci Math Québec, 9:79-90, 995