A pair of dual Hopf algebras on permutations

Abstract: Hopf algebras are important objects in algebraic combinatorics since they have strong stability. In particular, its dual space is an important tool to study the properties of the original Hopf algebra. Based on the classical shuffle Hopf algebra structure, we have proved that the shuffle product and deconcatenation coproduct on the standard factorizations of permutations define a graded shuffle Hopf algebra on permutations. In this paper, we figure out a new product and a new coproduct on permutations to get the duality of this graded shuffle Hopf algebra.


Introduction
In 1941, Hopf [1] put forward the concept of both algebraic structure and coalgebra structure in the study of cohomology algebra H * (G, K) of Lie group G. In 1965, Milnor and Moore [2] formally called them Hopf algebras, then Sweedler [3] and Artamonov [4] also studied it. Especially in recent two decades, the rise of quantum groups, the successful solution of Kaplansky's conjectures [5] and the development of Hopf algebra theory make it a new scientific system. Hopf algebras are widely used in various fields, for example, universal enveloping algebras in Lie theory and free Lie algebras [6,7], nonlinear control theory [8], geometry algebras [9], degenerate versions of Drinfeld-Jimbo quantum groups [10], generalized Clifford algebras in aforementioned categories [11] and even the genetic inheritance in biology [12].
In 1984, Michel Van den Bergh [13], Blattner and Montgomery [14] introduced the duality of Hopf algebras. It is proved that the group ring is a classical special case of Hopf algebra. But the dual space of a general Hopf algebra is only an algebra, not a Hopf algebra. Only when a Hopf algebra is graded, its graded dual space has a corresponding graded Hopf algebra structure.
Because the rigidity of Hopf algebra structure often reveals the deep nature and connection in combinatorics, Hopf algebra has gradually become one of the important research contents in algebraic combinatorics. In 1979, Joni and Rota [15] found that the discrete structures in combinatorics have natural Hopf algebraic structures. In 2006, Aguiar, Bergeron and Sottle [16] proposed the concept of combinatorial Hopf algebras and studied the category of them. More and more combinatorial Hopf algebras have been discovered and studied. In 2009, Bergeron and Li [17] gave the axioms on a tower of algebras to guarantee that its Grothendieck groups are dual graded Hopf algebras. Later, Bergeron, Lam and Li [18] analyzed the relationship between combinatorial Hopf algebras and dual graded graphs. The Hopf algebra of symmetric functions is a self-dual Hopf algebra, which plays an important role in algebraic combinatorics. In 2016, using the duality Li, Morse and Shields [19] provided a dual approach to the structure constants for K-theory of Grassmannians. It is well-known that symmetric functions are closed related to symmetric groups. They can be applied to algebraic number theory [20] and substochastic matrices [21][22][23][24].
In combinatorics, a permutation of degree n is an arrangement of n elements. The symmetric group of degree n, denoted by S n , contains all permutations of [n] = {1, 2,. . . , n}. Let KS n be the vector space spanned by S n over field K. Define KS := n≥0 KS n , the direct sum of KS n , where S 0 = { } and is the empty permutation. Then KS is graded and its nth component is KS n . In 1958, Ree [25] studied an algebra associated with shuffles, then the study of shuffles is further promoted [26,27]. In 1995, Malvenuto and Reutenauer [28] constructed the product * according to [29]. In fact, the product * is the shuffle product x (Eq 2.2 in Section 2). However, this product * is not commutative on permutations.
In 2005, Aguiar and Sottile [30] introduced global descents of permutations in symmetric group S n . It plays a crucial role in this paper. In 2018, Bergeron, Ceballos and Pilaud [31] introduced the concept of gaps on permutations of S n . Then Bergeron et al. linked the global descents and the shuffle product x together and defined a new shuffle product on permutations. The new shuffle product is based on the global descents of permutations, which requires that from a standard factorization to the original permutation we renumber the numbers in the atoms from right to left. This ensures the new shuffle product is commutative on permutations. We denote such a shuffle by x G and define a coproduct ∆ on KS such that (KS , x G , µ, ∆, ν) is a Hopf algebra [32]. There must be a graded Hopf algebra dual to the shuffle Hopf algebra (KS , x G , µ, ∆, ν), which could be seen from the definition of dual graded Hopf algebras.
The main result of this paper is to find the duality (KS , x * can be found in [28]. Here, we only briefly introduce these two graded Hopf algebras. A bialgebra (H, ψ, µ, ∆, ν) over a field K, where ψ is a product, µ is a unit map, ∆ is a coproduct and ν is a counit map, is a Hopf algebra if it admits a unique antipode θ satisfying the following identity (2.1) A graded connected bialgebra (H, ψ, µ, ∆, ν) admits a unique antipode θ so is a graded Hopf algebra. Let A be a set called an alphabet and the elements in it are called letters. A word f over alphabet A is composed of finite letters, i.e., f = f 1 f 2 · · · f n with f i ∈ A for all i. A word without any letters is called the empty word, denoted by .
The shuffle product x on words is defined by where f = f 1 f 2 · · · f n , g = g 1 g 2 · · · g m and f x = xf = f .
A sequence on [n] is obtained by selecting finite different elements from the set [n], denoted by β = β 1 β 2 · · · β m (1 ≤ m ≤ n) where β i ∈ [n]. When no element is selected, we have the empty sequence, also denoted by . Let alph(β) be the set that consists of all elements in β.
Here neither product * nor product * is commutative. For a permutation α ∈ S n and 0 ≤ i ≤ n, let α [i] consist of all elements of [i] in the one-line notation of α. Then α [i] is a permutation over [i], which is obtained by removing all digits greater than i in permutation α. For the permutation α = 3642751 we have α [5] = 34251 and For any permutation α ∈ S n , define the coproduct ∆ by For example, ∆ (2413) = ⊗ 2413 Then (KS , * , µ, ∆, ν) and (KS , * , µ, ∆ , ν) are both graded connected bialgebras, and so they are both Hopf algebras. In 2005, Aguiar and Sottile gave a detailed description of their antipodes [30,Theorem 5.4].
If there is a bilinear pairing <, >: H ⊗ H * gr → K satisfying the following identities: for any P, Q ∈ H, M, N ∈ H * gr , then H and H * gr are graded dual to each other. Define a bilinear pairing <, > on KS ⊗ KS by for any permutations α and β. For x, y, z ∈ KS , Thus the graded Hopf algebras (KS , * , µ, ∆, ν) and (KS , * , µ, ∆ , ν) are dual to each.

Concatenation product and draw coproduct
For a permutation α in S n in one-line notation, from left to right the positions in front of the first number, between two adjacent numbers and behind the last number indexed by {0, . . . , n} are called the gaps of α.
Consequently, for α in S n putting a bullet • at each global descent except 0 and n we get the factorization of α = α 1 • α 2 • · · · • α s . Denote the length of α by |α| = s, which is always less than or equal to its degree n. It is obvious that all st(α i )'s are atoms. Replacing all α i 's in the factorization of α by their standard form st(α i )'s, we get the standard factorization of α, i.e., Define the shuffle product x G on KS recursively by Obviously, the shuffle product x G is commutative and α x G β ∈ KS n+m . Thus, (KS , x G , µ) is a graded algebra.
From the example above, if we color the atoms of α blue and the atoms of β red then atoms 1 and 1 are different and we can consider all permutations in the shuffle product distinct even them are same, for example, 654312, 654312, 654312 and 654312 are distinct. In this paper, we color the atoms of different permutations in different colors.
For any permutaion α = α 1 • α 2 • · · · • α s , let I be a subset of [s]. Define α I be a permutation consisting of atoms index by I and keeping their relative positions in the factorization of α. Therefore, Define the deconcatenation coproduct ∆ on KS by where α = α 1 • α 2 • · · · • α s is its factorization with length |α| = s and ∆( ) = ⊗ . From (3.2), each term in the deconcatenation coproduct of a permutation is putting a tensor sign at one of its global descents.
The following lemma was first mentioned in [31] without a proof. Zhao and Li proved it and the result is published in a journal in Chinese [32]. Here we give a sketch of the proof. Lemma 1. The shuffle product x G and the deconcatenation coproduct ∆ are compatible, i.e., where α and β are permutations and ∆(αx Proof. Let α = α 1 •· · ·•α s and β = β 1 •· · ·•β t . Denote ∆(αx G β)| j=k to be the sum of terms in ∆(αx G β) with the number of atoms of β on the left side of the tensor sign is k, 0 ≤ k ≤ t. Then From Lemma 1 the deconcatenation coproduct ∆ is an algebra homomorphism, thus (KS , x G , µ, ∆, ν) is a bialgebra, where the unit µ and counit ν are the same in Section 2. A graded connected bialgebra is a Hopf algebra so the bialgebra (KS , x G , µ, ∆, ν) is a Hopf algebra.
Next, we would introduce another product and coproduct on KS .
Definition 1. Defined the concatenation product on KS by
The renumbering of the product is still from right to left and we notice that the concatenation product does not satisfy commutativity.
Example 3. Take α = 312 = 1 • 12 ∈ S 3 , β = 4213 = 1 • 213 ∈ S 4 , then It is easy to check that µ is also a unit for the concatenation product . In (KS , , µ), the product of any two permutations of degrees p and q is a permutation of degree p + q. So (KS , , µ) is a graded algebra.
Definition 2. The draw coproduct ∆ * on KS is defined by

Conditions (1)-(3) are obvious, so here we only prove Condition (4) which is equivalent to
for any two permutations α and β, i.e., the coproduct ∆ * is an algebra homomorphism.
Proof. Let α = α 1 • α 2 • · · · • α s and β = β 1 • β 2 • · · · • β t in standard factorization form. Then Denote ∆ * (α β)| y=k to be the sum of terms in ∆ * (α β) with the number of atoms of β on the left side of the tensor sign is k, 0 ≤ k ≤ t. Therefore, where X is a subset of [s] and the sum is over all subsets X of [s]. And Similarly, Proof. The bialgebra (KS , , µ, ∆ * , ν) is graded and connected, so it is a Hopf algebra.
Proof. According to the definition of the antipode, we are supposed to verify that all elements of the Hopf algebra (KS , , µ, ∆ * , ν) are all satisfy Eq (4.7).
Since the both sides of the equal signs in Eq (4.7) are symmetric and the coproduct ∆ * is cocommutative, we just need to prove one side of the formula (4.7) First, for a permutation α and |α| = t, then where the sum is over all subsets I of [t].
It is easy to check that Assume α . From the definition of coproduct ∆ * , we have (4.12) From Eq (4.12), when x = k − 1, we have (4.13) (4.14) It is easy to see that • (θ ⊗ id) • ∆ * (α)| x=k would be cancelled by part of When x = 1, we have (4.16) When From the results above (4.12-4.18) We notice that the closed-formula of the antipode is similar to the antipode in [7, Proposition 1.10], because for a graded connected Hopf algebra we always can compute its antipode recursively.

Duality of the shuffle Hopf algebra
A sub-permutation of a permutation α is a subsequence composed of some atoms of α with the same relative positions.
We can see the sub-permutations of any permutation from its contact map. Let α 1 = 1, α 2 = 12, α 3 = 123 and α = α Figure 1 for an illustration. In order to see the sub-permutations of a permutation clearly, we omit bullets • of the standardization factorization in the figures. Given two permutations α, β, the number of sub-permutations equal to β in α is called the binomial coefficient [15,27] of β in α, denoted by α β . Notice that α = 1 and α α = 1.
From the above examples, for any permutations α and β, α β is also a useful tool for determining whether β is a sub-permutation of α. If β is not sub-permutation of α, then α β = 0.
Hence, the draw coproduct ∆ * can be written as follows Here Example 7. We also can define draw coproduct ∆ * on words by binomial coefficients. For example, From the definition of ω α, β , we can think it as a top-down calculation, that is, it counts the number of ways that the upper permutation ω can be divided into the lower two sub-permutations α and β with β = ω\α. We can also think it as a bottom-up calculation, that is, it counts the number of ways that the lower permutations α and β compose the upper permutation ω with with β = ω\α.
Therefore, the shuffle product x G can be written as for any permutations α and β in S . Similarly, for a subword f of η, the remaining letters of η compose another subword g (keeping the relative positions), denoted by g = η\ f . Thus, we define that η f, g is the number of ways to chose f as a subword in η with g = η\ f . Then, Eq (2.2) can be rewritten as Thus Condition 4) is proved. From Eq (3.2) and Example 2 we can notice that in the result of ∆(P) each term is different. That is to say the coefficient of each term of ∆(P) all is 1. Thus, when P = M•N, we have < ∆(P), M⊗N >= 1; meanwhile, since M N = P, so < P, (M ⊗ N) >= 1. That proves Condition 5).
Here, we verified Conditions 4) and 5) by the example following.

Conclusions
Let A be a set called an alphabet and the elements in it are called letters. A finite sequence consisting of several letters is called a word over A. Denoted A * as the set of all such words with the empty word . Thus, if we replace all atoms by letters in A or all standard factorizations by words in A * , then (KA * , x G , µ, ∆, ν) and (KA * , , µ, ∆ * , ν) are also dual graded Hopf algebras.
Furthermore, let X = {x 1 , x 2 , . . .}. The set of polynomials on noncommutative variables X equipped shuffle product x G and deconcatenation coprodcut ∆, concatenation product and draw coprodcut ∆ * , respectively, is a pair of dual graded Hopf algebras. In particular, when X = {x}, the corresponding dual graded graphs [18] are the chain and weighed chain [35]. When X = {x 1 , x 2 }, the corresponding dual graded graphs are the binary tree and the weighed binary tree [35]. Similarly, when X = {x 1 , x 2 , . . . , x n }, the corresponding dual graded graphs are the n-ary tree and weighted n-ary tree.
Similarly, if we use global ascents [36] instead of global descents of permutations and renumber atoms from left to right, then we obtain dual graded Hopf algebras (KS , x G , µ, ∆, ν) and (KS , , µ, ∆ * , ν). In this paper we consider a pair of dual graded Hopf algebras. A graded Hopf algebra H has a direct sum decomposition H = i=0 H i , where the degree of elements in H i is i, satisfying that the product of two elements of degree p and degree q is an element of degree p + q. If a bialgebra has a direct sum decomposition H = i=0 H i but the product of two elements of degree p and degree q is not an element of degree p + q, then this biagebra is not graded. If we have a such bialgebra, we can study the following problems: 1. How to verify that a such bialgebra is a Hopf algebra or not? 2. If it is a Hopf algebra, how to figure out its antipode? 3. What is the duality of the bialgebra or Hopf algebra?