Powersum Bases in Quasisymmetric Functions and Quasisymmetric Functions in Non-commuting Variables

We introduce a new $P$ basis for the Hopf algebra of quasisymmetric functions that refine the symmetric powersum basis. Unlike the quasisymmetric power sums of types 1 and 2, our basis is defined combinatorially: its expansion in quasisymmetric monomial functions is given by fillings of matrices. This basis has a shuffle product, a deconcatenate coproduct, and has a change of basis rule to the quasisymmetric fundamental basis by using tuples of ribbons. We lift our quasisymmetric powersum $P$ basis to the Hopf algebra of quasisymmetric functions in non-commuting variables by introducing fillings with disjoint sets. This new basis has a shifted shuffle product and a standard deconcatenate coproduct, and certain basis elements agree with the fundamental basis of the Malvenuto-Reutenauer Hopf algebra of permutations. Finally we discuss how to generalize these bases and their properties by using total orders on indices.


Introduction
The Hopf algebra of symmetric functions, denoted Sym and indexed by partitions λ and µ, is a very well known space for its connections in representation theory and other areas of mathematics.Some of the well studied bases are the monomial basis m λ , powersum basis p λ , and Schur basis s λ .In [12] Zabrocki illustrates the change of basis from p λ to m µ by fillings.In this work, a filling F is a matrix with one non-zero entry in every row.The column (respectively, row) reading is a composition recording the sum of all entries in each column (respectively, row) denoted as col(F) (respectively, row(F)).Thus the change of basis is combinatorially defined as where A(λ) is the set of all the distinct fillings with row reading λ and the column reading is a partition.For example, p (2,2,1) = 2m (2,2,1) + 2m (3,2) + m (4,1) + m (5) since all the fillings of A(2, 2, 1) are .
One of the important properties of the powersum basis is the Murnaghan-Nakayama rule which illustrates the product rule of s λ and p µ expanded in terms of Schur functions.
A space that contains Sym is the Hopf algebra of quasisymmetric functions, QSym, and is indexed by compositions α and β.This space was defined in [7] by using P -partitions, which generates the fundamental quasisymmetric functions F α .In [4] the authors introduced two quasisymmetric powersum bases (i.e. a basis of QSym that refines p λ ) Φ and Ψ whose duals are the Φ and Ψ bases in NSym which was introduced in [6].These two bases aren't defined combinatorially, but by using series.The Ψ basis is most notable for the change of basis to the fundamental basis by using P -partitions as defined in [1].In [2] the authors introduced the Shuffle basis S α , which also refines p λ , is notable because S α is an eigenvector under the theta map Θ.
In this extended abstract we define a quasisymmetric powersum basis P α combinatorially by using fillings in an analog of Equation 1.Alternatively, P α can be defined using a subposet of the refinement poset P on compositions.Both the fillings and the subposet have generalizations that depend on a total order on the parts of the compositions.Hence we can define a whole family of quasisymmetric powersum bases, denoted as P α for different choices of , such that every basis has a shuffle product, a deconcatenate coproduct, and they refine the symmetric powersum basis.When is the usual order ≥, our quasisymmetric powersum basis is dual (up to scaling) to the Zassenhaus functions Z α in NSym, the algebra of non-commutative symmetric functions, as defined in [8].All these results are stated in Section 3. We note that the special cases P ≥ α and P ≤ α were independently defined in [3] using weighted generating functions of P-partitions.

NSym FQSym
Sym QSym NCSym NCQSym In Section 4, we extend the above results to non-commuting variables -that is, we define a family of powersum bases P Φ in NCQSym, which refine the symmetric powersum basis p φ of NCSym as defined in [10].These bases have a shifted shuffle product and a deconcatenate coproduct.Several particular bases in the family contains the fundamental basis F τ of FQSym.
Finally, in Section 5 we prove a Murnaghan-Nakayama change of basis rule from the quasisymmetric powersum basis to the quasisymmetric fundamental basis.
Figure 1 above summarizes the relationships between our new powersum bases for QSym and NCQSym and existing powersum bases in other algebras.

Preliminaries
Let X = {x 1 , x 2 . ..} be a set of variables and α = (a 1 , . . ., a k ) be a composition of n.Then a generating function is quasisymmetric if, for every k and i The set of all quasisymmetric functions is a graded Hopf algebra denoted as QSym.One of the more natural bases of QSym is the quasisymmetric monomial basis M α , defined as QSym is most notable for the quasisymmetric fundamental basis F α because it corresponds to the characters of the 0-Hecke algebra.To define this basis, let P be the refinement poset on compositions, i.e. with the cover relation α ⋖ β if β = (a 1 , . . ., a i + a i+1 , . . ., a j ) for some i.Then Functions in NCQSym are indexed by set compositions (also called ordered set partitions), which are obtained by replacing each integer part of a composition by a set.To be more exact, a set composition Φ = (B 1 , . . ., B k ) of n is a composition of subsets of [n] such that B i ∩ B j = ∅ where i = j and the size of Φ is In this extended abstract we will denote a set composition (B 1 , . . ., B k ) as B 1 | . . .|B k , for example the set composition ({5}, {1, 3}, {2}, {4}) is written as 5|13|2|4.The refinement poset P on set compositions has the cover relation Φ Denote ρ as the map from set compositions of n to compositions of n by Φ → Quasisymmetric function in non-commuting variables, denoted NCQSym, is the space spanned by M Φ where M Φ is defined as The map ρ from set compositions to compositions induces a Hopf morphism that we will also denote as ρ: ρ : NCQSym → QSym is given by ρ(M Φ ) = M ρ(Φ) .

Descending Quasisymmetric Powersum Basis
In this section we define a set of fillings analogous to A(λ) in order to define the quasisymmetric powersum basis.Let α = (a 1 , . . ., a k ) be a composition of n.Definition 1.A Strict Diagonal filling is a filling such that: 1. entry a 1 is in the upper left most corner of the matrix.
2. entry a i+1 is directly below a i or is in the southeast position of a i if a i ≥ a i+1 , or a i+1 is in the southeast position of a i otherwise.
Denote the set of strict diagonal fillings with row(F) = α as SD(α).Given any filling and any integer i, we denote by σ i a permutation of the rows with entries i.We will only consider σ i such that for any two entries a j = a k = i, σ i (a j ) = a k implies that a j and a k are not in the same column.Denote σ = (σ 1 , • • • , σ n ) as a tuple of row permutations of all the parts of α.The set of all such row permutations for a filling F is S F .Thus we define the descending quasisymmetric powersum basis Example 1.Let the first two fillings below be denoted as F 1 and F 2 , then SD(212) = {F 1 , F 2 }.The third and fourth fillings are σ(F 1 ) and σ(F 2 ) where σ = (id, (21)).Thus Theorem 1.The quasisymmetric powersum functions refine the symmetric powersum functions.In other words, The sketch of this proof is as follows: we define a new type of filling Q(λ) such that p λ = F∈Q(λ) M col(F) .Then we show that there is a bijection between Q(λ) and the fillings of α SD(α) where α are all compositions such that sort(α) = λ.

Scaled Quasisymmetric Powersum Basis
We first introduce a scaling factor so as to obtain cleaner formulas for the product and coproduct in the quasisymmetric powersum basis.
The symmetric powersums are self dual such that p λ , p µ = z λ δ λµ where z λ = i i m i (λ) m i (λ)! where m i (λ) is the number of parts of size i in λ.The scaled symmetric powersums are defined as pλ = 1 z λ p λ , and has the property that p λ , pµ = δ λµ .Analogously, we define the scaled quasisymmetric powersum basis as Pα = 1 zα P α , which by Theorem 1 implies that Pα refines pλ .Let Z α be the left Zassenhaus basis in NSym as defined in [8], then the dual of Pα is the left Zassenhaus basis in NSym, i.e.

Product and Coproduct
Let α = (a 1 , . . ., a j ) and β = (b 1 , . . ., b k ) be compositions of n and m respectively.Their concatenation is the composition α|β = (a 1 , . . ., a j , b 1 , . . ., b k ) of n + m, and their shuffle, a set of compositions of n + m, is defined recursively by Theorem 2. The scaled quasisymmetric powersum basis has a shuffle product and a deconcatenate coproduct, i.e.
This can be shown either using the product and coproduct of the left Zassenhaus basis, or by defining a product and coproduct on fillings.We give a rough sketch of the latter, as it can be generalized to the case of non-commuting variables.
For simplicity, consider the case where all parts c 1 , . . ., c k in γ are distinct, so Pγ = F∈SD(γ) M col(F) .Then To biject between the terms on the two right hand sides above, we consider the deconcatenation of fillings: for an SD filling F, we write F = F 1 |F 2 to mean a vertical line is drawn in between two columns of F to get two SD fillings F 1 and F 2 (after removing empty rows).Note that col(F 1 )|col(F 2 ) = col(F) and row(F 1 )|row(F 2 ) = row(F).In [5], the coproduct of matrices in MQSym is defined in the exact same way (except that our fillings are transposed to their matrices).
Example 2. The following are all four deconcatenations of one filling.
A similar bijective proof of the product rule relies on the quasishuffle of fillings, defined recursively as follows.Let F = F 1 | . . .|F j and G = G 1 | . . .|G k , where F i and G i are fillings with a single column.Then set 1. ∅¡F = F¡∅ = F where ∅ is the empty filling where F i + G i ′ is a filling with a single column whose entries are those in F i and G i ′ , arranged in descending order.Note that, for any quasishuffle of F and G, its column reading is a quaishuffle of col(F) and col(G), as in the product of monomial basis elements, and its row reading is a shuffle of row(F) and row(G), leading to a shuffle product for the scaled quasipowersum basis.
Example 3. The quasishuffle of two fillings.
The product of matrices in [5] is defined as the quasishuffle of rows; since the fillings described here are the transpose of their matrices, it is equivalent to our quasishuffle of columns.The only other difference is that the quasishuffle in this extended abstract additionally diagonalizes the fillings.

Generalizations of P α Using Total Orders
Notice that in Definition 1 we put two entries in the same column if a i ≥ a i+1 .However we can generalize a strict diagonal filling according to a total order where we put two entries in the same column if a i a i+1 , then denote the analogously defined quasisymmetric powersum function as P α .Then P α has a shuffle product, deconcatenate coproduct, and refines the symmetric powersums, as we can switch ≥ to in relevant proofs.Note that the scaled version of P ≤ α is the right Zassenhaus basis.An alternative definition of P α uses the subposet R of P, with cover relations of the form (a 1 , . . ., a j ) ⋖ R (a 1 , . . ., a i + a i+1 , . . ., a j ) where a i a i+1 .Then P α = α≤ R β C αβ M β for some numbers C αβ which we won't address here.

Descending Quasisymmetric Powersum Basis in Non-commuting Variables
To generalize the above results to non-commuting variables, we consider fillings containing sets instead of integers, which we denote by F. Analogous to above, the column (respectively, row) reading is a set composition recording the union of all sets in each column (respectively, row), denoted as col( F) (respectively row( F)).
To define a set analog of a SD filling, we need to replace the condition a i ≥ a i+1 in Definition 1 by a comparison of blocks in the set composition.To this end, let min(B i ) be the smallest integer in B i , and define A > D B if either |A| > |B|, or |A| = |B| and min(A) < min(B).Definition 2. A Labelled Diagonal Descending (LDD) filling of Φ is an assignment of the blocks B i to entries of a matrix such that 1. B i is in row i and B 1 is in the first column.

B i+1 can be in the same column as
Example 4. The first two fillings are all the fillings in LDD(34|15|2), the latter four fillings are all the fillings in LDD(15|34|2).
Definition 3. The descending quasisymmetric powersum basis in NCQSym is defined as Example 5.The previous example yields As mentioned in the Introduction, the analogs of Sym and QSym in non-commuting variables are NCSym and NCQSym.NCSym is indexed by set partitions φ.This space has a powersum basis denoted p φ ; further information about NCSym can be found in [10].Just as in Theorem 1, quasisymmetric powersum functions in non-commuting variables refines the symmetric powersum functions in non-commuting variables.Let sort denote the map from set compositions of n to set partitions of n by forgetting the order of the blocks.The proof of this theorem is analogous to the proof of Theorem 1.

Product and Coproduct
Consider disjoint subsets A 1 , . . ., A j , B 1 , . . ., B k of integers.Define the shifted shuffle of Let Ψ ↑ n denote the operation of adding n to every element of let the standardization of Φ, denoted st(Φ), be the set composition of m where the relative order of the elements across all the B i is preserved.
Theorem 4. The quasisymmetric powersum basis of NCQSym has a shifted shuffle product, and its coproduct is given by the standardization of the deconcatenation of blocks, i.e. if The proof is analogous to the sketch above for Theorem 2, the QSym case.

Generalizations of P Φ Using Total Orders
With careful work we can generalize a LDD filling by replacing > D by a different total ordering ⊲ on disjoint integer sets.The resulting powersum P ⊲ Φ refines the symmetric powersums in non-commuting variables, has a shifted shuffle product, and a deconcatenate coproduct.Let B and A be two disjoint sets of different sizes.If there exists a total ordering on integers ≻, such that B ⊲ A implies |B| ≻ |A|, then there exists an analog of Theorem 5 such that sums of P ⊲ Φ project to P ≻ α .

Image of FQSym
FQSym is the Hopf algebra of permutations first introduced in [9], which is a subalgebra of NCQSym and has a fundamental basis F τ .
Theorem 6.The image of the F basis on NCQSym is the quasisymmetric powersum function in non-commuting variables indexed by a singleton set composition, i.e. for a permutation τ Taking the image under ρ of both sides of Equation 6gives the equation ρ(P τ −1 (1|2|•••|n) ) = F des(τ ) , where des denotes the descent composition.It is then natural to ask for the expansion of ρ(P Φ ) into quasisymmetric fundamental functions when the blocks of Φ are not all singletons.To do so, we define two operations on set compositions.

Change of basis
The Murnaghan Nakayama rule has two perspectives in Sym: it is the change of basis rule from symmetric powersums to Schur functions, and it is the product rule of a powersum function and a Schur function.These two perspectives are related due to the fact that symmetric powersums are multiplicative.In QSym, bases are not multiplicative and so we will define the Murnaghan Nakayama rule based off of the first perspective according to [11].Let µ be a partition and k an integer, then the product is where λ/µ is a border strip of size r.This becomes a change of basis rule if we set µ to be the empty partition.Now we move onto the analog in QSym.
Recall that a ribbon is a tableau such that there are no 2 × 2 boxes; for this paper we will allow ribbons to be disconnected.Definition 4. Let α be a composition of n.A filling of the ribbon of α is a Standard Descent Ribbon if the Ribbon filling is standard (i.e. the numbers 1 to n each appear once) and increasing from left to right, and decreasing from top to bottom.
For this paper we will need to use tuples of standard descent ribbons, for example We denote the height in the algorithm as ht(β, α).Finally, denote by SDR(α, β) the set of all standard descent ribbon fillings of D(α, β).We define two more operations on compositions, the analog of ρ C and ρ I of the previous section, before getting to the main theorem.Let α = (a 1 , . . ., a k ) be a composition of n, then break α at its ascents such that α = γ 1 | • • • |γ l where γ i = (a j 1 , . . ., a j 1 +j 2 ) where a j 1 −1 < a j 1 ≥ . . .≥ a j 1 +j 2 < a

Figure 1 :
Figure 1: Diagram of some Hopf Algebras related to QSym

, 1 .
Let β = (b 1 , . . ., b l ) and α = (a 1 , . . ., a k ) ≤ β be compositions of n.D(β, α) = (R 1 , R 2 , . . ., R n ) is a tuple of ribbons made from the following algorithm.RT1 Let the height be equal to 0 initially and let R be the ribbon of β.RT2 For i in [1, . . ., k] one at a time, insert a box into R a i in one of the following three ways: RTa1 If a i−1 = a i (or if i = 1), then insert a box in the southeast position of the bottom right most box.RTa2 If a i−1 = a i and there exists a b j such that (b j , . . ., b l ) forms the ribbon R, then insert a box in the south position of the bottom right most box.RTa3 If a i−1 = a i and there doesn't exist a b j such that (b j , . . ., b l ) forms the ribbon R, then insert a box in the east position of the bottom right most box.RTb Remove the first a i boxes of R and set this as R. The number of rows removed from R, minus 1, is added to the height and becomes the new height.