6. Dendriform structures 149
6.5. Dendriform shuffle-compatibility
We have seen (in Proposition 5.3) that the kernelKstof a descent statistic st is an ideal of QSym if and only if st is shuffle-compatible. It is natural to ask whether similar combinatorial interpretations exist for when the kernelKst of a descent statistic st is a ź-ideal, a Á-ideal, a ≺-ideals or a -ideal. In this section, we shall prove such interpretations.
Now, let us define two further variants of LR-shuffle-compatibility (to be com-pared with those introduced in Definition 3.16):
Definition 6.17. Let st be a permutation statistic.
(a) We say that st is weakly left-shuffle-compatible if for any two disjoint nonempty permutationsπ andσ having the property that
each entry of π is greater than each entry ofσ, (91) the multiset{stτ | τ ∈S≺(π,σ)}multidepends only on stπ, stσ, |π|and|σ|. (b) We say that st is weakly right-shuffle-compatible if for any two disjoint nonempty permutationsπ andσ having the property that
each entry of π is greater than each entry ofσ,
the multiset{stτ | τ ∈S(π,σ)}multidepends only on stπ, stσ, |π|and|σ|. Then, the following analogues to the first part of Proposition 5.3 hold:
Theorem 6.18. Let st be a descent statistic. Then, the following three state-ments are equivalent:
• Statement A:The statistic st is left-shuffle-compatible.
• Statement B:The statistic st is weakly left-shuffle-compatible.
• Statement C:The set Kst is an ≺-ideal of QSym.
Theorem 6.19. Let st be a descent statistic. Then, the following three state-ments are equivalent:
• Statement A:The statistic st is right-shuffle-compatible.
• Statement B:The statistic st is weakly right-shuffle-compatible.
• Statement C:The set Kst is an -ideal of QSym.
Let us prove Theorem 6.18 directly, without using shuffle algebras:
Proof of Theorem 6.18 (sketched). The implication A=⇒B is obvious.
Proof of the implication B=⇒C:Assume that Statement B holds. Thus, the statis-tic st is weakly left-shuffle-compatible.
Let us show that the setKst is a ≺-ideal of QSym. Indeed, it suffices to show that every two st-equivalent compositionsJandKand every further composition L satisfy
(FJ−FK) ≺FL ∈ Kst and FL ≺(FJ−FK) ∈ Kst (92) (because of the definition ofKst). So let J and K be two st-equivalent composi-tions, and let L be a further composition. If J = K, then (92) follows immedi-ately from realizing that FJ −FK = 0; thus, we WLOG assume that J 6= K. But
|J|=|K|(sinceJ andK are st-equivalent). Hence,|J| =|K| >0 (since otherwise, we would have|J| =|K| =0, which would imply that both J andKwould be the empty composition, contradicting J 6= K). Thus, the power series FJ and FK are homogeneous of degree |J| = |K| > 0; consequently, ε(FJ) = 0 and ε(FK) = 0.
Hence,ε(FJ−FK) = ε(FJ)
| {z }
=0
−ε(FK)
| {z }
=0
=0.
The compositions J and K are nonempty (since |J| = |K| > 0). If L is empty, then (92) holds for easy reasons (indeed, we have FL =1 in this case, and there-fore (75) yields
(FJ −FK) ≺FL = (FJ −FK)−ε(FJ−FK)
| {z }
=0
=FJ−FK ∈ Kst,
and similarly (74) leads to FL ≺(FJ −FK)∈ Kst). Hence, we WLOG assume that L is nonempty.
Pick three disjoint permutations ϕ, ψand σhaving descent compositions J, K and L, respectively, and having the property that
each entry of ϕis greater than each entry ofσ and
each entry ofψ is greater than each entry ofσ.
(Such permutations ϕ, ψandσ exist, since the set Pis infinite.)
The permutations ϕand ψare st-equivalent (since their descent compositions J and Kare st-equivalent). In other words, |ϕ| =|ψ|and stϕ=stψ.
The statistic st is weakly left-shuffle-compatible. Thus, the multiset
{stτ | τ ∈ S≺(π,σ)}multi (where π is a nonempty permutation disjoint from σ and having the property that each entry of π is greater than each entry
of σ) depends only on stπ and |π| (by the definition of “weakly left-shuffle-compatible”)76. Therefore, the multisets {stτ | τ ∈ S≺(ϕ,σ)}multi and
{stτ | τ ∈ S≺(ψ,σ)}multi are equal (since |ϕ| = |ψ| and stϕ = stψ). Hence, there exists a bijection α : S≺(ϕ,σ) → S≺(ψ,σ) such that each χ ∈ S≺(ϕ,σ) satisfies
st(α(χ)) =stχ. (93)
Consider thisα. Clearly, each χ ∈S≺(ϕ,σ) satisfies (χand α(χ) are st -equivalent) (because of (93) and since|χ| = |ϕ|
|{z}
=|ψ|
+|σ|=|ψ|+|σ| =|α(χ)|) and therefore
(Compχ and Comp(α(χ)) are st -equivalent)
(since st is a descent statistic) and thus FCompχ−FComp(α(χ)) ∈ Kst (by the defi-nition ofKst) and therefore
FCompχ ≡FComp(α(χ))modKst. (94) The first claim of Corollary 6.10 yields
FCompϕ≺ FCompσ =
∑
χ∈S≺(ϕ,σ)
FCompχ and FCompψ≺ FCompσ =
∑
χ∈S≺(ψ,σ)
FCompχ. Hence,
FCompϕ≺ FCompσ =
∑
χ∈S≺(ϕ,σ)
FCompχ
| {z }
≡FComp(α(χ))modKst (by (94))
≡
∑
χ∈S≺(ϕ,σ)
FComp(α(χ))
=
∑
χ∈S≺(ψ,σ)
FCompχ
here, we have substituted χfor α(χ) in the sum, since the mapα : S≺(ϕ,σ) →S≺(ψ,σ) is a bijection
=FCompψ ≺FCompσmodKst.
Since Compϕ = J, Compψ = K and Compσ = L (by the definition of ϕ, ψ and σ), this rewrites as FJ ≺FL ≡ FK ≺FLmodKst. In other words, FJ ≺FL − FK ≺FL ∈ Kst. In other words, (FJ−FK) ≺FL ∈ Kst. This proves the first claim of (92). The second is proven similarly. Altogether, we thus conclude that Kst
76Recall thatσ is fixed here, which is why we don’t have to say that it depends on stσand|σ| as well.
is a ≺-ideal of QSym. In other words, Statement C holds. This proves the implication B=⇒C.
Proof of the implication C=⇒A: Assume that Statement C holds. Thus, the set Kst is an ≺-ideal of QSym.
LetXbe the codomain of the map st. LetQ[X]be the freeQ-vector space with basis ([x])x∈X. Then, we can define a Q-linear map st : QSym → Q[X], FJ 7→
[stJ]. This map st sends each of the generators of Kst to 0 (by the definition of Kst), and therefore sends the wholeKst to 0. In other words, st(Kst) =0.
Now, consider any two disjoint nonempty permutations π and σ having the property that π1 > σ1. Also, consider two further disjoint nonempty permuta-tions π0 and σ0 having the property that π10 > σ10 and satisfying stπ = st(π0), stσ =st(σ0), |π|=|π0|and |σ|=|σ0|. We shall show that
{stτ | τ ∈ S≺(π,σ)}multi =stτ | τ ∈ S≺ π0,σ0 multi.
This will show that the multiset{stτ | τ ∈ S≺(π,σ)}multidepends only on stπ, stσ, |π| and |σ|.
From stπ =st(π0)and|π| =|π0|, we conclude thatπandπ0are st-equivalent.
In other words, Compπ and Comp(π0) are st-equivalent. Hence, FCompπ − FComp(π0) ∈ Kst (by the definition of Kst), so that FCompπ ≡ FComp(π0)modKst. Similarly,FCompσ ≡ FComp(σ0)modKst. These two congruences, combined, yield FCompπ ≺ FCompσ ≡ FComp(π0) ≺FComp(σ0)modKst. (Indeed, we can conclude a≺c ≡b ≺dmodKst whenever we have a ≡bmodKst and c ≡dmodKst; this is because we know thatKst is a ≺-ideal of QSym.)
FromFCompπ ≺FCompσ ≡FComp(π0) ≺ FComp(σ0)modKst, we obtain st FCompπ ≺FCompσ
=st
FComp(π0) ≺FComp(σ0)
(95) (sincest(Kst) =0).
The first claim of Corollary 6.10 yields
FCompπ ≺ FCompσ =
∑
χ∈S≺(π,σ)
FCompχ. Applying the mapst to both sides of this equality, we find
st FCompπ ≺ FCompσ
=st
∑
χ∈S≺(π,σ)
FCompχ
=
∑
χ∈S≺(π,σ)
st FCompχ
| {z }
=[st(Compχ)]=[stχ]
=
∑
χ∈S≺(π,σ)
[stχ].
Similarly,
st
FComp(π0) ≺FComp(σ0)
=
∑
χ∈S≺(π0,σ0)
[stχ].
But the left-hand sides of the last two equalities are equal (because of (95));
therefore, the right-hand sides must be equal as well. In other words,
∑
χ∈S≺(π,σ)
[stχ] =
∑
χ∈S≺(π0,σ0)
[stχ].
This shows exactly that{stχ | χ∈ S≺(π,σ)}multi ={stχ | χ ∈S≺(π0,σ0)}multi. In other words,{stτ | τ ∈ S≺(π,σ)}multi ={stτ | τ ∈ S≺(π0,σ0)}multi. Thus, we have proven that the multiset {stτ | τ ∈ S≺(π,σ)}multi depends only on stπ, stσ, |π| and |σ|. Hence, the statistic st is left-shuffle-compatible. In other words, Statement A holds. This proves the implication C=⇒A.
Now that we have proven all three implications A=⇒B, B=⇒C and C=⇒A, the proof of Theorem 6.18 is complete.
Proof of Theorem 6.19. The proof of Theorem 6.19 is analogous to the above proof of Theorem 6.18.
Corollary 6.20. Let st be a permutation statistic that is LR-shuffle-compatible.
Then, st is a shuffle-compatible descent statistic, and the set Kst is an ideal and a ≺-ideal and a -ideal of QSym.
Proof of Corollary 6.20 (sketched). Proposition 3.18 yields that st is head-graft-compatible and shuffle-compatible. Proposition 3.17 shows that st is left-shuffle-compatible, right-shuffle-compatible and head-graft-compatible (since st is LR-shuffle-compatible).
Hence, Proposition 4.5 shows that st is a descent statistic. Thus, Theorem 6.18 yields that Kst is a ≺-ideal of QSym (since st is left-shuffle-compatible). Like-wise, Theorem 6.19 yields thatKstis a -ideal of QSym (since st is right-shuffle-compatible). Finally, Proposition 5.3 yields thatKstis an ideal of QSym (since st is a shuffle-compatible descent statistic). This proves Corollary 6.20.
A converse of Corollary 6.20 also holds:
Corollary 6.21. Let st be a descent statistic such that Kst is a ≺-ideal and a -ideal of QSym. Then, st is LR-shuffle-compatible and shuffle-compatible.
Proof of Corollary 6.21 (sketched). Theorem 6.18 yields that st is left-shuffle-compatible (sinceKstis an ≺-ideal of QSym). Likewise, Theorem 6.19 yields that st is right-shuffle-compatible (since Kst is an -ideal of QSym). Hence, Corollary 3.23 shows that st is LR-shuffle-compatible. Thus, Proposition 3.18 yields that st is head-graft-compatible and shuffle-compatible. This proves Corollary 6.21.
As a consequence of Theorem 6.18 and Theorem 6.19, we can see that any descent statistic that is weakly left-shuffle-compatible and weakly right-shuffle-compatible must automatically be shuffle-right-shuffle-compatible77. Note that this is only
77Proof. Let st be a descent statistic that is weakly left-shuffle-compatible and weakly right-shuffle-compatible. We must prove that st is right-shuffle-compatible.
The implication B=⇒C in Theorem 6.18 shows that the setKstis a ≺-ideal of QSym. Sim-ilarly, the setKstis a -ideal of QSym. Hence, Proposition 6.13 (applied toM=Kst) yields thatKstis an ideal of QSym. By Proposition 5.3, this shows that st is shuffle-compatible.
true for descent statistics! As far as arbitrary permutation statistics are con-cerned, this is false; for example, the number of inversions is weakly left-shuffle-compatible and weakly right-shuffle-left-shuffle-compatible but not shuffle-left-shuffle-compatible.
Recall that every permutation statistic that is left-shuffle-compatible and right-shuffle-compatible must automatically be LR-right-shuffle-compatible (by Corollary 3.23) and therefore also shuffle-compatible (by Corollary 3.22) and head-graft-compatible (again by Corollary 3.22) and therefore a descent statistic (by Propo-sition 4.5).
Corollary 6.22. The descent statistic Epk is left-shuffle-compatible and right-shuffle-compatible.
Corollary 6.22 follows by combining Theorem 3.12(c)with Theorem 3.17. But we can also give a proof using Theorem 6.18:
Proof of Corollary 6.22. To prove that Epk is left-shuffle-compatible, combine The-orem 6.18 with TheThe-orem 6.15. Similarly for right-shuffle-compatibility.
Using Theorem 6.3, we can state an analogue of Theorem 4.14. Let us first define the notion of dendriform algebras:
Definition 6.23. (a) A dendriform algebra over a field k means a k-algebra A equipped with two further k-bilinear binary operations ≺ and (these are operations, not relations, despite the symbols) from A×A to A that satisfy the four rules
a≺b+ab= ab;
(a≺b) ≺c= a≺(bc); (ab) ≺c= a(b ≺c); a(bc) = (ab) c
for all a,b,c ∈ A. (Depending on the situation, it is useful to also impose a few axioms that relate the unity 1 of the k-algebra A with the operations ≺ and . For example, we could require 1≺a = 0 for each a ∈ A. For what we are going to do in the following, it does not matter whether we make this requirement.)
(b)If Aand Bare two dendriform algebras overk, then adendriform algebra homomorphismfrom AtoBmeans ak-algebra homomorphismφ: A →B pre-serving the operations ≺ and (that is, satisfying φ(a≺b) = φ(a) ≺φ(b) and φ(ab) = φ(a)φ(b) for all a,b ∈ A). (Some authors only require it to be a k-linear map instead of being a k-algebra homomorphism; this boils down to the question whether φ(1) must be 1 or not. This does not make a difference for us here.)
Thus, QSym (with its two operations ≺ and ) becomes a dendriform algebra overQ.
Notice that if A and B are two dendriform algebras over k, then the kernel of any dendriform algebra homomorphism A → Bis an ≺-ideal and a -ideal of A. Conversely, if A is a dendriform algebra over k, and I is simultaneously a ≺-ideal and a -ideal of A, then A/I canonically becomes a dendriform algebra, and the canonical projection A → A/I becomes a dendriform algebra homomorphism.
Therefore, Theorem 6.18 and Theorem 6.19 (and theAst ∼=QSym /Kst isomor-phism from Proposition 5.3) yield the following:
Corollary 6.24. If a descent statistic st is left-shuffle-compatible and right-shuffle-compatible, then its shuffle algebraAst canonically becomes a dendri-form algebra.
We furthermore have the following analogue of Theorem 4.14, which easily follows from Theorem 6.18 and Theorem 6.19:
Theorem 6.25. Let st be a descent statistic.
(a) The descent statistic st is left-shuffle-compatible and right-shuffle-compatible if and only if there exist a dendriform algebra A with basis (uα) (indexed by st-equivalence classesα of compositions) and a dendriform alge-bra homomorphismφst : QSym→ A with the property that wheneverα is an st-equivalence class of compositions, we have
φst(FL) = uα for each L ∈ α.
(b)In this case, theQ-linear map
Ast → A, [π]st 7→ uα,
where α is the st-equivalence class of the composition Compπ, is an isomor-phism of dendriform algebrasAst → A.
Question 6.26. Can the Q-algebra PowN from Definition 2.20 be endowed with two binary operations ≺ and that make it into a dendriform algebra?
Can we then find an analogue of Proposition 2.25 along the following lines?
Let (P,γ), (Q,δ) and (PtQ,ε) be as in Proposition 2.25. Assume that each of the posets P and Q has a minimum element; denote these elements by minP and minQ, respectively. Define two posets P≺Q and PQ as in Proposition 6.8. Then, we hope to have
ΓZ (P,γ)≺ΓZ (Q,δ) = ΓZ (P≺Q,ε) and ΓZ (P,γ)ΓZ (Q,δ) = ΓZ (PQ,ε),
assuming a simple condition on minP and minQ (say, γ(minP) <Z δ(minQ)).
Ideally, this would be a generalization of Proposition 6.8.