On the Product of Generating Functions for Domino and Bi-Tableaux

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Let X = {x1, x2, ···} and Y = {y1,y2, ···} be two alphabets of commutative indeterminates. Denote also XY the product alphabet {xiyj }i,j ordered by the lexicographical order. Let FI (X) be the Gessel’s fundamental quasisymmetric function on X indexed by the subset I ⊂ [n − 1]:

FI (X)= xi1 xi2 ··· xin . (1)

i1≤···≤in k∈I⇒Xik

The power series FI (X) is not symmetric in X but verifies the property that for any strictly increasing sequence of indices i1 < i2 < ··· < ip the coefficient k k xk1 xk2 x p xk1 xk2 x p s X of 1 2 ··· p is equal to the coefficient of i1 i2 ··· ip . Let λ( ) be the Schur polynomial indexed by partition λ on alphabet X. Schur polynomials are the generating functions for semistandard Young tableaux. As a result,

T sλ(X)= X = FDes(T )(X). (2) T ∈SSYX T (λ) T ∈SYX T (λ)

Using for any π ∈ Sn Gessel’s formula for the coproduct of fundamental quasisymmetric functions in [4]

FDes(π)(XY )= FDes(σ)(X)FDes(ρ)(Y ), (3) σ,ρ S σρ π ∈ Xn; = and the Cauchy formula for Schur polynomials

sλ(X)sλ(Y )= s(n)(XY ), (4) λ n X⊢ one can derive

FDes(T )(X)FDes(U)(Y )= F∅(XY )= FDes(π)(X)FDes(π−1 )(Y ).

λ⊢n π∈Sn T,U∈XSY T (λ) X (5) This gives a direct analytical proof of one important property usually proved using the Robinson-Schensted correspondence, i.e. that permutations π in Sn are in bijection with pairs of standard Young tableaux T,U of the same shape such that Des(T )= Des(π) and Des(U)= Des(π−1).

1.3 Product of Schur polynomials

One may write a permutation α ∈ Sn as a word on the letters in [n], α = α1α2 ··· αn where αi = α(i). Given α ∈ Sn and β ∈ Sm, let α β be the set of permutations γ ∈ Sm+n obtained by shuﬄing the letters α and β = n + β1 n + β2 ··· n + βm such that the initial order of the letters of α (resp. of β) is preserved in γ. By extension if A and B are two sets of permutations, we

2 denote A B the set of all the shuﬄes of a permutation of A and a permutation of B. We have the product formula

FDes(α)(X)FDes(β)(X)= FDes(γ)(X). (6) γ α β ∈X Given a standard Young tableau T of shape λ, Equation (5) proves the existence −1 of classes of permutations CT such that for any π ∈ CT , Des(π ) = Des(T ) and there is a descent preserving bijection between CT and SYT (λ).

Remark 1. Classically the CT are the Knuth classes of permutations but here we derive these results directly without using plactic relations and/or the Robinson- Schensted correspondence. Equation (6) directly implies for integer partitions λ and µ and T,U ∈ SYT (λ)× SYT (µ):

sλ(X)sµ(X)= FDes(α)(X) FDes(β)(X)= FDes(γ)(X). (7) α C β C γ C C X∈ T X∈ U ∈ XT U ν For any ν ⊢ |λ| + |µ| define formally the coefficients cλµ as the coefficients of the expansion of sλ(X)sµ(X) in the Schur basis (sν )ν . Equation (7) shows for some Vν ∈ SYT (ν) that

ν ν cλµsν (X)= cλµFDes(γ)(X)= FDes(γ)(X). (8) γ C C ν⊢|Xλ|+|µ| ν⊢|Xλ|+|µ| ∈ XT U γ∈CVν As a result, we have the following proposition.

Proposition 1. For I ⊂ [|λ| + |µ|− 1] denote DI the set of permutations π such that Des(π)= I. Using the notations above for integer partitions λ, µ and for T,U in SYT (λ) × SYT (µ), there is a set of tableaux Vν ∈ SYT (ν) indexed by the set of integer partitions ν ⊢ |λ| + |µ| such that

ν cλµ|CVν ∩ DI | = | (CT CU ) ∩ DI |. (9) ν⊢|Xλ|+|µ| The following sections are dedicated to the extension of this method to weak composition quasisymmetric functions and Chow’s type B quasisymmetric functions.

2 Type B extension to weak composition quasisymmetric functions 2.1 Weak composition quasisymmetric functions and coloured permutations Weak composition quasisymmetric functions are a generalisation of Gessel’s quasisymmetric functions introduced in [5] and [12] in order to provide a framework

3 for the study of a question a G.-C. Rota relating Rota-Baxter algebras and symmetric type functions. We consider a ﬁnite alphabet X = {x1, x2, ...,xM }. For any non-negative integer n, let WC(n) be the set of weak compositions of n, i.e. the set of ordered tuples of non-negative integers that sum up to n. A weak composition α ∈ WC(n) is of the form

i1 i2 ik ik+1 α = (0 ,s1, 0 ,s2,..., 0 ,sk, 0 ) with ij ≥ 0, 1 ≤ j ≤ k + 1, si > 0, 1 ≤ j ≤ k and si = n. We further deﬁne ℓ(α)= k + i1 + ··· + ik+1 the length of α, ℓ0(α)= i1 + ··· + ik+1 as its P 0-length, |α| = n the weight of α and ||α|| = |α| + ℓ0(α) the total weight of α. The descent set of α is

D(α)= {a1,...,ak|aj = i1 + s1 + ··· + ij + sj , j =1,...,k}.

j1 j2 jk jk Deﬁnition 1. Given a weak composition α = (0 ,s1, 0 ,s2,..., 0 ,sk, 0 +1 ) with D(α) = {a1,a2,...,ak} deﬁne the weak composition fundamental quasisymmetric function indexed by α the formal power series

Fα(X) = e 0 0 0 0 X xi ··· xi xij +1 ··· xia ··· xia +j +1 ··· xia xi ··· xi (10) 1 j1 1 1 k−1 k k ak+1 ak+jk+1 1≤i ≤···≤i 1 ak+jk+1 j∈D(α)⇒ij 0. We look at chain posets Pπ = {π1

··· y ⇒ f(x)

δ 0 if π is an overlined letter, w(f)= x πi , where δ = i f(πi) πi π P (1 otherwise. iY∈ π

Denote A(π) the set of Pπ-partitions. We are interested in the generating functions Γ indexed by coloured permutations π

Γ(π)= w(f). (11) f∈AX(π) Li shows in [6] that the Γ(π) can be expressed as a Z-linear combination of the Fα but not always positive. In particular if the indices j1 < j2 < ··· < jp of the

4 overlined letters in a coloured permutation π are such that πj1 < πj2 < ··· < πjp , denote the weak composition

i1 i2 ik ik+1 comp(π)=(0 ,s1, 0 ,s2,..., 0 ,sk, 0 ) where (s1,s2,...,sk) is the tuple of lengths of the sequences of consecutive increasing non-overlined letters in π and ir is the quantity of overlined letters (p) between sequences r − 1 and r. We have ℓ0(comp(π)) = p and we denote Sn the set of such coloured permutations with |comp(π)| = n. As an example for π = 597126384, we have comp(π)=(2, 1, 02, 1, 0, 1, 0). As a direct consequence (p) of Equation (10), for n,p ≥ 0 and π ∈ Sn , one has ([6])

Fcomp(π) = Γ(π).

On the other hand, take π =32e1, then it is easy to show that Γ(π)= F(1,02) − (p) (q) F(1,0). Finally, as in Section 1.3, define for α, β ∈ Sn × Sr the set α β of coloured permutations obtained by shuffling the letters of α and β wheree the non-overlinede letters of α are shifted by q to get α and the overlined (resp. non- overlined) letters of β are shifted by p (resp. by p + n) to gete β. Forb instance, let α = 321, β = 12, then α = 431, β = 25 and e b α β = 43125+43215+42315+24315+43251+42351+24351+42531+24531+25431. e b (p) (q) One can see that despite the fact that α and β belong to Sn and Sr , not all the permutations in α β fulfil the conditions of proper order of the overlined (p) (q) letters. We have for α, β ∈ Sn × Sr ,

Fcomp(α)(X)Fcomp(β)(X)= Γ(γ). (12) γ α β ∈X e e In particular for n = r =0 p j p p + q − j F p (X)F q (X)= (−1) F p+q−j (X). (13) (0 ) (0 ) j p (0 ) j=0 X In some sense,e weake composition fundamental quasisymmetrice functions are a specialisation of Poirier quasisymmetric functions (see [9]), a classical type B generalisation of Gessel’s quasisymmetric functions that we do not detail here. In [1], the authors provide a connexion between the generating functions for Young bi-tableaux and the fundamental quasisymmetric functions of Poirier. As a result, there should be a natural bi-tableaux interpretation of weak composition quasisymmetric functions. This is the focus of the next section.

2.2 Tableau-theoretic interpretation Let λ be an integer partition and p a non-negative integer. We consider pairs of Young tableau (T −,T +) where T − is of shape (p) and T + has shape λ. We call such a bi-tableau a (p)-tableau. We proceed with the formal deﬁnition.

5 Definition 2. For any couple of non-negative integer p,n and an integer partition λ ⊢ n, a semistandard (p)-tableau T = (T −,T +) is a pair of Young diagrams of bi-shape ((p), λ) whose boxes are filled with the integers in [M] such that the entries are strictly increasing down the columns and non-decreasing along the rows. The shape of T is noted sh(T ) = ((p), λ) and we denote SSBT (p)(λ) the set of all (p)-tableaux of shape ((p), λ). If the boxes are filled with integers in [n+p] all used exactly once, we call T a standard (p)-tableau and we denote SBT (p)(λ) the set of such tableaux. We also need the following notions. The standardisation T st of a semistandard (p)-tableau T is the standard (p)-tableau of shape ((p), λ) obtained by relabelling the boxes of T with all the integers in [|λ| + p] such that the order of the labels is preserved. When a label in T is used both in T − and T +, the boxes of T − are relabelled first. Finally, given a standard (p)-tableau T ∈ SBT (p)(λ) with |λ| = n denote comp(T ) the weak composition of n, i1 i2 ik ik+1 comp(T ) = (0 ,s1, 0 ,s2,..., 0 ,sk, 0 ), where i1 is the number of con- − + secutive labels k = 1, 2 ...,i1 in T (i1 = 0 if 1 is in T ). Then s1 is the + number of consecutive labels k = i1 +1, ··· ,i1 + s1 in T such that no label k is in a higher row than k +1. Then i2 is the number of consecutive labels − k = i1 + s1 +1, 2 ...,i1 + s1 + i2 in T , etc... Example 1. The following semistandard (3)-tableau belongs to SSBT (3)((2, 2, 1)).

1 4 T = T − = 2 4 4 ,T + = 5 5   9        Furthermore, its standardisation is:

1 5 T st = T − = 2 3 4 ,T + = 6 7   8        Finally the weak composition of T st is comp(T st)=(1, 03, 1, 2, 1). (p) For λ ⊢ n and p ≥ 0 denote sλ the generating function for such tableaux. We have

(p) T + sλ (X)= X . (14) (p) T ∈SSBTX (λ) Recall the notation of the Schur symmetric function sλ indexed by λ. The following lemma is immediate. Lemma 1. For any λ ⊢ n and p ≥ 0, one has M + p − 1 s(p)(X)= s (X). (15) λ p λ

6 We can now state the main result of this section. Proposition 2. For non-negative integers p and n and an integer partition λ ⊢ n the generating function for (p) − tableau of shape ((p), λ) is related to weak composition fundamental quasisymmetric functions through

(p) sλ (X)= Fcomp(T )(X). (16) (p) T ∈SBTX (λ) e Proof. (sketch) Starting from Equation (14), sort all the semi-standard (p)- tableaux according to their standardisation. Then, given a standard (p)-tableau T0 show that T + Fcomp(T0)(X)= X . (p) st T ∈SSBT X(λ);T =T0 e Finally, sum on SBT (p)(λ) to get the result.

2.3 Product formulas We use Proposition 2 to derive new results. We show the following formula. Theorem 1. Let p,q,n,m be non-negative integers and λ, µ integer partitions ν of n and m. For ν ⊢ |λ| + |µ| recall the coeﬃcients cλ,µ introduced in Section 1.3. The following identity holds.

p p + q − j F (X)F (X)= (−1)j cν F (X). comp(T ) comp(U) j p λµ comp(V ) T ∈SBT (p)(λ) 0≤j≤p X(q) ν⊢|Xλ|+|µ| U∈SBT (µ) e e p q−j e V ∈SBT ( + ) (ν) (17) Proof. (sketch) First, note that for any integer p

(p) M + p − 1 s = F p = . ∅ (0 ) p e ν Then use Equation (13) and the deﬁnition of the coeﬃcients cλµ to write

M + p − 1 M + q − 1 s(p)s(q) = cν s , λ µ λµ p q ν ν⊢|Xλ|+|µ| p p p + q − j M + p + q − j − 1 = cν (−1)j s , λµ j p p + q − j ν j=0 ν⊢|Xλ|+|µ| X p p p + q − j = cν (−1)j s(p+q−j), λµ j p ν j=0 ν⊢|Xλ|+|µ| X and apply Proposition 2 to get the result.

7 There is an interesting corollary of Theorem 1 in terms of generating functions for Pπ-partitions. For a (classical) standard Young tableau T , recall the permutation class CT of Section 1.3. Given a non-negative integer p, we denote εp the coloured permutation εp = 12 ··· p. The result can be stated as follows. Corollary 1. For non-negative integers p, q, integers partitions λ and µ and T ∈ SYT (λ), U ∈ SYT (µ), one has:

p p + q − j Γ(γ)= (−1)j cν F (X). j p λµ comp(V ) γ∈(εp CT ) (εq CU ) 0≤j≤p X ν⊢|Xλ|+|µ| − e V ∈SBT (p+q j) (ν) (18) Proof. We omit the proof in this extended abstract.

3 Type B extension to Chow’s quasisymmetric functions 3.1 Signed permutations and domino tableaux

Let Bn be the hyperoctahedral group of order n. Bn is composed of all signed permutations π on {-n, ··· , -2, -1, 0, 1, 2, ··· ,n} such that π(−i)= −π(i) for all i. The total colour of π ∈ Bn is the number tc(π)= |{i ≥ 1; π(i) < 0}|. Its descent set is the subset of {0} ∪ [n − 1] equal to Des(π)= {0 ≤ i ≤ n − 1 | π(i) > π(i + 1)}. The main difference with respect to the case of the symmetric group is the possible descent in position 0 when π(1) is a negative integer. For λ ⊢ 2n,a standard domino tableau T of shape λ is a Young diagram of shape shape(T )= λ tiled by dominoes, i.e. 2 × 1or1 × 2 rectangles filled with the elements of [n] such that the entries are strictly increasing along the rows and down the columns. In the sequel we consider only the set P0(n) of empty 2-core partitions λ ⊢ 2n that fit such a tiling. A standard domino tableau T has a descent in position i> 0 if i + 1 lies strictly below i in T and has descent in position 0 if the domino filled with 1 is vertical. We denote Des(T ) the set of all its descents. A semistandard domino tableau T of shape λ ∈ P0(n) and weight w(T ) = µ = (µ0,µ1,µ2, ··· ) with µi ≥ 0 and i µi = n is a tiling of the Young diagram of shape λ with horizontal and vertical dominoes labelled with integers in {0, 1, 2, ···} such that labels are non decreasingP along the rows, strictly increasing down the columns and exactly µi dominoes are labelled with i. If the top leftmost domino is vertical, it cannot be labelled 0. Denote SDT (λ) (resp. SSDT (λ)) the set of standard (resp. semistandard) domino tableaux of shape λ. Finally, we denote sp(T ), the spin of (semi-) standard domino tableau T , i.e. half the number of its vertical dominoes. Remark 2. The possible labelling of top leftmost horizontal dominoes in semistandard domino tableaux with 0 was first introduced by the authors in [8] and

8 is essential to connect their generating functions with Chow’s quasisymmetric functions.

5 3 0 0 1 2 3 1 2 5 5 6 4 2 2 5 T1 = 4 8 T2 = 6 8 T3 = 5 7 7 7 5

Figure 1: Two standard domino tableaux T1 and T2 of shape (5, 5, 4, 1, 1), descent set {0,3,5,6}, a semistandard tableau T3 of shape (5, 5, 4, 3, 1) and weight µ = (2, 0, 2, 0, 0, 4, 0, 1). All of the tableaux have a spin of 2.

3.2 Chow’s type B quasisymmetric functions Chow deﬁnes in [3] another Type B extension of Gessel’s algebra of quasisymmetric functions that is dual to Solomon’s descent algebra of type B. Let X = {··· , x-i, ··· , x-1, x0, x1, ··· , xi, ···} be an alphabet of totally ordered commutative indeterminates with the assumption that x-i = xi and let I be a subset of {0} ∪ [n − 1], he deﬁnes a type B analogue of the fundamental quasisymmetric functions

B FI (X)= xi1 xi2 ...xin .

0=i0≤i1≤i2≤...≤in j∈I⇒Xij

Note the particular rle of the variable x0. Given two signed permutations α ∈ Bn and β ∈ Bm, denote α β the set of signed permutations in Bm+n obtained by shuﬄing the letters of α and β where we shift the absolute value of the βi to get β. Chow shows in [3, Prop. 2.2.6]

B B B FDes(α)(X)FDes(β)(X)= FDes(γ)(X). (19) γ α β ∈X Furthermore, we introduce in [8] modiﬁed generating functions for domino tableaux called domino functions. Given a semistandard domino tableau T T µ0 µ1 µ2 0 of weight µ, denote X the monomial x0 x1 x2 ··· . For λ ∈P (n) and an ad- ditional parameter q we deﬁne the domino function indexed by λ on alphabet X as sp(T ) T Gλ(X; q)= q X . (20) T ∈SSDTX (λ) These functions are related to Chow’s quasisymmetric functions through

9 Lemma 2 ([7]). For λ ∈ P0(n), the q-domino function indexed by λ can be expanded in the basis of Chow’s quasisymmetric functions as

sp(T ) B Gλ(X; q)= q FDes(T )(X). (21) T ∈SDTX (λ) 3.3 Type B Littlewood-Richardson coeﬃcients There is a natural analogue of the RSK-correspondence for signed permutations involving domino tableaux. Barbash and Vogan ([2]) built a bijection between signed permutations of Bn and pairs of standard domino tableaux of equal shape in P0(n). An independent development on the subject is due to Garﬁnkle. Ta¸skin ([11, Prop. 26]) showed that the two standard domino tableaux T and U associated to a signed permutation π by the algorithm of Barbash and Vogan have respective descent sets Des(T ) = Des(π−1) and Des(U) = Des(π) while Shimozono and White showed in [10] the colour-to-spin property i.e. tc(π)= sp(T )+ sp(U). Let X, Y be two alphabets of indeterminates. In [7], we show that Equation (21) implies the following proposition. Proposition 3 ([7]). For any integer n, we have

tc(π) B B q FDes(π)(X)FDes(π−1 )(Y )= π B X∈ n sp(T )+sp(U) B B q FDes(T )(X)FDes(U)(Y ). (22) λ∈P0(n) T,U∈XSDT (λ)

Proposition 3 provides a direct analytical proof of the properties showed in a bijective fashion by Barbash, Vogan, Ta¸skin, Shimonozo and White. More formally it proves the existence for any standard domino tableau T of a class B B −1 of signed permutations CT such that for any π ∈ CT , Des(π ) = Des(T ) B and there is a descent preserving bijection CT → SDT (λ), π 7→ U such that tc(π) = sp(T )+ sp(U). In a similar fashion as in Section 1.3, we use Equation (22) to show some properties for the type B Littlewood-Richardson coeﬃcients.

First we focus on the case q = 1. There is a well known (not descent preserving) bijection between semistandard domino tableaux of weight µ and − + + − semistandard bi-tableaux of respective weights µ and µ such that µi + µi = µi for all i. The respective shapes of the two Young tableaux depend only on the shape of the initial semistandard domino tableau. Denote (T −,T +) the bi- tableau associated to a semistandard domino tableau T of shape λ and (λ−, λ+) their respective shapes. (T −,T +) (resp. (λ−, λ+)) is the 2-quotient of T (resp. ∗ 0 λ). Denote X = X \ x0. For λ ∈P (n) one has

∗ Gλ(X;1) = sλ− (X )sλ+ (X)

10 and for λ, µ ∈P0(n) ×P0(m),

− ν ν+ Gλ(X; 1)Gµ(X;1) = cλ−µ− cλ+µ+ Gν (X; 1). (23) 0 ν∈PX(n+m) The following result extends Proposition 1 to signed permutations. Theorem 2. Let λ, µ ∈ P0(n) ×P0(m). For I ⊂ [0] ∪ [n + m − 1] denote B B DI the set of signed permutations π such that Des(π) = I and ∆I the set of standard domino tableaux of descent set I. Using the notations above for integer partitions λ, µ and T,U in SDT (λ) × SDT (µ)

− ν ν+ B B B B cλ−µ− cλ+µ+ |SDT (ν) ∩ ∆I | = CT CU ∩ DI . (24) ν∈P0(n+m) X

Proof. (sketch) Theorem 2 is a consequence of Equations (23) and (13). On the other hand, the general case for q remains open. Indeed, using Proposition 3, one can show for λ ∈P0(n) and T ∈ SDT (λ)

tc(π)−sp(T ) B Gλ(X; q)= q FDes(π). π∈CB XT Then, if µ ∈P0(m) and U ∈ SDT (µ)

−(sp(T )+sp(U)) tc(γ) B Gλ(X; q)Gµ(X; q)= q q FDes(γ). γ∈CB CB XT U B B However, even if we showed that the set of signed permutations CT CU is in − ν ν+ descent preserving bijection with a set of cardinality cλ−µ− cλ+µ+ |SDT (ν)|, we cannot say a priori how this bijection preserves or transforms the statistic tc.

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