HAL Id: hal-03092817
https://hal.archives-ouvertes.fr/hal-03092817
Preprint submitted on 2 Jan 2021
HAL
is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or
L’archive ouverte pluridisciplinaire
HAL, estdestinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires
On the product of generating functions for domino and bi-tableaux
Ekaterina Vassilieva
To cite this version:
Ekaterina Vassilieva. On the product of generating functions for domino and bi-tableaux. 2019.
�hal-03092817�
On the product of generating functions for domino and bi-tableaux
Ekaterina A. Vassilieva November 23, 2019
Abstract. The connection between the generating functions of various sets of tableaux and the appropriate families of quasisymmetric functions is a signifi- cant tool to give a direct analytical proof of some advanced bijective results and provide new combinatorial formulas. In this paper we focus on two kinds of type B Littlewood-Richardson coefficients and derive new formulas using weak com- position quasisymmetric functions and Chow’s quasisymmetric functions. In the type A case these coefficients give the multiplication table for Schur functions, i.e. the generating functions for classical Young tableaux. We look at their generalisations involving a set of bi-tableaux and domino tableaux.
1 Type A quasisymmetric functions
1.1 Permutations, Young tableaux and descent sets
For any positive integer nwrite [n] = {1,· · ·, n}, Sn the symmetric group on [n]. A partition λ of n, denoted λ ` n is a sequence λ = (λ1, λ2,· · ·, λp) of `(λ) = p parts sorted in decreasing order such that |λ| = P
iλi = n. A partitionλis usually represented as a Young diagram ofn=|λ|boxes arranged in `(λ) left justified rows so that the i-th row from the top contains λi boxes.
A Young diagram whose boxes are filled with positive integers such that the entries are increasing along the rows and strictly increasing down the columns is called a semistandard Young tableau. If the entries of a semistandard Young tableau are restricted to the elements of [n] and strictly increasing along the rows, we call it astandard Young tableau. The partitionλis theshape of the tableau and we denote SY T(λ) (resp. SSY T(λ)) the set of standard (resp. semistandard) Young tableaux of shapeλ.
One important feature of a permutation π in Sn is its descent set, i.e. the subset of [n−1] defined asDes(π) ={1≤i≤n−1|π(i)> π(i+ 1)}. Similarly, define thedescent set of a standard Young tableau T as Des(T) ={1 ≤ i≤n−1|iis in a strictly higher row thani+ 1}.
1.2 Quasisymmetric expansion of Schur polynomials
LetX ={x1, x2,· · · } and Y ={y1, y2,· · · } be two alphabets of commutative indeterminates. Denote alsoXY the product alphabet{xiyj}i,jordered by the lexicographical order. LetFI(X) be the Gessel’s fundamental quasisym- metric functiononX indexed by the subsetI⊂[n−1]:
FI(X) = X
i1≤···≤in
k∈I⇒ik<ik+1
xi1xi2· · ·xin. (1)
The power series FI(X) is not symmetric inX but verifies the property that for any strictly increasing sequence of indicesi1< i2<· · · < ip the coefficient ofxk11xk22· · ·xkpp is equal to the coefficient of xki1
1xki2
2 · · ·xkip
p. Let sλ(X) be the Schur polynomial indexed by partitionλon alphabetX. Schur polynomials are the generating functions for semistandard Young tableaux. As a result,
sλ(X) = X
T∈SSY T(λ)
XT = X
T∈SY T(λ)
FDes(T)(X). (2)
Using for anyπ ∈Sn Gessel’s formula for the coproduct of fundamental qua- sisymmetric functions in [4]
FDes(π)(XY) = X
σ,ρ∈Sn;σρ=π
FDes(σ)(X)FDes(ρ)(Y), (3) and the Cauchy formula for Schur polynomials
X
λ`n
sλ(X)sλ(Y) =s(n)(XY), (4) one can derive
X
λ`n T ,U∈SY T(λ)
FDes(T)(X)FDes(U)(Y) =F∅(XY) = X
π∈Sn
FDes(π)(X)FDes(π−1)(Y).
(5) This gives a direct analytical proof of one important property usually proved using theRobinson-Schensted correspondence, i.e. that permutations π in Sn
are in bijection with pairs of standard Young tableauxT, U of the same shape such thatDes(T) =Des(π) andDes(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+β1n+β2· · · n+βmsuch that the initial order of the letters ofα(resp. of β) is preserved inγ. By extension ifAandB are two sets of permutations, we
denoteA Bthe set of all the shuffles of a permutation ofAand a permutation ofB. We have the product formula
FDes(α)(X)FDes(β)(X) = X
γ∈α β
FDes(γ)(X). (6)
Given a standard Young tableauT of shapeλ, Equation (5) proves the existence of classes of permutationsCT such that for any π∈CT, Des(π−1) =Des(T) and there is a descent preserving bijection betweenCT andSY T(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 partitionsλandµandT, U ∈SY T(λ)×
SY T(µ):
sλ(X)sµ(X) = X
α∈CT
FDes(α)(X) X
β∈CU
FDes(β)(X) = X
γ∈CT CU
FDes(γ)(X). (7) For any ν ` |λ|+|µ| define formally the coefficients cνλµ as the coefficients of the expansion ofsλ(X)sµ(X) in the Schur basis (sν)ν. Equation (7) shows for someVν∈SY T(ν) that
X
ν`|λ|+|µ|
cνλµsν(X) = X
ν`|λ|+|µ|
γ∈CVν
cνλµFDes(γ)(X) = X
γ∈CT CU
FDes(γ)(X). (8)
As a result, we have the following proposition.
Proposition 1. For I ⊂ [|λ|+|µ| −1] denote DI the set of permutations π such thatDes(π) =I. Using the notations above for integer partitionsλ, µ and forT, U in SY T(λ)×SY T(µ), there is a set of tableaux Vν ∈SY T(ν)indexed by the set of integer partitionsν` |λ|+|µ|such that
X
ν`|λ|+|µ|
cνλµ|CVν∩DI|=|(CT CU)∩DI|. (9) 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 qua- sisymmetric functions
2.1 Weak composition quasisymmetric functions and coloured permutations
Weak composition quasisymmetric functions are a generalisation of Gessel’s qua- sisymmetric functions introduced in [5] and [12] in order to provide a framework
for the study of a question a G.-C. Rota relatingRota-Baxter algebrasand sym- metric 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 ton. A weak composition α∈ WC(n) is of the form
α= (0i1, s1,0i2, s2, . . . ,0ik, sk,0ik+1)
with ij ≥ 0, 1 ≤ j ≤ k+ 1, si > 0, 1 ≤ j ≤ k and Psi = n. We further define`(α) =k+i1+· · ·+ik+1 thelengthofα, `0(α) =i1+· · ·+ik+1 as its 0-length,|α|=ntheweightof αand||α||=|α|+`0(α) thetotal weightof α. The descent set ofαis
D(α) ={a1, . . . , ak|aj =i1+s1+· · ·+ij+sj, j= 1, . . . , k}.
Definition 1. Given a weak composition α= (0j1, s1,0j2, s2, . . . ,0jk, sk,0jk+1) with D(α) = {a1, a2, . . . , ak} define the weak composition fundamental quasisymmetric functionindexed byαthe formal power series
Feα(X) = X
1≤i1≤···≤iak+jk+1 j∈D(α)⇒ij<ij+1
x0i1· · ·x0i
j1xij
1 +1· · ·xia1· · ·xiak−1 +jk+1· · ·xiakx0i
ak+1· · ·x0i
ak+jk+1 (10)
Weak composition fundamental quasisymmetric functions may also be de- scribed in terms ofP-partitions indexed by coloured permutations. Define the total order
1<2<3<· · ·<1<2<3· · ·
and denote |i| =i for i > 0. We look at chain posets Pπ = {π1 <Pπ π2 <Pπ
· · · <Pπ πl} indexed by coloured permutations π for any non-negative integer l where the πi are distinct letters in [l]∪[l] and such that |π1||π2| · · · |πl| has no repeated letters. A Pπ-partition is a map f : Pπ → [M] such that for x <Pπ y,f(x)≤f(y) withx > y⇒f(x)< f(y). Theweightoff,w(f) is the monomial
w(f) = Y
πi∈Pπ
xδf(ππi
i), whereδπi =
(0 ifπi is an overlined letter, 1 otherwise.
DenoteA(π) the set ofPπ-partitions. We are interested in the generating func- tions Γ indexed by coloured permutationsπ
Γ(π) = X
f∈A(π)
w(f). (11)
Li shows in [6] that the Γ(π) can be expressed as aZ-linear combination of the Fαbut not always positive. In particular if the indicesj1< j2<· · ·< jp of the
overlined letters in a coloured permutationπare such thatπj1< πj2<· · ·< πjp, denote the weak composition
comp(π) = (0i1, s1,0i2, s2, . . . ,0ik, sk,0ik+1)
where (s1, s2, . . . , sk) is the tuple of lengths of the sequences of consecutive increasing non-overlined letters inπ andir is the quantity of overlined letters between sequencesr−1 andr. We have`0(comp(π)) =pand we denote Sn(p)
the set of such coloured permutations with|comp(π)| =n. As an example for π= 597126384, we havecomp(π) = (2,1,02,1,0,1,0). As a direct consequence of Equation (10), forn, p≥0 andπ∈Sn(p), one has ([6])
Fecomp(π)= Γ(π).
On the other hand, takeπ= 321, then it is easy to show that Γ(π) =Fe(1,02)− Fe(1,0). Finally, as in Section 1.3, define forα, β ∈ Sn(p)×Sr(q) the setα β of coloured permutations obtained by shuffling the letters ofαe and βbwhere the non-overlined letters ofαare shifted byqto getαeand the overlined (resp. non- overlined) letters ofβ are shifted byp(resp. byp+n) to getβ. For instance,b letα= 321,β = 12, thenαe= 431, βb= 25 and
α β= 43125+43215+42315+24315+43251+42351+24351+42531+24531+25431.
One can see that despite the fact thatαandβ belong toS(p)n andSr(q), not all the permutations inα β fulfil the conditions of proper order of the overlined letters. We have forα, β∈Sn(p)×Sr(q),
Fecomp(α)(X)Fecomp(β)(X) = X
γ∈α β
Γ(γ). (12)
In particular forn=r= 0 Fe(0p)(X)Fe(0q)(X) =
p
X
j=0
(−1)j p
j
p+q−j p
Fe(0p+q−j)(X). (13) In some sense, weak composition fundamental quasisymmetric functions are a specialisation ofPoirier 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 Youngbi-tableauxand the fundamental quasisymmetric functions of Poirier. As a result, there should be a natural bi-tableaux interpretation of weak composi- tion quasisymmetric functions. This is the focus of the next section.
2.2 Tableau-theoretic interpretation
Letλbe an integer partition and p a non-negative integer. We consider pairs of Young tableau (T−, T+) whereT− is of shape (p) andT+ has shapeλ. We call such a bi-tableau a (p)-tableau. We proceed with the formal definition.
Definition 2. For any couple of non-negative integer p, nand an integer par- tition 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 callT astandard(p)-tableau and we denoteSBT(p)(λ)the set of such tableaux.
We also need the following notions. Thestandardisation Tst of a semistan- dard (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 or- der of the labels is preserved. When a label in T is used both in T− and T+, the boxes ofT− are relabelled first. Finally, given a standard (p)-tableau T ∈ SBT(p)(λ) with |λ| = n denote comp(T) the weak composition of n, comp(T) = (0i1, s1,0i2, s2, . . . ,0ik, sk,0ik+1), 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 labelsk =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+i2in T−, etc...
Example 1. The following semistandard(3)-tableau belongs toSSBT(3)((2,2,1)).
T =
T−= 2 4 4 , T+= 1 4 5 5 9
Furthermore, its standardisation is:
Tst=
T−= 2 3 4 , T+= 1 5 6 7 8
Finally the weak composition ofTst is comp(Tst) = (1,03,1,2,1).
Forλ`nandp≥0 denotes(p)λ the generating function for such tableaux. We have
s(p)λ (X) = X
T∈SSBT(p)(λ)
XT+. (14)
Recall the notation of the Schur symmetric function sλ indexed by λ. The following lemma is immediate.
Lemma 1. For any λ`nandp≥0, one has s(p)λ (X) =
M+p−1 p
sλ(X). (15)
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
s(p)λ (X) = X
T∈SBT(p)(λ)
Fecomp(T)(X). (16) 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
Fecomp(T0)(X) = X
T∈SSBT(p)(λ);Tst=T0
XT+.
Finally, sum onSBT(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, mbe non-negative integers and λ, µinteger partitions of n andm. For ν ` |λ|+|µ| recall the coefficientscνλ,µ introduced in Section 1.3. The following identity holds.
X
T∈SBT(p)(λ) U∈SBT(q)(µ)
Fecomp(T)(X)Fecomp(U)(X) = X
0≤j≤p ν`|λ|+|µ|
V∈SBT(p+q−j)(ν)
(−1)j p
j
p+q−j p
cνλµFecomp(V)(X).
(17) Proof. (sketch) First, note that for any integerp
s(p)∅ =Fe(0p)=
M+p−1 p
.
Then use Equation (13) and the definition of the coefficientscνλµ to write s(p)λ s(q)µ = X
ν`|λ|+|µ|
cνλµ
M+p−1 p
M +q−1 q
sν,
= X
ν`|λ|+|µ|
cνλµ
p
X
j=0
(−1)j p
j
p+q−j p
M +p+q−j−1 p+q−j
sν,
= X
ν`|λ|+|µ|
cνλµ
p
X
j=0
(−1)j p
j
p+q−j p
s(p+q−j)ν ,
and apply Proposition 2 to get the result.
There is an interesting corollary of Theorem 1 in terms of generating func- tions forPπ-partitions. For a (classical) standard Young tableau T, recall the permutation classCT of Section 1.3. Given a non-negative integerp, we denote εp the coloured permutationεp= 12· · ·p. The result can be stated as follows.
Corollary 1. For non-negative integers p, q, integers partitions λ and µ and T ∈SY T(λ),U ∈SY T(µ), one has:
X
γ∈(εp CT) (εq CU)
Γ(γ) = X
0≤j≤p ν`|λ|+|µ|
V∈SBT(p+q−j)(ν)
(−1)j p
j
p+q−j p
cνλµFecomp(V)(X).
(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 permutationsπon{-n,· · ·,-2,-1,0,1,2,· · · , n}such thatπ(−i) =−π(i) for alli. Thetotal colourofπ∈Bn is the numbertc(π) =|{i≥1;π(i)<0}|.
Itsdescent setis the subset of{0} ∪[n−1] equal toDes(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 whenπ(1) is a negative integer.
Forλ`2n, astandard domino tableauT of shapeλis a Young diagram of shapeshape(T) =λtiled bydominoes, i.e. 2×1 or 1×2 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 setP0(n) ofempty 2-core partitions λ ` 2n that fit such a tiling. A standard domino tableau T has a descent in positioni >0 ifi+ 1 lies strictly belowiin T and has descent in position 0 if the domino filled with 1 is vertical. We denoteDes(T) the set of all its descents. Asemistandard domino tableauT of shape λ∈ P0(n) and weightw(T) =µ= (µ0, µ1, µ2,· · ·) withµi ≥0 andP
iµi =n is a tiling of the Young diagram of shapeλwith horizontal and vertical dominoes labelled with integers in{0,1,2,· · · }such that labels are non decreasing along the rows, strictly increasing down the columns and exactlyµi 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 tableauT, i.e. half the numberof its vertical dominoes.
Remark 2. The possible labelling of top leftmost horizontal dominoes in semi- standard domino tableaux with0 was first introduced by the authors in [8] and
is essential to connect their generating functions with Chow’s quasisymmetric functions.
T1=
1 2 3
4
5 6
7
8 T2=
1 2 3 4
5
6
7
8 T3=
0 0
2 2
5 5
5 5
7
Figure 1: Two standard domino tableaux T1 and T2 of shape (5,5,4,1,1), descent set
{0,3,5,6}, a semistandard tableauT3of shape (5,5,4,3,1) and weightµ= (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 quasisym- metric functions that is dual to Solomon’s descent algebra of type B.
LetX ={· · ·, 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
FIB(X) = X
0=i0≤i1≤i2≤...≤in
j∈I⇒ij<ij+1
xi1xi2. . . xin.
Note the particular rle of the variablex0.
Given two signed permutations α∈Bn andβ ∈ Bm, denote α β the set of signed permutations inBm+n obtained by shuffling the letters ofαandβwhere we shift the absolute value of theβi to getβ. Chow shows in [3, Prop. 2.2.6]
FDes(α)B (X)FDes(β)B (X) = X
γ∈α β
FDes(γ)B (X). (19) Furthermore, we introduce in [8] modified generating functions for domino tableaux called domino functions. Given a semistandard domino tableau T of weightµ, denoteXT the monomialxµ00xµ11xµ22· · ·. Forλ∈ P0(n) and an ad- ditional parameterqwe define thedomino functionindexed byλon alphabet X as
Gλ(X;q) = X
T∈SSDT(λ)
qsp(T)XT. (20)
These functions are related to Chow’s quasisymmetric functions through
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
Gλ(X;q) = X
T∈SDT(λ)
qsp(T)FDes(TB )(X). (21)
3.3 Type B Littlewood-Richardson coefficients
There is a natural analogue of the RSK-correspondence for signed permutations involvingdomino tableaux. Barbash and Vogan ([2]) built a bijection between signed permutations ofBnand 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 tableauxT and U associated to a signed permutationπby the algorithm of Barbash and Vogan have respective descent setsDes(T) =Des(π−1) and Des(U) =Des(π) while Shimozono and White showed in [10] thecolour-to-spinproperty 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 integern, we have X
π∈Bn
qtc(π)FDes(π)B (X)FDes(πB −1)(Y) = X
λ∈P0(n) T ,U∈SDT(λ)
qsp(T)+sp(U)FDes(TB )(X)FDes(U)B (Y). (22)
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 tableauT of a class of signed permutations CTB such that for any π ∈ CTB, Des(π−1) = Des(T) and there is a descent preserving bijection CTB → 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 thetype 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 weightsµ−andµ+such thatµ+i +µ−i = 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 tableauT of shapeλand (λ−, λ+) their respective shapes. (T−, T+) (resp. (λ−, λ+)) is the 2-quotientofT (resp.
λ). DenoteX∗=X\x0. Forλ∈ P0(n) one has Gλ(X; 1) =sλ−(X∗)sλ+(X)
and forλ, µ∈ P0(n)× P0(m),
Gλ(X; 1)Gµ(X; 1) = X
ν∈P0(n+m)
cνλ−−µ−cνλ++µ+Gν(X; 1). (23) 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 DBI the set of signed permutationsπ such that Des(π) =I and∆BI the set of standard domino tableaux of descent setI. Using the notations above for integer partitionsλ, µ andT, U inSDT(λ)×SDT(µ)
X
ν∈P0(n+m)
cνλ−−µ−cνλ++µ+|SDT(ν)∩∆BI |=
CTB CUB
∩DBI
. (24) 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 forλ∈ P0(n) andT ∈SDT(λ)
Gλ(X;q) = X
π∈CTB
qtc(π)−sp(T)FDes(π)B .
Then, ifµ∈ P0(m) andU ∈SDT(µ)
Gλ(X;q)Gµ(X;q) =q−(sp(T)+sp(U)) X
γ∈CTB CUB
qtc(γ)FDes(γ)B .
However, even if we showed that the set of signed permutationsCTB CUB is in descent preserving bijection with a set of cardinalitycνλ−−µ−cνλ++µ+|SDT(ν)|, we cannot saya priorihow this bijection preserves or transforms the statistictc.
References
[1] R.M. Adin et al. “Character formulas and descents for the hyperoctahedral group”. In:Advances in Applied Mathematics 87 (2017), pp. 128–169.
[2] D. Barbash and D. Vogan. “Primitive ideals and orbital integrals in com- plex classical groups”. In:Math. Ann.259 (1982), pp. 153–199.
[3] C.O. Chow. “Noncommutative symmetric functions of type B”. PhD the- sis. MIT, 2001.
[4] I. Gessel. “Multipartite P-partitions and inner products of skew Schur functions”. In:Contemporary Mathematics 34 (1984), pp. 289–317.
[5] L. Guo, H. Yu, and J. Zhao. “Rota–Baxter algebras and left weak composi- tion quasi-symmetric functions”. In:The Ramanujan Journal 44.3 (2017), pp. 567–596.
[6] Y. Li. “On weak peak quasisymmetric functions”. In: Journal of Comb.
Theory, Series A158 (2018), pp. 449–491.
[7] A.R. Mayorova and E.A. Vassilieva. “Aq-deformed type B Cauchy iden- tity and Chow’s quasisymmetric functions”. In: S´eminaire Lotharingien de Combinatoire82B.87 (2019).
[8] A.R. Mayorova and E.A. Vassilieva. “On the structure constants of the descent algebra of the hyperoctahedral group”. In: Electronic Notes in Discrete Mathematics 61 (2017), pp. 847–853.
[9] S. Poirier. “Cycle type and descent set in wreath products”. In:Discrete Math.180 (1998), pp. 315–343.
[10] M. Shimozono and D. White. “A color-to-spin domino Schensted algo- rithm”. In:The Electronic Journal of Combinatorics 8 (2001), R2.
[11] M. Ta¸skin. “Plactic relations for r-domino tableaux”. In:The Electronic Journal of Combinatorics 19 (2012), P38.
[12] H. Yu, L. Guo, and J.Y. Thibon. “Weak quasi-symmetric functions, Rota–
Baxter algebras and Hopf algebras”. In: Advances in mathematics 344 (2019), pp. 1–34.
