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A new link between the descent algebra of type B, domino tableaux and Chow’s quasisymmetric functions
Ekaterina Vassilieva, Alina Mayorova
To cite this version:
Ekaterina Vassilieva, Alina Mayorova. A new link between the descent algebra of type B, domino
tableaux and Chow’s quasisymmetric functions. Discrete Mathematics, Elsevier, 2019, 342 (6),
pp.1658-1673. �10.1016/j.disc.2019.01.021�. �hal-02410964�
A new link between the descent algebra of type B, domino tableaux and Chow’s quasisymmetric functions
Ekaterina A. Vassilievaa, Alina R. Mayorovaa,b
aLaboratoire d’Informatique de l’Ecole Polytechnique, 91128 Palaiseau Cedex, France
bDepartment of Higher Algebra, Moscow State University, Moscow, Russia
Abstract
Introduced by Solomon in his 1976 paper, the descent algebra of a finite Coxeter group received significant attention over the past decades. As proved by Gessel, in the case of the symmetric group its structure constants give the comultipli- cation table for the fundamental basis of quasisymmetric functions. We show that this latter property actually implies several well known relations linked to the Robinson-Schensted-Knuth correspondence and some of its generalisations.
This provides a new link between these results and the theory of quasisymmet- ric functions and allows to derive more advanced formulas involving Kronecker coefficients. Using the theory of type B quasisymmetric functions introduced by Chow, we extend this connection to the hyperoctahedral group and derive new formulas relating the structure constants of the descent algebra of type B, the numbers of domino tableaux of given descent set and the Kronecker coefficients of the hyperoctahedral group.
Keywords: descent algebra, type B quasisymmetric functions, Kronecker coefficients, RSK-correspondence, hyperoctahedral group, domino tableaux.
1. Introduction
For any positive integer nwrite [n] = {1,· · · , n}, Sn the symmetric group on [n] and idn the identity permutation of Sn. A composition α n is a sequence of positive integersα= (α1,· · · , αp)such that α1+α2+· · · =n. A compositionλofnwith its parts sorted in decreasing order is called apartition and denotedλ`nor|λ|=n. A partitionλis usually represented as a Young diagram ofn=|λ|boxes arranged in`(λ)left justified rows so that thei-th row from the top containsλi boxes. A standard Young tableau T is a Young diagram whose boxes are filled with the elements of[n]such that the entries are strictly increasing along the rows and down the columns. The partition given by the number of boxes in each row is itsshapeand denotedshape(T).
Email addresses: ekaterina.vassilieva@lix.polytechnique.fr(Ekaterina A.
Vassilieva),alina.r.m@yandex.ru(Alina R. Mayorova)
Example 1.1. The following diagrams are standard Young tableaux of shape λ= (6,4,2,1,1).
T1=
1 2
3 4
5 6 7
8 9 10
11 12
13
14
T2=
1 2
3 4
5
6 12 14 7
13 10
8
11
9
Compositions of n are in natural bijection with subsets of [n−1]. For α n and I ={i1, i2,· · ·, im} a subset of [n−1] such that i1 < i2 < · · ·im, we denote Des(α) = {α1, α1 +α2,· · · , α1+α2 +· · ·αp−1} and comp(I) = (i1, i2−i1,· · · , im−im−1, n−im)this bijection and its inverse. As a result the number of compositions ofnis2n−1.
One important feature of a permutationπofSnis itsdescent setDes(π) = {1 ≤ i ≤ n−1 | π(i) > π(i+ 1)}. Similarly define the descent set of a standard Young tableauT as the subset of[n−1]defined byDes(T) ={1≤ i≤n−1|iis in a strictly higher row thani+ 1}. For instance the descent set of the tableaux in Example 1.1 is{1,4,9,10,12}. We denotedλI the number of standard Young tableaux of shapeλand descent setI.
LetDI (resp. BI) be the element of the algebraCSn defined as the formal sum of all the permutationsπsuch thatDes(π) =I(resp. Des(π)⊆I). In the more general context of finite Coxeter groups, Solomon showed in [19] that theDI’s (resp. BI’s) generate a subalgebra ofCSnof dimension2n−1usually referred to as theSolomon’s descent algebra. More specifically, he showed that there exist (non negative integer valued)structure constants(aKIJ)I,J,K⊆[n−1] and (bKIJ)I,J,K⊆[n−1] that verify:
DIDJ = X
K⊆[n−1]
aKIJDK, BIBJ = X
K⊆[n−1]
bKIJBK.
As a result the number of ways to write a fixed permutation π of DK as the ordered product of two permutationsπ =στ such that σ ∈ DI and τ ∈ DJ depends only on Des(π) = K and is equal to aKIJ. If we require instead σ and τ to be respectively in BI and BJ then the number of such products is P
K0⊇KbKIJ0.
Remark 1.2. The two families of structure constants are linked through the formulas
X
I0⊆I,J0⊆J
aKI0J0 = X
K0⊇K
bKIJ0. (1)
Because of its important combinatorial and algebraic properties the de- scent algebra received significant attention afterwards. In particular, Garsia and Reutenauer in [11] provide a decomposition of its multiplicative structure and a new combinatorial interpretation of the coefficientsbKIJ in terms of the
number of non-negative integer matrices with specified constraints. Bergeron and Bergeron give analogous results in [4] when the symmetric group is re- placed by the hyperoctahedral group (Coxeter group of type B). As shown by Norton in [16], the descent algebra of a finite Coxeter group is also strongly related to its0-Hecke algebra. In particular, she proves that the dimension of each left principal indecomposable module of the 0-Hecke algebra is equal to the cardinality of the analogue of one the DI’s. She further shows that the analogues of the a∅IJ are the entries of the Cartan matrix giving the number of times each irreducible module is a composition factor of each indecompos- able module. In the specific case of the symmetric groupSn, Carter in [6] uses the celebratedRobison-Schensted (RS) correspondence to explain the following relation obtained by computation of the Cartan matrix
a∅IJ =X
λ`n
dλIdλJ. (2)
Finally, Solomon’s descent algebra is dual to the Hopf algebra ofquasisymmetric functions(see Section 2) introduced by Gessel in [12]. In particular, he shows that the comultiplication table for their fundamental basis is given by theaKIJ’s.
In Section 2, we show that Equation (2) and its generalisations are a direct consequence of Gessel’s result. Then we use the theory of quasisymmetric func- tions of type B introduced by Chow ([7]) in Section 3 to provide new analogous formulas in the case of the hyperoctahedral group. Section 4 provides additional results involving more general coefficients.
2. Descent algebra of the symmetric group 2.1. Quasisymmetric functions
Let X = {x1, x2,· · · } be an alphabet of commutative indeterminates and I⊆[n−1]. A quasisymmetric function is a bounded degree formal power series in C[X] such that for any composition(α1,· · ·αp) and any strictly increasing sequence of distinct indicesi1< i2<· · ·< ip the coefficient ofxα11xα22· · ·xαppis equal to the coefficient ofxαi11xαi22· · ·xαipp. Quasisymmetric functions admit the monomial and fundamental quasisymmetric functions as classical bases:
MI(X) = X
i1≤···≤in
k∈I⇔ik<ik+1
xi1xi2· · ·xin, FI(X) = X
i1≤···≤in
k∈I⇒ik<ik+1
xi1xi2· · ·xin.
These two bases are related through FI(X) = X
I⊆J⊆[n−1]
MJ(X). (3)
Remark 2.1. In Gessel’s paper [12], quasisymmetric functions are indexed by compositions and not subsets. However, the bijection between subsets and compositions given in Section 1 makes the definition equivalent.
Forλ`ndenote alsopλ(X)andsλ(X)the power sum and Schur symmetric functions in the alphabetX indexed byλ. In what follows, we may remove the reference toX when there is no confusion.
A semistandard Young tableau is a Young diagram whose entries are strictly increasing down the column and non-decreasing along the rows. A tableau has weight µ = (µ1, µ2,· · ·) if it has µi entries equal to i. Denote Kλµ the number of semistandard Young tableaux of shape λ and weight µ.
The decomposition of Schur symmetric functions in the fundamental (e.g. [20, 7.19.7]) and the monomial bases is
sλ(X) = X
I⊆[n−1]
dλIFI(X) (4)
= X
I⊆[n−1]
Kλ comp(I)MI(X). (5)
2.2. Gessel’s relation and its consequences
For two commutative sets of variables X = {x1, x2,· · ·, xi,· · · } and Y = {y1, y2,· · · , yi,· · · }, we denoteXY the set of indeterminates{xiyj;xi∈X, yj∈ Y}ordered by the lexicographical order. Gessel shows in [12, Thm. 11] that for any subsetK⊆[n−1]
FK(XY) = X
I,J⊆[n−1]
aKIJFI(X)FJ(Y). (6)
We can say thatFK(XY)is the generating series for the coefficients aKIJ. As stated in introduction, Equation (2) and some of its generalisations are a direct consequence of Equation (6). More precisely, denote byχλthe irreducible char- acter ofSnindexed by partitionλand for anyλ, µ, ν`ndefine theKronecker coefficientsg(λ, µ, ν) = 1/n!P
ω∈Snχλ(ω)χµ(ω)χν(ω). We have the following theorem.
Theorem 2.2. ForI, J, K ⊂[n−1] leta∅I,J be the structure constant defined in introduction and leta∅I,J,K = [D∅]DIDJDK be the number of triples of per- mutationsσ1,σ2,σ3 ofSn such thatDes(σ1) =I,Des(σ2) =J,Des(σ3) =K andσ1σ2σ3=idn. Equation (6) directly implies
a∅IJ=X
λ`n
dλIdλJ, (7)
a∅I,J,K= X
λ,µ,ν`n
g(λ, µ, ν)dλIdµJdνK. (8)
Proof. According to Equation (4) F∅ =s(n). Then using the Cauchy identity for Schur functions
s(n)(XY) = X
λ`n
sλ(X)sλ(Y) (9)
and applying Equation (4) once again one gets:
X
I,J⊆[n−1]
a∅IJFI(X)FJ(Y) =F∅(XY) =s(n)(XY)
= X
λ`n
sλ(X)sλ(Y)
= X
I,J⊆[n−1]λ`n
dλIdλJFI(X)FJ(Y).
This can be generalised to prove Equation (8) by noticing that the structure constants verifya∅I,J,K=P
La∅I,LaLJ,K.As a result, Equation (6) extends to F∅(XY Z) = X
I,J,K
a∅I,J,KFI(X)FJ(Y)FK(Z).
Finally use the generalised version of Equation (9) sν(XY) =X
λ,µ
g(λ, µ, ν)sλ(X)sµ(Y) (10) to decomposes(n)(XY Z) =F∅(XY Z).
Remark 2.3. Denote λ0 the partition corresponding to the transposed Young diagram ofλ. For I, J⊆[n−1], the numbersa[n−1]IJ are given by
a[n−1]IJ =X
λ`n
dλIdλ0J. (11)
Proof. According to Equation (4) F[n−1] = s(1n). Then use the identity for Schur functions
s(1n)(XY) =X
λ`n
sλ(X)sλ0(Y) and apply Equation (4) tosλand sλ0.
2.3. Extension to the RSK-correspondence
Generalising the RS-correspondence to matrices with non-negative integral entries, Knuth ([14]) proved that forr, cnthe number mr,c of such matrices with row and column sums equal respectively tor andcis given by
mr,c=X
λ`n
KλrKλc. (12)
We have the following corollary to Theorem 2.2.
Corollary 2.4. Equation (12) is also a consequence of Equation (6). Further- more let p, qand rbe three compositions of nand denotemp,q,r the number of three-dimensional arrays M = (Mi,j,k) with non-negative integer entries such that pk = P
i,jMi,j,k, qj = P
i,kMi,j,k and ri = P
j,kMi,j,k. Equation (6) implies
mp,q,r= X
λ,µ,ν`n
g(λ, µ, ν)KλpKµqKνr. (13)
Proof. We proceed with the proof of Equation (12).
DenoteAKI,J =P
I0⊆I,J0⊆JaKI0,J0. According to Equation (3) X
I0,J0⊆[n−1]
aKI0J0FI0(X)FJ0(Y) = X
I0,J0⊆[n−1]
aKI0J0
X
I⊇I0,J⊇J0
MI(X)MJ(Y)
= X
I,J⊆[n−1]
MI(X)MJ(Y) X
I0⊆I,J0⊆J
aKI0J0
= X
I,J⊆[n−1]
AKI,JMI(X)MJ(Y).
As a result,A∅I,J is the coefficient in MI(X)MJ(Y)ofF∅(XY)which we know is equal toP
λ`nsλ(X)sλ(Y)(see above). Finally use Equation (5) to get A∅I,J =X
λ`n
Kλ comp(I)Kλ comp(J). (14)
It remains to prove thatA∅I,J =mcomp(I),comp(J). The combinatorial interpre- tation of [11] states that bKIJ is the number of non-negative integer matrices M with row and column sums equal tocomp(I)andcomp(J)respectively and with the word obtained by reading the entries ofM row by row from top to bottom equal toK(zero entries being omitted). But according to Equation (1) A∅I,J =P
K⊆[n−1]bKIJ. We get Equation (12).
In order to prove Equation (13), it suffices to notice that the interpretation of [11] generalises well to three dimensional arrays.
A bijective proof of Equation (13) is provided in [2]. The proof involves semi- standard Young tableaux and Littlewood-Richardson tableaux. Our approach allows us to recover these results immediately.
3. Descent algebra of the hyperoctahedral group 3.1. Signed permutations and domino tableaux
Let Bn be the hyperoctahedral group of order n. Bn is composed of all permutationsπon the set{-n,· · · ,-2,-1,0,1,2,· · ·, n}such that for all iin {0}∪[n],π(−i) =−π(i)(in particularπ(0) = 0). As a result, such permutations usually referred to as signed permutations are entirely described by their
restriction to[n]. Thedescent setofπis the subset of{0} ∪[n−1]defined by 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 position0whenπ(1) is a negative integer. We denoteDBI (resp. BIB) the formal sum inCBn of the signed permutationsπwith Des(π) =I (resp. Des(π)⊆I). LetcKI1,I2,···,Ip = [DKB]Qp
j=1DBIj (resp. eKI1,I2,···,Ip = [BKB]Qp
j=1BIBj) be the structure constants of Solomon’s descent algebra of type B forI1,· · · , Ip, K⊆ {0} ∪[n−1].
Remark 3.1. These constants verify X
I0⊆I,J0⊆J
cKI0J0 = X
K0⊇K
eKIJ0. (15)
Forλ`2n, astandard domino tableauT of shapeλis a Young diagram of shape λ tiled by dominoes, 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. Denote SDT(λ) the set of standard domino tableaux of shapeλ. We denoteP0(n)the set of partitionsλ`2nwhose associated Young diagram may be fully tiled by dominoes. Such partitions are usually called empty2-core partitions. A standard domino tableauT has a descent in position i >0ifi+ 1lies strictly belowiinT and has descent in position0if the domino filled with 1 is vertical. We denote Des(T) the set of all its descents. For λ in P0(n) and I ⊆ {0} ∪[n−1], denote dBλI the number of standard domino tableaux of shapeλand descent set I.
Example 3.2. The following standard domino tableaux have shape (5,5,4,1,1) and descent set {0,3,5,6}.
T1=
1 2 3
4
5 6
7
8 T2=
1 2 3 4
5
6
7
8
There is a natural analogue of the RSK-correspondence for signed permuta- tions involving domino tableaux. Barbash and Vogan ([3]) built a bijection between signed permutations ofBn and pairs of standard domino tableaux of equal shape in P0(n). An independent development on the subject is due to Garfinkle in [8, 9, 10]. Van Leeuwen shows in [23] that the two approaches are actually equivalent. See also Stanton and White in [21] for a more gen- eral treatment of rim hook tableaux. Taşkin ([22, Prop. 26]) showed that the two standard domino tableaux associated to a signed permutationπby the algo- rithm of Barbash and Vogan have respective descent setsDes(π−1)andDes(π).
As a result of this descent preserving property, we have the following analogue to Equation (2).
c∅IJ = X
λ∈P0(n)
dBλIdBλJ. (16)
3.2. Quasisymmetric functions of type B and main results
Chow defines in [7] an analogue of Gessel’s algebra of quasisymmetric func- tions that is dual to the Solomon’s descent algebra of type B.
LetX ={· · ·, x−i,· · ·, x−1, x0, x1,· · ·, xi,· · · }be an alphabet of commutative indeterminates withx−i=xi andI be a subset of{0} ∪[n−1], he defines:
MIB(X) = X
0≤i1≤i2≤...≤in
j∈I⇔ij<ij+1
xi1xi2. . . xin, FIB(X) = X
0≤i1≤i2≤...≤in
j∈I⇒ij<ij+1
xi1xi2. . . xin.
wherei0= 0. Note the particular role played by variable x0.
Example 3.3. Let n= 2and X ={x−2, x−1, x0, x1, x2} then F∅ =x20+x21+ x22+x0x1+x0x2+x1x2,F{1} =x0x1+x0x2+x1x2,F{0} =x21+x22+x1x2
andF{0,1}=x1x2.
Remark 3.4. Prior to Chow, Poirier ([18]) introduced an alternative type B analogue of the quasisymmetric functions used for example in [1]. However Poirier’s quasisymmetric functions are dual to the Mantaci-Reutenauer algebra and not Solomon’s descent algebra of type B. See [17] for further details. The question of finding a similar development as in the present paper for Poirier’s quasisymmetric functions and the Mantaci-Reutenauer algebra remains open.
LetY be a second copy ofX. Chow uses the theory ofP-partitions of type B to show in [7, Thm. 2.3.4] that forK⊆ {0} ∪[n−1]
FKB(XY) = X
I,J⊆{0}∪[n−1]
cKI,JFIB(X)FJB(Y). (17) Theorem 3.5. Equation (16) is a consequence of Equation (17).
The irreducible characters of Bn are naturally indexed by partitions of P0(n) (see e.g. [15, I, Appendix B]). Denote ψλ the character indexed by λ ∈ P0(n) and define the Kronecker coefficients of Bn, gB(λ, µ, ν) = 1/|Bn|P
ω∈Bnψλ(ω)ψµ(ω)ψν(ω). A more general form of Theorem 3.5 can be stated as follows.
Theorem 3.6. Let I, J andK ⊆ {0} ∪[n−1]. The following formula holds c∅I,J,K= X
λ,µ,ν∈P0(n)
gB(λ, µ, ν)dBλIdBµJdBνK. (18) The proof of Theorems 3.5 and 3.6 uses Equation (17) and generating func- tions for domino tableaux. We detail it in the two following sections.
3.3. Domino functions 3.3.1. Definition
Generating functions for domino tableaux sometimes called domino func- tionsare well studied objects (see e.g. [13]). We introduce a modified definition
to get an analogue of Schur functions verifying Equations (20) and (22) that we need to prove Theorem 3.6. Our development differs by the addition of "0"
entries to the domino tableaux in some cases.
A semistandard domino tableau T of shape λ ∈ P0(n) and weight w(T) = µ = (µ0, µ1, µ2,· · ·) with µi ≥ 0 and P
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 labelled0 (we leave it to the reader to check that the only possible sub-pattern of dominos with label0 in a semistandard domino tableau is a row composed of horizontal dominos).
DenoteSSDT(λ)the set of semistandard domino tableaux of shapeλandKλµB the number of semistandard domino tableaux of shapeλand weight µ.
Thestandardisation Tst of a semistandard domino tableau T of weight µis the standard domino tableau obtained by relabelling the dominoes of T with 1,2,· · ·, n such that the dominoes labelled with im = min{i | µi > 0} are relabelled with1,2,· · ·, µim from left to right and so on.
Example 3.7. Figure 5 shows a semistandard domino tableau of weight µ = (2,0,2,0,0,4,0,1) and its standardisation.
T =
0 0
2 2
5 5
5 5
7
−−−→ Tst=
1 2
3 4
5 6
7 8
9
Figure 1: A semistandard domino tableau and its standardisation.
Definition 3.8. Given alphabet X and a semistandard domino tableau T of weightµ, denote XT the monomialxµ00xµ11xµ22· · ·. For λ∈ P0(n) we call the domino functionindexed byλthe function defined in the alphabet X by
Gλ(X) = X
T∈SSDT(λ)
XT. (19)
3.3.2. Expansion into Chow’s fundamental and monomial bases We have the following proposition.
Proposition 3.9. Domino functions admit the following expansion into Chow’s fundamental and monomial bases.
Gλ= X
I⊆{0}∪[n−1]
dBλIFIB = X
I⊆{0}∪[n−1]
Kλ comp(IB )MIB. (20)
In order to prove Proposition 3.9 we need the following lemma.
Lemma 3.10. Let T0 be a standard domino tableau. Then the fundamental quasisymmetric function FDes(TB
0) is the generating function for semistandard domino tableauxT such thatTst=T0, i.e.
FDes(TB
0)= X
Tst=T0
XT. (21)
Proof. Given a fixed standard domino tableau T0, we reverse the standardis- ation operation to obtain the set of semistandard domino tableaux with stan- dardisationT0. Since the standardisation operation is shape and domino tiling preserving, any such semistandard tableauT may be identified by the sequence of its entriesi1, i2, . . . in.
T0=
1 2
3 4 5
−−−−−−→
relabeling T =
i1 i2
i3
i4
i5
Figure 2: A standard domino tableauT0and the template for its semistandard preimages.
Relabel the entries ofT0with the sequencei1, i2, . . . , insuch thatkis mapped toik, as depicted on Figure 2. According to the standardisation procedure, we have 0 ≤ i1 ≤ i2 ≤ . . . ≤ in and 0 < i1 if the top leftmost domino is vertical. Furthermore such a sequence needs to verify additional conditions to give a valid semistandard domino tableau with standardisation equal toT0. Consider each constraint ik ≤ ik+1 separately. If ik < ik+1 then locally the sequence gives a valid semistandard domino tableau (entries are increasing along the row and down the columns) and the standardisation process provides the expected result.
We look at the caseik=ik+1and show two properties.
• First, ifik=ik+1thenk /∈Des(T0). Indeed assumek∈Des(T0). Domino khas to lay in a higher row thank+ 1as in Figure 3.
But dominoik+1 may not be outside the green area defined in Figure 3 as the entries in the tableau are strictly increasing down the columns. As a consequence, domino ik+1 has to be on the left of and below domino ik, as in Figure 3, picture (c). This is a contradiction since according to the definition of the standardisation operation, we should relabel the domino ik+1 before the domino ik with such a pattern and T0 is not the standardisation of such a tableau. As a result k ∈ Des(T0)implies ik < ik+1.
• Secondly, when k /∈Des(T0)any sequence with ik =ik+1 locally gives a valid semistandard tableau whose standardisation is equal toT0. Indeed,
ik
ik+1
(a)
ik
ik+1
(b)
ik
ik+1
(c)
Figure 3: Layouts with a descent in position k. Green area corresponds to positions of dominoik+1which do not violate the property of increasing entries along the rows and down the columns whenik=ii+1. In pictures (a) and (b) dominoik+1lays outside the green area.
Picture (c) shows a valid layout of dominoesikandik+1.
ifk /∈Des(T0), dominok does not lay in a higher row thank+ 1.
As a result, domino ik+1 cannot be outside the green area defined in Figure 4 as entries are increasing along the rows and down the columns, i.e. ik+1 is to the right of and above domino ik. This makes sure that ik is relabelled beforeik+1 in the standardisation procedure.
ik ik+1
(a)
ik ik+1
(b)
ik ik+1
(c)
Figure 4: Layouts with no descent in positionk. Green area corresponds to positions of dominoik+1which do not violate the property of increasing entries along the rows and down the columns whenik=ii+1. In pictures (a) and (b) dominoik+1lays outside the green area.
Picture (c) shows a valid layout of dominoesikandik+1.
As a consequence the preimages of T0 can be identified as the sequences 0≤i1≤i2≤. . .≤in such thatk∈Des(T0)⇒ik < ik+1 fork≥0andi0= 0.
Consider the constraints all together to ensure that such a sequence globally gives a valid semistandard tableau whose standardisation is equal toT0.
We are now ready to prove Proposition 3.9.
Proof of Proposition 3.9. Classify the set of semistandard domino tableaux ac- cording to their standardisation and add the monomials corresponding to the
tableaux in the same class. Using Lemma 3.10 we obtain the first part of the Proposition.
Gλ(X) = X
T∈SSDT(λ)
XT
= X
T0∈SDT(λ)
X
T∈SSDT(λ), Tst=T0
XT
(21)= X
T0∈SDT(λ)
FDes(TB 0)(X)
= X
I⊆{0}∪[n−1]
dBλIFIB(X).
To get the second part of the proposition, relabel the entries of a semis- tandard tableauT with successive integers0,1,2, . . . such that inequalities and equalities between entries inT are preserved after relabelling. If the top left- most domino is vertical skip0. This operation removes the zeros in the weight of the tableau except, possibly, the first one. Denote Te the resulting tableau and forλ∈ P0(n),SSDT^(λ)the set of semistandard domino tableaux of shape λsuch thatTe=T. The second part of the statement is a consequence of the fact that the monomial quasisymmetric functionMset(w(TB
0))is precisely the gen- erating function for all semistandard domino tableauxT mapped to the same T0∈SSDT^(λ)by this operation.
Gλ(X) = X
T∈SSDT(λ)
XT
= X
T0∈SSDT(λ)^
X
T∈SSDT(λ),T=Te 0
XT
= X
T0∈SSDT(λ)^
Mset(w(TB
0))
= X
I⊆{0}∪[n−1]
Kλ comp(I)B MIB(X).
Equation (20) is fully proven.
3.3.3. Cauchy identities
There is a well known (not descent preserving) bijection between semistan- dard domino tableaux of weightµandbi-tableaux, i.e. pairs of semistandard Young tableaux of respective weights µ− andµ+ such that µ+i +µ−i =µi for all i (see e.g. [5, Algorithm 6.1]). 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 shapeλand (λ−, λ+)their respective shapes. (T−, T+) (resp. (λ−, λ+)) is the2-quotient of T (resp. λ). Semistandard Young tableaux T− and T+ are built by filling each box ofT (a domino is composed of two boxes) by a ’–’
or a ’+’ sign such that the top leftmost box is filled with ’–’ and two adjacent boxes have opposite signs. T− (resp. T+) is obtained from the sub tableau of T composed of the dominoes with top rightmost box filled with ’–’ (resp. ’+’).
Remark 3.11. If the top leftmost domino of a semistandard domino tableauT is vertical (resp. horizontal), its label is used forT− (resp. T+). Therefore not labelling the top leftmost domino with 0 if it is vertical is sufficient to ensure that onlyT+ may have entries equal to0 and any bi-tableau withT+ containing entries equal to 0 gives a valid semistandard domino tableau according to our constraints. As a result, in our setup, the bijection is actually between semistan- dard domino tableaux indexed by labels in{0,1,2,· · · }and pairs of semistandard Young tableaux indexed respectively by{1,2,· · · } and{0,1,2,· · · }.
Example 3.12. Figure 5 shows a semistandard domino tableau of weightµ= (2,0,2,0,0,4,0,1) and its 2-quotient.
T =
– 0 + – 0 + +
2 –
– 2 +
+ 5 –
+ 5 – – 5 +
– 5 +
– 7 +
−−−→
T− = 5 5 5 , T+=
0 0 2 2 5 7
Figure 5: A semistandard domino tableau and its 2-quotient.
Denote X− ={x−i}i>0 and X+ ={xi}i≥0 (note thatX− =X+\ {x0} as x−i=xi).
Proposition 3.13. For λ ∈ P0(n), the domino function Gλ and the Schur symmetric functions are related through
Gλ(X) =sλ−(X−)sλ+(X+). (22) Proof. According to the definition of the 2-quotient above and Remark 3.11, one can get
Gλ(X) = X
shape(T−)=λ− shape(T+)=λ+
X−T
−
X+T
+
,
where the sum is on all pairs of semistandard Young tableau T− and T+ of shapeλ− andλ+ such that no entry ofT− is equal to0.
Finally we provide analogues of the Cauchy identities for domino functions.
Proposition 3.14. The domino functions Gλ(XY)verify the following identi- ties
G(2n)(XY) = X
λ∈P0(n)
Gλ(X)Gλ(Y), Gλ(XY) = X
µ,ν∈P0(n)
gB(λ, µ, ν)Gµ(X)Gν(Y).
Proof. Note that
(XY)+={xiyj}(i,j)≥(0,0)={x0yj}j≥0∪ {xiyj}i>0,j∈Z=X+Y+∪X−Y−. Using Proposition 3.13 we get the first identity.
G(2n)(XY) =s(n)((XY)+)
=s(n)(X−Y−∪X+Y+)
=
n
X
k=0
s(k)(X−Y−)s(n−k)(X+Y+)
=
n
X
k=0
X
λ−`k
sλ−(X−)sλ−(Y−) X
λ+`n−k
sλ+(X+)sλ+(Y+)
= X
λ∈P0(n)
sλ−(X−)sλ+(X+)sλ−(Y−)sλ+(Y+)
= X
λ∈P0(n)
Gλ(X)Gλ(Y).
Using Proposition 3.13 and the theory of symmetric functions on wreath prod- ucts we prove the second identity. Define as in [1] for any set of indeterminates U and V and any partitionλofn:
p+λ(U, V) =Y
i
[pλi(U) +pλi(V)], p−λ(U, V) =Y
i
[pλi(U)−pλi(V)]. Note that if U ∩V = ∅, p+λ(U, V) = pλ(U ∪V) and if V ⊆ U, p−λ(U, V) = pλ(U\V). Then for partitionsλ∈ P0(n)andµ∈ P0(n)of 2-quotient(µ−, µ+), denoteψλµthe common value of characterψλon all signed permutations ofcycle type(µ−, µ+). According to [15, I, Appendix B],
p−µ−(U, V)p+µ+(U, V) = X
λ∈P0(n)
ψµλsλ−(V)sλ+(U).
SetU =X+ andV =X− and use Proposition 3.13 to get for anyµ∈ P0(n) pµ+(X)(x0)n−|µ+|= X
λ∈P0(n)
ψλµGλ(X).
Butpµ+(XY)(x0y0)n−|µ+|=pµ+(X)(x0)n−|µ+|pµ+(Y)(y0)n−|µ+|. We get X
λ∈P0(n)
ψλGλ(XY) = X
µ,ν∈P0(n)
ψµψνGµ(X)Gν(Y).
That yields the second identity.
3.4. Proof of Theorems 3.5 and 3.6
Both Theorem 3.5 and Theorem 3.6 are consequences of Equation (17) and Proposition 3.14.
Proof of Theorem 3.5. Decompose the domino functionsG(2n)(XY)in two ways.
G(2n)(XY) =F∅B(XY) =X
I,J
c∅I,JFI(X)FJ(Y) G(2n)(XY) = X
λ∈P0(n)
Gλ(X)Gλ(Y) =X
I,J
X
λ`n
dBλIdBλJFIB(X)FJB(Y).
Proof of Theorem 3.6. Similarly, G(2n)(XY Z) =F∅B(XY Z) = X
I,J,K
c∅I,J,KFIB(X)FJB(Y)FKB(Z) G(2n)(XY Z) = X
ν∈P0(n)
Gν(XY)Gν(Z)
= X
λ,µ,ν∈P0(n)
gB(λ, µ, ν)Gλ(X)Gµ(Y)Gν(Z)
= X
I,J,K
X
λ,µ,ν∈P0(n)
gB(λ, µ, ν)dBλIdBµJdBνKFIB(X)FJB(Y)FKB(Z).
Remark 3.15. Theorem 3.5 is a special case of Theorem 3.6. However, we felt that an independent proof that does not use Kronecker coefficients but only the elementary first identity of Proposition 3.14 was of interest.
3.5. A corollary to Theorem 3.6
ForI⊆ {0} ∪[n−1], denote|I|the number of elements inI. The structure constantseKIJ also count templates M with non-negative integer entries of the following type.
M = (mi,j) =
a0,0 a0,1 . . . a0,|I|
b1,1 . . . b1,|I|
a1,0 a1,1 . . . a1,|I|
... ...
b|J|,1 . . . b|J|,|I|
a|J|,0 a|J|,1 . . . a|J|,|I|
Given qs =as,0+P
k(as,k+bs,k)and q0 =a0,0+P
ka0,k we denoteq(M) = (q0, q1, . . . q|I|) the row sums. The column sums p(M) = (p0, p1, . . . p|J|) verifypj =a0,j+P
k(ak,s+bk,s)andp0=a0,0+P
kak,0. Successive reading of a-lines from left to right, andb-lines from right to left yields thereading word.
DenotenKI,J the number of such templates with row sumsI, column sumsJ and reading wordK andnI,J =P
KnKI,J. Bergeron and Bergeron in [4] prove that nKI,J =eKIJ. We get as a corollary to Theorem 3.6:
Corollary 3.16. Let I, J andK ⊆ {0} ∪[n−1]. The numbers nI,J andnKI,J verify
nI,J =X
λ
Kλ comp(IB )Kλ comp(J)B
X
R,D⊆{0}∪[n−1]
nDI,JnRD,K= X
λ,µ,ν`n
gB(λ, µ, ν)Kλ comp(I)B Kµ comp(J)B Kν comp(K)B . Proof. Considering relations (16) and (18) in the monomial basis one gets
X
I,J,
X
I0⊆I,J0⊆J
c∅I0,J0MIB(X)MJB(Y) = X
I,J
X
λ`n
Kλ comp(I)B Kλ comp(J)B MIB(X)MJB(Y), and
X
I,J,K
X
I0⊆I,J0⊆J,K0⊆K
c∅I0,J0,K0MIB(X)MJB(Y)MKB(Z) = X
I,J,K
X
λ,µ,ν`n
gB(λ, µ, ν)Kλ comp(I)B Kµ comp(J)B Kν comp(K)B MIB(X)MJB(Y)MKB(Z).
The first equation of the statement is a consequence of the result of Bergeron and Bergeron and Equation (15).
nI,J =X
K
eKIJ= X
I0⊆I,J0⊆J
c∅I0J0 =X
λ`n
Kλ comp(I)B Kλ comp(J)B .
To get the second identity compute X
R,D
nDI,JnRD,K =X
R,D
eDI,JeRD,K
=X
R
eRI,J,K
= X
I0⊆I,J0⊆J,K0⊆K
c∅I0,J0,K0
= X
λ,µ,ν`n
gB(λ, µ, ν)Kλ comp(I)B Kµ comp(J)B Kν comp(K)B .
4. Skew shape partitions 4.1. Type A
Let n be a non-negative integer and λ and µ, two integer partitions such that the diagram of µ is included in the diagram of λ and |λ| − |µ| = n. A standard (resp. semistandard) skew Young tableau of shape λ/µ is a Young diagram of skew shapeλ/µwhose boxes are filled with the elements of[n]
(resp. with positive integers) such that the entries are strictly increasing down the columns and along the rows (resp. strictly increasing down the columns and non decreasing along the rows). The descent set of a standard Young tableau is defined similarly as in the case of classical shapes. Theskew Schur function sλ/µ is the generating function for semistandard skew Young tableaux of shape λ/µ
sλ/µ(X) = X
shape(T)=λ/µ
XT,
where the sum is on all semistandard skew Young tableaux of shapeλ/µ. Given a skew shapeλ/µdenotekλµν theLittlewood-Richardson coefficientsi.e. the structure constants of the algebra of symmetric functions in the Schur basis.
They also appear in the decomposition of skew Schur functions in terms of non-skew Schur functions
sλsµ=X
ν
kλµν sν, sλ/µ =X
ν
kµνλ sν. (23)
Let I and J be two subsets of [n−1]. Gessel focuses in [12, Thm. 14] on the numberaλ/µIJ of couples of permutations α, βsuch thatαhas descent setI, β has descent set J and αβ is compatiblewith the skew shape λ/µ (i.e. αβ is a correct reading word for the skew shapeλ/µ). We need some more notions introduced in [12] to define this properly.
Given any partial order P on[n], denoteL(P)the set of permutations π∈ Sn, such that the corresponding linear order π(1) < π(2) < · · · < π(n) is a linearisation ofP. To obtain a partial orderPλ/µ from the skew shape|λ/µ|=n we fill the corresponding skew Young diagram by integers[n] from bottom to top and from left to right and then consider relations i > k if k lay directly below or to the left of i. These relations generate the corresponding partial orderPλ/µ.
Example 4.1. Let λ/µ = (4,4,3)/(2,1). Figure 6 shows the corresponding partial order. Firstly fill the corresponding skew Young diagram by integers [7]from bottom to top and from left to right and then rotate it by 135 degrees counterclockwise and take its vertical mirror image to get the plane presentation of the orderingP(4,4,3)/(2,1).
Due to Gessel [12, p. 295],L(Pλ/µ)may be identified with Young tableaux of shapeλ/µ. Givenπ∈L(Pλ/µ)construct the corresponding skew Young tableau by filling the skew shapeλ/µ with the letters of the wordπ−1 from bottom to top and from left to right, as in Example 4.2. The permutations fromL(Pλ/µ) are calledcompatiblewith the skew shapeλ/µ.
f ill
−−−−−→
7 8 4 5 6 1 2 3
rotate/mirror
−−−−−−−−−−→
3 6
2 5 8
1 4 7
Figure 6: A partial orderP(4,4,3)/(2,1)obtained from the skew shape(4,4,3)/(2,1).
Example 4.2.
π= 74581263 (π−1= 56823714)
f ill with π−1
−−−−−−−−−→
1 4 2 3 7 5 6 8
Using the bijection above, one can notice that aλ/µIJ = X
K⊆[|λ|−|µ|−1]
dλ/µKaKIJ
wheredλ/µK is the number of standard Young tableaux of skew shapeλ/µand descent setK.
Proposition 4.3. Let λand µ two integer partitions such thatλ/µ is a skew shape andI, J⊆[|λ|−|µ|−1]. The coefficientsaλ/µIJ verify the following equality involving the Littlewood-Richardson coefficients, the Kronecker coefficients of the symmetric group and the numbers of standard Young tableaux of given shape and descent
aλ/µIJ = X
ν,ρ,ε
kµνλ g(ν, ρ, ε)dρIdεJ.
Proof. Gessel showed in [12, Thm. 14] that sλ/µ(XY) =X
I,J
aλ/µIJ FI(X)FJ(Y). (24) Use Equation (23) to decompose skew Schur functionsλ/µ(XY)using Littlewood- Richardson coefficients. Then use the generalised Cauchy identity (10) and decomposition (4) to prove the proposition.
sλ/µ(XY) =X
ν
kλµνsν(XY)
=X
ν,ρ,ε
kλµνg(ν, ρ, ε)sρ(X)sε(Y)
=X
I,J
X
ν,ρ,ε
kµνλ g(ν, ρ, ε)dρIdεJFI(X)FJ(Y).
In the following section, we prove that there is a similar formula for the hyperoctahedral group provided some constraints on the skew-shapes.
4.2. Type B
4.2.1. Skew domino tableaux
Letλandµbe two integer partitions such thatλ/µis a skew shape,|λ|−|µ|= 2n and such that the Young diagrams of shape µ and λ/µ (and by extension λ) may be tiled by horizontal and vertical dominoes. By abuse of notation, we also writeλ/µ∈ P0(n)when all these conditions are fulfilled. Astandard skew domino tableauis a tiling of the skew Young diagram of shapeλ/µwith horizontal and vertical dominoes labelled with integers of[n] such that labels are strictly increasing along the rows and down the columns.
The bijection described in Carre and Leclerc [5] extends to the case of skew shapes. As a result, there is a bijection between standard skew domino tableaux of shape λ/µ and standard skew bi-tableaux of shape (λ−/µ−, λ+/µ+). As noticed in Section 3.3.3, µ− and µ+ depends only on µ. One may apply the algorithm of Carre and Leclerc with an arbitrary tiling of the deleted part of shape µ. Note that the bijection is not descent preserving as in the case of non-skew shapes.
Example 4.4. Decompose the following standard skew domino tableauT.
T =
– +
+ –
– +
+ –
– + + –
– 2 + +
3 –
– 1 +
– 4 + +
6 – + 5 –
– 7 +
– 9 +
– 8 +
−−−−→
T− = 1 7 9
5
, T+=
2 3 8 4 6
Given a standard skew domino tableau T, we call in the sequel negative (resp. positive) domino, a domino that goes toT− (resp. T+) according to the bijection above. We have the following definition.
Definition 4.5(Descent set of a standard skew domino tableau). A standard skew domino tableauT has a descent in positioni >0ifi+ 1lies strictly belowi inT and has descent in position0if the domino filled with1is negative. Denote Des(T)the set of all the descents of T. For λ/µ∈ P0(n)andI⊆ {0} ∪[n−1]
denote alsodBλ/µ I the number of standard skew domino tableaux of shape λ/µ and descent setI.
Example 4.6. In Example 4.4 we have Des(T) ={0,3,4,8}.
A semistandard skew domino tableauis a tiling of the skew Young dia- gram of shapeλ/µwith horizontal and vertical dominoes labelled with integers in {0,1,2,· · · } such that labels are non decreasing along the rows, strictly in- creasing down the columns, and negative dominoes cannot be labelled0. The bi- jection of Carre and Leclerc [5] also generalises to semistandard domino tableaux and pairs of semistandard Young tableaux such that dominoes labelled0 may appear only inT+.
