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A new link between domino tableaux and Chow’s type B quasisymmetric functions
This chapter plays a supporting rôle. We make an attempt to generalise all the main notions we used in Chapter 1 to the case of type B, trying to preserve the properties and links between them. We suggest a modified definition of domino tableaux and a definition of domino functions instead of Young tableaux and Schur functions. We show how to decompose the domino functions using type B Chow’s fundamental and monomial functions and investigate the restrictions we needed to have such a decomposition.
Signed permutations and domino tableaux
Hyperoctahedral group and descent set
Let Bn be the hyperoctahedral group of order n, i.e. the Coxeter group of type B. Bn is composed of all permutations on the set f-n; ; -2; -1; 0; 1; 2; ; ng such that for all i in f0g [ [n], ( i) = (i) (in particular (0) = 0). As a result, such permutations usually referred to as signed permutations are entirely described by their restriction to [n].
Further we suggest an overview on the notions of descent set of a signed permutation. We start from the most essential definition.
Definition 18. The descent set of a signed permutation of Bn is the subset of f0g[[n 1] defined by
Des( ) = f0 i n 1 j (i) > (i + 1)g:
The main difference with respect to the case of the symmetric group is the possible descent in position 0 when (1) is a negative integer. Further in the thesis we use bars instead the sign » » for the convenience.
Another approach to define descent set in the hyperoctahedral group uses a different order on the elements of the set fn; : : : 1; 1; : : : ; ng. Define: 1 <r 2 <r : : : n <r 1 <r 2 <r : : : n:
One may define another notion of descent set Desr using the less essential order >r: Desr( ) = f0 i n 1 j (i) >r (i + 1)g:S
The descent set in this case is also a subset of 0 [n 1].
A third option is the signed descent. Denote by B(n) the set consisting of all pairs (S; « ), called signed sets, where S is subset of [n] and » is map from S to f ; +g.
Definition 19. Denote by sDes( ) the signed descent of the signed permutation , i.e. the signed set (S; « ), such that S contains all s 2 [n 1] for which (s) >r (s + 1) or (s) is barred and (s + 1) is not.
S contains n.
For every s 2 S we denote « (s) = if (s) is barred and « (s) = + otherwise.
Given a signed set = (S; « ) 2 B(n) we will define two statistics which relies on it. Denote by wDes( ) the set of elements si 6= sn; si 2 S such that « (si) = « (si+1) or « (si) = +; « (si+1) = . Denote also
wDes0( ) = 8 wDes( ); g if « 0(1) = +; : wDes( ) [ f ; if « 0(1) = : < 0
In case of signed permutations one can give a definition of wDes( ) and wDes0( ) without the notion of signed descent, as follows wDes( ) = wDes(sDes( )) = f1 i n 1 j (i) >r (i + 1)g; and wDes0( ) = wDes0(sDes( )) = Desr( ) = f0 i n 1 j (i) >r (i + 1)g:
So, the signed descent sDes refines Desr.
One may note that the signed descent sDes refines also Des. Indeed, it is possible to continue the sign map » from the set S to [n]. In particular, in the case of the descent set of a permutation it coincides with the map i ! sign( (i)). Denote by » 0 such a map. Further, any two permutations with the same signed descent set (S; « ) have also the same Solomon descent set, which may be obtained with the following rule:
0 2 Des( ) if and only if « 0(1) = ,
k 2 Des( ) if and only if
– k 2 S and « 0(i) = « 0(i + 1) = +, or
– k 62S and « 0(i) = « 0(i + 1) =
Domino tableaux
One way to generalise the notion of Young tableau to the case of type B is domino tableaux.
Recall the definition from [Gar90].
Definition 20. For ‘ 2n, a standard domino tableau T of shape is a Young diagram of shape 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 by SDT ( ) (SDT (n)) the set of standard domino tableaux of shape (of n dominoes).
We consider only the set P0(n) of empty 2-core partitions ‘ 2n, i.e. partitions that fit such a tiling. Domino tableaux are related to Chow’s quasisymmetric functions. Another advantage is the notion of the descent set, which is similar to the one in type A.
Definition 21. A standard domino tableau T has a descent in position i > 0 if i + 1 lies strictly below i in T and has a descent in position 0 if the domino filled with 1 is vertical. We denote by Des(T ) the set of all its descents.
For in P0(n) and I f0g [ [n 1], denote by dBI the number of standard domino tableaux of shape and descent set I.
Example 2.1. The following standard domino tableaux have shape (5; 5; 4; 1; 1) and de-scent set {0,3,5,6}.
Definition 22. A semistandard domino tableau T of shape 2 P0(n) and weight P w(T ) = = ( 0; 1; 2; ) with i 0 and i i = n is a tiling of the Young diagram of shape with horizontal and vertical dominoes labelled with integers of the set f0; 1; 2; g such that labels are non decreasing along the rows, strictly increasing down the columns and exactly i dominoes are labelled with i. If the top leftmost domino is vertical, it cannot be labelled 0.
Our notion of semistandard domino tableau differs from the usual one (which is without zeroes). The reason is that we need zeroes in the tableau to make valid links to Chow’s quasisymmetric functions.
Remark 1. The only possible sub-pattern of dominoes with label 0 in a semistandard domino tableau is a row composed of horizontal dominoes.
Example 2.2. The following semistandard tableau of shape (5; 5; 4; 3; 1) has weight = (2; 0; 2; 0; 0; 4; 0; 1).
Denote by SSDT ( ) (SSDT (n)) the set of semistandard domino tableaux of shape (of n dominoes) and KB the number of semistandard domino tableaux of shape and weight .
Stanton and White bijection
Bi-tableaux
A second way to generalise the notion of the Young tableau is to consider bi-tableaux. Bi-tableaux are closely related to Poirier’s quasisymmetric functions. At the same time, they are related to domino tableaux via the Stanton and White bijection.
Definition 23. Denote a pair of shapes ( ; +) a bi-shape of n if j j + j +j = n. A standard Young bi-tableau of bi-shape ( ; +) is a pair of Young diagrams (T ; T +)
with shape(T ) = and shape(T +) = +, whose boxes are filled with the elements of [n] such that the entries are strictly increasing along the rows and down the columns for each of them.
For a standard Young bi-tableau it is also possible to denote a notion of a descent set. However, it differs from the usual one used in this thesis. In fact, one uses signed descent sets for the case of bi-tableaux. Recall, B(n) is the set consisting of all pairs (S; « ), where S is subset of [n] and » is map from S to f ; +g.
Definition 24. The signed descent set sDes((Q+; Q )) of a bi-tableau (Q+; Q ) 2 SY T ( ; ) is the signed set (S; « ) defined as follows:
S contains all s 2 [n 1] for which either both s and s+1 appear in the same tableau and s + 1 is in a lower row than s, or s and s + 1 appear in different tableaux.
S contains n.
For every s 2 S we denote « (s) = if s appears in Q and « (s) = + otherwise.
Recall, that in the case of permutations the notion of signed descent sDes generalises both the notions Des and Desr. Given a signed descent sDes we can find the corresponding Desr. In the case of bi-tableaux, this turns into the following definition for Desr.
Definition 25. The descent set Desr((Q+; Q )) of a bi-tableau (Q+; Q ) 2 SY T ( ; ) S
each s 2 [n 1] such that both s and s + 1 appear in the same tableau and s + 1 is in a lower row than s each s 2 [n 1] such that s 2 Q+ and s + 1 2 Q 0 if 1 2 Q .
Example 7. The following standard bi-tableau has bi-shape ((3); (2; 2; 2)) and descent set Desr equal to f2; 4; 8g.
Definition 26. A semistandard Young bi-tableau (T ; T +) is a pair of Young diagrams of bi-shape ( ; +) whose boxes are filled with nonnegative integers such that the entries
are strictly increasing down the columns and non-decreasing along the rows for each of them. The additional constraint is that zeroes may appear only in T +.
Example 8. The following standard bi-tableau has bi-shape ((3); (2; 2; 2)).
Description of the bijection
In [CL95] Carré and Leclerc studied a bijection due to Stanton and White [SW85]. They introduced an easier description of the bijection between semistandard domino tableaux (without zeroes) of n dominoes and pairs of semistandard Young tableaux with n cells in common. This bijection restricts to the sets of shapes, it does not depend on the tilling and numbers in cells. So, jP0(n)j = X
p(k)p(n k):
0 k n
In this section, we further describe the idea of the algorithm and apply it to prove the similar statements for our definitions of a domino tableau and a bi-tableau.
Given a domino tableau T we consider dl the diagonal line (i; j) : i = j 2l. Such a line may intersect a domino in four different ways (see Figure 2.1).
Table of contents :
Abstract
Résumé
Acknowledgements
1 Definitions and results for type A Coxeter groups
1.1 Permutations and Young tableaux
1.2 RSK correspondence
1.3 Ring of symmetric functions
1.4 Schur functions
1.4.1 Skew Schur functions and Littlewood-Richardson coefficients
1.4.2 Cauchy identity
1.5 Gessel’s quasisymmetric functions
1.6 Link between Schur functions and quasisymmetric functions
1.7 Solomon’s descent algebra
1.7.1 Descent algebra of a Coxeter group
1.7.2 Descent algebra of the symmetric group
1.8 Computation of structure constants
1.8.1 Gessel’s relation and its consequences
1.8.2 Extension to the RSK-correspondence
1.8.3 Skew shapes
1.9 Descent preservation property of the Cauchy formula
1.10 Schur positivity
1.11 Material of the thesis
2 A new link between domino tableaux and Chow’s type B quasisymmetric functions
2.1 Signed permutations and domino tableaux
2.1.1 Hyperoctahedral group and descent set
2.1.2 Domino tableaux
2.2 Stanton and White bijection
2.2.1 Bi-tableaux
2.2.2 Description of the bijection
2.3 BV and Garfinkle bijections
2.4 Quasisymmetric functions of type B
2.5 Modified domino functions
2.6 Decomposition in Schur functions
2.7 Decomposition in Chow fundamental and monomial functions
2.8 Skew shape decomposition
2.8.1 Skew domino tableaux
2.8.2 Skew domino functions
2.9 Admissible shapes
2.9.1 Admissible and strictly admissible shapes
2.9.2 Allowed transitions
2.9.3 Patterns
2.9.4 Detailed information about possible patterns
2.9.5 Estimations
3 Structure constants of the descent algebra of type B, type B Kronecker and Littlewood Richardson coefficients
3.1 Type B Descent algebra
3.2 Cauchy formula and Kronecker coefficients
3.3 Structure constants of Solomon’s descent algebra of the hyperoctahedral group
3.3.1 A corollary to Theorem 3.5
3.3.2 Skew shapes
4 Type B Schur positivity
4.1 A new definition of type B Schur-positivity
4.1.1 Basic examples of G-positivity
4.2 Application to signed arc permutations
4.2.1 Description of signed arc permutations
4.2.2 Explicit bijections for all types
4.3 Another proof of Theorem 4.5
4.3.1 Poirier’s quasisymmetric functions
4.3.2 Bi-tableaux
4.3.3 Proof
5 Refined statistics with an additional parameter q
5.1 Type B q-Cauchy identity
5.2 Analytical proof for the bijections
5.2.1 Barbash and Vogan correspondence
5.2.2 Two equidistributed statistics on pairs of domino tableaux
5.3 Type B q-Schur positivity
A Admissible shapes
B More type B Schur positive sets
C Spin and negative number statistics
Bibliography
