Skew key polynomials and a generalized Littlewood-Richardson rule
Sami Assaf, Stephanie van Willigenburg
TL;DR
This paper introduces a new partial order called the key poset on weak compositions, generalizes skew Schur functions to skew key polynomials, and provides a nonnegative Littlewood-Richardson rule for their expansion.
Contribution
It defines the key poset and skew key polynomials, extending classical combinatorial objects and rules to a nonsymmetric setting with a new Littlewood-Richardson rule.
Findings
Defined the key poset on weak compositions.
Introduced skew key polynomials as a generalization of skew Schur functions.
Provided a nonnegative Littlewood-Richardson rule for skew key polynomial expansion.
Abstract
Young's lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing Young's lattice, we introduce a new partial order on weak compositions that we call the key poset. Saturated chains in this poset correspond to standard key tableaux, the combinatorial objects that generate the key polynomials, a nonsymmetric polynomial generalization of the Schur basis. Generalizing skew Schur functions, we define skew key polynomials in terms of this new poset. Using weak dual equivalence, we give a nonnegative weak composition Littlewood-Richardson rule for the key expansion of skew key polynomials, generalizing the flagged Littlewood-Richardson rule of Reiner and Shimozono.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · Algebraic structures and combinatorial models
Skew key polynomials and a generalized Littlewood–Richardson rule
Sami Assaf
Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, U.S.A.
and
Stephanie van Willigenburg
Department of Mathematics, University of British Columbia, 1984 Mathematics Rd., Vancouver, BC V6T 1Z2, Canada
Abstract.
Young’s lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing Young’s lattice, we introduce a new partial order on weak compositions that we call the key poset. Saturated chains in this poset correspond to standard key tableaux, the combinatorial objects that generate the key polynomials, a nonsymmetric polynomial generalization of the Schur basis. Generalizing skew Schur functions, we define skew key polynomials in terms of this new poset. Using weak dual equivalence, we give a nonnegative weak composition Littlewood–Richardson rule for the key expansion of skew key polynomials, generalizing the flagged Littlewood–Richardson rule of Reiner and Shimozono.
Key words and phrases:
composition poset, Demazure characters, key polynomials, skew Schur functions, weak compositions,Young’s lattice
2010 Mathematics Subject Classification:
Primary 05E05; Secondary 06A07, 14M15, 14N15
S.A. supported in part by NSF DMS-1763336. S.v.W. supported in part by NSERC
1. Introduction
Schur polynomials are central to the study of the representation theories of the general linear group and of the symmetric group, as well as to understanding the geometry of the Grassmannian. The combinatorics of Young tableaux, the ubiquitous objects that generate the Schur polynomials, often sheds light on important representation theoretic or geometric properties such as tensor products, induction and restriction of modules, and intersection multiplicities.
The celebrated Littlewood–Richardson rule [15] gives a combinatorial description for the Schur expansion of a product of two Schur polynomials or, equivalently, for the Schur expansion of a skew Schur polynomial as
[TABLE]
where is the number of saturated chains in Young’s lattice from to satisfying certain conditions depending on . Here Young’s lattice is the partial order on integer partitions given by containment of Young diagrams. These so-called Littlewood–Richardson coefficients arise in representation theory as the irreducible multiplicities for the tensor product of two irreducible representations for the general linear group and as the irreducible multiplicities for the induced tensor product of two irreducible representations for the symmetric group. They also appear geometrically, giving the number of points lying in a suitable intersection of three Grassmannian Schubert varieties.
The key polynomials are nonsymmetric polynomial generalizations of Schur polynomials first studied by Demazure [8] in connection with Schubert varieties. Key polynomials are irreducible characters of Demazure modules for the general linear group [9] and represent Schubert classes for vexillary permutations [14]. They form an important basis for the polynomial ring, and so we may consider their structure constants parallel to those for the Schur basis of symmetric polynomials. However, in stark contrast with the Schur case, the structure constants of key polynomials are not, in general, nonnegative, though Kouno [12] has partial results where nonnegativity holds and Assaf and Quijada [3] have made progress on understanding the signs in the Pieri case.
Nevertheless, key polynomials appear in many of the myriad generalizations of the Littlewood–Richardson rule. One such rule is the flagged Littlewood–Richardson rule of Reiner and Shimozono [18]. Flagged Schur polynomials [13] arise as those polynomials occuring both as key polynomials and as Schubert polynomials. Reiner and Shimozono [18] considered the flagged skew Schur polynomials and gave a nonnegative rule for their expansion into the key basis. Assaf [6] considered a diagram containment-based skew analog of key polynomials along the same lines, but obtained nonnegativity results only in very special cases.
Another nonnegative rule is the quasisymmetric Littlewood–Richardson rule. The quasisymmetric Schur functions of Haglund, Luoto, Mason and van Willigenburg [10] are a quasisymmetric generalization of Schur polynomials whose combinatorics shares many nice properties with that for Schur polynomials. These authors [11] use the quasisymmetric Schur functions to derive a nonnegative refined Littlewood–Richardson rule for the product of a key polynomial and a Schur polynomial with sufficiently many variables. Related to this, Bessenrodt, Luoto and van Willigenburg [7] define a partial order on strong compositions that gives rise to a nonnegative Littlewood–Richardson rule for a skew analog of the quasisymmetric Schur functions. For details on these results and quasisymmetric Schur functions in general, see the book by Luoto, Mykytiuk, and van Willigenburg [16].
In this paper, we generalize the flagged Littlewood–Richardson rule to skew key polynomials in the largest possible setting where nonnegativity prevails. To do so, we begin in Section 2 by generalizing Young’s lattice to a partial order on weak compositions that we call the key poset. In contrast with the generalization to strong compositions in [7], we give explicit cover relations as well as explicit criteria for comparability in the poset, and remark on the latter hence filling this gap, though as with the strong composition poset, the key poset is not a lattice. In Section 3, we relate the poset with the tableaux combinatorics for key polynomials. Using these paradigms together, in Section 4 we re-define skew key polynomials with respect to the poset and give a general nonnegative Littlewood–Richardson rule for skew key polynomials, vastly generalizing the nonnegativity results of [6, 18]. The relations in the key poset are more restrictive than mere containment used to define skew key polynomials considered in [6]. Under the more general containment definition [6], all key polynomial coefficients are nonnegative if and only if the two indexing shapes are comparable in the key poset. Thus the key poset is precisely the right notion to capture positivity.
2. Posets
An integer partition is a weakly decreasing sequence of positive integers, . The rank of a partition , denoted by , is the sum of the parts,
[TABLE]
and we call its length.
The Young diagram of a partition is the collection of unit cells left-justified in row indexed from the bottom (French notation). Abusing notation, we use interchangeably for the integer partition and for its diagram.
A partially ordered set, or poset, is a set together with a partial order comparing certain elements of the set. We turn integer partitions into a poset by the containment relation, setting if for all or, equivalently, if the diagram for is a subset of the diagram for . We call this poset Young’s lattice.
For in a poset , we say covers , denoted by , if and for any for which , either or . The cover relations for Young’s lattice may be described by if and only if is obtained from by incrementing a single part for which by or, equivalently, by adding a single cell to the end of a row for which the row with one smaller index is strictly longer.
A poset is a lattice if every pair of elements have a unique least upper bound and a unique greatest lower bound. For Young’s lattice, these constructions are given by the set-theoretic union and the set-theoretic intersection of the diagrams, respectively.
Young’s lattice is a prominent tool in algebraic combinatorics, used to study symmetric functions, representations of finite and affine Lie groups, and intersection numbers for finite and affine Grassmannians. We generalize the construction from integer partitions to weak compositions in such a way that maintains the connection to representation theory and geometry.
A weak composition is a sequence of nonnegative integers, that is, . A weak composition is a strong composition if all . Given a weak composition we denote the strong composition obtained by removing its zeros by . Extending notation, the rank of a weak composition , denoted by , is the sum of the parts,
[TABLE]
and we call its length.
The key diagram of a weak composition is the collection of unit cells left-justified in row indexed from the bottom. As above, we use interchangeably for the weak composition and for its key diagram.
Definition 2.1**.**
The key poset is the partial order on weak compositions of length defined by the relation if and only if for and for any indices for which and , we have .
If , then the collection of cells in but not in is called a skew key diagram denoted by . We will also consider more general skew diagrams whenever , or for all .
Definition 2.1 has the following interpretation in terms of key diagrams that we will use often: if and only if and whenever a cell of lies above a cell of , the lower row is strictly longer in ; see Fig. 1.
Notice this partial order is not given simply by containment of key diagrams.
Example 2.2**.**
Let and . Then for , showing . However, since and but . See Fig. 2.
As with Young’s lattice, this partial order on weak compositions is ranked by the number of cells, and we may describe the covering relations in terms of adding a cell subject to certain conditions.
Theorem 2.3**.**
The key poset on weak compositions is ranked by with covering relation if and only if is obtained from by incrementing by where for any we have .
Proof.
Let be two comparable weak compositions of length . If , then and letting denote the row index of the unique cell of , then and for any for which , we must have . In particular, we must have either or , satisfying the stated cover relation.
Conversely, given and a row index for which or for all , the weak composition defined by for and satisfies the condition that for any row index for which , we have . Thus the cover relation implies .
Finally, suppose with . Let denote the shortest, then lowest if tied, row of containing an element of . Set for and . Then clearly . We claim . Given row indices , if and , then we also have , and so since , we have . This ensures by choice of as the shortest row of for , and so showing . If and , then and by choice of the shortest then lowest row for , and so . In particular, is not covered by , proving the poset is ranked with cover relations as stated. ∎
Remark 2.4*.*
The composition poset [7, Definition 2.3] on strong compositions can be described as being ranked by with covering relation if and only if is obtained from by incrementing by where for any we have , or appending a 1 at the front. Therefore using a proof similar to the first two paragraphs of the above proof we have a description of it analogous to Definition 2.1: The composition poset is the partial order on strong compositions of length (prepending zeros if lengths differ) defined by the relation if and only if for and for any indices for which and , we have .
In terms of key diagrams, if and only if , there is a single cell of , and this cell does not sit above any cell lying at the end of its row. Fig. 3 depicts the key poset up to rank .
While containment is not sufficient for covering in general, it is for the partition case. In this way, the key poset generalizes Young’s lattice.
Proposition 2.5**.**
For a weakly increasing weak composition , and for any weak composition, we have if and only if .
Proof.
For weakly increasing, we never have for , making the latter condition of Definition 2.1 vacuously true. Thus containment is comparability. ∎
Corollary 2.6**.**
Any finite subposet of Young’s lattice is a subposet of the key poset.
Proof.
For a partition of length , let be the weakly increasing weak composition of length whose nonzero parts rearrange to . Given any weak composition of length , by Proposition 2.5 if and only if . In particular, if and only if . ∎
We have the following relations for intersections and unions of diagrams.
Lemma 2.7**.**
Given weak compositions of length ,
- (1)
if and , then ; 2. (2)
if and , then .
Proof.
Suppose and . By definition of the poset, we have and , whence . For any cell in , if sits above a cell of , then necessarily the row of is strictly shorter than that of in and in since . Thus, the row of is strictly shorter than that of in , and so .
Suppose and . Then and , whence . For any cell in , if sits above a cell of , then necessarily the row of is strictly shorter than that of in or in . Thus, the row of is strictly shorter than that of in since , and so . ∎
In particular, by Lemma 2.7, if , then is the unique least upper bound of and , and if , then is the unique greatest lower bound of and . However, neither of these conditions needs be the case.
Proposition 2.8**.**
The key poset is not a lattice.
Proof.
Let and . By Lemma 2.7, any greatest lower bound for and must be contained in , but by Example 2.2. Thus any greatest lower bound must have rank at most . Let . Then one can check (visually from Fig. 4) , so since has rank , it is a greatest lower bound for and . Now let . Once again one can check, . However, since and but . Thus is not unique.
Dually, Lemma 2.7 ensures any least upper bound for and must be contained in , but we have just seen . Thus any least upper bound has rank at least , and so the earlier check confirms as a least upper bound. Then as is likewise also a common upper bound and , we see is not unique. ∎
The set-theoretic unions and intersections are natural candidates for least upper bounds and greatest lower bounds, though as Proposition 2.8 shows, they are not necessarily above or below their constituent parts in the key poset. While this problem cannot be overcome completely, there are derived diagrams that do always lie above or below their constituent parts and which are contained in or contain all other least upper or greatest lower bounds.
Proposition 2.9**.**
Given weak compositions and , let denote together with all cells such that the cell immediately left of sits under a cell of and some cell left of either lies in with the cell above in or lies in with the cell above in ; see Fig. 5.
Then , and for any for which , we have .
Proof.
By definition, . Moreover, if some (resp. ) lies above some (resp. ), then by construction of , the lower row is strictly longer, so .
Let be some weak composition for which . By Lemma 2.7, we have , so, in particular, . By construction of , any lies in a row for which there is a higher row that is strictly shorter in and in . Since , we must have . In particular, . ∎
Example 2.10**.**
We now interpret Fig. 5 as a concrete example: Let and , so , and .
Proposition 2.11**.**
Given weak compositions and , let denote the subset of cells such that for any and any lying above , the row of is strictly longer than that of in and of in ; see Fig. 6.
Then , and for any for which , we have .
Proof.
By definition, . Moreover, if some (resp. ) lies above some , then by construction of , the lower row is strictly longer, so .
Let be some weak composition for which . By Lemma 2.7, we have , so, in particular, . By construction of , any must have either a cell or above it for which the row of in or the row of in is weakly longer than that of , and so we cannot have . In particular, . ∎
Example 2.12**.**
We now interpret Fig. 6 as a concrete example: Let and , so , and .
3. Polynomials
Based on the quasi-Yamanouchi tableaux of Assaf and Searles [5], Assaf defined standard key tableaux [6, Definition 3.10]. We generalize this naturally from key diagrams to skew key diagrams as follows.
Definition 3.1** ([6]).**
A standard (skew) key tableau is a bijective filling of a (skew) key diagram with such that rows decrease left to right and if some entry is above and in the same column as an entry with , then there is an entry immediately right of , say , and .
Remark 3.2*.*
There is a natural semistandard analog of Definition 3.1 present in [1] where it is shown that these objects coincide exactly with Mason’s semiskyline augmented fillings [17] defined by a triple rule arising from nonsymmetric Macdonald polynomials.
We denote the set of standard key tableaux of shape by and extend this naturally to standard skew key tableaux.
A saturated chain in a poset is a sequence of elements . We may identify standard Young tableaux of shape with saturated chains in Young’s lattice from to .
Parallel to the case for Young’s lattice, saturated chains from to in the key poset precisely correspond to standard key tableaux of shape . Look ahead to Theorem 4.1 for the skew analog.
Theorem 3.3**.**
For a saturated chain in the key poset, the standard filling of the key diagram for defined by placing into the unique cell of is a standard key tableaux.
Conversely, given with cells, setting and, for , setting to be the diagram containing cells labeled results in a saturated chain from to in the key poset.
Proof.
Suppose is a saturated chain in the key poset. Then by Theorem 2.3, and has one cell. Thus we may indeed set to be the bijective filling of with into the unique cell of , for . The cover relations ensure cells are added only to the right end of a row, ensuring row entries decrease from left to right. Suppose are in the same column with above . Restricting our attention to , the last cell added corresponds to the cell with entry . By Theorem 2.3, no row below that containing entry can have the same length. In particular, the row of must be strictly longer. Thus there exists some entry immediately right of , and since was not the most recently added cell, we have . Therefore is a standard key tableau.
Suppose now is a standard key tableau of size . Since entries decrease from left to right, the shape of the restriction of to entries must be a key diagram. Therefore we may define a nested sequence of weak compositions by setting to be the diagram containing cells of labeled . Note restricted to these entries still satisfies the row and column conditions for a standard key tableau. To see that , the column condition for key tableaux ensures the cell of with smallest entry, which necessarily corresponds to , cannot lie above another cell that ends its row, since that cell necessarily has a larger entry. Thus by Theorem 2.3, the sequence is a saturated chain in the key poset. ∎
For example, Fig. 7 shows the two saturated chains in the key poset from to .
The key polynomials, indexed by weak compositions, form an important basis for the full polynomial ring. Key polynomials arise as characters of Demazure modules [8] for the general linear group and coincide with Schubert polynomials [13] in the vexillary case [14]. Key polynomials are nonsymmetric generalizations of Schur functions, studied combinatorially by Reiner and Shimozono [18] and later by Mason [17], though our perspective follows that of Assaf and Searles [5] and Assaf [6] who define them as the fundamental slide generating polynomial for standard key tableaux.
Assaf and Searles introduced the fundamental slide polynomials [4], indexed by weak compositions, that form a basis for the full polynomial ring. Given strong compositions , we say refines if there exist indices , where is the length of , such that for all
[TABLE]
For example, refines but does not refine .
Definition 3.4** ([4]).**
For a weak composition , the fundamental slide polynomial is
[TABLE]
where the sum is over weak compositions such that refines and dominates .
For a standard key tableau , say is a descent of if lies weakly right of . We assign a weak descent composition for , defined in [6, Definition 3.12], that will index the corresponding fundamental slide polynomial.
Definition 3.5** ([6]).**
For a standard key tableau , let be the partitioning of the decreasing word broken between and precisely whenever is a descent of . Set to be the lowest row index in of a letter in . Set and, for , set . Define the weak descent composition of , denoted by , by and all other parts are zero if all ; and otherwise.
Example 3.6**.**
The left tableau in Fig. 7 has a descent only at , so the partitioning is . Thus , the row index of , , the row index of and , and . These determine the positions of the descents, and so the weak descent composition is .
The right tableau in Fig. 7 has a descent only at , so the partitioning is now . Thus , the row index of and , , the row index of , and . Thus the weak descent composition is .
We take [6, Corollary 3.16] as our definition for key polynomials.
Definition 3.7** ([6]).**
For a weak composition , the key polynomial is
[TABLE]
where the sum is over all standard key tableaux of shape for which the weak descent composition is not .
For example, from Example 3.6, we have
[TABLE]
Note working with semistandard in place of standard objects would replace the fundmental slide polynomials with their monomial expansions, giving a far less compact notation.
Composing the bijective correspondence defined by Assaf and Searles between quasi-Yamanouchi Kohnert tableaux and quasi-Yamanouchi Young tableaux [5, Theorem 4.6] with the bijective correspondence defined by Assaf [6, Theorem 3.15] between those and the corresponding standard tableaux yields a bijective proof that for of length , we have
[TABLE]
Thus key polynomials generalize Schur polynomials. Moreover, key polynomials stabilize to Schur functions as
[TABLE]
where is with zeros prepended and is the partition reordering of .
4. Littlewood–Richardson rules
Assaf considers standard skew key tableaux in [6, Definition 4.7], defined for any pair of weak compositions . However, the positivity result for the corresponding skew key polynomials [6, Theorem 4.10], is only proved for the limited case when the smaller weak composition is a partition. Examples show these are not the only cases where nonnegativity holds, with many overlooked examples arising naturally from geometric contexts. As we shall see, the key to positivity lies in the key poset. We begin by generalizing Theorem 3.3 to skew key diagrams.
Theorem 4.1**.**
For a saturated chain in the key poset, the standard filling of the skew key diagram for defined by placing into the unique cell of is a standard skew key tableaux.
Conversely, for with cells, setting and, for , setting to be the diagram with cells of skewed and containing cells labeled gives a saturated chain from to in the key poset.
Proof.
Both the poset cover relations and the key tableaux row and column conditions are local, so this follows from Theorem 3.3 by restricting attention to shapes from onward and to cells with entries . ∎
Following [6], extend Definition 3.5 directly to standard skew key tableaux of shape by regarding the cells of as labeled , where and .
Definition 4.2**.**
For weak compositions , the skew key polynomial is
[TABLE]
where the sum is over all standard skew key tableaux of shape for which the weak descent composition is not .
Note unlike [6, Definition 4.8], we define skew key polynomials only for comparable elements of the key poset. However, the special case of skewing by an increasing composition, for which the positivity in [6, Theorem 4.10] holds, conforms with this more restrictive definition by Proposition 2.5.
Since key polynomials are a basis for all polynomials, define weak composition Littlewood–Richardson coefficients as the key expansion of skew key polynomials,
[TABLE]
A priori, these coefficients are integers. In fact, we will show that they are nonnegative integers, and so skew key polynomials are key positive.
To prove nonnegativity of the weak composition Littlewood–Richardson coefficients, we utilize weak dual equivalence [6], a polynomial generalization of dual equivalence that consolidates standard key tableaux into equivalence classes, each of which corresponds to a single key polynomial.
Extending earlier notation, given a weak composition of rank and integers , let be the weak composition obtained by deleting the first and last pieces from .
Example 4.3**.**
Let . Then , corresponding to deleting the first pieces both of which come from and last pieces one of which comes from and the remaining from .
Definition 4.4** ([6]).**
Let be a finite set, and let be a map from to weak compositions of rank . A weak dual equivalence for is a family of involutions on such that
- (i)
For all and , there exists a weak composition such that
[TABLE]
where is the equivalence class generated by . 2. (ii)
For all and all , we have .
Define the simple involutions for on standard fillings of rank that interchange and . Define the braid involutions for on standard fillings of rank for which exactly one of or lies in the same row as by cycling entries in the unique way that maintains this condition. For example, will exchange the two standard key tableaux of shape shown in Fig. 7. We combine the braid involutions with the simple involutions to give the elementary weak dual equivalence involutions in [6, Definition 3.21].
Definition 4.5** ([6]).**
Define elementary weak dual equivalence involutions, denoted by , on standard skew key tableaux that act by
[TABLE]
where we take the reading order to be taking the entries in each column, bottom to top and left to right.
For examples of the elementary weak dual equivalence involutions, see Fig. 8.
Assaf [6, Theorem 3.25] showed these are well-defined involutions on standard key tableaux, that all standard key tableaux of fixed shape fall into a single equivalence class, and that this gives an example of a weak dual equivalence.
Theorem 4.6** ([6]).**
Given a weak composition , the involutions give a weak dual equivalence for consisting of a single equivalence class.
Moreover, under certain stability conditions, the converse holds. That is, by [6, Theorem 3.29], any weak dual equivalence is essentially this and, on the level of generating polynomials, we have the following.
Theorem 4.7** ([6]).**
If there exists a weak dual equivalence for for which for every , then
[TABLE]
is key positive.
The condition that the weak descent composition is nonempty for every element can often be circumvented if the polynomials under consideration stabilize. Assaf uses this along with the elementary weak dual equivalence involutions to prove the special case of the following when is weakly increasing [6, Theorem 4.10], which follows from Theorem 4.8 by Proposition 2.5.
Theorem 4.8**.**
For in the key poset, the weak composition Littlewood–Richardson coefficient is the number of weak dual equivalence classes of isomorphic to . In particular, skew key polynomials are key positive.
Proof.
By Theorem 4.6, the elementary weak dual equivalence involutions of Definition 4.3 give a weak dual equivalence on . For in the key poset, by Theorem 4.1, we have . Since Definition 4.4 is completely local, these same involutions restrict to a weak dual equivalence on .
By Definition 3.5, it is clear there exists some nonnegative integer for which for every (the minimal such is given explicitly in [5, Theorem A.6]). Therefore also satisfies the hypotheses of Theorem 4.7, and so we conclude is a nonnegative sum of key polynomials. Setting the first variables to [math], we have that is also key positive. ∎
In fact, we can use the poset structure to prove this result is tight.
Theorem 4.9**.**
For such that , there exists a weak composition for which the weak composition Littlewood–Richardson coefficient is negative.
Proof.
Suppose for all but . Let be the largest index for which there exists an index such that but, contrary to Definition 2.1, , and take largest among these. First consider the minimal case when both and have at most nonzero parts. Generically, we may assume for and where and , as illustrated in Fig. 9. In this case, using Definition 4.1, we have the expansion
[TABLE]
where the lower terms do not use the variable . Since , the two terms with do not cancel, and so the key expansion is not nonnegative.
In the general case, the term appears with coefficient where , and the term appears with coefficient where for and and . Since , these terms do not cancel, and so is negative. ∎
Taken together, Theorems 4.8 and 4.9 show the key poset precisely characterizes the skew key polynomials with nonnegative key polynomial expansion.
The flagged Schur polynomials are another polynomial generalization of Schur functions originally defined by Lascoux and Schützenberger [13] and studied further by Wachs [19]. Given partitions and a flag , the flagged skew Schur polynomial is the sum of the monomials corresponding to semistandard Young tableaux of skew shape with entries in row bounded above by . Taking gives the (nonskew) flagged Schur polynomial.
In addition to the tableaux definition, Assaf and Bergeron [2] describe flagged (skew) Schur functions in terms of flagged -partitions, a description that immediately gives the fundamental slide expansion as well.
Reiner and Shimozono [18, Theorem 23] show every flagged Schur polynomial is equal to a single key polynomial for some weak composition , though the converse does not hold. That is, key polynomials are more general than flagged Schur polynomials.
Moreover, Reiner and Shimozono [18, Theorem 20] give a flagged Littlewood–Richardson rule showing any flagged skew Schur polynomial is a positive sum of key polynomials, that is
[TABLE]
where counts the number of -compatible tableaux whose left-nil key with respect to is ; see [18] for definitions and details.
By Theorems 4.8 and 4.9, it follows that each flagged skew Schur polynomial is equal to a skew key polynomial for some weak compositions , though once again the converse does not hold. That is, skew key polynomials are more general than flagged skew Schur polynomials, and so we obtain a maximal generalization of the flagged Littlewood–Richardson rule.
5. Acknowledgments
The authors would like to thank the referees for useful suggestions and careful reading.
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