Combinatorial expansions in -theoretic bases
We study the class of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials, -atoms, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of , Grothendieck functions represent the -theory of the same space. In this paper, we give a combinatorial description of the coefficients when any element of is expanded in the -basis or the basis dual to .
Schubert calculus uses intersection theory to convert enumerative problems in projective geometry into computations in cohomology rings. In turn, the representation of Schubert classes by Schur polynomials enables such computations to be carried out explicitly. The combinatorial theory of Schur functions is central in the application of Schubert calculus to problems in geometry, representation theory, and physics.
In a similar spirit, a family of power series called Grothendieck polynomials were introduced by Lascoux and Schützenberger in [LS83] to explicitly access the -theory of . In [FK94], Fomin and Kirillov first studied the stable limit of Grothendieck polynomials as . When indexed by Grassmannian elements, we call these limits the -functions. Grothendieck polynomials and -functions are connected to representation theory and geometry in a way that leads to a generalization of Schubert calculus where combinatorics is again at the forefront. Moreover, fundamental aspects of the theory of Schur functions are contained in the theory of -functions since each is an inhomogeneous symmetric polynomial whose lowest homogeneous component is the Schur function .
Parallel to the study of -functions is the study of a second family of functions that arise by duality with respect to the Hall inner product on the ring of symmetric functions. In particular, results in [Len00, Buc02] imply that the -functions form a Schauder basis for the completion of with respect to the filtration by the ideals . The dual Hopf algebra to this completion is isomorphic to . Therein lies the basis of -functions, defined by their duality to the -basis. Lam and Pylyavskyy first studied these functions directly in [LP07] where they were called dual stable Grothendieck polynomials. By duality, each is inhomogeneous with highest homogeneous component equal to .
Strictly speaking, the - and -functions do not lie in the same space and there is no sensible way to write -functions in terms of -functions. However, any element of can be expanded into both the - and the -functions and it is such expansions that are of interest here. Motivated by the many families of symmetric functions whose transition matrices with Schur functions have combinatorial descriptions and encode representation theoretic or geometric information, our focus is on functions with what we refer to as tableaux Schur expansions. A symmetric function, , is said to have a tableaux Schur expansion if there is a set of tableaux and a weight function so that
Among the classical examples is the family of Hall-Littlewood polynomials [Hal59, Lit61], whose tableaux Schur expansion gives the decomposition of a graded character of into its irreducible components [GP92]. A more recent example is given by the -atoms [LLM03]. These are conjectured to represent Schubert classes for the homology of the affine Grassmannian when and their very definition is a tableaux Schur expansion.
In this paper, we give combinatorial descriptions for the - and the -expansion of every function with a tableaux Schur expansion. Our formulas are in terms of set-valued tableaux and reverse plane partitions; -functions are the weight generating functions of the former and -functions are the weight generating functions of the latter. More precisely, for any given set of semistandard tableaux, we describe associated sets and of set-valued tableaux and reverse plane partitions, respectively. Given also any function on , we define an extension of to and . In these terms, we prove that any function satisfying (1) can be expanded as
Since a Schur function has a trivial tableaux Schur expansion, the simplest application of (2) describes the transition matrices between and Schur functions in terms of certain reverse plane partitions and set-valued tableaux. These transition matrices were alternatively described by Lenart in [Len00] using certain skew semistandard tableaux. Although our description is not obviously equinumerous, we give a bijective proof of the equivalence in section 5. As a by-product, we show that Lenart’s theorem follows from (2).
In section 6, we show how the description of a -expansion given by (2) may lead to a more direct combinatorial interpretation for the expansion coefficients. For example, we show that the Hall-Littlewood functions can be defined by extending the notion of charge to reverse plane partitions and set-valued tableaux. We also show that the /-expansions of a product of Schur functions can be described by certain Yamanouchi reverse plane partitions and set-valued tableaux. Note, this is not the -expansion of a product of -functions which was settled in [Buc02].
2. Definitions and notation
2.1. Symmetric function basics
The Ferrers diagram of a partition is a left- and bottom-justified array of square cells in the first quadrant of the coordinate plane, with cells in the row from the bottom.
The Ferrers diagram of the partition is .
Given any partition , the conjugate is the partition obtained by reflecting the diagram of about the line . For example, the conjugate of is .
A semistandard tableau of shape is a filling of the cells in the Ferrers diagram of with positive integers, such that the entries
are weakly increasing while moving rightward across any row, and
are strictly increasing while moving up any column.
A semistandard tableau of shape is .
Throughout this paper, the letter will generally refer to a tableau, and will typically denote a set of tableaux.
The evaluation of a semistandard tableau is the sequence where is the number of cells containing . The evaluation of the tableau in Example 2.4 is (it is customary to omit trailing ’s). We use to denote the set of all semistandard tableaux of shape , and to denote the set of all semistandard tableaux of shape and evaluation .
A word is a finite sequence of positive integers. The reading word of a tableau , which we denote by , is the sequence obtained by listing the elements of starting from the top-left corner, reading across each row, and then continuing down the rows.
We have .
We use the fundamental operations jeu-de-taquin [Sch77] and -insertion [dBR38, Sch61, Knu70] on words. The reader can find complete details and definitions of these operations in [LS81, Sta99, Ful91]. A key property of -insertion is that
for any tableau . When two words insert to the same tableau under the RSK map, they are said to be Knuth equivalent.
The weight generating function of semistandard tableaux can be used as the definition of Schur functions. For any tableau , let , where is the evaluation of .
The Schur function is defined by
The Schur functions are elements of , the power series ring in infinitely many variables, and are well known to be a basis for the symmetric functions (i.e., those elements of which are invariant under any permutation of their indices).
The Schur function is
corresponding to the tableaux
Another basis for the symmetric functions is given by the monomial symmetric functions.
The monomial symmetric function is defined by
summing over all distinct sequences which are a rearrangement of the parts of . (Here is thought of as having finitely many non-zero parts, followed by infinitely many parts.)
The monomial symmetric function is
The Kostka numbers give the change of basis matrix between the Schur and monomial symmetric functions. For two partitions , we define the number to be the number of semistandard tableaux of shape and weight . From the previous definitions, one can see that a consequence of the symmetry of the Schur functions is that
There is a standard inner product on the vector space of symmetric functions (known as the Hall inner product), defined by setting
The following proposition is a basic, but very useful, fact of linear algebra.
If and are two pairs of dual bases for an inner-product space, and
Buch introduced the combinatorial notion of set-valued tableaux in [Buc02] to give a new characterization for -functions and to prove an explicit formula for the structure constants of the Grothendieck ring of a Grassmannian variety with respect to its basis of Schubert structure sheaves. It is this definition of -functions we use here.
A set-valued tableau of shape is a filling of the cells in the Ferrers diagram of with sets of positive integers, such that
the maximum element in any cell is weakly smaller than the minimum element of the cell to its right, and
the maximum element in any cell is strictly smaller than the minimum entry of the cell above it.
Another way to view this definition is by saying that the selection of a single element from each cell (in any possible way) will always give a semistandard tableau.
A set-valued tableau of shape is
We have omitted the set braces, ‘’ and ‘’, here and throughout for clarity of exposition.
The evaluation of a set-valued tableau is the composition where is the total number of times appears in . The evaluation of the tableau in Example 2.13 is . The collection of all set-valued tableaux of shape will be denoted , and the subset of these with evaluation will be denoted . We write for the number of set-valued tableaux of shape and evaluation . We will typically denote a set-valued tableau with the letter . Finally, we define the sign of a set-valued tableau, , to be the number elements minus the number of cells:
A multicell will refer to a cell in that contains more than one letter. Note that when has no multicells we view , as a usual semistandard tableau. In this case, , and .
For any partition , the Grothendieck function is defined by
For terms where , since there are no multicells. Hence equals plus higher degree terms. Since the are known to be symmetric functions, they therefore form a basis for the appropriate completion of .
Applying Proposition 2.11 to this definition gives rise to the basis that is dual to by way of the system,
over partitions . These -functions were studied explicitly by Lam and Pylyavskyy in [LP07] where they showed that can be described as a certain weight generating function for reverse plane partitions.
A reverse plane partition of shape is a filling of the cells in the Ferrers diagram of with positive integers, such that the entries are weakly increasing in rows and columns.
A reverse plane partition of shape is .
Following Lam and Pylyavskyy (and differing from some other conventions) we define the evaluation of a reverse plane partition to be the composition where is the total number of columns in which appears. The evaluation of the reverse plane partition in Example 2.16 is . The collection of all reverse plane partitions of shape will be denoted and the subset of these with evaluation will be denoted . We will typically use the letter to refer to a reverse plane partition.
Theorem 2.17 (Lam-Pylyavskyy).
The polynomials have the expansion
We note that when , the entries must be strictly increasing up columns; hence is equal to plus lower degree terms.
3. General formula for -theoretic expansions
Our combinatorial formula for the - and the -expansion of any function with a tableaux Schur expansion is in terms of reverse plane partitions and set-valued tableaux, respectively. The formula relies on a natural association of these objects with semistandard tableaux which comes about by a careful choice of reading word for set-valued tableaux and reverse plane partitions.
The reading word of a set-valued tableau , denoted by , is the sequence obtained by listing the elements of starting from the top-left corner, reading each row according to the following procedure, and then continuing down the rows. In each row, we first ignore the smallest element of each cell, and read the remaining elements from right to left and from largest to smallest within each cell. Then we read the smallest element of each cell from left to right, and proceed to the next row.
The reading word of the set-valued tableau in Example 2.13 is .
Given a reverse plane partition , circle in each column only the bottommost occurrence of each letter. The reading word of , which we denote by , is the sequence obtained by listing the circled elements of starting from the top-left corner, and reading across each row and then continuing down the rows.
The reverse plane partition in Example 2.16 has .
This given, for a set of semistandard tableaux, we define sets and of set-valued tableaux and reverse plane partitions, respectively, by
Similarly, we can extend any function defined on to and by setting
for any in or .
It is in terms of these definitions that we express the - and -expansions for functions with a tableaux-Schur expansion.
Given with a tableaux Schur expansion,
Writing the on the right hand side of equation (8) as the sum over tableaux gives
Since every semistandard tableau can be viewed as a set-valued tableau and as a reverse plane partition, every monomial term in (11) also appears as a term in (12). Thus to prove that (11) equals (12), it suffices to show that the terms in (12) not occurring in (11) sum to zero.
From these observations, we can simultaneously prove that (11) equals (12) and (13) by producing a single sign-reversing and weight-preserving involution. To be precise, in the next section we introduce a map and prove that it is an involution on the set of pairs of , where is a set-valued tableau and is a reverse plane partition of the same shape, satisfying the properties:
if and only if and are both semistandard tableaux,
when is not a semistandard tableau, and
4. The involution
We introduce basic operations on set-valued tableaux and reverse plane partitions called dilation and contraction and will then define the involution in these terms. To this end, first setting some notation for set-valued tableaux and reverse plane partitions will be helpful.
Given a set-valued tableau , let be the highest row containing a multicell. Let denote the subtableau formed by taking only rows of lying strictly higher than row . For a reverse plane partition , let denote the highest row containing an entry that lies directly below an equal entry. We use the convention that when has no multicell, and when no column of has a repeated entry, .
Given a set-valued tableau , let be the rightmost multicell in and define to be the largest entry in . The dilation of , , is constructed from by removing from and inserting it, via , into .
Since and ,
For any set-valued tableau , is a set-valued tableau.
Let be the rightmost multicell in row and let . Rows weakly lower than row in form a set-valued tableau since is set-valued to start. For rows above row , first note that the cell above is empty or contains a letter strictly greater than . Thus, the insertion of into row puts in a cell that is weakly to the left of . Moreover, all entries in row of that are weakly to the left of cell are strictly smaller than since is a multicell in . Thus, in , is strictly larger than all entries in the cell below it. The claim then follows from usual properties of RSK insertion. ∎
For any set-valued tableaux and ,
Let . For some word , can be factored as since is the first letter in the reading word of row . The definition of dilation then gives that the word of is . Thus the Knuth equivalence classes of and are the same since RSK insertion preserves Knuth equivalence. ∎
We remark that it is Property 4.4 which motivated our definition for the reading word of a set-valued tableau.
Given a reverse plane partition , let and let be the rightmost cell in row that contains the same entry as the cell below it. The contraction of , , is constructed by replacing with a marker and using reverse jeu-de-taquin to slide this marker up and to the right until it exits the diagram.
For any reverse plane partitions and ,
Let and note that the portion of the reading word of obtained by reading rows weakly below row is unchanged by contraction. Moreover, the rows of higher than form a semistandard tableau and thus the jeu-de-taquin moves in these rows preserve Knuth equivalence. In row , any initial rightward slide of the marker does not change the reading word since every letter to the right of the marker is strictly greater than the letter below it (and hence, strictly greater than the letter below it and to its left). It thus suffices to check that the move taking the marker from row to row preserves Knuth equivalence. To this end, let denote the stage of the jeu-de-taquin process at which the next move takes the empty marker from row to . Consider the subtableau consisting only of the letters in rows and which contribute to the reading word. In general, the form of will be
where are weakly increasing words with and each , is a letter with , and are weakly increasing words with (since an entry of may not be the lowest in its column and thus not part of the reading word whereas all entries of contribute to reading word since they are in row ). It remains to show the Knuth equivalence of the words and . We first note that the insertion of the word is
for some decomposition of into disjoint subwords . With this observation, it is not hard to verify that both words and insert to
and hence these words are Knuth equivalent. ∎
For a set-valued tableau and a reverse plane partition of the same shape as , define the map
according to the following four cases (where , and the dilation and contraction of a pair are defined below):
if , the pair is a fixed point
if , the pair is dilated
if , the pair is contracted
if , the pair is dilated when and is otherwise contracted.
In case (2), and is constructed from by replacing the cell of in position with an empty marker and sliding this marker to the south-west using jeu-de-taquin. When the marker reaches row , we replace it by the entry in the cell directly above it. In case (3), . Construct from by deleting the cell of in position and reverse RSK bumping its entry until the entry is bumped from row . Finally, add entry to the unique cell of row where is maximal in its cell and the non-decreasing row condition is maintained. Case (4) reduces to case (2) or (3), determined by comparing the entry to the number described in the definition of dilation.
The involution exchanges the two pairs below:
The pair on the left has and , implying that is dilated under . The pair on the right has and and is thus contracted. Note that and that
Thus, reverses the sign and preserves the weight of this pair.
The map is a sign-reversing and weight-preserving involution on the set of pairs of , where is a set-valued tableau and is a reverse plane partition of the same shape. The fixed points of are pairs where and are both semistandard tableau.
We first verify that is in fact a pair where is a set-valued tableau and is a reverse plane partition. When requires dilation, Property 4.3 assures that is a valid set-valued tableaux and it is straightforward to verify that is a valid reverse plane partition by properties of jeu-de-taquin.
In the case that involves contraction, is a reverse plane partition again by properties of jeu-de-taquin. Since , is a semistandard tableau implying by RSK that is as well and that is well-defined. It remains to check that there is a unique cell in row of into which can be placed so that it is maximal in this cell and row maintains the non-decreasing condition. We claim that this cell is the rightmost cell of row whose entries are all strictly less than . Note that exists since the cell of directly below the cell from which was bumped has only entries smaller than . We claim that is the unique cell into which can be placed; namely, no cell to the right of contains an element strictly less than . This is clear when since then has no multicells in row . Otherwise, conditions of case (4) imply that the largest element of a multicell in row is . Hence there are no multicells to the right of and the claim follows. Finally, we observe that the entry in the cell directly above (if it exists) must be strictly greater than , since this cell is weakly to the right of the cell from which was bumped.
We now show that is indeed an involution by proving that if is obtained by dilation then will require contraction, and vice versa. Consider the case that requires dilation. The definition of implies that given . Further, the reversibility of the RSK algorithm and jeu-de-taquin on semistandard tableaux give that . Therefore, applying to requires the contraction case. The cases in which requires contraction to start follow similarly.
That is sign-reversing and has the appropriate fixed point set is easy to verify from the definition and it remains only to show that is weight-preserving. From the definition of , and the fact that is an involution, we have either:
and , or
5. Schur expansions and an alternate proof
Lenart proved in [Len00] that the transition matrices between and Schur functions have a beautiful combinatorial interpretation in terms of objects that have since been called elegant fillings in [LP07]. Here we show how the simplest application of Theorem 3.5 gives rise to a new interpretation for these transition matrices in terms of certain reverse plane partitions and set-valued tableaux.
Alternatively, we give a bijection between the elegant fillings and these reverse plane partitions/set-valued tableaux. As a by-product, we have an alternate proof for Lenart’s result following from Theorem 3.5 and vice versa.
5.1. A new approach to Schur and /-transitions
An elegant filling is a skew semistandard tableaux with the property that the numbers in row are no larger than . An elegant filling whose entries are strictly increasing across rows is called strict. We let denote the number of elegant fillings of shape and denote the number of strict elegant fillings of shape .
[Len00] The transition matrices between the Schur functions and the -functions are given by the following:
Note that the transition between and Schur functions follows immediately by duality:
The simplest application of Theorem 3.5 provides a new combinatorial description for the and coefficients.
Fix a partition and a semistandard tableau of shape . is the number of set-valued tableaux of shape whose reading word is equivalent to and is the number of reverse plane partitions of shape whose reading word is equivalent to .
Consider the simple case that consists of just one tableau of shape . We can then apply Theorem 3.5 to the trivial expansion
to find that
where is the set of all set-valued tableaux whose reading word is Knuth equivalent to and is the set of all reverse plane partitions whose reading word is Knuth equivalent to . The result on then follows by (17) and the interpretation for follows from (14). ∎
Here we describe bijections between elegant fillings and certain reverse plane partitions, and between strict elegant fillings and certain set-valued tableaux. The bijections lead to alternate proofs for Proposition 5.3, Theorem 5.2, and Theorem 3.5.
We first consider a map on strict elegant fillings. Recall that is the highest row of with a multicell.
For any fixed tableau , we define the map
where is the filling of that records the sequence of set-valued tableaux
by putting in cell , the difference between the row index of this cell and .
Given and ,
is constructed by recording the sequence of dilations
For any tableau , is a bijection.
Fix a tableau and let denote its shape. Consider a set-valued tableau of shape whose reading word is equivalent to .
We start by showing that is a strict elegant filling of shape . Let
Note that this procedure indeed ends with since each dilation preserves the Knuth equivalence class of the reading word. Since each , and , is an elegant filling of the correct shape by construction. To ensure that is a strict elegant filling, it is enough to know that the bumping paths created by successive dilations starting in the same row do not terminate in the same row. Since dilation starts with the largest entry in a row, such successive dilations involve bumping from row , a letter after an where . Therefore, the bumping path created by must be weakly inside the bumping path of and in particular, terminates in a higher row.
It remains to show that is invertible. For the inverse map, consider a strict elegant filling of shape . In the elegant filling, we first replace each entry by where is the row index of the cell containing . The resulting filling consists of “destination rows” for the corresponding entries in our fixed tableau . We proceed by performing contraction on the entries of which are outside of the inner shape , stopping the reverse-bumping procedure when we get to the destination row. The order in which these contractions are performed is determined first by the destination rows (smallest to largest) and then by the height of the original cell (highest to lowest). This concludes the proof. ∎
Recall that is the row index of the highest cell in a reverse plane partition which contains the same entry as the cell immediately above it.
For any fixed tableau , we define the map
where is the filling of that records the sequence of reverse plane partitions
by putting in cell .