Experimental Mathematics
ISSN: 1058-6458 (Print) 1944-950X (Online) Journal homepage: https://www.tandfonline.com/loi/uexm20
Kohnert Polynomials
Sami Assaf & Dominic Searles
To cite this article: Sami Assaf & Dominic Searles (2019): Kohnert Polynomials, Experimental Mathematics, DOI: 10.1080/10586458.2019.1588180 To link to this article: https://doi.org/10.1080/10586458.2019.1588180
Published online: 23 Apr 2019.
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Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=uexm20 EXPERIMENTAL MATHEMATICS https://doi.org/10.1080/10586458.2019.1588180
Kohnert Polynomials
Sami Assafa and Dominic Searlesb aDepartment of Mathematics, University of Southern California, Los Angeles, CA, USA; bDepartment of Mathematics and Statistics, University of Otago, Dunedin, New Zealand
ABSTRACT KEYWORDS We associate a polynomial to any diagram of unit cells in the first quadrant of the plane Schubert polynomials; using Kohnert’s algorithm for moving cells down. In this way, for every weak composition Demazure characters; key one can choose a cell diagram with corresponding row-counts, with each choice giving rise polynomials; fundamental slide polynomials to a combinatorially-defined basis of polynomials. These Kohnert bases provide a simultan- eous generalization of Schubert polynomials and Demazure characters for the general linear 2010 MATHEMATICS group. Using the monomial and fundamental slide bases defined earlier by the authors, we SUBJECT show that Kohnert polynomials stabilize to quasisymmetric functions that are nonnegative CLASSIFICATION on the fundamental basis for quasisymmetric functions. For initial applications, we define Primary 14M15; secondary and study two new Kohnert bases. The elements of one basis are conjecturally Schubert- 14N15; 05E05 positive and stabilize to the skew-Schur functions; the elements of the other basis stabilize to a new basis of quasisymmetric functions that contains the Schur functions.
1. Introduction Kohnert defined an algorithmic process on cell dia- grams that moves the rightmost cell of a row down to Certain homogeneous bases of the ring of polynomials the first available position below. The Kohnert dia- are of central importance in representation theory and grams for a are the cell diagrams that may be geometry. Foremost among these are the Schubert obtained by a (possibly empty) sequence of these polynomials [Lascoux and Schutzenberger€ 82], which Kohnert moves on DðaÞ: Kohnert proved that the are characters of Kraskiewicz–Pragacz modules Demazure character for a is the generating function [Kraskiewicz and Pragacz 87, Kraskiewicz and Pragacz of the Kohnert diagrams of DðaÞ: 04] and represent Schubert basis classes in the coho- Kohnert conjectured that the Schubert polynomials mology of the complete flag variety, and the arise by applying the exact same algorithm to different Demazure characters [Demazure 74] (also known as initial cell diagrams, namely, the Rothe diagrams of key polynomials), which are the characters of permutations. Proofs were given by Winkel [Winkel Demazure modules for the general linear group. We 99, Winkel 02], though were not fully accepted due to are motivated by the question of finding other bases the very technical nature of the arguments; a recent of polynomials that exhibit close connections to and more direct proof was given by Assaf [Assaf 17] using share key properties with these important bases. Such the expansion of Schubert polynomials into bases may be used to understand Schubert polyno- Demazure characters. mials and Demazure characters and moreover may be In this work, we study the polynomials arising of independent representation-theoretic or geomet- from application of Kohnert’s algorithm to any cell ric interest. diagram in N  N; we call these polynomials Kohnert Kohnert [Kohnert 90] introduced a combinatorial polynomials. By definition, Kohnert polynomials model for the monomial expansion of a Demazure expand positively in monomials, and simultaneously character. Let a be a weak composition, that is, a generalize both Schubert polynomials and Demazure sequence of nonnegative integers. Kohnert’s model characters. Given a weak composition a, there are sev- begins with the diagram DðaÞ of a, the cell diagram eral different (though finitely many) Kohnert polyno- N N in  which has ai cells in row i, left-justified. mials for a: in creating an initial cell diagram one
CONTACT Sami Assaf [email protected] Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, USA. To the memory of Axel Kohnert. ß 2019 Taylor & Francis Group, LLC 2 S. ASSAF AND D. SEARLES
must place ai cells in row i, but one may choose in and Shimozono 95, Reiner and Shimozono 98]in which columns the cells are placed. If one Kohnert their study of Specht modules associated to diagrams, polynomial is chosen for every weak composition a, suggesting a possible connection between flagged we call the resulting set of polynomials a Kohnert Weyl modules and Kohnert polynomials. basis of the polynomial ring. Each Kohnert polyno- There are several natural choices of ways to associ- mial in a Kohnert basis has a unique monomial that ate a two-dimensional cell diagram to a weak compos- is minimal in dominance order, hence a Kohnert basis ition. Demazure characters arise from left-justification, is lower uni-triangular with respect to the basis of Schubert polynomials arise from choosing the Rothe monomials. Thus, Kohnert bases are bases of the diagram of the associated permutation. Using the con- polynomial ring, justifying the nomenclature. struction of Kohnert bases, we believe that other nat- Kohnert bases thus comprise a vast collection of ural choices of diagram for a weak composition will combinatorially-defined bases of polynomials, which yield several new and interesting combinatorial includes the Schubert and Demazure character bases. objects, from Kohnert bases of the polynomial ring to To motivate and facilitate further investigation of families (or bases) of quasisymmetric functions. Initial Kohnert bases, we prove that every Kohnert polyno- computer experiments lead us to introduce, as a first mial expands positively in the monomial slide polyno- application, two new Kohnert bases with nice com- mials introduced in [Assaf and Searles 17]. An binatorial properties and surprising connections to immediate application is that every Kohnert polyno- representation theory and geometry. mial has a stable limit, which, in fact, is quasisymmet- The skew polynomials are the Kohnert polynomials ric and expands positively in the monomial basis of associated to diagrams arising from certain rightward quasisymmetric functions. shifts of contiguous rows of cells. As predicted by our We define necessary and sufficient conditions on demazure condition, we prove that skew polynomials cell diagrams for the corresponding Kohnert polyno- expand positively into Demazure characters. Based on mial to expand positively in the fundamental slide computer evidence, we conjecture they also expand basis [Assaf and Searles 17], a polynomial ring analog positively in Schubert polynomials, suggesting a hid- of Gessel’s basis of fundamental quasisymmetric func- den connection with geometry. Stable limits of skew tions [Gessel 84]. While not every Kohnert polynomial polynomials are symmetric, and in fact are the skew- expands positively in the fundamental slide basis, we Schur functions. prove that, surprisingly, the stable limit of any The lock polynomials are the Kohnert polynomials Kohnert polynomial expands positively in fundamen- associated to right-justified diagrams. Lock polyno- tal quasisymmetric functions. For example, the stable mials expand positively in fundamental slide polyno- limits of Schubert polynomials are Stanley symmetric mials (as do the Schubert polynomials and Demazure functions [Macdonald 91] and the stable limits of characters), and coincide with Demazure characters Demazure characters are Schur polynomials [Assaf and Searles 18, Lascoux and Schutzenberger€ 90]; each when the nonzero entries of a are weakly decreasing of these is known to expand positively in fundamental (which is also the only case when the diagrams satisfy quasisymmetric functions. Thus, by taking stable lim- our conjectured demazure condition). The stable lim- its of Kohnert bases, one obtains new and recovers its of lock polynomials, which we call the extended known families of fundamental-positive quasisymmet- Schur functions, are a new basis of quasisymmetric ric functions. These families may or may not be bases functions. By the theory of Kohnert bases, the of quasisymmetric functions; for example, the stable extended Schur functions expand positively in the limits of Schubert polynomials and Demazure charac- fundamental basis. Per the name, the extended Schur ters are not. function basis contains the Schur functions as a sub- We define a simple condition on diagrams that we set, thus is a lifting of the Schur basis from symmetric conjecture characterizes those diagrams for which the to quasisymmetric functions. The description in terms corresponding Kohnert polynomial expands non- of cell diagrams naturally gives rise to families of negatively as a sum of Demazure characters. Both key tableaux generating the lock polynomials and the diagrams, indexing Demazure characters, and Rothe extended Schur functions, similar to the definition of diagrams, indexing Schubert polynomials, satisfy the Kohnert tableaux for Demazure characters in [Assaf stated condition. In further support of the conjecture, and Searles 18]. The tableau description enables us to the demazure condition is exactly the same as the give explicit formulas for the expansion of an northwest condition of Reiner and Shimozono [Reiner extended Schur function in terms of fundamental EXPERIMENTAL MATHEMATICS 3
Figure 1. Four diagrams of weight ð0; 2; 1; 2Þ:
quasisymmetric polynomials, and extract further inter- For example, Figure 2 shows all 16 Kohnert dia- esting properties. grams for the key diagram D ð0; 2; 1; 2Þ: For compari- We expect these bases and others arising as son, the second diagram in Figure 1 gives rise to 26 Kohnert bases may, like the Schubert and Demazure Kohnert diagrams shown in Figure 3 and the third character bases, have deep connections to representa- gives rise to 9 Kohnert diagrams shown in Figure 17. tion theory and geometry. Definition 2.2. The Kohnert polynomial indexed by D is X 2. Kohnert polynomials ðÞ K wt T 1 wtðÞT n : – D ¼ x1 ÁÁÁxn (2 1) Let N denote the natural numbers f1; 2; :::g: In T2KDðÞD Section 2.1, we review Kohnert’s algorithm that gener- For example, from Figure 2, we see that ates a polynomial from a cell diagram in N Â N and K 2 2 2 2 2 2 2 use this to define Kohnert polynomials. We review the DðÞ0;2;1;2 ¼ x1x2x3 þ x1x2x4 þ x1x2x3 þ 2x1x2x3x4 motivating examples of Demazure characters in 2 2 2 2 2 2 2 2 þ x1x2x4 þ x1x3x4 þ x1x3x4 þ x1x2x3 Section 2.2 and Schubert polynomials in Section 2.3, þ 2x x2x x þ x x2x2 þ x x x2x presenting both in the context of Kohnert 1 2 3 4 1 2 4 1 2 3 4 2 2 2 2 2: polynomials. þ x1x2x3x4 þ x2x3x4 þ x2x3x4
Note that the diagram of a Kohnert polynomial is 2.1. Kohnert diagrams not necessarily unique. For instance, if two diagrams A diagram is an array of finitely many cells in N Â N: differ by insertion or deletion of empty columns, then The weight of a diagram D, denoted by wtðDÞ; is the they necessarily give the same Kohnert polynomial. weak composition whose ith part is the number of However, as demonstrated by Theorem 6.12 below, cells in row i. For example, four diagrams with weight this is not sufficient. It is an interesting question to ð0; 2; 1; 2Þ are shown in Figure 1. ask for necessary and sufficient conditions for two A diagram is called a key diagram if the rows are diagrams to give the same Kohnert polynomial. left justified. For each weak composition a, there is a Given weak compositions a and b, say that b domi- 6 ; 6 unique key diagram of weight a which we call the key nates a, denoted by a b if a1 þÁÁÁþak b1 þÁÁÁþ diagram for a and denote by DðaÞ: For example, the bk for all k. Since Kohnert polynomials have a unique leftmost diagram in Figure 1 is the key diagram leading term that is minimal in dominance order, for ð0; 2; 1; 2Þ: they provide a simple mechanism for constructing In his thesis, Kohnert [Kohnert 90] described an interesting bases of the polynomial ring. algorithm for generating a Demazure character, which Theorem 2.3. Given any set of diagrams fDag, one for he called a key polynomial after Lascoux and every weak composition, such that wtðDaÞ¼a, the cor- € € K Schutzenberger [Lascoux and Schutzenberger 90], responding Kohnert polynomials f Da g form a basis of from a key diagram by iteratively applying certain the polynomial ring. Kohnert moves to the diagram. Proof. For any weak composition a and any diagram Definition 2.1 ([Kohnert 90]). A Kohnert move on a D such that wtðD Þ¼a; the corresponding Kohnert diagram selects the rightmost cell of a given row and a polynomial KD expands as moves the cell to the first available position below in X K a1 an b1 bn the same column (if such a position exists), jumping D ¼ x1 ÁÁÁxn þ ca;bx1 ÁÁÁxn over other cells in its way as needed. Given a diagram b>a D, let KDðDÞ denote the set of all diagrams that can for some nonnegative integers ca;b; where the sum is be obtained by applying a series of Kohnert moves over weak compositions b that strictly dominate a.In to D. particular, any set of Kohnert polynomials of the form 4 S. ASSAF AND D. SEARLES
Figure 2. Kohnert diagrams for Dð0; 2; 1; 2Þ:
Figure 3. Kohnert diagrams for Dð143625Þ:
K f Da g where wtðDaÞ¼a is lower uni-triangular with 2.2. Demazure characters respect to monomials, and thus is also a basis. w Kohnert’s original motivation for studying key dia- As this concept is central to the current study, we grams arose from characters of Demazure modules introduce the following terminology. for the general linear group [Demazure 74], which may be regarded as truncations of irreducible char- Definition 2.4. A basis fB g for polynomials is a a acters [Demazure 74]. These polynomials were Kohnert basis if each element B can be realized as a a studied combinatorially by Lascoux and Kohnert polynomial for some diagram D Schutzenberger€ [Lascoux and Schutzenberger€ 90], with wtðDÞ¼a: who call them key polynomials.Foranicesurveyof Two important examples of Kohnert bases are the combinatorial aspects, see [Reiner and Demazure characters and Schubert polynomials, dis- Shimozono 95]; for a recent treatment from cussed below. In addition to proving general positivity Kohnert’s perspective, see [Assaf and Searles 18]. results for Kohnert polynomials, we demonstrate the The original definition for Demazure characters is power of this paradigm by giving a new example of a in terms of divided difference operators, denoted by @i; Kohnert basis in Section 6. defined on a polynomial f by EXPERIMENTAL MATHEMATICS 5
f Àsi Á f character. Therefore, Kohnert polynomials are a gen- @if ¼ ; (2–2) xi À xiþ1 eralization of Demazure characters. Macdonald [Macdonald 91] noted that when a is where s is the simple transposition interchanging i i weakly increasing of length n, we have j ¼ and i þ 1 and it acts on polynomials by interchanging a s ðx ; :::; x Þ; where sk is the Schur polynomial that x and x : Extending this, we may define a linear revðaÞ 1 n i iþ1 gives the irreducible characters for the general linear operator p on polynomials by i group. This follows as well from Demazure’s perspec- p @ : – if ¼ iðÞxif (2 3) tive [Demazure 74] since the Demazure modules interpolate between the highest weight space and the Given a permutation w, we may define full irreducible highest weight module. The Demazure @ @ :::@ w ¼ s1 sk characters are obtained from the irreducible characters p p :::p w ¼ s1 sk by truncating, and so they are, in general, only par- for any expression s1:::sk ¼ w with k minimal. It can tially symmetric. However, they are well-defined under stabilization and in the limit converge to the be shown that both @w and pw are independent of the choice of reduced expression. Schur functions. This result is implicit in [Lascoux and Schutzenberger€ 90] and explicit in [Assaf and Definition 2.5. Given a weak composition a, the Searles 18]. j Demazure character a is Proposition 2.7. For a weak composition a, we have j p sortðÞa ; – j ; :::; ; ; :::; ; ; ::: ; a ¼ waðÞx (2 4) lim 0mÂaðÞx1 xm 0 0 ¼ ssortðÞa ðÞx1 x2 m!1 (2–6) where sortðaÞ is the weakly decreasing rearrangement of a, w(a) is the shortest permutation that sorts a and where 0m  a denotes the weak composition obtained b b1 b2 bn : x denotes the monomial x1 x2 ÁÁÁxn by prepending m zeros to a. For example, for a ¼ð0; 2; 1; 2Þ; we have sortðaÞ¼ We will see below that Kohnert polynomials also ð2; 2; 1; 0Þ and w(a) ¼ 2431, and so stabilize, though not, in general, to symmet- ÀÁ 2 2 ric functions. j ; ; ; ¼ p p p p x x x ðÞ0 2 1 2 1 2 3ÀÁ2 1 2 3 ¼ p p p x2x2x þ x2x x2 1 2 3 1 2 3 1 2 3 2.3. Schubert polynomials p p 2 2 2 2 2 2 ¼ 1 2ðx1x2x3 þ x1x2x4 þ x1x2x3 Schubert polynomials were introduced by Lascoux þ x2x x x þ x2x x2Þ 1 2 3 4 1 2 4 and Schutzenberger€ [Lascoux and Schutzenberger€ 82] p 2 2 2 2 2 2 ¼ 1ðx1x2x3 þ x1x2x4 þ x1x2x3 as polynomial representatives for Schubert classes in 2 2 2 2 2 2 2 the cohomology ring of the flag manifold for the gen- þ 2x1x2x3x4 þ x1x2x4 þ x1x3x4 þ x1x3x4Þ eral linear group. That is, they are polynomials ¼ x2x2x þ x2x2x þ x2x x2 1 2 3 1 2 4 1 2 3 indexed by permutations whose structure constants 2 2 2 2 2 2 2 þ 2x1x2x3x4 þ x1x2x4 þ x1x3x4 þ x1x3x4 precisely correspond to those for the distinguished lin- 2 2 2 2 2 þ x1x2x3 þ 2x1x2x3x4 þ x1x2x4 ear basis of the cohomology ring. They are defined by 2 2 2 2 2 2 the divided difference operators, which Fulton [Fulton þ x1x2x x4 þ x1x2x3x þ x x x4 þ x x3x : 3 4 2 3 2 4 92] showed have deep connections to modern inter- section theory. Notice that the final computation agrees with Definition 2.8 ([Lascoux and Schutzenberger€ 82]). KD computed earlier. ðaÞ Given a permutation w, the Schubert polynomial Sw
Theorem 2.6 ([Kohnert 90]). The Demazure character is given by j K D ÀÁ a is equal to the Kohnert polynomial ðaÞ, i.e. nÀ1 nÀ2
S @ À1 ; –
w ¼ w w0 x1 x2 ÁÁÁxnÀ1 (2 7) j K D ; –
a ¼ ðÞa (2 5) D where w0 ¼ n ÁÁÁ21 is the longest permutation of where ðaÞ is the key diagram for the indexing com- n length : position a. 2 À1 Kohnert’s algorithm for key diagrams precisely For example, for w ¼ 143625, we have w w0 ¼ gives the monomial expansion of a Demazure 462351; and so 6 S. ASSAF AND D. SEARLES ÀÁ S @ @ @ @ @ @ @ @ @ @ 5 4 3 2 143625 ¼ 1 2 3 4 5 4 2 3 1 2 x1x2x3x2x1 Winkel 02], though given the obscure and intricate 3 3 3 2 2 2 2 nature of the arguments, they are not widely accepted. ¼ x x2x3 þ x x2x4 þ x x3x4 þ 2x x x3 þ 2x x x4 1 1 1 1 2 1 2 A direct, bijective proof by Assaf [Assaf 17] utilizes 2 2 2 þ x1x2x3 þ 3x1x2x3x4 the expansion of Schubert polynomials into 2 2 2 2 2 2 3 þ x1x2x4 þ x1x3x4 þ x1x3x4 þ x1x2x3 Demazure characters. 3 2 2 2 þ x1x2x4 þ x1x2x3 þ 3x1x2x3x4 Theorem 2.10 ([Assaf 17, Winkel 99, Winkel 02]). 2 2 2 2 3 The Schubert polynomial S is given by the Kohnert þ x1x2x4 þ x1x2x3x4 þ x1x2x3x4 þ x2x3x4 w 2 2 2 2 polynomial þ x x x4 þ x x3x : 2 3 2 4 S K ; –
w ¼ DðÞw (2 9) For a permutation w with a unique descent at pos- D where ðwÞ is the Rothe diagram for the indexing per- ition k, we have S ¼ skðx ; :::; x Þ; where k is the w 1 k mutation w. partition given by kkÀiþ1 ¼ wiÀk: In particular, Schubert polynomials contain the Schur polynomials While the Schubert polynomials contain the Schur as a special case. In certain cases, including this so- polynomials, and so are also a polynomial generaliza- called grassmannian case, a Schubert polynomial is tion of Schur polynomials, we argue that this fact has equal to a Demazure character. more to do with the result that Schubert polynomials The Rothe diagram of a permutation w, denoted by expand as nonnegative sums of Demazure characters, DðwÞ; is given by and the latter naturally contains Schur polynomials. ÈÉ Macdonald [Macdonald 91] showed that Schubert DðÞw ¼ ðÞi; wj ji
polynomial. D ticular, when a Kohnert polynomial is quasisymmetric. When S ¼ j , we have a ¼ wtðÞðÞw : w a In Section 3.3, we use the stable limit of monomial Kohnert observed that his algorithm can be used slide polynomials to define Kohnert quasisymmetric on the Rothe diagram of a vexillary permutation to functions, which are the well-defined stable limits of compute the Schubert polynomial, and he asserted Kohnert polynomials. that his rule worked for Schubert polynomials in gen- eral. For example, Figure 3 gives the Kohnert dia- 3.1. Monomial slide polynomials grams for Dð143625Þ; where we have deleted the empty column on the left since doing so does not A polynomial f 2 Z½x1; :::; xn is quasisymmetric if for affect the Kohnert polynomial. Note that the corre- every (strong) composition (i.e., sequence of positive sponding Kohnert polynomial is precisely the integers) a ¼ða ; :::; a‘Þ with ‘ 6 n; we have 1 hi Schubert polynomial for 143625. ÂÃa a a a x 1 ÁÁÁx ‘ jf ¼ x 1 ÁÁÁx ‘ jf (3–1) Two proofs of Kohnert’s rule for Schubert polyno- i1 i‘ j1 j‘ mials appear in the literature by Winkel [Winkel 99, EXPERIMENTAL MATHEMATICS 7
Figure 4. The set MKDðDÞ for D the leftmost diagram above.
M ; ; ; 2 2 2 2 2 2: ðÞ2;0;1;2 ðÞx1 x2 x3 x4 ¼ x1x2x3 þ x1x2x4 þ x1x3x4 As this example illustrates, monomial slide polyno- mials are not, in general, quasisymmetric. The follow- ing result characterizes when a monomial slide D Figure 5. The set QYKDð Þ of quasi-Yamanouchi Kohnert dia- polynomial is quasisymmetric. grams for D the leftmost diagram above. Proposition 3.3 ([Assaf and Searles 17]). For a weak composition a of length n, Ma is quasisymmetric in ; :::; for any two sequences 1 6 i1< ÁÁÁ
Assaf and Searles [Assaf and Searles 17] introduced Proof. For any diagram D of weight (0, 2), we claim K M : ; ; a new basis for the polynomial ring, called monomial D 6¼ ð0;2Þ If wtðDÞ¼ð0 2Þ then KDðDÞ must slide polynomials, that gives a natural polynomial gen- have a diagram of weight (1, 1) by pushing the right- eralization of monomial quasisymmetric functions. most box in row 2. Therefore KD will contain the ; M Definition 3.2 ([Assaf and Searles 17]). The mono- monomial x1x2 which does not appear in ð0;2Þ ¼ 2 2: M mial slide polynomial indexed by a is x2 þ x1 Therefore ð0;2Þ is not a X Kohnert polynomial. w M b1 bn ; – a ¼ x1 ÁÁÁxn (3 3) bPa flatðÞb ¼flatðÞa 3.2. Kohnert polynomials are monomial where bPa means b1 þÁÁÁþbkPa1 þÁÁÁþak for slide positive ; :::; : all k ¼ 1 n P ; For every row index r 1 define an operator Upr on For example, taking a ¼ð2; 0; 1; 2Þ; which implies 4 diagrams that raises all cells in row r up to row r þ 1. variables, we compute While this is not, in general, well-defined, we will 8 S. ASSAF AND D. SEARLES
Figure 6. The Kohnert diagrams for the leftmost diagram above, for which the Kohnert polynomial is not fundamental slide positive.
each of which contains exactly one element of MKDðDÞ: To complete the proof, we need to show that X wtðÞU M : x ¼ wtðÞT U:UpðÞU ¼T Figure 7. An example (left) and non-example (right) of the : fundamental property. Suppose UpðUÞ¼T Since Upi moves an entire row without consolidating rows, we have flatðUÞ¼ only apply Upr when no cell in row r sits immediately flatðTÞ for any such U. Moreover, since rows move below a cell in row r þ 1. The following definition up, we also have wtðUÞPwtðTÞ: Finally, for any weak allows us to relate Kohnert polynomials with mono- composition b such that flatðbÞ¼flatðwtðTÞÞ and mial slide polynomials. bPwtðTÞ; we may construct U 2 KDðDÞ such that Definition 3.6. For a diagram D, define the subset wtðbÞ¼U and UpðUÞ¼T uniquely as follows. First MKDðDÞ of Kohnert diagrams for D by suppose T has only one row. Then this row can be È moved down, cell by cell from right to left until all MKDðÞD ¼ T 2 KDðÞD jUprðÞT 62 KDðÞD 8r cells sit in the row indexed by the nonzero part of b; this is a sequence of Kohnert moves. Now suppose T such that ðÞr þ 1; c 62 T 8cg: (3–5) has k > 1 rows, and assume inductively that U can be For example, for D the third diagram in Figure 1, constructed whenever T has fewer than k rows. Begin Figure 4 shows the set MKDðDÞ: Notice that by moving the lowest row of T down to the row indexed by the leftmost nonzero part of b. Let b0 be K ¼ M ; ; ; þ M ; ; ; þ M ; ; ; þ M ; ; ; þ M ; ; ; ; D ðÞ0 2 1 2 ðÞ1 1 1 2 ðÞ2 1 1 1 ðÞ1 2 0 2 ðÞ1 2 1 1 the weak composition obtained by setting the leftmost 0 which corresponds precisely to the weights of the dia- nonzero entry of b equal to zero, and T the diagram grams in MKDðDÞ: obtained by deleting the lowest row of T. By induc- tion, we may construct a diagram U0 such that Theorem 3.7. Given any diagram D, we have 0 0 0 0: X wtðU Þ¼b and UpðU Þ¼T Since, we begin by K M : – moving the lowest row of T down to the row of the D ¼ wtðÞT (3 6) T2MKDðÞD leftmost nonzero entry of b, we may follow by con- structing U0 from the remaining T0; the result is a dia- In particular, Kohnert polynomials expand non- gram U of weight b0 such that UpðUÞ¼T; as negatively into monomial slide polynomials. required. w Proof. For U 2 KDðDÞ; define UpðUÞ to be the dia- Combining Theorem 3.7 and Proposition 3.3,we gram resulting from applying Upi operators sequen- have the following characterization of when a Kohnert tially to U, under the restriction that at each step the polynomial is quasisymmetric. ; resulting diagram is in KDðDÞ until no Upi operator Proposition 3.8. Let D be a diagram with highest can be applied without leaving KDðDÞ: The diagram occupied row r. The polynomial K0mÂD is quasisymmet- UpðUÞ is well-defined: Upi only moves the cells in ric for all mP0 if and only if the set of columns con- row i to the empty row i þ 1, so rows of cells retain taining cells of D in row i is a subset of the set of ; their relative order, and if both UpiðUÞ UpjðUÞ2 columns containing cells of D in row i þ 1, for KDD then UpiUpjðUÞ2KDD and UpjUpiðUÞ¼ all i Figure 8. The elements of MKDðDÞ for D the leftmost diagram, which is not fundamental. KD is not quasisymmetric. If row i þ 1ofD is non- their stable limits are precisely the monomial quasi- empty, consider 0  D: Then from right to left, per- symmetric functions. form a single Kohnert move on all cells in row i þ 2 Theorem 3.9 ([Assaf and Searles 17]). For a weak of 0  D (the reason for passing to 0  D is to ensure composition a, we have that these Kohnert moves can be performed). Let N M ; :::; ; ; :::; ; ; ::: ; lim 0mÂaðÞx1 xm 0 0 ¼ MflatðÞa ðÞx1 x2 be the associated monomial of the resulting Kohnert m!1 diagram. Then the monomial obtained from N by (3–7) replacing x with x for all j 6 i does not belong to j jþ1 where 0m  a denotes the weak composition obtained K ; hence K is not quasisymmetric. 0ÂD 0ÂD by prepending m 0’stoa. Conversely, if the rows of D are nested with smaller Let 0m  D denote the diagram of D shifted up ver- rows below larger ones, then each T 2 MKDð0m  DÞ tically by m rows. For example, once again taking D will satisfy the condition that if there is a cell in row to be the third diagram in Figure 1, we may compute i Figure 9. The three quasi-Yamanouchi Kohnert diagrams for the leftmost diagram (left), which is not itself fundamental, and the two quasi-Yamanouchi Kohnert diagrams for the fourth diagram (right), which is fundamental. Definition 3.6 is a local condition about the cells in below row m, and similarly for KD0 ðx1; :::; xm; 0; 0; :::Þ: relative positions. Slightly abusing notation, let flatðDÞ be the diagram For any diagram D,ifT 2 KDð0m  DÞ has no obtained by deleting all empty rows from D. Then any cells in row j < m and row j À 1 is nonempty, then Kohnert diagram of 0m  D whose highest cell is weakly m ; m : UpjÀ1ðTÞ2KDð0  DÞ so by Definition 3.6, T 62 below row m is also a Kohnert diagram of 0  flatðDÞ MKDð0m  DÞ: In particular, if the bottom row of a That is, we have an injective map from the subset of diagram T 2 MKDð0m  DÞ is occupied, then so are KDð0m  DÞ with no cell above row m to KDð0m  all rows j 6 m: Theonlypossibletermsnotinthe flatðDÞÞ: By definition flatðDÞ¼flatðD0Þ; therefore m image of the injection MKDð0  DÞ! KDðx1; :::; xm; 0; 0; :::Þ¼KD0 ðx1; :::; xm; 0; 0; :::Þ: The MKDð0mþ1  DÞ are those T 2 MKDð0mþ1  DÞ with statement then follows by letting m !1: w at least one cell in the bottom row. Therefore, if D is a diagram with m cells, then the injection MKDð0m  DÞ!MKDð0mþ1  DÞ must be 4. Fundamental slide expansions abijection. Strengthening the results of the previous section, we Therefore, the result follows from Theorem 3.7 and next investigate the fundamental slide expansion of w Theorem 3.9. a Kohnert polynomial, which is not always nonnega- We review two motivating examples of this stabil- tive. In Section 4.1, we review the basis of funda- ity. For D aRothediagramforw, the Kohnert poly- mental slide polynomials introduced in [Assaf and nomial is a Schubert polynomial and the stable Searles 17]. In Section 4.2, we characterize those limit, as shown by Macdonald [Macdonald 91], is diagrams for which the corresponding Kohnert the Stanley symmetric function [Stanley 84]intro- polynomial expands nonnegatively into the funda- duced by Stanley to enumerate reduced expressions mental slide basis. Further, we conjecture a simple for a permutation. Implicit in the work of Lascoux condition on diagrams that ensures the correspond- and Schutzenberger€ [Lascoux and Schutzenberger€ ing Kohnert polynomial expands nonnegatively into 90] and explicit in [Assaf and Searles 18], the the Demazure character basis. In Section 4.3,we Kohnert polynomial of a key diagram stabilizes to prove the surprising fact that, while some Kohnert the Schur function indexed by the partition to polynomials are not positive on the fundamental which the weak composition sorts. Both of these slide basis, every Kohnert quasisymmetric function examples have stable limits that are symmetric func- is positive on the fundamental quasisymmetric func- tions, but Kohnert quasisymmetric functions are not tion basis. always symmetric. In Section 6,weconsideranew Kohnert basis that gives rise to an interesting new 4.1. Fundamental slide polynomials basis for quasisymmetric functions. As remarked earlier, two diagrams that differ by Gessel [Gessel 84] introduced another basis for quasi- insertion or deletion of empty columns give rise to symmetric functions that is closely related to the same Kohnert polynomial. In the stable limit, we Schur functions. a a ; :::; a b can strengthen this with the following. Given two compositions ¼ð 1 ‘Þ and ¼ ðb ; :::; b Þ such that a1 þÁÁÁþa‘ ¼ b þÁÁÁþb ; Proposition 3.12. Given two diagrams D and D0 that 1 m 1 m say that b refines a if there exist indices 1 ¼ differ by insertion or deletion of empty rows, we < < < 6 <‘ : i0 i1 ÁÁÁ i‘ such that for all 0 j we have have KD ¼KD0 b b a : ijþ1 þÁÁÁþ ijþ1 ¼ jþ1 For example, (1, 2, 2) refines (3, 2) but does not refine (2, 3). Proof. Fix a positive integer m.Then Definition 4.1 KDðx1; :::; xm; 0; 0; :::Þ is the weighted sum of all ([Gessel 84]). The fundamental quasi- Kohnert diagrams of D whose highest cell is weakly symmetric polynomial indexed by a is EXPERIMENTAL MATHEMATICS 11 X X ; :::; ; :::; F F F ; – FaðÞx1 xn ¼ MbðÞx1 xn a b ¼ ½cja b c (4 3) b refines a c X (4–1) b1 bn ; ¼ x1 ÁÁÁxn where ½cja b is the coefficient of c in the slide prod- flatðÞb refines a uct a b. In particular, ½cja b is a nonnega- where the latter sum is over all weak compositions of tive integer. length n whose flattening refines a. As was the case for the monomial slide polyno- For example, taking a ¼ð2; 1; 2Þ and restricting to mials, the fundamental slide polynomials are also not 4 variables, we have a Kohnert basis. ; ; ; ; ; ; Proposition 4.5. FðÞ2;1;2 ðÞx1 x2 x3 x4 ¼ MðÞ2;1;2 ðÞx1 x2 x3 x4 The fundamental slide polynomials ; ; ; are not a Kohnert basis. þ MðÞ1;1;1;2 ðÞx1 x2 x3 x4 ; ; ; ; þ MðÞ2;1;1;1 ðÞx1 x2 x3 x4 Proof. For any diagram D of weight (0, 2, 1), we claim 2 2 2 2 2 2 K F ; ; : K ; ¼ x1x2x3 þ x1x2x4 þ x1x3x4 D 6¼ ð0 2 1Þ If this is a Kohnert polynomial D then 2 2 2 2 D must have two cells in row 2 and one in row 3. Let þ x x3x þ x1x2x3x þ x x2x3x4: 2 4 4 1 D be any such diagram. Since there is only one cell, say c in row 3 of D, this cell is the rightmost in its The fundamental quasisymmetric functions row. If c is in the same column as some cell in row 2 inspired the fundamental slide polynomials of Assaf then applying a Kohnert move to c yields a diagram and Searles [Assaf and Searles 17], analogous to the of weight (1, 2, 0), otherwise applying a Kohnert relationship between the monomial slide polynomials move to c yields a diagram of weight (0, 3, 0). Thus and the monomial quasisymmetric polynomials. K 2 3 D must have one of x1x2 or x2 appear as a term. F Definition 4.2 ([Assaf and Searles 17]). The funda- However, neither of these terms appears in ð0;2;1Þ ¼ 2 2 2 ; F mental slide polynomial indexed by a is x2x3 þ x1x3 þ x1x2x3 þ x1x2 and so ð0;2;1Þ cannot be X a Kohnert polynomial. w F b1 bn ; – a ¼ x1 ÁÁÁxn (4 2) bPa flatðÞb refines flatðÞa 4.2. Fundamental diagrams where bPa means b1 þÁÁÁþbkPa1 þÁÁÁþak for all k ¼ 1; :::; n: One motivation for defining and studying the funda- mental slide polynomials is a refined expansion of For example, taking a ¼ð2; 0; 1; 2Þ; which implies 4 Schubert polynomials [Assaf and Searles 17]. This for- variables, we compute mula utilizes the pipe dream model for the monomial F M M 2 2 2 2 expansion of Schubert polynomials given by Bergeron ðÞ2;0;1;2 ¼ ðÞ2;0;1;2 þ ðÞ2;1;1;1 ¼ x1x2x3 þ x1x2x4 and Billey [Bergeron and Billey 93] based on the com- þ x2x x2 þ x2x x x : 1 3 4 1 2 3 4 patible sequences model due to Billey, Jockusch, and The fundamental slide polynomials generalize the Stanley [Billey et al. 93]. fundamental quasisymmetric polynomials. Theorem 4.6 ([Assaf and Searles 17]). For w a permu- Proposition 4.3 ([Assaf and Searles 17]). For a weak tation, we have X composition a of length n, F is quasisymmetric in a S ¼ F ; (4–4) x ; :::; x if and only if a 6¼ 0 whenever a 6¼ 0 for w wtðÞP 1 n j i P2QPDðÞw some i < j. Moreover, in this case we F ; :::; : have a ¼ FflatðaÞðx1 xnÞ where the sum is over quasi-Yamanouchi pipe dreams for w. The fundamental slide polynomials give a basis for For example, the Schubert polynomial for the per- the polynomial ring [Assaf and Searles 17]. mutation 143625 is Remarkably, their structure constants are nonnegative S F F F F and generalize the shuffle product of Eilenberg and 143625 ¼ ðÞ0;2;1;2 þ ðÞ1;2;0;2 þ ðÞ0;2;2;1 þ ðÞ0;3;1;1 F F F : Mac Lane [Eilenberg and Lane 53]. þ ðÞ1;2;1;1 þ ðÞ1;3;0;1 þ ðÞ2;2;0;1 Theorem 4.4 ([Assaf and Searles 17]). The fundamen- Assaf and Searles [Assaf and Searles 18] also show F tal slide polynomials f ag are a basis of the polynomial that the Demazure characters have a natural decom- ring with structure constants position into fundamental slide polynomials. This 12 S. ASSAF AND D. SEARLES Figure 11. The left diagram is not southwest; the right is southwest. Figure 10. A split diagram U (left) and the diagram dropðUÞ (right). In each diagram the threading is given by labeling of Yamanouchi Kohnert diagram, so that the funda- cells with x, y, and z. mental slide expansion of the corresponding formula utilized Kohnert’s model for Demazure char- Kohnert polynomials is precisely given by the quasi- acters [Kohnert 90]. Yamanouchi Kohnert diagrams. However, unlike the case with monomial slide polynomials, Kohnert pol- Theorem 4.7 ([Assaf and Searles 18]). For a weak ynomials are not, in general, fundamental slide posi- composition a, we have X tive. For example, taking D to be the left diagram in j F ; – Figure 6, the corresponding Kohnert polynomial a ¼ wtðÞT (4 5) T2QKTðÞa expands as K M M F F F : where the sum is over all quasi-Yamanouchi Kohnert D ¼ ðÞ0;1;1 þ ðÞ0;2;0 ¼ ðÞ0;1;1 þ ðÞ0;2;0 À ðÞ1;1;0 tableaux for a. The impediment to fundamental slide positivity is For example, the Demazure character for the weak captured by the following notion. ; ; ; composition ð0 2 1 2Þ decomposes as Definition 4.9. A diagram is split if there exist rows j F F F F : ðÞ0;2;1;2 ¼ ðÞ0;2;1;2 þ ðÞ1;2;0;2 þ ðÞ0;2;2;1 þ ðÞ1;2;1;1 r1 – diagrams in QYKDð D ð143625ÞÞ nor in (4 6) D QYKDð ð0; 2; 1; 2ÞÞ is split. Note that QYKDðDÞMKDðDÞ: For example, for D the third diagram in Figure 1, Figure 5 shows the Lemma 4.10. Let U 2 KDðDÞ be such that both S ¼ : Up ÁÁÁUp ðUÞ and T ¼ Up ÁÁÁUp ðUÞ raise rows set QYKDðDÞ Notice that ik i1 jl j1 only when all cells of the row above lie strictly to the K ¼ F ; ; ; þ F ; ; ; ; D ðÞ0 2 1 2 ðÞ1 2 0 2 left. If S; T 2 QYKDðDÞ and S 6¼ T, then at least one which corresponds precisely to the weights of the dia- of S, T is split. grams in QYKDðDÞ: Similarly, seven of the Kohnert diagrams in Figure Proof. Suppose T is non-split. Since each raising of a 3 are in QYKDð D ð143625ÞÞ and four of the Kohnert row either moves a row up or consolidates two rows, diagrams in Figure 2 are in QYKDð D ð0; 2; 1; 2ÞÞ: The for U 2 KDðDÞ; if U has a path to T 2 QYKDðDÞ; fundamental slide generating polynomials of these two then the rows of T are unions of the rows of U, taken S j ; sets are 143625 and ð0;2;1;2Þ respectively. in order and moved weakly up. Suppose that U has Similar to the proof of Theorem 3.7,wewishto another raising path to S. Consider the highest row, consolidate Kohnert diagrams into equivalence say r2, in which S and T differ, say with T having t classes, each of which contains a unique quasi- cells and S having s cells in row r2. Without loss of EXPERIMENTAL MATHEMATICS 13 generality, we may assume s < t. Then T must have with the exception of certain diagrams with nonempty consolidated more rows of U into its row r2. rows close to the x-axis for which split diagrams can- Therefore, row r2 of S must consist of the s leftmost not arise given insufficient room to move rows down. cells of row r2 of T. Set c1 to be the column of the sth To prove this, we begin with the following. cell from the left in row r of T (equivalently, S), and 2 Lemma 4.12. If D has a split quasi-Yamanouchi let c be the column of the next cell to its right. Let 2 Kohnert diagram, then D has a split quasi- r mþjDjþ1 : then, from right to left, move the first bijÀ1þ1 cells KDmð0  UpiðDÞÞ To see the other contain- ; mþjDj down to row ijÀ1 þ 1 the next bijÀ1þ2 cells down to ment, observe that 0  UpiðDÞ is a Kohnert dia- mþjDjþ1 row ijÀ1 þ 2; and so on. Each of these moves is a valid gram of 0  D; formed by dropping all cells in Kohnert move with no cells jumping over any others. row i þ 1ofD down to row i and dropping all cells Existence is proved, and uniqueness follows from the not in rows i or i þ 1 down one row. w lack of choice at every step. w For example, letting D be the leftmost diagram in Both Schubert polynomials and Demazure characters Figure 8, we may compute the fundamental quasisym- expand non-negatively into fundamental slide polyno- metric function expansion of the Kohnert quasisym- mials, with the former indexed by quasi-Yamanouchi metric function by pipe dreams and the latter by quasi-Yamanouchi Kohnert tableaux. Since both Rothe diagrams and key KD ¼ FflatðÞ 0;1;0;2;1 þ FflatðÞ 0;2;0;2;0 þ FflatðÞ 0;1;1;2;0 ÀFflatðÞ 1;1;0;2;0 : diagrams are fundamental, the expansion in (4–8) gives ¼ FðÞ1;2;1 þ FðÞ2;2 a common generalization of these results. Notice that each of the split quasi-Yamanouchi Kohnert diagrams is canceled in the limit. 4.3. Kohnert quasisymmetric functions are Compare this with the fourth diagram, say D0; in fundamental positive Figure 9, which is fundamental. For this diagram Assaf and Searles [Assaf and Searles 17] showed that there are two quasi-Yamanouchi Kohnert diagrams, the fundamental slide polynomials stabilize and that and so we have their stable limits are precisely the fundamental quasi- ; symmetric functions. KD0 ¼ FflatðÞ 0;0;1;2;1 þ FflatðÞ 0;0;2;2;0 ¼ FðÞ1;2;1 þ FðÞ2;2 Theorem 4.15 ([Assaf and Searles 17]). For a weak which coincides with KD computed above, illustrating 0 : composition a, we have Lemma 4.16 since D ¼ Up2ðDÞ F ; :::; ; ; :::; ; ; ::: ; Despite the restriction of Theorem 4.14 to funda- lim 0mÂaðÞx1 xm 0 0 ¼ FflatðÞa ðÞx1 x2 m!1 mental diagrams, positivity in the stable limit holds in (4–9) general (Figure 10). where 0m  a denotes the weak composition obtained Definition 4.17. For any diagram U, the threading of by prepending m 0’stoa. U is the partitioning of the cells of U into equivalence In order to remove extraneous redundancy from classes called threads, defined as follows. Let c1; c2; :::; cm the stable limits, we say that a diagram is flat if there be the cells of U read left to right, starting at the highest is no empty row below a nonempty row. For each dia- > row. For i 1, ci is in the same thread as ciÀ1 if and gram D, we define flatðDÞ to be the diagram obtained only if ci lies strictly to the right of ciÀ1 in U. by removing empty rows. For any diagrams C, D such We refer to the length of a thread is the number of that flatðCÞ¼flatðDÞ; we have KDðXÞ¼KCðXÞ: In particular, in the stable limit, it is enough to consider cells in that thread. ; flat diagrams. It is an interesting, though difficult, By construction, if ciÀ1 ci are consecutive (in read- question to characterize when two Kohnert quasisym- ing order) cells of U that lie in different threads, then : metric functions are equal. We offer the following ci must lie in a strictly lower row than ciÀ1 In par- partial solution that allows us to consider only flat- ticular, the rightmost cells of all threads lie in distinct tened, fundamental diagrams. rows, making the following well-defined. Lemma 4.16. Given a diagram D and a row index i Definition 4.18. For any diagram U, define dropðUÞ such that all cells in row i þ 1 lie strictly left of all cells to be the diagram obtained by placing all cells of each K ð Þ¼K ð Þ: thread in the same row as the rightmost cell in that in row i, we have D X UpiðDÞ X thread. For example, see Figure 10. K ; ::: Proof. It is enough to show 0mþjDjþ1ÂDðx1 xmÞ¼ Lemma 4.19. K mþjDjþ1 ðx1; :::xmÞ for all m. Specifically, let Let D be a diagram and 0 ÂUpiðDÞ KDiðDÞ denote the subset of KDðDÞ consisting of dia- U 2 QYKDðDÞ. Then grams having no cells in row i þ 1 or higher. We will mþjDjþ1 mþjDjþ1 : ; show KDmð0  DÞ¼KDmð0  UpiðDÞÞ 1. dropðUÞ2QYKDðDÞ Since 0mþjDjþ1  D is a Kohnert diagram of 2. the sequence of thread lengths (from highest thread mþjDjþ1 ; mþjDjþ1 ; 0  UpiðDÞ we have KDmð0  DÞ to lowest thread) is the same in U and dropðUÞ 16 S. ASSAF AND D. SEARLES 3. the threads in dropðUÞ are exactly the rows vice versa. Therefore, by Lemma 4.19, the sequence of of dropðUÞ; thread lengths of E is the same as that of dropðUÞ: 4. U is non-split if and only if dropðUÞ¼U: Now suppose E has every cell weakly below the lowest row of dropðUÞ: We construct a raising path ; Proof. Notice that if ciÀ1 ci are consecutive (in reading from E to dropðUÞ as follows. Select any thread of E order) cells of U that lie in the same thread, then ci that uses cells in more than one row. Raise all cells in : must lie strictly to the right of and weakly below ciÀ1 the row of the rightmost cell of the thread. Continue Suppose that ciÀ1 lies strictly above ci. Then from the this process until all cells in the thread have been reading order, there is no cell right of ciÀ1 in its row raised to the row of the leftmost cell in the thread. nor left of ci in its row nor in a row strictly between Now perform this process with all remaining threads those of ciÀ1 and ci. Therefore, we may apply a of E that use cells in more than one row. In the 0 Kohnert move to ciÀ1 resulting in it lying exactly one resulting diagram E ; the threads are exactly the rows, row lower with no cell to its left. In particular, this and the sequence of thread lengths is the same as in move preserves the reading order, and as such pre- dropðUÞ: Since all rows of E0 lie below all rows of serves the threading. Iterating this process one cell at dropðUÞ; we may now raise rows one by one to a time until all cells of each thread lie in the same obtain dropðUÞ: Every step of this procedure is legit- row results in dropðUÞ: The threads are maintained at imate: all such moves are (a sequence of) reverse each step, proving (2), and, in the terminal case, prov- Kohnert moves, hence any intermediate diagram at ing (3). Since each move is a Kohnert move, we have any stage in this process may be obtained from dropðUÞ2KDðDÞ and shows (2). Moreover, by con- dropðUÞ2QYKDðDÞ via Kohnert moves; in particu- struction the rows that have a cell of dropðUÞ are a lar, we never leave KDðDÞ during this procedure. w subset of the rows that have a cell of U. Since U 2 QYKDðDÞ; no row of U can be raised, and since the Theorem 4.21. For any diagram D and any m at least rows of dropðUÞ are a subset of those of U, no row of as great as the number of cells of D, we have dropðUÞ can be raised, proving (1). X : – Finally, to see (4), we have dropðUÞ 6¼ U if and KD ¼ FflatðÞ wtðÞT (4 10) only if a thread uses two consecutive cells in different T 2 QYKDðÞ 0m  D T non–split rows, which is true if and only if the lower cell is strictly southeast of the higher cell and there are no In particular, Kohnert quasisymmetric functions cells in between them in reading order, i.e., if and expand non-negatively into fundamental quasisymmet- only if U is split. w ric functions. In order to obtain a nonnegative formula for the Proof. From Definition 4.8, for m at least the number of fundamental quasisymmetric expansion of the cells of D,noT 2 QYKDð0mþ1  DÞ has a cell in the Kohnert quasisymmetric function for a diagram D,we bottom row. So QYKDð0m  DÞ is stable in the sense will sum over only the non-split quasi-Yamanouchi that QYKDð0mþ1  DÞ consists of exactly the elements diagrams. To justify this, we will show that all contri- of QYKDð0m  DÞ with every cell raised by one row. butions from a split quasi-Yamanouchi diagram U 2 Partition the Kohnert diagrams of 0m  D into QYKDðDÞ are subsumed in the limit by dropðUÞ; m m m KDð0  DÞ¼KDNSpð0  DÞtKDSpð0  DÞ; where which, by Lemma 4.19, is non-split. m KDNSpð0  DÞ are those diagrams that have a raising Lemma 4.20. Let D be a diagram. Suppose there is a path to a non-split element of QYKDð0m  DÞ; and m raising path for a diagram E 2 KDðDÞ to U 2 KDSpð0  DÞ are those that can only raise to split ele- : ments of QYKDð0m  DÞ: Thus, we have QYKDðDÞ If all cells of E are weakly below the lowest X X cell of dropðUÞ; then there is a raising path from E wtðÞE wtðÞV KD ¼ x þ x : to dropðUÞ: m m E2KDNSpðÞ0 ÂD V2KDSpðÞ0 ÂD – Proof. The key point is that if there is a raising path (4 11) m from E to U, then the sequence of thread lengths is First consider KDNSpð0  DÞ: By Lemma 4.10,ifa the same in E and U. This follows because when cells Kohnert diagram of 0m  D raises to a non-split element are moved in a raising path, the position of cells in T 2 QYKDð0m  DÞ then it cannot raise to any other reading order is unchanged, and no cell is moved non-split element of QYKDð0m  DÞ: Hence m from weakly left to strictly right of any other cell or KDNSpð0  DÞ is partitioned further into the sets of EXPERIMENTAL MATHEMATICS 17 Figure 12. Illustration of construction of the skew diagram Sð1; 0; 3; 2; 0; 3Þ: Kohnert diagrams that can raise to each non-split QYKD: Moreover, as in the proof of Theorem 4.14, the monomials corresponding to the set of all Kohnert dia- grams of 0m  D that can raise to a given non-split elem- ent T 2 QYKDð0m  DÞ (equivalently, those Kohnert diagrams obtained from T by Kohnert moves without jumping or moving a cell into a row containing another Figure 13. Illustration of the partitions k ¼ð6; 4; 4; 1Þ and F : cell) comprise the fundamental slide polynomial wtðTÞ l ¼ð3; 2; 1Þ associated to the skew diagram Sð1; 0; 3; 2; 0; 3Þ: Therefore, we can rewrite the left sum in (4–11) as X X wtðÞE F : – functions on which all Kohnert quasisymmetric func- x ¼ wtðÞT (4 12) m T 2 QYKDðÞ 0m  D tions are nonnegative. E2KDNSpðÞ0 ÂD T non–split By Theorem 4.15, in the limit this becomes a sum of fundamental quasisymmetric functions indexed by 5. Demazure character expansions the flattened weights of the non-split QYKDs. Our two motivating examples for general Kohnert poly- m : Now consider KDSpð0  DÞ By Lemma 4.19, for nomials are Schubert polynomials and Demazure char- any U 2 QYKDð0m  DÞ we have dropðUÞ2 acters. Generalizing Proposition 2.9 that characterizes QYKDð0m  DÞ and dropðUÞ is non-split, hence when a Schubert polynomial is equal to a Demazure m m KDSpð0  DÞ consists of Kohnert diagrams of 0  character, Lascoux and Schutzenberger€ [Lascoux and D that have a raising path to a split U but no raising Schutzenberger€ 90] proved that Schubert polynomials path to dropðUÞ: By Lemma 4.20, any diagram that always expand as a nonnegative integral sum of m raises to U 2 QYKDð0  DÞ but not to dropðUÞ Demazure characters. Thus, it is natural to explore the must have a cell above the lowest row of dropðUÞ: In question of when a general Kohnert polynomial expands m general, therefore, all diagrams in KDSpð0  DÞ have as a nonnegative integral sum of Demazure characters. a cell above the lowest row that is occupied by some element of QYKDð0m  DÞ: By Definition 4.8, this index of this row is greater than mÀjDj: Therefore, if 5.1. Demazure diagrams m wtðVÞ V 2 KDSpð0  DÞ then x is divisible by some Inspired by these two important bases, Schubert poly- P : variable xr where r mÀjDj Since jDj is constant, r nomials and Demazure characters, we have the follow- : grows without bound as m !1 Therefore, in the ing simple condition on diagrams. limit, every term in the second sum in (4–11) van- Definition 5.1. ishes, leaving only (4–12). The theorem follows. w A diagram D is southwest if whenever a pair of cells (r2,c1) and (r1,c2) are in D, where Since the fundamental quasisymmetric expansion is r1 K K : Figure 14. Skew diagrams illustrating the skew polynomial expansion of Sð2;0;2Þ Á Sð0;2;0Þ K K : Figure 15. Skew diagrams illustrating the skew polynomial expansion of Sð1;0;1Þ Á Sð0;0;1Þ positive and shows the potential power of Kohnert polynomials. 5.2. Skew polynomials We now give a new and nontrivial example of a Figure 16. The skew diagram Sð1; 0; 3; 2; 0; 3Þ and Rothe dia- southwest Kohnert basis that further supports gram Dð216539478Þ: Conjecture 5.2. Definition 5.3. For a weak composition a, the skew Figure 1 are fundamental, but the third is diagram SðaÞ is constructed as follows: not southwest. Our motivation for the southwest condition comes left justify ai cells in row i, from Schubert polynomials and Demazure characters, for j from 1 to n such that aj>0; take i < j max- since both composition diagrams and Rothe diagrams imal such that ai>0; and if ai > aj, then shift rows are southwest. Moreover, extensive computer experi- kj rightward by aiÀaj columns, mentation supports the following conjecture that we shift each row j rightward by #fi Kohnert polynomial K expands non-negatively into ð1 0 3 2 0 3Þ from the composition diagram D D ; ; ; ; ; P Demazure characters. ð1 0 3 2 0 3Þ by shifting rows k 4 rightward by 1 column since a3Àa4 ¼ 1; then shifting rows 3, 4 right- In further support of Conjecture 5.2, the southwest ward by one since a2 ¼ 0 and row 6 rightward by two condition is exactly the same as the northwest condi- since a2 ¼ a5 ¼ 0: These steps are illustrated in tion of Reiner and Shimozono [Reiner and Figure 12. Shimozono 95, Reiner and Shimozono 98] in their Proposition 5.4. study of Specht modules associated to diagrams. This Skew diagrams are southwest. suggests a connection between flagged Weyl modules Proof. Failure of the southwest condition ensures the and Kohnert polynomials, conjecturally that Kohnert existence of a pair of cells (r , c ) and (r , c )inD, polynomials give the characters of flagged 1 2 2 1 where r Conjecture 5.2 implies that skew polynomials Then the stable limit of the skew polynomial indexed should expand nonnegatively in Demazure characters. by a is Indeed, this fact follows using the machinery of weak K KSðÞa ¼ lim SðÞa ¼ sk=l dual equivalence developed in [Assaf 17]. m!1 Theorem 5.5. K Skew polynomials f SðaÞg form a basis In particular, skew polynomials are a polynomial of Z½x1; x2; :::; xn, expand nonnegatively in demazure generalization of skew Schur functions. characters, and stabilize to Schur positive symmet- For example, Figure 13 illustrates the computation ric functions. of k and l k in establishing the following limit, : Proof. By Theorem 2.3, since skew polynomials are a KSðÞ1;0;3;2;0;3 ¼ sðÞ6;4;4;1 nðÞ3;2;1 Kohnert basis, they are lower uni-triangular with respect to monomials, and as such form a basis for the polynomial ring. 5.3. Applications of skew polynomials In [Assaf 17] (Theorem 4.10), Assaf proves a gener- Unlike Schubert polynomials, whose structure con- – alized Littlewood Richardson rule that expands a skew stants enumerate points in a suitable intersection of key polynomial, indexed by a composition with a par- Schubert varieties and as such as known to be non- tition shape removed from the northwest corner, as a negative, Demazure characters often have negative nonnegative integral sum of Demazure characters structure constants. For example, (therein called key polynomials). The definition for j j j j j j these skew key polynomials, [Assaf 17] (Definition ðÞ2;0;2 ðÞ0;2;0 ¼ ðÞ2;2;2 þ ðÞ3;1;2 þ ðÞ4;0;2 þ ðÞ2;3;1 j j j : 4.7), uses standard key tableaux. The bijection À ðÞ3;2;1 þ ðÞ2;4;0 À ðÞ4;2;0 between standard key tableaux and quasi-Yamanouchi Kohnert tableaux stated in [Assaf 17] (Definition 3.14) Interestingly, the structure constants for skew poly- and proved in [Assaf 17] (Theorem 3.15) together nomials are often nonnegative. For example, Figure 14 with the bijection from the latter to quasi- illustrates the following expansion, K K K K K Yamanouchi Kohnert diagrams stated in [Assaf and SðÞ2;0;2 Á SðÞ0;2;0 ¼ SðÞ2;2;2 þ SðÞ3;1;2 þ SðÞ4;0;2 Searles 18](Definition 2.5) and proved in [Assaf and þ KS ; ; þ KS ; ; : Searles 18] (Theorem 2.8) establishes the equivalence ðÞ2 3 1 ðÞ2 4 0 of skew key polynomials with the Kohnert polyno- In light of Corollary 5.6, this gives the following mials defined by the indexing shape. Thus, each skew nonnegative expansion of a product of skew Schur polynomial is a skew key polynomial, and so expands function into skew Schur functions, nonnegatively into Demazure characters. s ; = Á s ¼ s 2;2;2 þ s ; ; = ; þ s ; = þ s ; ; = þ s 4;2 : Finally, by Proposition 2.7 Demazure characters ðÞ3 2 ðÞ1 ðÞ2 ðÞ ðÞ4 3 3 ðÞ2 2 ðÞ5 4 ðÞ3 ðÞ3 3 2 ðÞ2 ðÞ stabilize to Schur functions, hence the stable limit of a Since skew Schur functions over determine a basis skew polynomial is a Schur-positive symmet- for symmetric functions, this expansion is surprising. ric function. w However, such a nice expansion does not always For example, the skew diagram Sð1; 0; 3; 2; 0; 3Þ can hold. For example, Figure 15 illustrates the following be realized as the composition diagram D ð1; 0; 4; 4; 3; 6Þ signed expansion, skewed by the partition ð0; 0; 1; 2; 3; 3Þ; and so the poly- K K K K K K : SðÞ1;0;1 Á SðÞ0;0;1 ¼ SðÞ1;0;2 þ SðÞ1;1;1 þ SðÞ2;0;1 À SðÞ3;0;0 nomial equivalence states In this case, Corollary 5.6 still applies and gives the KS ; ; ; ; ; ¼ j ; ; ; ; ; ; ; ; ; ; : ðÞ1 0 3 2 0 3 ðÞ1 0 4 4 3 6 nðÞ0 0 1 2 3 3 following signed expansion, Thanks to the stability results for Kohnert polyno- : sðÞ2;1 =ðÞ1 Á sðÞ1 ¼ sðÞ3;1 =ðÞ1 þ sðÞ1;1;1 þ sðÞ3;2 =ðÞ2 ÀsðÞ3 mials, we have the following. Despite the signs, the canonical expansion of a Corollary 5.6. Let a be a weak composition. Let k be product of skew Schur functions into skew Schur the partition given by the flattening of the weight of the functions is still interesting, and signs appearing can skew diagram for a, i.e. be natural (e.g. see [Assaf and McNamara 11]). k S ; ¼ revðÞ flatðÞ wtðÞðÞa Therefore, exploring the structure constants for skew and let l be the partition given by polynomials is a worthwhile endeavor. Shifting to a more positive direction, skew polyno- l ¼ k ÀflatðÞa : i i i mials correspond with Demazure characters in the 20 S. ASSAF AND D. SEARLES Figure 17. Kohnert diagrams for Dð0; 2; 1; 2Þ: case when a is weakly increasing, in which case both 6.1. Lock polynomials are Schur polynomials. Skew polynomials also corres- We now define a new Kohnert basis. Using the pond with Schubert polynomials in certain cases, even machinery of Kohnert polynomials, this requires only outside of the above coincidence with Schur polyno- that for each weak composition a, we make a choice mials. For instance, we have the following nonobvious for the columns in which we place the ai boxes in coincidence, row i. For each weak composition a, there is a unique right-justified diagram of weight a which we call the K S : D : SðÞ1;0;3;2;0;3 ¼ 216539478 lock diagram for a and denoted by ðaÞ For example, the third diagram in Figure 1 is the lock diagram for ð0; 2; 1; 2Þ: The Kohnert diagrams for this diagram are Comparing the skew diagram Sð1; 0; 3; 2; 0; 3Þ and shown in Figure 17. Rothe diagram Dð216539478Þ as in Figure 16, the dia- grams themselves are somewhat different yet the Definition 6.1. The lock polynomial indexed by a is resulting Kohnert polynomials coincide. L K D ; – a ¼ ðÞa (6 1) While this coincidence of skew polynomials and Schubert polynomials does not hold in general, we where D ðaÞ is the right justified diagram of weight a. conjecture that Schubert polynomials are, in fact, For example, from Figure 17, we see that L 2 2 2 2 2 2 2 2 2 nested between Demazure characters and skew poly- ðÞ0;2;1;2 ¼ x1x2x3 þ x1x2x3x4 þ x1x2x4 þ x1x3x4 þ x1x2x3 2 2 2 2 2 2: nomials in the following sense. þx1x2x3x4 þ x1x2x4 þ x1x2x3x4 þ x2x3x4 Conjecture 5.7. Skew polynomials expand as nonnega- By Theorem 2.3, since lock polynomials are a tive sums of Schubert polynomials. Kohnert basis, they are, in particular, a basis of Conjecture 5.7 has been verified for degree up to polynomials. 10 in up to 6 variables. If true, this conjecture is Corollary 6.2. The lock polynomials form a basis for highly suggestive that skew polynomials are a com- the polynomial ring that is lower uni-triangular with binatorial shadow of representation-theoretic and geo- respect to monomials. metric objects yet to be discovered. By Theorem 3.7, we may express lock polynomials more compactly in the monomial slide basis as follows. 6. Extending Schur functions to the ring of quasisymmetric functions Corollary 6.3. Lock polynomials expand non-negatively into monomial slide polynomials by We now demonstrate the construction of another X L M : – a ¼ D (6 2) Kohnert basis, this one not southwest, with interesting T2MKDðÞðÞa wtðÞT properties. In Section 6.1 we define the new Kohnert basis of lock polynomials and apply our previous For the previous example, refined to MKD in results to give explicit formulas for the monomial and Figure 4, we have fundamental slide expansions. In Section 6.2, we dem- L ; ; ; ¼ M ; ; ; þ M ; ; ; þ M ; ; ; þ M ; ; ; þ M ; ; ; : onstrate how to generate a tableaux model from the ðÞ0 2 1 2 ðÞ0 2 1 2 ðÞ1 1 1 2 ðÞ2 1 1 1 ðÞ1 2 0 2 ðÞ1 2 1 1 corresponding Kohnert diagrams and use this to Even more powerful, by Theorem 4.14 we have prove a special case when lock polynomials and the following. Demazure characters coincide. In Section 6.3, we con- Corollary 6.4. sider the stable limits of lock polynomials, which we Lock polynomials expand non-negatively into fundamental slide polynomials by term extended Schur functions, since they contain X Schur functions and give a basis for quasisymmet- L F : – ¼ D (6 3) a T2QYKD ðÞa wtðÞT ric functions. ðÞ EXPERIMENTAL MATHEMATICS 21 Figure 18. The nine lock tableaux for Dð0; 2; 1; 2Þ: Figure 19. The set QLTð0; 3; 2; 1Þ of quasi-Yamanouchi lock tableaux of content ð0; 3; 2; 1Þ: Proof. Given a lock diagram D ðaÞ; for every column readily if a given diagram can arise as a Kohnert dia- c 6 maxðaÞ and for any nonempty collection of rows, gram for a key diagram. We extend this procedure to there are at least as many cells in those rows in col- lock diagrams below. – – umn c as there are in column c 1. Therefore (4 7) Definition 6.5. Given a weak composition a of length is always satisfied, and so lock diagrams are funda- n, a lock tableau of content a is a diagram filled with mental by Definition 4.11. Thus Theorem entries 1a1 ; 2a2 ; :::; nan , one per cell, satisfying the fol- 4.14 applies. w lowing conditions: Returning again to our example, with the funda- mental diagrams shown in Figure 5, we have i. there is exactly one i in each column from maxðaÞÀa þ 1 through maxðaÞ; L F F : i ðÞ0;2;1;2 ¼ ðÞ0;2;1;2 þ ðÞ1;2;0;2 ii. each entry in row i is at least i; Unlike Schubert polynomials and, trivially, iii. the cells with entry i weakly descend from left Demazure characters, the lock polynomials do not to right; expand non-negatively into Demazure characters. For iv. the labeling strictly decreases down columns. Denote the set of lock tableaux of content a example, by LTðaÞ: L j j j j : ðÞ0;2;1;2 ¼ ðÞ0;2;1;2 À ðÞ0;2;2;1 þ ðÞ1;2;2;0 À ðÞ2;2;1;0 For example, the lock tableaux of content ð0; 2; 1; 2Þ are shown in Figure 18. Compare this with In light of Conjecture 5.2, this comes as no surprise D ; ; ; since lock diagrams are not, in general, southwest. the Kohnert diagrams for ð0 2 1 2Þ shown in We remark that lock polynomials do not have non- Figure 17. negative expansions into other familiar bases of the The definition of Kohnert tableaux in [Assaf and Searles 18], the analogous model for key dia- polynomial ring, including quasi-key polynomials grams, differs from Definition 6.5 only in condi- [Assaf and Searles 18] and Demazure atoms [Lascoux tion (iv). For the Kohnert tableaux case, condition and Schutzenberger€ 90]. (iv) allowed for an inversion in a column, i.e. a pair i < j with i above j inthesamecolumn,only 6.2. Lock tableaux if there is an i in the column immediately to the right of and strictly above j.Condition(iv)for Kohnert’s rule allows for easy computations, but the lock tableaux is far simpler. Moreover, this simpli- potential redundancy of two different sequences of fication is forced in the following sense. Kohnert moves arriving at the same diagram can be problematic. In [Assaf and Searles 18], the authors Proposition 6.6. Given a weak composition a, let T be gave a static description of Kohnert tableaux for key any filling of a diagram with entries 1a1 ; 2a2 ; :::; nan , diagrams by tracking from where each cell in a one per cell, satisfying the conditions (i), (ii), and (iii) Kohnert diagram came in the key diagram, and when of Definition 6.5 and the following Kohnert’s algorithm gives multiple possibilities for (iv)’ if i < j appear in a column with i above j, then this, fixing a canonical choice. This was done via a there is an i in the column immediately to the canonical labeling of a Kohnert diagram coming from right of and strictly above j. a key diagram, resulting in a simple rule to determine Then T is a lock tableau. 22 S. ASSAF AND D. SEARLES Proof. Suppose T satisfies conditions (i), (ii), and (iii) labeling algorithm for Kohnert diagrams coming from of Definition 6.5 and condition (iv)’ above. Let c be a key diagram. the rightmost column of T such that there exist D Definition 6.8. Given D 2 KDð ðaÞÞ, define the lock entries i < j in column c with i above j. By condition labeling of D with respect to a, denoted by L ðDÞ,by (iv)’, there is an i in column c þ 1. However, by con- a placing the labels fija Pjg to cells of column dition (i), since there is an i in column c þ 1, there i maxðaÞÀj þ 1 in increasing order from bottom to top. must also be a j in column c þ 1. By condition (iii), the j in column c þ 1 lies weakly below the j in col- The lock labeling algorithm is well-defined and umn c which, by condition (iv’), lies strictly below the establishes the following. i in column c þ 1, contradicting the choice of c. Theorem 6.9. The labeling map La is a weight-preserv- Therefore, columns of T must be strictly decreasing ing bijection between KDð D ðaÞÞ and LTðaÞ. In particu- top to bottom. w lar, we have X We mirror the results of [Assaf and Searles 18] for wtðÞT La ¼ x ; (6–4) Kohnert tableaux to prove that lock tableaux precisely T2LTðÞa characterize Kohnert diagrams of lock diagrams. where wtðTÞ is the weak composition whose ith part is Lemma 6.7. For T 2 LTðaÞ, the diagram of T is a D the number of cells in row i of T. Kohnert diagram for ðaÞ: Proof. Suppose that D 2 KDð D ðaÞÞ: The lock labeling Proof. Fix T 2 LTðaÞ: We claim one can perform map La is well-defined on D since the number of cells reverse Kohnert moves on T to obtain the lock dia- per column is preserved by Kohnert moves. No filling gram of a. Reading the cells of T left to right along of D other than LaðDÞ can give an element of LTðaÞ rows, starting at the top row, find the first cell, say C, by condition (iv). Therefore, by Lemma 6.7, removing whose label is greater than its row index. Any cell the labels gives an inverse map provided LaðDÞ is a above C must have had larger label by condition (iv) Kohnert tableau. of Definition 6.5, so by choice of C there cannot be a Condition (i) of Definition 6.5 is manifest from the cell immediately above it, so we may raise C to the selection of entries, and condition (iv) follows imme- row above. To show this reverse move is valid, we diately from the lock labeling. For condition (ii), note 0 need to show there is no cell C to the right of the that every cell in any given column of D must be position in which C lands. Any cell of the landing weakly below where it started, since D is a Kohnert row must have entry equal to its row index by the diagram. In particular, for any index i the number of choice of C, and by condition (ii) C has entry at least boxes of D appearing below row i is weakly larger that large, so the entry of C is at least as great as the than the number of boxes of D ðaÞ appearing below 0: entry of C However, by condition (i), there must be row i, so since the columns are labeled in increasing a cell with entry the same as that of C in the column order from bottom to top with labels given by the 0; of C and by condition (iii) that cell must be strictly row indices of the original positions of the cells, the lower. This creates a violation of condition (iv) in the 0 label of every cell of D must be weakly larger than its column of C in T, a contradiction. row index. This procedure preserves condition (i) of Condition (iii) holds for the lock labeling of D ðaÞ; Definition 6.5. Since this procedure moves the top left so it is enough to check this condition is preserved cell having a given label greater than its row number, under Kohnert moves. Let D be a Kohnert diagram of conditions (ii) and (iii) are preserved. Since cells do D; assume that (iii) is satisfied under the lock labeling not change their order within a column, condition of D. When we make a Kohnert move on D and (iv) is preserved. Therefore, the result is in LTðaÞ: relabel according to the lock labeling, the overall effect Iterating this procedure, one eventually obtains the is to move a cell C and the interval of cells immedi- lock diagram with all entries in row i equal to i, and ately below C down one space, retaining their labels. each move is a valid reverse Kohnert move. Hence T D Since all labels in the column of C move downwards, with its labels removed is in KDð ðaÞÞ: w this does not introduce any violation of (iii) with To establish the converse of Lemma 6.7, we give a entries to the left of the column of C. Since we started canonical labeling of a Kohnert diagram. Once again, with a lock diagram, all labels appearing in the inter- the algorithm for Kohnert diagrams coming from a val of cells below C must also appear in the column lock diagram is far simpler than the analogous to the right of C, and since we performed a Kohnert EXPERIMENTAL MATHEMATICS 23 move on C, the cell immediately right of C in D is flatðbÞ refines flatðwtðSÞÞ: We show there is a unique empty. Hence C’s label appears strictly lower than C T 2 LTðaÞ with wtðTÞ¼b and dstðTÞ¼S: To recon- ; :::; ; in the column right of C, and by the lock labeling and struct T from b and U, for j ¼ 1 n if wtðSÞj ¼ ; the fact that the cells below C form an interval, the bijÀ1þ1 þÁÁÁþbij then, from right to left, move the ; same must be true for all cells in the interval below C. first bijÀ1þ1 cells down to row ijÀ1 þ 1 the next bijÀ1þ2 Hence (iii) is preserved on moving all these cells cells down to row ijÀ1 þ 2; and so on. By construction (with their labels) down one space. w T is a Kohnert diagram of S (so T 2 LTðaÞ), wtðTÞ¼ ; : Recall the quasi-Yamanouchi condition for Kohnert b and dstðTÞ¼S Uniqueness follows from the lack tableaux in [Assaf and Searles 18]. of choice at every step. w Definition 6.10. A lock tableau is quasi-Yamanouchi For example, from Figure 19 we may quickly com- if for each nonempty row i, one of the following holds: pute L F F F F ðÞ0;3;2;1 ¼ ðÞ0;3;2;1 þ ðÞ1;3;1;1 þ ðÞ2;2;1;1 þ ðÞ2;3;0;1 1. there is a cell in row i with entry equal to i,or þ F ; ; ; þ F ; ; ; þ F ; ; ; : 2. there is a cell in row i þ 1 that lies weakly right of ðÞ1 3 2 0 ðÞ2 2 2 0 ðÞ3 1 2 0 a cell in row i. While we noted that lock polynomials are not, in Denote the set of quasi-Yamanouchi lock tableaux general, nonnegative on Demazure characters, there is of content a by QLTðaÞ: a case where lock polynomials and Demazure charac- ters coincide. Letting Conjecture 5.2 be our guide, we For example, the quasi-Yamanouchi lock tableaux D of content ð0; 3; 2; 1Þ are shown in Figure 19. notice that a lock diagram ðaÞ is southwest if and Quasi-Yamanouchi lock tableaux allow for the fol- only if flatðaÞ is weakly decreasing. Thus, the follow- lowing re-characterization of the fundamental slide ing result lends more weight to Conjecture 5.2. expansion of lock polynomials. Theorem 6.12. Given a weak composition a such that Theorem 6.11. Lock polynomials expand non-nega- flatðaÞ is weakly decreasing, we have tively into fundamental slide polynomials by L ¼ j : (6–6) X a a L F : – a ¼ wtðÞT (6 5) T2QLTðÞa Proof. We utilize the machinery of weak dual equiva- lence [Assaf 17] (Definition 3.20) to establish that La Proof. We define a de-standardization map from LTðaÞ expands nonnegatively into Demazure characters to QLTðaÞ by sending a lock tableau T to the quasi- when flatðaÞ is weakly decreasing. For a a weak com- w ; :::; w Yamanouchi lock tableau dstðTÞ constructed as follows. position of n, we must define involutions 2 nÀ1 w w w w For each row, say i,ifeverycellinrowi lies strictly on QLTðaÞ such that i jðTÞ¼ j iðTÞ whenever P 6 ; right of every cell in row i þ 1 and the leftmost cell of jiÀjj 3 and for iÀh 3 there exists a weak compos- ition b of iÀh þ 3 such that row i has label larger than i,thenmoveeverycellin X row i up to row i þ 1. Repeat until no such row exists. F j ; wt ; ðÞU ¼ b To see that the de-standardization map maintains the ðÞhÀ1 iþ1 U2½T ðÞh;i lock tableau conditions, note that the labels within each column are maintained, proving (i). De-standardization where ½Tðh;iÞ is the equivalence class generated by w ; :::; w ; does not move cells to a row higher than their label, so h i and wtðh;iÞðUÞ is the weak composition of (ii) is maintained. No cell is moved from weakly below iÀh þ 1 obtained by deleting the first h – 1 and last n to strictly above any other, and no cell moves upward if – i nonzero parts from wtðUÞ: Once such involutions there is a cell to its right in the row above, so conditions are defined, by [Assaf 17] (Theorem 3.29), the gener- (iii) and (iv) are maintained. Finally, by definition de- ating polynomial over each equivalence class under standardization terminates if and only if the quasi- the involutions will be a nonnegative sum of Yamanouchi condition is met. Demazure characters. Let T 2 LTðaÞ and suppose dstðTÞ¼S 2 QLTðaÞ: To define the desired involutions, we first relabel the Since dst moves cells upwards we have wtðTÞPwtðSÞ; cells of T 2 QLTðaÞ with 1; 2; :::; n from the bottom and since dst moves all cells in row i to row i þ 1, we rowup,labelingeachrowrighttoleft,thenraisethe have flatðwtðTÞÞ refines flatðwtðSÞÞ: Hence xT is a cells to D ðaÞ maintaining their relative order. The result F : ; D ; ; :::; monomial of wtðSÞ Conversely, let S 2 QLTðaÞ and is a bijective filling of ðaÞ with 1 2 n such that let b be a weak composition such that bPwtðSÞ and rows decrease left to right and columns decrease top to 24 S. ASSAF AND D. SEARLES bottom. This process is reversible by lowering cells of a Schur functions (Prop 6.15) and, moreover, form a bijective filling of D ðaÞ such that 1 is the lowest, 2 the basis for quasisymmetric functions (Theorem 6.20). next lowest, and so on, and then applying the de-stand- Thus, we may regard them as extending the Schur ardization map from the proof of Theorem 6.11.Infact, functions to a full basis of quasisymmetric functions, if we allow cells to fall below the x-axis, then this estab- and so call the stable limits the extended lishes a bijection between QLTðaÞ and bijective fillings Schur functions. of D ðaÞ decreasing rows and columns. Definition 6.13. Given a (strong) composition a, the For 2 6 i 6 nÀ1; let wi act on T 2 QLTðaÞ by D extended Schur polynomial indexed by a is given by instead acting on bijective fillings of ðaÞ with ; :::; L ; :::; ; ; :::; ; – decreasing rows and columns as follows. If, in reading EaðÞx1 xm ¼ 0mÂaðÞx1 xm 0 0 (6 7) 6 entries right to left from the top row down, i 1 lies a 7 ; w 7 ; and the extended Schur function indexed by is given between i and i 1 then i exchanges i and i 1 w by otherwise i acts by the identity. To see that this is L D : – EaðÞX ¼ lim 0mÂa ¼K a (6 8) well-defined, there are two cases to check. If i þ 1 lies m!1 ðÞ above i in the same column, then i – 1 lies between them in the previous sense only if it lies right of i. For example, we can compute the extended Schur Since D ðaÞ is right justified, this forces an entry j function Eð2;1;2ÞðXÞ by – L F ; right of i þ 1 and above i 1. The decreasing rows ðÞ2;1;2 ¼ ðÞ2;1;2 and columns forces j ¼ i, which is a contradiction. L F F ; ðÞ0;2;1;2 ¼ ðÞ0;2;1;2 þ ðÞ1;2;0;2 Similarly, if i lies above i – 1 in the same column, L F F F ; then i þ 1 lies between them in the previous sense ðÞ0;0;2;1;2 ¼ ðÞ0;0;2;1;2 þ ðÞ0;1;2;0;2 þ ðÞ1;1;2;0;1 . only if it lies left of i. If flatðaÞ is weakly decreasing, . then this forces an entry j below i þ 1 and left of i – : 1. The decreasing rows and columns forces j ¼ i, EðÞ2;1;2 ¼ FðÞ2;1;2 þ FðÞ1;2;2 þ FðÞ1;1;2;1 which is again a contradiction. Therefore when flatðaÞ By Proposition 3.12, we have the following state- w is weakly decreasing, i is well-defined. ment showing that the extended Schur functions w ; ; Given the local nature of i, since fiÀ1 i i þ 1g\ include all Kohnert quasisymmetric functions for ; ; P ; fjÀ1 j j þ 1g¼; whenever jiÀjj 3 we have the lock diagrams. commutativity relation w w ðTÞ¼w w ðTÞ: The i j j i Corollary 6.14. second condition is local, requiring between three and Given a weak composition a, we have D : six consecutively labeled cells that must fit inside a K ðÞa ¼EflatðÞa staircase diagram. Therefore, there are finitely many cases to check, which can be verified easily by direct To justify the name extended Schur functions,we (and tedious) enumeration or by computer. We have have the following result. done both, so Demazure positivity follows by [Assaf Proposition 6.15. k 17] (Theorem 2.29). For a partition, we have To see that this is a single Demazure character, we EkðÞX ¼ skðÞX : (6–9) note that bijective fillings of D ðaÞ with decreasing rows and columns are in bijection with bijective fillings of Proof. By Theorem 6.12, we have L m k ¼ j m k since partition shape flatðaÞ with increasing rows and col- 0  0  k is weakly decreasing. By Proposition 2.7, we have umns, the latter of which are standard Young tableaux lim j m k ¼ skðXÞ: The result now follows from that generate a Schur function. Therefore, in the stable m!1 0  Definition 6.13. w limit we have a single term in the Schur expansion, so the non-negativity together with Proposition 2.7 implies Note that Demazure characters do not expand the Demazure expansion must have a single term as non-negatively into lock polynomials. For example, well. By the unique leading term for lock polynomials j L L L : ðÞ0;2;1;2 ¼ ðÞ0;2;1;2 þ ðÞ0;2;2;1 À ðÞ1;2;2;0 and Demazure characters, we must have La ¼ ja: w Conversely, Proposition 6.15 shows that the stable limits of Demazure characters do expand nonnega- 6.3. The extended Schur basis tively into the stable limits of lock polynomials. By Theorem 3.11, we may consider the stable limits of We can use lock tableaux to give a tableaux model lock polynomials. These stable limits contain the for extended Schur functions as follows. EXPERIMENTAL MATHEMATICS 25 Figure 20. The set SSET4ð2; 1; 2Þ of semi-standard extended tableaux of shape (2, 1, 2) with entries in f1; 2; 3; 4g: Definition 6.5 condition (i) is equivalent to T hav- ing shape D ðaÞ; condition (ii) and the restriction to lock tableaux with all entries weakly below row m is Figure 21. The set SET 2; 1; 2 of standard extended tableaux ð Þ equivalent to T having labels in f1; 2; :::; mg; condi- of shape (2, 1, 2). tion (iii) is equivalent to rows of T weakly decreasing, and condition (iv) is equivalent to columns of T Definition 6.16. Given a (strong) composition a,a strictly decreasing. Moreover, wtðDÞ¼wtðTÞ: semi-standard extended tableau of shape a is a filling Therefore, this gives a weight-preserving bijection D m of ðaÞ with positive integers such that rows weakly between LTð0  aÞ with all cells weakly below row a ; decrease left to right and columns strictly decrease top m and SSETmð Þ so the first formula follows. The to bottom. Denote the set of semi-standard extended second follows from the first by letting m go to infin- w tableaux of shape a by SSETðaÞ; or by SSETnðaÞ if we ity. ; ; :::; : restrict the integers to f1 2 ng Campbell, Feldman, Light, Shuldiner, and Xu For example, the semi-standard extended tableaux [Campbell et al. 14] defined the same class of tableaux of shape (2, 1, 2) are shown in Figure 20. Compare of composition shape, which they termed shin- these with the lock tableaux for ð0; 2; 1; 2Þ shown in tableaux, when they introduced the shin functions, Figure 18. which are a basis of symmetric functions in non-com- For T a semi-standard extended tableau, let wtðTÞ muting variables that generalize the Schur functions. be the weak composition whose ith part is the number As these tableaux characterize the expansion of non- of entries of T equal to i. Then extended Schur func- commutative homogeneous symmetric functions into tions are the generating function for shin functions, they also characterize the expansion of extended tableaux. the dual basis, which are quasisymmetric functions, into monomial quasisymmetric functions. In other Theorem 6.17. For a a (strong) composition, we have X words, the extended Schur functions are the dual basis ðÞ ; :::; wt T 1 wtðÞT n ; EaðÞx1 xn ¼ x1 ÁÁÁxn to the noncommutative shin functions. In [Campbell a T2SSETXnðÞ et al. 14], the authors observe this and state the posi- wtðÞT EaðÞX ¼ X ; tive expansion into monomial quasisymmetric func- T2SSETðÞa tions. Below we develop further properties of wtðTÞ this basis. where XwtðTÞ is the monomial xwtðTÞ1 ÁÁÁx n when 1 n Just as the fundamental slide expansion of lock pol- wtðTÞ has length n. ynomials is a more compact formula, we may trans- Proof. By Theorem 6.9 and Definition 6.13,we late Theorem 6.17 into a fundamental quasisymmetric have that Eaðx1; :::; xmÞ is the generating polynomial function expansion using the following. for lock tableaux of content ð0m  aÞ with no cells Definition 6.18. A standard extended tableau of shape above row m. Given such a lock tableau D,wemay a is a semi-standard extended tableau of shape a that define a semi-standard extended tableau T as follows. uses each of the integers 1; 2; :::; n exactly once. For every cell x of D, place an entry equal to the row Denote the set of standard extended tableaux of shape index of x into the cell of T in the same column as x a by SETðaÞ, and call the element of SETðaÞ whose and in the row given by m minus the entry of x. For entries in row i þ 1 are the first a integers larger example, the lock tableaux in Figure 18 map to the iþ1 than a þ ::: þ a the super-standard extended tableau semi-standard extended tableau in Figure 20, respect- 1 i of shape a. ively. To reverse the procedure, given a semi-standard extended tableau T, we may construct a lock tableau For example, the standard extended tableaux of D by raising T up m rows then moving each cell of T shape (2, 1, 2) are shown in Figure 21. The leftmost is down to the row equal to its entry. the super-standard one. 26 S. ASSAF AND D. SEARLES Table 1. A table of the fundamental expansion of the composition such that flatðbÞ refines DesðSÞ: We show extended Schur functions. there is a unique T 2 SSETðaÞ with wtðTÞ¼b that F Eð1Þ ¼ ð1Þ F standardizes to S. To reconstruct T from b and S, for Eð2Þ ¼ ð2Þ F ; :::; ; ; Eð11Þ ¼ ð11Þ j ¼ 1 n if DesðSÞj ¼ bijÀ1þ1 þÁÁÁþbij then, from F Eð3Þ ¼ ð3Þ right to left, set the first bi þ1 cells to have entry F F jÀ1 Eð21Þ ¼ ð12Þ þ ð21Þ ; ; F ijÀ1 þ 1 the next bijÀ1þ2 cells to have entry ijÀ1 þ 2 Eð12Þ ¼ ð12Þ F Eð111Þ ¼ ð111Þ and so on. This maintains the extended tableaux con- F Eð4Þ ¼ ð4Þ ditions, wtðTÞ¼b; and T standardizes to S. F F F Eð31Þ ¼ ð13Þ þ ð22Þ þ ð31Þ F Uniqueness follows from the lack of choice at Eð13Þ ¼ ð13Þ F F Eð22Þ ¼ ð121Þ þ ð22Þ every step. w F F F Eð211Þ ¼ ð112Þ þ ð121Þ þ ð211Þ F F Eð121Þ ¼ ð112Þ þ ð121Þ Using this characterization, we now justify the ter- F Eð112Þ ¼ ð112Þ F minology extended Schur basis. Eð1111Þ ¼ ð1111Þ F Eð5Þ ¼ ð5Þ Theorem 6.20. F F F F The extended Schur functions form a Eð41Þ ¼ ð14Þ þ ð23Þ þ ð32Þ þ ð41Þ F Eð14Þ ¼ ð14Þ basis for the ring of quasisymmetric functions. F F F F F Eð32Þ ¼ ð23Þ þ ð122Þ þ ð131Þ þ ð221Þ þ ð32Þ F F Eð23Þ ¼ ð23Þ þ ð122Þ F F F F F F Proof. The descent composition of the super-standard Eð311Þ ¼ ð113Þ þ ð122Þ þ ð131Þ þ ð212Þ þ ð221Þ þ ð311Þ F F F extended tableau of shape a is larger in lexicographic Eð131Þ ¼ ð113Þ þ ð122Þ þ ð131Þ F Eð113Þ ¼ ð113Þ order than the descent composition of any other F F F F F Eð221Þ ¼ ð122Þ þ ð1121Þ þ ð212Þ þ ð1211Þ þ ð221Þ a : F F F element of SETð Þ Hence by Theorem 6.19, the Eð212Þ ¼ ð122Þ þ ð1121Þ þ ð212Þ F F Eð122Þ ¼ ð122Þ þ ð1121Þ extended Schur functions are upper uni-triangular F F F F Eð2111Þ ¼ ð1112Þ þ ð1121Þ þ ð1211Þ þ ð2111Þ with respect to lexicographic order on the fundamen- F F F Eð1211Þ ¼ ð1112Þ þ ð1121Þ þ ð1211Þ F F tal quasisymmetric functions. w Eð1121Þ ¼ ð1112Þ þ ð1121Þ F Eð1112Þ ¼ ð1112Þ F The extended Schur basis exhibits many nice prop- Eð11111Þ ¼ ð11111Þ erties and should have interesting applications to sym- metric and quasisymmetric functions. We close our For T a standard extended tableau, define the des- introduction of this basis with two such properties. cent composition of T, denoted by DesðTÞ; to be the (strong) composition given by increasing runs of the Proposition 6.21. Let a be a (strong) composition and entries 1; 2; :::; n when read right to left in T. For let b be obtained from a by exchanging two adjacent example, the descent compositions for the standard parts ai Proof. Define a map from SETðaÞ to SETðbÞ by drop- Theorem 6.19. a D For a (strong) composition, we have a a a X ping the leftmost iþ1À i cells of row i þ 1of ð Þ : – EaðÞX ¼ FDesðÞT ðÞX (6 10) down one row, retaining all entries. This map is well- T2SETðÞa defined since all cells retain their relative order within columns, and since in any element of SETðaÞ the left- Proof. Similar to the proof of Theorem 6.11, we con- most cell of row i has smaller entry than the cell struct a standardization map from SSETðaÞ to SETðaÞ immediately above it, which in turn has smaller entry by reading entries from smallest to largest and reading than the cell immediately left of it (which is the right- cells with entry i from right to left, change the entries most cell to drop down). The map is injective, and to 1; 2; 3; :::; n: It is easy to see that this maintains the moreover preserves Des since we only move entries extended tableau conditions, so the result is a stand- within their original column. w a ; ; a ard extended tableau. For example, taking ¼ð2 1 2Þ and exchanging 2 a a b ; ; ; Let T 2 SSETð Þ and suppose that T standardizes and 3 to get ¼ð2 2 1Þ we have to S 2 SETðaÞ: By construction and the definition of E ; ; ÀE ; ; ¼ F ; ; þ F ; ; ; : the descent composition, we have flatðwtðTÞÞ refines ðÞ2 2 1 ðÞ2 1 2 ðÞ2 2 1 ðÞ1 2 1 1 : T : DesðSÞ Hence x is a monomial of FDesðSÞ For further examples, compare entries for the Conversely, let S 2 SETðaÞ; and let b be a weak extended Schur functions in Table 1. EXPERIMENTAL MATHEMATICS 27 Sup. (4) 7(1974), 53–88, Collection of articles dedicated Proposition 6.22. The extended Schur function Ea is to Henri Cartan on the occasion of his 70th birthday, I. “ equal to a single fundamental quasisymmetric function [Demazure 74]. M. Demazure. Une nouvelle formule des D caracteres.” Bull. Sci. Math. (2) 98:3 (1974), 163–172. Fa if and only if ðaÞ is a (reverse) hook shape, i.e. [Edelman and Greene 87]. P. Edelman and C. Greene. a k;‘ ‘: ¼ð1 Þ for some k and “Balanced Tableaux.” Adv. in Math. 63:1 (1987), 42–99. [Eilenberg and Lane 53]. S. Eilenberg and S. M. Lane. Proof. If a is a reverse hook shape, there is only one “On the Groups of HðP; nÞ: I.” Ann. of Math. (2) 58 standard extended tableau of shape a, specifically the (1953), 55–106. a > [Fulton 92]. W. Fulton. “Flags, Schubert Polynomials, super-standard one. Conversely, suppose i 1 and ” a > : a Degeneracy Loci, and Determinantal Formulas. Duke iþ1 0 Then a second element of SETð Þ may be Math. J. 65:3 (1992), 381–420. obtained from the super-standard one by swapping [Gessel 84]. I. M. Gessel. Multipartite P-Partitions and Inner the entry of the leftmost cell of row i and the right- Products of Skew Schur Functions, Combinatorics and Algebra most cell of row i þ 1. w (Boulder,CO,1983),Contemp.Math.,vol.34,pp.289–317. Providence, RI: American Mathematical Society, 1984. [Hoffman 00]. M. E. Hoffman. “Quasi-Shuffle Products.” J. Algebraic Comb. 11:1 (2000), 49–68. Acknowledgments [Kohnert 90]. A. Kohnert. “Weintrauben, Polynome, ” – The authors thank Per Alexandersson, Nantel Bergeron, Tableaux. Bayreuth. Math. Schr. 38 (1991), 1 97, € and Vic Reiner for helpful comments and illuminating Dissertation, Universitat Bayreuth, Bayreuth, 1990. discussions. [Kraskiewicz and Pragacz 87]. W. Kraskiewicz and P. Pragacz. “Foncteurs de Schubert.” C. R. Acad. Sci. Paris Ser. I. Math. 304 (1987), 209–211. Funding [Kraskiewicz and Pragacz 04]. W. Kraskiewicz and P. Pragacz. “Schubert Functors and Schubert Polynomials.” S. Assaf was support by the Simons Foundation Eur. J. Comb. 25 (2004), 1327–1344. Collaboration Grant for Mathematicians (Award 524477). [Lascoux and Schutzenberger€ 82]. A. Lascoux and M.-P. Schutzenberger.€ “Polynomes^ de Schubert.” C. R. Acad. Sci. Paris Ser. I Math. 294:13 (1982), 447–450. [Lascoux and Schutzenberger€ 85]. A. Lascoux and M.-P. References Schutzenberger.€ “Schubert Polynomials and the Littlewood-Richardson Rule.” Lett. Math. Phys. 10:2–3 [Assaf and McNamara 11]. S. H. Assaf and P. R. W. (1985), 111–124. McNamara. “A Pieri Rule for Skew Shapes.” J. Combin. [Lascoux and Schutzenberger€ 90]. A. Lascoux and M.-P. Theory Ser. A 118:1 (2011), 277–290. Schutzenberger.€ Keys & Standard Bases, Invariant Theory [Assaf and Searles 17]. S. Assaf and D. Searles. “Schubert and Tableaux (Minneapolis, MN, 1988), IMA Volumes Polynomials, Slide Polynomials, Stanley Symmetric in Mathematics and its Applications, vol. 19, Functions and Quasi-Yamanouchi Pipe Dreams.” Adv. – – pp. 125 144. New York: Springer, 1990. Math. 306 (2017), 89 122. [Macdonald 91]. I. G. Macdonald. Notes on Schubert polyno- [Assaf and Searles 18]. S. Assaf and D. Searles. “Kohnert ” mials, LACIM, Univ. Quebec a Montreal, Montreal, PQ, 1991. Tableaux and a Lifting of Quasi-Schur Functions. J. [Reiner and Shimozono 95]. V. Reiner and M. Shimozono. – Combin. Theory Ser. A 156 (2018), 85 118. “Key Polynomials and a Flagged Littlewood-Richardson “ [Assaf 17]. S. Assaf. Combinatorial Models for Schubert Rule.” J.Combin.TheorySer.A70:1 (1995), 107–143. ” Polynomials. arXiv:1703.00088, 2017. [Reiner and Shimozono 95]. V. Reiner and M. “ [Assaf 17]. S. Assaf. Weak Dual Equivalence for Shimozono. “Specht Series for Column-Convex ” Polynomials. arXiv:1702.04051, 2017. Diagrams.” J. Algebra 174:2 (1995), 489–522. “ [Bergeron and Billey 93]. N. Bergeron and S. Billey. RC- [Reiner and Shimozono 98]. V. Reiner and M. Shimozono. ” Graphs and Schubert Polynomials. Exp. Math. 2:4 “Percentage-Avoiding, Northwest Shapes and Peelable (1993), 257–269. Tableaux.” J.Combin.TheorySer.A82:1 (1998), 1–73. [Billey et al. 93]. S. C. Billey, W. Jockusch, and R. P. [Stanley 84]. R. P. Stanley. “On the Number of Reduced Stanley. “Some Combinatorial Properties of Schubert Decompositions of Elements of Coxeter Groups.” Eur. J. Polynomials.” J. Algebraic Combin. 2:4 (1993), 345–374. Comb. 5:4 (1984), 359–372. [Campbell et al. 14]. J. Campbell, K. Feldman, J. Light, P. [Winkel 99]. R. Winkel. “Diagram Rules for the Shuldiner, and Y. Xu. “A Schur-Like Basis of NSym Generation of Schubert Polynomials.” J. Comb. Theory Defined by a Pieri Rule.” Electron. J. Combin. 21:3 Ser. A 86:1 (1999), 14–48. (2014), 19, Paper 3.41. [Winkel 02]. R. Winkel. “A Derivation of Kohnert’s [Demazure 74]. M. Demazure. “Desingularisation des var- Algorithm from Monk’s Rule.” Sem. Lothar. Comb. 48 ietes de Schubert generalisees.” Ann. Sci. Ecole Norm. (2002), 14 Art. B48f.