arXiv:2109.05651v1 [math.CO] 13 Sep 2021 AN INSERTION ALGORITHM FOR MULTIPLYING DEMAZURE CHARACTERS BY SCHUR POLYNOMIALS SAMI ASSAF Abstract. Certain polynomial generalizations of Schur polynomials, such as Demazure characters and Schubert polynomials, have structure constants with geometric significance, thus motivating the search for combinatorial formulas for these numbers. In this paper, we introduce an insertion algorithm on Kohnert’s combinatorial model for these polynomials, generalizing Robinson– Schensted–Knuth insertion on tableaux. This new insertion algorithm yields an explicit, nonnegative formula expressing the product of a Demazure charac- ter and a Schur polynomial as a sum of Schubert characters, partially resolving a conjecture of Polo. Moreover, we lift this expression to a nonnegative, com- binatorial formula for the Demazure character expansion of the product of a Schubert polynomial and a Schur polynomial providing new progress toward a combinatorial formula for Schubert structure constants. 1. Introduction 1.1. Schur polynomials. Schur polynomials are a basis for symmetric polyno- mials of fundamental importance due to their role as characters for irreducible polynomial representations of the general linear group, Frobenius characters for irreducible representations of the symmetric group, and polynomial representatives for cohomology classes of Schubert cycles in Grassmannians. Their structure con- stants, the Littlewood–Richardson numbers c λ μ,ν triply indexed by partitions, satisfy (1.1) s μ (x 1 ,...,x k )s ν (x 1 ,...,x k )= λ c λ μ,ν s λ (x 1 ,...,x k ). These numbers c λ μ,ν give the multiplicity of the irreducible decomposition of tensor products in the general linear group, describe the decomposition of restrictions of modules in the symmetric group, and enumerate points in a triple intersection of Grassmannian Schubert varieties. These contexts ensure c λ μ,ν are nonnegative and motivate finding combinatorial rules to compute their values. The first such rule, stated by Littlewood and Richardson [17] and proved by Thomas [29] and Sch¨ utzenberger [27] based on ideas of Robinson [24], interprets c λ μ,ν as the cardinality of LYT(λ/µ, ν ), the set of lattice semistandard Young tableaux of skew shape λ/µ and weight ν . A beautiful proof of this comes via the elegant Robinson–Schensted–Knuth insertion algorithm [24, 26, 13] giving a bijection (1.2) SSYT(µ) × SSYT(ν ) ∼ −→ λ (SSYT(λ) × LYT(λ/µ, ν )) , where SSYT(λ) denotes the semistandard Young tableaux of partition shape λ. For a thorough and self-contained treatment, see [10, §5]. 1
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AN INSERTION ALGORITHM FOR MULTIPLYING DEMAZURE
CHARACTERS BY SCHUR POLYNOMIALS
SAMI ASSAF
Abstract. Certain polynomial generalizations of Schur polynomials, such asDemazure characters and Schubert polynomials, have structure constants withgeometric significance, thus motivating the search for combinatorial formulasfor these numbers. In this paper, we introduce an insertion algorithm onKohnert’s combinatorial model for these polynomials, generalizing Robinson–Schensted–Knuth insertion on tableaux. This new insertion algorithm yieldsan explicit, nonnegative formula expressing the product of a Demazure charac-ter and a Schur polynomial as a sum of Schubert characters, partially resolvinga conjecture of Polo. Moreover, we lift this expression to a nonnegative, com-binatorial formula for the Demazure character expansion of the product of aSchubert polynomial and a Schur polynomial providing new progress towarda combinatorial formula for Schubert structure constants.
1. Introduction
1.1. Schur polynomials. Schur polynomials are a basis for symmetric polyno-mials of fundamental importance due to their role as characters for irreduciblepolynomial representations of the general linear group, Frobenius characters forirreducible representations of the symmetric group, and polynomial representativesfor cohomology classes of Schubert cycles in Grassmannians. Their structure con-stants, the Littlewood–Richardson numbers cλµ,ν triply indexed by partitions, satisfy
(1.1) sµ(x1, . . . , xk)sν(x1, . . . , xk) =∑
λ
cλµ,νsλ(x1, . . . , xk).
These numbers cλµ,ν give the multiplicity of the irreducible decomposition of tensorproducts in the general linear group, describe the decomposition of restrictions ofmodules in the symmetric group, and enumerate points in a triple intersection ofGrassmannian Schubert varieties. These contexts ensure cλµ,ν are nonnegative andmotivate finding combinatorial rules to compute their values.
The first such rule, stated by Littlewood and Richardson [17] and proved byThomas [29] and Schutzenberger [27] based on ideas of Robinson [24], interpretscλµ,ν as the cardinality of LYT(λ/µ, ν), the set of lattice semistandard Young tableauxof skew shape λ/µ and weight ν. A beautiful proof of this comes via the elegantRobinson–Schensted–Knuth insertion algorithm [24, 26, 13] giving a bijection
(1.2) SSYT(µ)× SSYT(ν)∼−→
⊔
λ
(SSYT(λ)× LYT(λ/µ, ν)) ,
where SSYT(λ) denotes the semistandard Young tableaux of partition shape λ. Fora thorough and self-contained treatment, see [10, §5].
1.2. Demazure characters. Demazure [7] considered submodules of irreduciblemodules generated by extremal weight spaces under the action of a Borel subal-gebra. These Demazure modules originated in connection with Schubert varietiesand have deep connections with specializations of nonsymmetric Macdonald poly-nomials [25, 3], a multi-parameter generalization of Schur polynomials. Demazure[8] generalized the Weyl character formula to Demazure modules, with proof gapsnoted by Kac and completed by Joseph [12] using excellent filtrations, nested se-quences of submodules whose quotients are (isotypical) Demazure modules.
The Demazure characters are indexed by a highest weight and a Weyl groupelement, which, for the general linear group, is a partition and a permutation.Letting the permutation act on the partition, we change the indexing to weakcompositions. When the permutation is the longest element, the Demazure moduleis the full irreducible module and the indexing composition is a partition. In thiscase, the corresponding Demazure character is a Schur polynomial. Since Demazurecharacters form a basis of the polynomial ring, it is natural to consider structureconstants for Demazure characters analogous to (1.1), namely
(1.3) κα(x1, . . . , xn)κβ(x1, . . . , xk) =∑
γ
cγα,βκγ(x1, . . . , xmax(n,k)).
Unfortunately, the coefficients cγα,β are not, in general, nonnegative.
Polo [21] considered which tensor products of Demazure modules admit excellentfiltrations and conjectured every tensor product admits a Schubert filtration. Onthe level of characters, this would imply the character is the sum of Demazureatoms whose indexing set forms a lower order ideal in Bruhat order. Lascouxand Schutzenberger [16] introduced Demazure atoms, under the name standardbases, as the minimal non-intersecting pieces of Demazure characters. Thus Polo’sconjecture implies the product of Demazure characters expands as a nonnegativesum of Demazure atoms, which was also directly conjectured by Pun [22].
In this paper, we define an insertion algorithm for Kohnert’s combinatorial modelfor Demazure characters [14] to obtain an explicit bijection
(1.4) KD(α)× SSYT(ν)∼−→
⊔
γ
(AKD(γ)× LAT(γ/α, ν)) ,
where KD(α) is the set of Kohnert diagrams of shape α, AKD(γ) is the set ofatomic Kohnert diagrams of shape γ, and LAT(γ/α, ν) is the set of lattice atomictableaux of skew shape γ/α and weight ν. Thus we have explicit formulas for
(1.5) κα(x1, . . . , xn)sν(x1, . . . , xk) =∑
γ
aγα,νAγ =∑
γ
cγα,νκγ ,
where aγα,ν gives the nonnegative expansion into Demazure atoms. We furthermoreshow this sum over Demazure atoms can naturally be indexed by lower order ideals,proving the stronger implication of Polo’s conjecture for Schubert characters.
Our new rule for (1.5) specializes to two previously known cases. Mathieu [20]proved the tensor product of a Demazure module with a full irreducible moduleadmits an excellent filtration. Taking characters, this ensures the product of aDemazure character with a Schur polynomial in at least as many variables (i.e.n ≤ k) expands nonnegatively into Demazure characters. Haglund, Luoto, Masonand van Willigenburg [11] use RSK insertion to give an explicit nonnegative rulefor the structure contants in this case. Assaf and Quijada [4] give a signed rule for
MULTIPLICATION BY SCHUR POLYNOMIALS 3
(1.5) in the Pieri case (i.e. ν has a single part) in any number of variables. Thoughnot explicitly stated, their formula implies a nonnegative expansion into Demazureatoms and, moreover, into Schubert characters as implied by Polo’s conjecture.
The insertion algorithm presented in this paper holds in a more general sensethan is needed to prove (1.4), suggesting that this new approach might be used toresolve the general the character version of Polo’s conjecture and to give an explicit,general formula for Demazure structure constants in (1.3).
1.3. Schubert polynomials. Lascoux and Schutzenberger [15] introduced Schu-bert polynomials Sw as polynomial representatives of Schubert classes of the co-homology ring of the complete flag manifold with nice algebraic and combinatorialproperties. In the Grassmannian case, Schubert polynomials are Schur polyno-mials, hence the geometric interpretation of cλµ,ν . More generally, the generalizedLittlewood–Richardson coefficients cwu,v triply indexed by permutations satisfy
(1.6) SuSv =∑
w
cwu,vSw,
and count points in a triple intersection of Schubert varieties, and so are nonnega-tive. Despite over a century of interest, giving a nonnegative combinatorial formulafor cwu,v, even for v Grassmannian, remains an important open problem.
Lascoux and Schutzenberger [16] gave a formula, with proof details supplied byReiner and Shimozono [23], for the nonnegative expansion of a Schubert polynomialinto Demazure characters using a generalization of RSK insertion to reduced wordsdeveloped by Edelman and Greene [9]. Assaf gave an alternative formulation ofthis positivity using rectification of Kohnert diagrams [6, 2] as
(1.7) Sw =∑
α
aw,ακα,
where aw,α is the number of Yamanouchi Kohnert diagrams for w with weight α.Using this formulation together with the insertion algorithm establishing (1.4), wegive an explicit formula for the nonnegative integer coefficients cγu,ν in the product
(1.8) Susν(x1, . . . , xk) =∑
γ
cγu,νκγ .
While the nonnegativity of coefficients cγu,ν follows the nonnegativity of coeffi-cients in (1.6) and (1.7), there is no combinatorial proof of the nonnegativity of cwu,νmuch less an explicit combinatorial formula. Thus this new, explicit combinato-rial result marks new progress toward a general formula for the Schubert structureconstants cwu,ν for u any permutation and ν any partition of any length.
2. Kohnert diagrams
In this section, we give combinatorial models for polynomial bases using dia-
grams, finite collections of cells in the first quadrant of Z×Z, and we introduce anew combinatorially defined basis of pinned polynomials.
2.1. Kohnert’s rule. Kohnert’s elegant combinatorial model for Demazure char-acters [14] uses the following operation on diagrams.
Definition 2.1.1 ([14]). A Kohnert move on a diagram selects the rightmostcell of a given row and moves the cell down within its column to the first availableposition below, if it exists, jumping over other cells in its way as needed.
4 S. ASSAF
The set of Kohnert diagrams for D, denoted by KD(D), consists of all dia-grams obtainable from D by some (possibly empty) sequence of Kohnert moves; seeFig. 1. The composition diagram D(α) for a weak composition α has αi cells,left-justified in row i. We compress notation by KD(α) = KD(D(α)).
❣❣❣
❣❣❣
❣
❣❣
❣❣❣
❣❣❣
3 22
3 12
3 11
2 11
2 21
Figure 1. The set KD(0, 2, 1) of Kohnert diagrams for (0, 2, 1)and their images under the injection with reverse tableaux.
Definition 2.1.2 ([14]). The Demazure character κα for α ∈ Nm is
(2.1) κα(x1, . . . , xm) =∑
T∈KD(α)
xwt(T )11 · · ·xwt(T )m
m ,
where wt(T )r is the number of cells in row r of T .
Not all diagrams can arise as Kohnert diagrams for some weak composition.
Definition 2.1.3. A diagram T is rectified if T ∈ KD(α) for some α.
A partition is a weakly increasing weak composition. A semistandard re-
verse tableau of partition shape λ, denoted by T ∈ SSRT(λ), is a filling of thecells of D(λ) with integers 1 ≤ i ≤ ℓ(λ) such that entries weakly decrease left toright within rows and strictly decrease top to bottom within columns.
Definition 2.1.4. The Schur polynomial indexed by the partition λ ∈ Nk is
(2.2) sλ(x1, . . . , xk) =∑
T∈SSRT(λ)
xwt(T )11 · · ·x
wt(T )kk ,
where k = ℓ(λ) and wt(T )i is the number of entries of T equal to i.
To relate the diagram model with the familiar paradigm of tableaux, Assaf andSearles [5, Thm 4.6] prove the following, rephrased here for reverse tableaux.
Proposition 2.1.5 ([5]). For α ∈ Nm and λ(α) the unique partition in the Sm
orbit of α, the map ϕ : KD(α) → SSRT(λ(α)) that assigns entry i to each cell inrow i and lifts the cells within their columns is a weight-preserving, injective mapthat is surjective if and only if α = λ(α).
In particular, every Schur polynomial is a Demazure character. Moreover, underthis injection, our new constructions align with classical operations on tableaux.
MULTIPLICATION BY SCHUR POLYNOMIALS 5
2.2. Threading algorithm. Demazure atoms were introduced by Lascoux andSchutzenberger [16] as a refinement of Demazure characters. Mason [19] gave atableau model for Demazure atoms, and Searles [28] gave a diagram model basedon the threading algorithm of Assaf and Searles [5, Def 3.5]; see Fig. 2.
Definition 2.2.1 ([5]). The thread decomposition of a diagram partitions cellsinto threads as follows. Begin a new thread with the rightmost then lowest un-threaded cell. After threading a cell in column c+1, thread the lowest unthreadedcell in column c weakly above it, and continue left until column 1 or until no cellcan be threaded. Continue threading until all cells are threaded.
Assaf and Searles prove every thread of T ends in the first column if and only ifT is rectified [5, Lemma 2.2], showing the following is well-defined.
Definition 2.2.2 ([5]). The thread weight of a rectified diagram T , denoted byθ(T ), has θ(T )r equal to the number of cells in the thread in column 1, row r.
Assaf and Searles [5] use threads to partition the set KD(α) corresponding tothe decomposition of a Demazure character into quasi-key polynomials. Searles [28]refines this to reflect the decomposition of a Demazure character into atoms.
Definition 2.2.3. A rectified diagram T is an atomic Kohnert diagram for αif θ(T ) = α. Denote the set of atomic Kohnert diagrams for α by AKD(α).
As this will be important for deriving formulas from the combinatorics to follow,notice for α 6= β, the sets AKD(α) and AKD(β) are disjoint.
Combining [5, Thm 3.7] and [28, Thm 3.6] gives the following.
Definition 2.2.4 ([5, 28]). The Demazure atom Aα for α ∈ Nm is
(2.3) Aα(x1, . . . , xm) =∑
T∈AKD(α)
xwt(T )11 · · ·xwt(T )m
m
where wt(T )r is the number of cells in row r of T .
Following [5], a left swap on a weak composition α exchanges two parts αr < αs
with r < s. Write α � β whenever α is obtainable via some (possibly empty)sequence of left swaps on β. The following is implicit in [5, Thm 3.7].
Lemma 2.2.5 ([5]). For T rectified, T ∈ KD(α) if and only if θ(T ) � α.
Thus we can refine Kohnert diagrams into their atomic subsets by
(2.4) KD(β) =⊔
α�β
AKD(α).
Taking generating polynomials gives the familiar decomposition
(2.5) κβ =∑
α�β
Aα.
That is, Demazure characters are sums over Bruhat intervals of Demazure atoms.Generalizing this, for I any lower order ideal in Bruhat order, define
(2.6) κI =∑
α∈I
Aα.
Then Polo’s conjecture [21] for characters can be stated as follows.
6 S. ASSAF
Conjecture 2.2.6 ([21]). There exist nonnegative integers aIβ,α such that
(2.7) κβκα =∑
I
aIβ,ακI ,
where the sum is over lower order ideal in Bruhat order.
Since these Schubert characters κI defined in (2.6) over determine a basis, thereis no computational test for Conjecture 2.2.6. However, by (2.6), the right handside of (2.7) expands nonnegatively into Demazure atoms, which are a basis. Thismotivates the following weaker conjecture, which first appears in [22, Conj. 1], wherePun gives a proof for compositions of length at most 3 with at most 2 nonzero parts.
Conjecture 2.2.7 ([22]). There exist nonnegative integers aγβ,α such that
(2.8) κβκα =∑
γ
aγβ,αAγ .
In Theorem 4.3.2, we give an explicit, combinatorial formula for aγβ,α for α a
partition. In Theorem 5.1.4, we give an explicit, combinatorial formula for aIβ,αwhen α is a partition, resolving this case of Polo’s conjecture with an explicit setof lower order ideals determining the Schubert characters.
2.3. Labeled diagrams. Assaf and Searles give a canonical labeling [5, Def 2.3]of cells of a rectified diagram by a composition α; see Fig. 2.
Definition 2.3.1 ([5]). Given a diagram, a semi-proper labeling of shape αassigns positive integers to the cells of the diagram such that
(1) column c consists of distinct entries {r | αr ≥ c};(2) each entry in row r is at least r;(3) cells with entry r weakly descend from left to right.
A labeling is proper if in addition
(4) if r < s appear in a column with r above s, then there is an r in the columnimmediately to the right of and strictly above s.
Combining [5, Thm 2.8] and [6, Lemma 5.2.1] gives the following.
Lemma 2.3.2 ([5, 6]). For a diagram T , the following are equivalent
(i) T ∈ KD(α);(ii) T has a (unique) proper labeling, denoted by Lα(T ), of shape α;(iii) T has a semi-proper labeling of shape α.
450250
6 65 5
63 5 5 52 2 2 6
3 2 2
450520
6 65 5
63 5 5 52 3 3 6
2 3 3
540520
6 65 5
63 5 6 62 3 3 5
2 3 3
550420
6 65 5
63 5 6 62 3 3 5
2 3 5
Figure 2. Pinned, proper labelings of a rectified diagram T ; theleftmost is the atomic labeling showing the thread decomposition.
MULTIPLICATION BY SCHUR POLYNOMIALS 7
Assaf and Searles give an explicit algorithm for constructing Lα(T ) [5, Def 2.5].The atomic labeling of T , denoted by Lθ(T ), in which each cell along a threadending in row r is labeled r, is proper and pinned as defined below.
Definition 2.3.3. A labeling is pinned if each entry in column 1 equals its row.Given α, denote the set of Kohnert diagrams of shape α with pinned, proper labelsby PKD(α) = {T ∈ KD(α) | Lα(T ) is pinned}.
A pinned left swap on β exchanges two nonzero parts 0 < βi < βj for i < j.Write α �0 β whenever α results from some sequence of pinned left swaps on β.
Lemma 2.3.4. We have AKD(α) ⊆ PKD(β) if and only if α �0 β.
Proof. If α �0 β, then α � β so, by Lemma 2.2.5, KD(α) ⊆ KD(β). Thus T ∈AKD(α) ⊆ KD(β) and since α �0 β, T is also pinned for β.
Conversely, suppose AKD(α) ⊆ PKD(β). Then αi = 0 if and only if βi = 0.Since D(α) ∈ PKD(β) ⊆ KD(β), by Lemma 2.2.5, α = θ(D(α)) � β as well. �
The generating polynomial for pinned Kohnert diagrams provides a useful inter-mediate basis between Demazure characters and Demazure atoms.
Definition 2.3.5. The pinned polynomial Pα for α ∈ Nm is
(2.9) Pα(x1, . . . , xm) =∑
T∈PKD(α)
xwt(T )11 · · ·xwt(T )m
m .
Proposition 2.3.6. The set {Pβ}β∈Nm is a Z-basis for Z[x1, . . . , xm]. Moreover,for α, β, γ compositions of length m, we have nonnegative expansions
(2.10) κγ =∑
β�γ
β�0α�γ⇒α=β
Pβ and Pβ =∑
α�0β
Aα.
Proof. The right side of (2.10) follows from Lemma 2.3.4. The left expansion followsfrom this and (2.4) together with the observation that for any α, there exists aunique maximal β for which α �0 β � γ. In particular, the pinned polynomials areupper uni-triangular with respect to Demazure atoms, which are a Z-basis. �
The product of a Demazure atom with a Schur polynomial does not alwaysexpand nonnegatively into Demazure atoms. The motivation for introducing thisnew basis is that the product of a pinned polynomial and a Schur polynomial does.
3. Insertion algorithm
In this section, we define our algorithm for insertion of cells at specific rows intolabeled Kohnert diagrams where we bound the insertion by a parameter k, and weshow this is equivalent to RSK via the injection ϕ when k is sufficiently large.
3.1. Label exchanges. We restrict our inserted cell by a parameter k and takeinto account the pinned labeling of the diagram into which it is inserted.
Definition 3.1.1. In a proper labeling, s touches r at column c if s lies above rin column c and s in column c+ 1 lies weakly below r in column c. In this case, scrosses r if there is an r above s in column c+ 1; otherwise s abuts r.
8 S. ASSAF
32 2 2
3
32 3 3
2
Figure 3. Labels that cross (left) and abut (right) at column 1.
The following procedure turns a proper labeling into a semi-proper labeling withsmaller labels raised higher and the shape related by a pinned left swap.
Definition 3.1.2. If s abuts r in column c of a semi-properly labeled diagram, theexchange labeling for r, s at column c, denoted by Lcr,s, is obtained by simulta-neously changing r to s and s to r in columns c+ 1, . . . , d, where d is the leftmostcolumn right of c in which s touches r, if it exists, or else d = αs.
Lemma 3.1.3. Exchange labelings of semi-proper labelings are semi-proper.
Proof. For condition (1), if s touches r after column c or if αs ≤ αr, then the setof labels within each column is unchanged, and so the weight of the labeling is α.Otherwise, labels s terminate at min(αr, αs) and labels r extend to max(αr , αs),changing the weight to tr,sα if αr < αs, but still maintaining (1).
For condition (2), the leftmost affected labels occur weakly below a row q occu-pied by a label r, and so q < r. By (3) for the original labeling, every affected labeloccurs weakly below row q, and so (2) holds for the exchange labeling.
Finally, condition (3) holds in columns b ≤ c and b > d since those columns areunaffected, and in columns c + 1 < b ≤ d since the labels are swapped in thosecolumns. Thus we need only check at columns c, c + 1 and d, d + 1. From thedefinition of touching, both r, s in column c+1 (if they exist) lie weakly below bothr, s in column c as required. Similarly, if s touches r at column d, then both r, s incolumn d + 1 (if they exist) lie weakly below both r, s in column d as required. Ifs never again touches r, then s terminates at min(αr, αs) ≤ d, leaving nothing toshow for s. In this case, if αr ≤ αs = d, there is also nothing to show for r, andif αr > αs, then since s, r never again touched, s lies above r in column d, and soabove r in column d + 1 by condition (3) for the original labeling. Thus (3) holdsfor the exchange labeling as well, and so it is semi-proper. �
r
s tx
L3s,t
−−→ r
s tx
L1r,s
−−−→ r
s tx
Figure 4. Exchanging labels of an r-exchange sequence for x.
We can iterate label exchanges using the following; see Fig. 4.
Definition 3.1.4. A sequence of labels r0, r1, . . . , rm in columns c0 < c1 < · · · < cmof a properly labeled diagram is an r0-exchange sequence if each ri abuts ri−1
in column ci−1 and rj does not touch ri in any column ci < c ≤ cj .
Theorem 3.1.5. For r0, r1, . . . , rm in columns c0 < c1 < · · · < cm an r0-exchangesequence of a semi-properly labeled diagram, the composition of exchange labelingsLc0r0,r1 · · · L
cm−1rm−1,rm is semi-proper, and the ri in column ci is re-labeled to r0.
MULTIPLICATION BY SCHUR POLYNOMIALS 9
Proof. We proceed by induction on m. The base case m = 0 is trivial. Assumethe theorem for exchange sequences of length less than m. Since rm abuts rm−1
at cm−1, the exchange labeling Lcm−1rm−1,rm applies and, by Lemma 3.1.3, results in
a semi-proper labeling. Since rm does not touch rm−1 at any column c ≤ cm,the column d in Definition 3.1.2 must satisfy d ≥ cm, and so the label rm incolumn cm changes to rm−1. Since rm does not touch ri for i < m in any columnci < c ≤ cm, the re-labeled rm−1’s still do not touch any ri for i < m − 1 in anycolumn ci < c ≤ cm. Thus omitting the label rm and the column cm−1 results inan r0-exchange sequence of length m− 1. The result now follows by induction. �
3.2. Labeled insertion. Our insertion procedure is similar to RSK in that aninserted cells may bump other cells within its column, though when no bump ispossible, the cell may also slide left to pass other cells within its row; see Fig. 5.
Definition 3.2.1. For T ∈ PKD(α) and x 6∈ T a vacant position weakly belowrow k, the rectification of T labeled by α with x restricted by k, denoted by
Rect(k)α (T, x), is defined as follows. Let R denote the set of labels r ≤ k for which
x abuts r after exchanging labels of an r-exchange sequence. Then
(1) if some r ∈ R has no re-labeled r in the column of x, then return T ∪ {x};
(2) else if R 6= ∅, then return Rect(k)α (T ∪ {x} \ {y}, y) for y the lowest cell in
the column of x re-labeled to r for some r ∈ R;
(3) else return Rect(k)α (T, y) for y the rightmost vacant position left of x.
We refer to case (2) as a bump and to case (3) as a pass. The bumping path
is the set of distinguished cells, and the landing cell is the final distinguished cell.
6 65 5
63 5 5 52 2 2 6 x
3 2 2
(3)−−→
6 65 5
63 5 5 52 2 2 6 x
3 2 2
(2)−−→
6 65 5
63 5 5 52 2 2 2 2
3 6 x
(3)−−→
6 65 5
63 5 5 52 2 2 2 2
3 x 6
(1)−−→
6 65 5
63 5 5 52 2 2 2 2
3 x 6
Figure 5. Rectification of a labeled diagram with x restricted by 4.
Theorem 3.2.2. For T ∈ PKD(β) and x 6∈ T a vacant position weakly below row
k, Rect(k)β (T, x) is a well-defined, rectified diagram.
Proof. Suppose case (3) occurs, say with x in some row r. Since x lies weakly belowrow k, we must have r ≤ k. Suppose, for contradiction, there is no vacant positionleft of x within its row. From left to right, let r, r1, . . . , rm be the distinct labels,say with each occurring last in column c0 < c1 < · · · < cm. We claim this is anr-exchange sequence, contradicting that R = ∅. For i > 0, each ri abuts ri−1 atci−1 since, by Definition 2.3.1(3), any ri−1 in column ci−1 + 1 lies weakly lower,and hence lower since ri is the label in row r, column ci−1 +1. Moreover, for i < j,all labels ri in columns ci < c lie strictly below row r, and all labels rj in columnsc ≤ cj lie weakly above row r. Thus for i < j, rj cannot touch ri in any columnci < c ≤ cj . We have a contradiction, and so a vacant position exists.
Suppose case (2) occurs, with y the lowest cell and r its label after an r-exchangelabeling. Since x abuts r, y lies below x and hence below row k. Label x with r
10 S. ASSAF
and, after removing y, we claim this is semi-proper. The exchange labeling is semi-proper by Theorem 3.1.5, and since x abuts r, it lies weakly below the last r left ofy. Since x lies above y, all subsequent r’s lie weakly below it, and so the labeling issemi-proper. By Lemma 2.3.2, T ∪{x}\{y} is a Kohnert diagram for the exchangedlabeling α �0 β, and so by Lemma 2.3.4, T ∪ {x} \ {y} ∈ PKD(β).
Suppose case (1) occurs for some r ≤ k. We claim labeling x with r is semi-proper. Indeed, the exchange labeling for the r-exchange sequence is semi-properby Theorem 3.1.5, and since x abuts r, it lies weakly below the last r. Thus byLemma 2.3.2, T ∪ {x} is a Kohnert diagram for α + er for some α �0 β. Finally,since the bumping path during rectification moves either strictly left or strictlydown, the process must eventually terminate with a rectified diagram. �
Definition 3.2.3. For T ∈ KD(β) and 1 ≤ r ≤ k integers, the insertion of r
restricted by k into T labeled by β, denoted by Tβ,k←−− r, is
(3.1) Tβ,k←−− r = Rect
(k)α (T, (c+ 1, r)).
where c = max(βi) and α is maximal such that θ(T ) �0 α � β.
To persuade the skeptic, note restricted rectification generalizes RSK.; see Fig. 6.
6 65 5 ←x
63 ←5 5 52 2 2 6
←3 2 2
Rect(6)
(052054)−−−−−−−−→
6 65 5 6
53 3 5 52 2 2 61 2 2
6 6 4 3 35 5 3 2 13 2 2 12 1
5
RSK−−−→
6 6 5 3 35 5 4 2 13 3 2 12 2
1
Figure 6. The insertion of 5 restricted by 6 into a diagram T ∈KD(052054) and RSK insertion of 5 into ϕ(T ) ∈ SSRT(002455).
Theorem 3.2.4. For T ∈ KD(α) and k ≥ ℓ(α), we have
(3.2) ϕ(Tα,k←−− r) = ϕ(T )
RSK←−−− r.
where ϕ is the injective, weight-preserving map KD(α)→ SSRT(λ(α)).
Proof. If T ∪ {x} is a rectified diagram, then x must abut some label since x canbe labeled consistently with T . Thus when k ≥ ℓ(α), case (3) of Definition 3.2.1occurs if and only if T ∪ {x} is not a rectified diagram, and in this case there isno cell immediately to the left of x. Therefore Definition 3.2.1 reduces to Assaf’soriginal rectification [6, Def 4.2.4], a procedure that maps an arbitrary diagram toa rectified one of the same weight. The theorem now follows from [4, Thm 4.2.7], inwhich Assaf and Quijada prove rectification, in the unrestricted sense, of a diagramwith one additional cell appended is equivalent to RSK insertion. �
In general, the label used for restricted rectification affects the result; see Fig 7.
3.3. Iterated insertion. To multiply a Demazure character by Schur polynomials,we must consider successive insertions, each with the same parameter k, determinedby the Schur polynomial, though the labels must change as the diagram grows.
Definition 3.3.1. Write β ⊂·k tr,sm · · · tr,s1β+ er whenever r ≤ k < s1 < . . . < smand 0 < βr < βs1 < · · · < βsm . Denote the transitive closure by β ⊆k γ.
MULTIPLICATION BY SCHUR POLYNOMIALS 11
43 x
3 34
Rect(3)
(0032)−−−−−−→
43 3x 3
4
43 x
4 43
Rect(3)
(0023)−−−−−−→
43 4
3 4x
Figure 7. An example showing labels matter for rectification.
We call r the extended row and βsm + 1 the added column .
Proposition 3.3.2. For T ∈ PKD(β) and x 6∈ T a vacant position weakly below
k, we have Rect(k)β (T, x) ∈ PKD(γ) for some β ⊂·k γ.
Proof. Let U = Rect(k)β (T, x), and let y denote the landing cell and c the landing
column. In any label exchange during rectification, the first column is unaffectedand every affected column has the larger label. Thus the change in the shape of thelabeling, if any, is a pinned left swap, and so, by Lemma 2.3.4, U \ {y} ∈ PKD(β).
Since y lands, it abuts some label r ≤ k for which there is no label r in columnc even after applying any necessary r-exchange sequence. We claim there is alsono ri in column c for any label ri on the r-exchange sequence. Indeed, if ri is thelargest label on the r-exchange sequence for which there is an ri in column c, thenthis ri must be changed to ri−1, and so on, until at last there is an r in column c,contradicting that this is the landing column. Thus the claim is proved.
There is an r-exchange sequence with all other labels greater than k, since ifr < s ≤ k, then by the previous claim there is no s in column c, so y abutss via the s-exchange subsequence. Thus we have an r-exchange sequence withr ≤ k < r1 < · · · < rm where βr < βr1 ≤ · · · ≤ βrm . Omitting transposition whichact trivially, after exchanging labels the shape becomes
Allowing y to take label r adds er, and the result follows from Lemma 2.3.2. �
Proposition 3.3.2 guides how we relabel cells after insertion; see Fig. 9. We havethe following definitions from [4, Def 3.2.6] and [4, Def 3.2.3].
Definition 3.3.3 ([4]). A position (c, r) with r ≤ k is k-addable for β if
• βr < c and if βr < c− 1, then there exists some t > k such that βt = c− 1;• for all r < s ≤ k, either βs < βr or βs ≥ c.
For a k-addable position, the k-addition of (c, r) to β is
β +(k) (c, r) = tr0,rq · · · tr0,r1β + er,
where r = r0 < r1 < · · · < rq is the unique sequence of row indices such thatβr0 < · · · < βrq = c− 1 and if ri−1 < s < ri, then βs ≤ βri−1 or βs > βri .
In particular, β ⊂·k β +(k) (c, r) for any k-addable position; see Fig. 8.
Lemma 3.3.4. For T ∈ KD(β) and r ≤ k, we have
(Tβ,k←−− r) ∈ KD(β +(k) (c, s))
for c the landing column and (c, s) a k-addable position for β.
12 S. ASSAF
+
+
+
+
++
Figure 8. The six possible 4-additions to the composition (013012).
Proof. Let U = Tβ,k←−− r. By Proposition 3.3.2, θ(U) � α+ ej for some α � β and
j ≤ k. Thus Lemma 2.2.5, U ∈ KD(α+ ej). By [4, Thm 3.2.10], we have
(3.3)⋃
α�β, j≤k
KD(α+ ej) =⋃
(c,r) k−addable
KD(β +(k) (c, r)).
In particular, α + ej � β +(k) (c, s) for some k-addable position (c, s) for β. By
transitivity, we have θ(U) � β +(k) (c, s). Therefore, by Lemma 2.2.5, there is atleast one k-addable position (c, s) for β for which U ∈ KD(β +(k) (c, s)). �
Note there may be multiple k-addable positions for an addable column c.
Definition 3.3.5. For T ∈ KD(β) and r1, . . . , rm ≤ k, the iterated insertion of
r1, . . . , rm restricted by k into T labeled by β, denoted by Tβ,k←−− r1, . . . , rm, is
(3.4) (· · · ((Tβ,k←−− r1)
β(1),k←−−−− r2) · · · )
β(m−1),k←−−−−−− rm,
where the rectified label is β(i) = β(i−1)+(k) (ci, si) for ci the landing column of ri
and si the minimal row index for which θ(Tβ,k←−− r1, . . . , ri) � β(i−1) +(k) (ci, si).
Figure 9. Iterated labeled insertion restricted by 4 for r = 3, 3, 1, 4, 2.
Theorem 3.3.6. For T ∈ KD(β) and r1, . . . , rm ≤ k, the iterated insertion
U = (Tβ,k←−− r1, . . . , rm)
is a well-defined, rectified diagram and α ⊆k α(m) where α and α(m) are maximalsuch that θ(T ) �0 α � β and θ(U) �0 α(m) � β(m), respectively.
Proof. Let Ui denote successive insertion of r1, . . . , ri. By Lemma 3.3.4, Ui ∈KD(β(i)) for i ≥ 1, and so by Theorem 3.2.2, each Ui is a well-defined rectifieddiagram. By Proposition 3.3.2, there exists α′ �0
k α and s ≤ k such that U1 ∈
PKD(α′ + es). Choose s with α′s − αs is minimal, and set α(1) = α′ + es. By
definition, we have α ⊂·k α(1). By Lemma 2.3.4, θ(U1) �0 α(1), and by (3.3),
α(1) � β(1). Moreover, the choice of s and β(1) coincide, ensuring α(1) � β(1) isthe maximal composition with these two properties. Therefore, by Definition 3.2.3,
MULTIPLICATION BY SCHUR POLYNOMIALS 13
U2 = Rect(k)
α(1)(U1, (c, r2)). Thus we may iterate this argument, obtaining a sequence
α(i−1) ⊂·k α(i) for which θ(Ui) �0 α(i) and α(i) � β(i). �
Therefore we have a well-defined combinatorial model for taking the product ofa Demazure character and a Schur polynomial. It remains to prove this correspon-dence is injective and to gather the objects in the image into Demazure atoms.
4. Product rule for Demazure atoms
In this section, we reverse the iterated insertions into a Kohnert diagram byrecording the insertions with an atomic tableau, thus proving our bijection.
4.1. Recording tableaux. As in RSK, we use a recording procedure to track theorder in which cells were added. For this, we require skew diagrams.
Definition 4.1.1. Suppose β ⊆k γ results from the k-addition of cells in columnsc1, . . . , cm extending rows r1, . . . , rm, respectively. For α � γ, the skew diagram
of shape α/β is set of cells {x1, . . . , xm} ⊆ D(α) where xi is the cell with label riin column ci of Lγ(D(α)). We draw this with non-skewed cells labeled by Lβ .
⊆4 + ⊆4 + + ⊆4 + + + ⊆4 + + +
+
⊆4 + + ++
+
Figure 10. Skew diagrams for a saturated chain in ⊆4.
To insert a tableau into a diagram, we convert the tableau into a two-line array,where we insert the bottom line and use the top line to record landing cells.
Definition 4.1.2. For T ∈ KD(β) and
(
q1 · · · qmr1 · · · rm
)
a two-line array with
each ri ≤ k, the insertion diagram is U = (Tβ,k←−− r1, . . . , rm) and the recording
tableau has shape θ(U)/β where the cell added when ri is inserted has entry qi.
To characterize the recording tableaux that can arise from insertion, we have thefollowing analog of Schensted’s row bumping lemma [26]; see Fig. 11.
Lemma 4.1.3. For S ∈ PKD(α) and s, t ≥ 1, let T = Sα,k←−− s with landing cell x
and rectified labeling β, and let U = Tβ,k←−− t with landing cell y.
(i) If s < t, then the bumping path for T lies weakly right of the bumping pathfor U ; i.e. x lies weakly right of y and below y if in the same column.
(ii) If s ≥ t, then the bumping path for T lies strictly left of the bumping pathfor U ; i.e. x lies strictly left of y and weakly above if in adjacent columns.
Proof. Suppose s < t, and follow the bumping paths of x for S and y for T , whichmove left and down. If x lands in column max(α)+1, then since y is inserted above,y passes to column max(α) + 1, at which point it lands above x or strictly to theleft if it bumps. Thus we may assume x passed at some point. Following the pathfor y, suppose, for contradiction, there is some column strictly right of the landing
14 S. ASSAF
6 65 5
63 5 5 5 y
2 2 2 6 x
3 2 2
s<t−−→
6 65 5
63 3 5 5 52 2 2 2 2y x 6
6 65 5
63 5 5 5 x
2 2 2 6 y
3 2 2
s≥t−−→
6 65 5
63 3 3 3 32 2 2 6 y
x 2 2
Figure 11. Insertion of 2 then 3 (left) and 3 then 2 (right) into apinned, properly labeled diagram with restriction parameter 4.
column for x in which either y lands or bumps a cell z below the bumping pathfor x. Take c to be the rightmost such column. Then y must abut some r ≤ k incolumn c, possibly after an r-exchange sequence. Since x lands strictly to the leftof c, x also encountered this column, and either passed through or bumped out ofit. However, since the same r was available for x and the bumped cell z lies belowthe exit row for x, x could have used this r-exchange sequence to land or to bump zas well, contradicting that x lands strictly to the left. Therefore the bumping pathfor y stays above that for x until the landing column of x, after which y eventuallyland in a column weakly to the left of x and above x if in the same column.
Suppose s ≥ t, and again follow the paths. If x lands in column max(α) + 1,then y abuts x and so lands in column max(α) + 2, which is strictly to the right.Otherwise, suppose, for contradiction, there is some column in which x bumps z,say in row r, or lands and y passes into or through the column in row s with s > rif x bumps z. Take c to be the rightmost such column. Since to the right of c thepath for x lies strictly above the path for y, the path for x entered column c abovethe path for y. In particular, since z, if it exists, lies below the path for y, the cellin column c + 1, row s (or the prior position of y if it passed from column c + 1)abuts x at column c, and so y did not pass. Therefore the bumping path for y staysstrictly below and lands in a column strictly to the right of the path for x. �
Definition 4.1.4. For α � γ and β ⊆k γ, an atomic tableau of skew shape
α/β is a filling of the skew diagram α/β with positive integers such that
(1) entries weakly decrease left to right within rows;(2) entries within a column are distinct;(3) if t > r appear in the same column with t above r, then there is an s > r
in the same row as and immediately to the right of t.
Here we ignore cells above row k, labeled cells in column 1 have entry ∞, andlabeled cells weakly below k have the same entry as their row neighbor to the left.
The weight haswt(T )i equal to the number of skew cells with entry i. A tableauis lattice if the weight of the columns weakly to the right of column c is a partition.
❣6❣6
❣4❣3 ❣3
❣6❣1 ❣3 ❣6
×4 3 23 1 =
❣6❣6
❣4❣3 ❣3
❣6❣1 ❣3 ❣6
4←−
(
4 4 4 3 33 3 1 4 2
)
=
66
4 43 3 4 4
6 41 1 3 3
, 4 4 4
3
3
Figure 12. The insertion diagram and recording tableau for T ∈AKD(10413) ⊆ PKD(10314) ⊆ KD(01314) and R ∈ SSRT(0023).
MULTIPLICATION BY SCHUR POLYNOMIALS 15
For λ a partition, let Rλ ∈ SSRT(λ) be the tableau with every entry in row requal to r. Given R ∈ SSRT(λ), there is a unique two-line array with top row(kλk , . . . , 1λ1), where k = ℓ(λ), mapping to the pair (R,Rλ) under RSK (see [10,§4]). The product of a rectified diagram and a semistandard reverse tableau is theinsertion of this two line array into the diagram; see Fig. 12.
Theorem 4.1.5. The recording tableau of the k-restricted insertion of a tableauR ∈ SSRT(λ) into a labeled diagram T ∈ KD(β) is a lattice atomic tableau of skewshape α/β for some α � γ where β ⊆k γ and weight λ.
Proof. By Theorem 3.3.6, the map sending the pair (T,R) ∈ KD(β) × SSRT(λ)defined by iterated insertion of the word for R into T restricted by k = ℓ(λ) iswell-defined and results in a diagram in AKD(α) for some α � γ where β ⊆k γ.Thus the recording tableau Q is well-defined, and by definition, wt(Q) = λ.
Since the recorded entries weakly decrease and cells are added at the end oftheir rows, new entries within a row of Q weakly decrease. Since cells are droppedonly from above row k, the dropped cells do not change this property, provingcondition (1) of Definition 4.1.4. From properties of RSK (see [10]), if two successiveinserted letters, say r, s, have the same recording value, say q, then r ≥ s. Thusby Lemma 4.1.3((ii)), the landing cells for the insertions lie in different columns,proving condition (2). Furthermore, by Definition 3.3.3, if r < t occur in the samecolumn with r below t (which could be a cell with label at most k), then the rowof t cannot end in this column, else r was not a k-extendable row. Therefore thereis an entry to the right of t, say s, which was recorded prior to r (or is labeled).Therefore s > r, and so condition (3) holds as well.
For the lattice property, we need only show the column-restricted weights haveat least as many q’s as they have q − 1’s for all q. Suppose r1 ≥ · · · ≥ rm arerecorded by q and s1 ≥ · · · ≥ sn are recorded by q − 1. Since RSK insertion of thetwo-line array records all ri’s in row q and all si’s in row q − 1, we have m ≥ nand indices i1 < . . . < in such that rij < sj . By Lemma 4.1.3((ii)), the bumpingpaths for the ri’s are disjoint, and by Lemma 4.1.3((i)), the landing cell for sj willlie weakly left of the landing cell for rij . In particular, to the right of any columnc, there are never more entries q − 1 than entries q, so Q is lattice. �
4.2. Single cell excision. By Proposition 3.3.2, we can uniquely identify the land-
ing column of U = (Tβ,k←−− t) from U . To reverse the insertion, we must locate the
landing cell within this column. To begin, we observe passing is a last resort.
Lemma 4.2.1. Let U ∈ KD(β), z 6∈ U a vacant position (or z = ∅ regarded inrow 0), and x ∈ U a cell for which T = U ∪ {z} \ {x} ∈ KD(β) and, for α as in
Definition 3.2.3, suppose Rect(k)α (T, x) bumps z. Let y be any cell of U weakly above
z and strictly below x for which S = U ∪ {z} \ {y} is rectified. Then for u 6∈ U thenearest vacant position to the right of y within its row, there exists r ≤ k for whichu abuts r after exchanging labels for some r-exchange sequence.
Proof. Consider U with its rectified labeling, and let r ≤ k be the label assigned
to x. Since Rect(k)α (T, x) bumps z (or lands), there must be an r in the column
immediately to the right of z and, by Definition 2.3.1(3), it lies weakly below therow of z (or no r’s strictly to the right of x). Let v be the cell of S immediatelyto the right of the vacant position y, which lies weakly left of u; see Fig. 13. Since
16 S. ASSAF
xy v ··· u
z
Rect(3)α−−−−→
xy v ··· u
z
xy v ··· u
z
6Rect
(3)α−−−−→
xy v ··· u
z
T=U∪{z}\{x}, x U, z S=U∪{z}\{y}, y S=U∪{z}\{y}, u
Figure 13. An illustration for the No Passing Lemma 4.2.1.
U \ {y} is rectified and y lies above z, we may label S the same as U except forthe cells right of y that share its label, which might be re-labeled. In particular,the cells with labels r in U still have label r in S, and so v since v lies below x andweakly above z, it abuts r. Since u lies in the same row as v with no empty positionsbetween, this ensures u abuts r after an r-exchange sequence, as desired. �
Using Lemma 4.2.1, we now prove restricted insertion is reversible.
Theorem 4.2.2. If (Sβ,k←−− s) = (T
β,k←−− t) for some S, T ∈ KD(β) and some
1 ≤ s, t ≤ k, then S = T and s = t.
Proof. By Proposition 3.3.2, both insertions have the same landing column. Ifeither insertion results in an immediate landing (case (1)), then the landing celllies in column max(βi) + 1, and so the insertion rows must coincide. Thus s = t.Since the final diagrams are the same and no adjustments were made to S or to T ,this also forces S = T as well. Since the original insertion occurs in a column notoccupied by any cell of S, T , the first nontrivial step in rectification must be a pass.In particular, both insertions include a pass at some point.
Suppose rectifications for S and T have different landing cells within the samecolumn. Since both rectifications must include a pass at some point, we may con-sider the cells xS and xT which mark the first position within the landing column
for S and T , respectively. If xS = xT , then since (Sβ,k←−− s) = (T
β,k←−− t) and this
is the landing column, both would have the same landing cell. Thus none of thebumped cells within this column may coincide. Without loss of generality, say xS
lies below xT . We may continue with the rectification of T until the point at whicha cell lands above xS or bumps a cell below xS . By Lemma 4.2.1, this contradictsthe possibility that xS entered the column by a pass. Thus both insertions havethe same landing cell, and so S, T ∈ PKD(α) for the same rectification label α.
Reading the rectification process in reverse, consider the diagram U with vacantposition xU that marks the last position before the two processes diverge. Thatis, assume for diagrams S, T ∈ PKD(α), both properly labeled of shape α, andvacant positions xS 6∈ S and xT 6∈ T , either xS 6= xT or S 6= T (or both) and one
nontrivial step (a bump or a pass) of the rectification process for Rect(k)α (S, xS) and
for Rect(k)α (T, xT ) results in the diagram U with vacant position xU .
If both Rect(k)α (S, xS) and Rect
(k)α (T, xT ) result in a pass, then both xS , xT must
be the nearest vacant position not in U to the right of xU within its row, so xS = xT
and S = U ∪{xS} \ {xU} = U ∪{xT } \ {xU} = T , a contradiction to this being thepoint of divergence. Thus at least one of the steps must have been a bump.
By Lemma 4.2.1, we cannot have one of the two passing to xU while the otherbumps, so both must bump, and so both xS , xT lie above xU within its column. IfxT = xS , then S = T as well, so this isn’t the case now, or in any preceding stepthat also resulted in a bump. Since both rectifications must include a pass at some
MULTIPLICATION BY SCHUR POLYNOMIALS 17
point, we may consider the lowest cell, say coming from S, that entered the columnby a pass. Rename this cell xS , and name xT the nearest position above this whichcame from T . Then xT must bump a cell strictly below xS , so Lemma 4.2.1 onceagain applies, contradicting that xS entered the column from a pass. All casesresult in contradiction, so we must have xS = xT and S = T throughout. �
Corollary 4.2.3. If U = Tβ,k←−− r for some T ∈ KD(β) and some r ≤ k, then, for
α as in Definition 3.2.3, the landing cell x is the highest cell weakly below row k in
the landing column for which U \ {x} ∈ KD(α) and Rect(k)α (Lα(U \ {x}), x) lands.
4.3. Expansion into Demazure atoms. We now show the necessary conditionon the threads of diagrams in Theorem 3.3.6 is also sufficient.
Lemma 4.3.1. If β ⊆k γ and U ∈ KD(γ), then U = (Tβ,k←−− r1, . . . , rm) for some
T ∈ KD(β) and some r1, . . . , rm ≤ k.
Proof. We proceed by induction on m = |γ| − |β|, noting the case m = 0 is trivial.Suppose β ⊆k β′ ⊂·k γ. By Lemma 2.3.2, since U ∈ KD(γ), U can be properlylabeled by γ. Let r ≤ k such that γ − er �
0 β′, and let x be the rightmost cellof Lγ(U) with label r. Set U ′ = U \ {x}. We claim U ′ ∈ KD(β′). The labelingon U ′ inherited from U is semi-proper of shape γ − er since x was the rightmostcell with label r. Thus by Lemma 2.3.2, U ′ ∈ KD(γ − er), and by Lemma 2.3.4,KD(γ − er) ⊆ KD(β′), proving the claim.
By Lemma 2.3.2, there exists a proper labeling of U ′ of shape β′. Moreover, inthis labeling x abuts r through the same label exchanges by which γ − er �
0 β′,
and no cell weakly to the right of x has label r. Thus Rect(k)β′ (U ′, x) = U .
Therefore we may take y to be the highest cell in the column of x, weakly below
row k for which U \ {y} ∈ KD(β′) and Rect(k)β′ (U \ {y}, y) = U . Now pass y to
the right to nearest vacant position not in U \ {y}. Continuing, replace y with thehighest cell within its column which could have bumped y, or, if none is found,pass y to the right. The process terminates once the vacant position passes intoa column with no cells, and this is the row r′ ≤ k, now excised from a diagram
T ′ ∈ KD(β′) for which, by Corollary 4.2.3, U = T ′ β′,k←−− r′. By induction, there
exists T ∈ KD(β) and r1, . . . , rm−1 ≤ k such that T ′ = (Tβ,k←−− r1, . . . , rm−1). The
theorem follows by inserting r′ into T ′ to obtain U . �
All that remains is to account for the multiplicities using recording tableaux.
Theorem 4.3.2. Restricted insertion gives a weight-preserving bijection
(4.1) KD(β) × SSRT(λ)∼−→
⊔
β⊆kγα�γ
(AKD(α)× LAT(α/β, λ)) ,
where LAT(α/β, λ) is the set of lattice atomic tableaux of skew shape α/β andweight λ. In particular, we have a nonnegative expansion
(4.2) κβsλ =∑
α
aαβ,λAα,
where aαβ,λ is the cardinality of LAT(α/β, λ) and so is nonnegative.
18 S. ASSAF
Proof. By Theorem 3.3.6, the map sending the pair (T,R) ∈ KD(β) × SSRT(λ)defined by iterated insertion of the word for R into T restricted by k = ℓ(λ) iswell-defined and results in a diagram in KD(γ) for some β ⊆k γ. By Lemma 2.2.5,AKD(α) ⊆ KD(γ) for all α � γ. By Theorem 4.1.5, the recording tableau is alattice atomic tableau of skew shape α/β and weight λ. Moreover, the union forthe image since threads are well-defined; that is, AKD(α)∩AKD(α′) = ∅ if α 6= α′.
Given a pair (U,Q) ∈ AKD(α) × LAT(α/β, λ), by Lemma 4.3.1 and (2.4), Ulies in the image of the map, though we must reverse with respect to Q. LetQ′ = Q \ {x} for x the rightmost instance of the smallest entry of Q. Then Q′ isa lattice atomic tableau condition since x is the smallest entry, necessarily takenfrom the end of a row. Let c be the column of x and r the label in Lγ(Q). Then
there exists β ⊆k β′ ⊂·k γ such that β′ +(k) (c, r) = γ. By Lemma 4.3.1, there exists
T ′ ∈ KD(β′) and r′ ≤ k such that U = (T ′ β′,k−−→ r′). Moreover, by Theorem 4.2.2,
both T ′ and r′ are unique. By induction, the pair (T ′, Q′) uniquely corresponds to(T,R) ∈ KD(β)×SSRT(λ−eq), where q is the entry of x. The theorem follows. �
5. Product rule for Demazure character
In this section, we lift the nonnegative expansion into Demazure atoms to asigned expansion into Demazure characters which we show is nonnegative whenmultiplying a Schubert polynomial by a Schur polynomial.
5.1. Expansion into Demazure characters. Given the simple expansion of aDemazure character into Demazure atoms (2.5), we can lift the expansion in (4.2)to Demazure characters, though this introduces signs in many cases.
Definition 5.1.1. For β ⊆k γ, a key tableau of skew shape γ/β is a filling ofthe skew diagram with positive integers such that
(1) entries weakly decrease left to right within rows;(2) entries within a column are distinct;(3) if t ≥ r appear in adjacent columns with r <∞ strictly below and right of
t, then there is an entry s > r immediately right of t.
Here we ignore cells above row k, labeled cells in column 1 have entry ∞, andlabeled cells weakly below k have the same entry as their neighbor to the left.
Note for β ⊆k γ, an atomic tableau of shape γ/β is usually not a key tableau.However, there is a bijection between atomic tableau and key tableau.
Lemma 5.1.2. For α � γ where β ⊆k γ, the map Rγα : LAT(α/β) → LKT(γ/β)
that assigns recorded entries to the added cells is a bijection.
Proof. Consider T ∈ LAT(α/β) and let U = Rγα(T ). Since cells are appended to
the ends of rows, the rows of U weakly decrease left to right. Since Rγα preserves
column sets, columns of U have distinct entries and U is lattice. In order for (c, r)to be k-addable, for any row r < r′ ≤ k, either there is no cell nor entry yet recordedin row r′, column c − 1 or there is a cell or prior entry in row r′, column c whichnecessarily has larger entry. Thus U is a lattice key tableau.
Conversely, let U ∈ LKT(γ/β). Let (c, r) be the cell of the rightmost instanceof the smallest entry, which is well-defined by condition (2). Then r ≤ k and, bycondition (1), γr = c, and by condition (3), (c, r) is a k-addable cell for γ− er, andso there exists β ⊆k β′ ⊂·k γ such that β′ +(k) (c, r). Since (c, r) has the smallest
MULTIPLICATION BY SCHUR POLYNOMIALS 19
entry and lies at the end of its row, U \ {(c, r)} is a lattice key tableau. Given anyα � γ, by Theorem 4.3.2, there exists a unique lattice atomic tableau T of skewshape α/β′ with the unique skew cell in column c. Let α′ denote α with this cellremoved. Then α′ � β′, and by induction we are done. �
We have the following description of intersections of sets of Kohnert diagrams.
Lemma 5.1.3. Given β ⊆k γ(1), . . . , γ(m) each with the same multiset of addedcolumns taken in the same order, there exists γ � γ(1), . . . , γ(m) such that
(5.1) KD(γ(1)) ∩ · · · ∩KD(γ(m)) = KD(γ).
Proof. By [4, Lem 3.3.4], for (c, r1), . . . , (c, rm) k-addable positions for β,
where indices are taken so that r1 < · · · < rm. Let β′ = tr1,r2 · · · trm−1,rm(β +k
(c, rm)). If (c′, r′) is k-addable for all β+k (c, ri), then it is k-addable for β′ as well.The result follows by induction on the length of the chain from β to any γ(i). �
We now prove Conjecture 2.2.6 for a Demazure character and Schur polynomial.
Theorem 5.1.4. Given a sequence c = (c1, . . . , cn) of successively k-addable columnsfor β, let Ic be the union of all compositions α � γ for some γ obtainable by succes-sive k-addition of cells in these columns. Then Ic is a lower order ideal in Bruhatorder, and we have a nonnegative expansion into Schubert characters as
(5.3) κβsλ =∑
c
aIc
β,λκIc,
where aIc
β,λ is the cardinality of LAT(α/β, λ) for any α ∈ Ic.
Proof. It follows from the definition that α ∈ Ic whenever α � γ ∈ Ic, so Ic is alower order ideal. By Lemma 5.1.3, for any γ, γ′ ∈ Ic, there exists α � γ, γ′. Thusby Lemma 5.1.2, we have #LAT(γ/β, λ) = #LAT(α/β, λ) = #LAT(γ′/β, λ) forall λ. In particular, the cardinality of LAT(α/β, λ) is the same for all α ∈ Ic. Theresult now follows from Theorem 4.3.2. �
Finally, we derive the explicit Demazure character expansion from (5.3).
Theorem 5.1.5. We have the cancellation-free, signed expansion
(5.4) κβsλ =∑
β⊆kγ(1),...,γ(m) distinct
(−1)m−1cγ(1)
β,λ κγ ,
where γ is maximal such that γ � γ(i) and cγβ,λ is the cardinality of LKT(γ/β, λ),
the set of lattice key tableaux of skew shape γ/β and weight λ.
Proof. By (2.4) and Lemma 5.1.2, for β ⊆k γ we have a bijection⊔
α�γ
AKD(α)× LAT(α/β, λ)id×Rγ
α−−−−→ KD(γ)× LKT(γ/β, λ).
Expanding as in (4.1) gives a union that is no longer disjoint. Thus (5.4) followsfrom Lemma 5.1.3 by inclusion–exclusion. �
20 S. ASSAF
5.2. Positive expansions. We can identify many instances in which the union in(4.1) remains disjoint under the bijection in Lemma 5.1.2, and so the Demazurecharacter expansion in (5.4) is nonnegative.
Definition 5.2.1. A composition β is k-positive if whenever βr ≤ βt for somer ≤ k < t, then βs ≥ βr for all r < s ≤ k.
Theorem 5.2.2. For any k-positive β (in particular, for k ≥ ℓ(β)), we have
(5.5) κβsλ =∑
γ
cγβ,λκγ ,
where cγβ,λ is the cardinality of LKT(γ/β, λ) and so is nonnegative.
Proof. A composition β has at most one k-addable position for each column ifand only if whenever βr ≤ βt for some r ≤ k < t, then either βs = βr for somer < s ≤ k or βs > βr for all r < s ≤ k. Thus if β is k-positive, then it has atmost one k-addable position. Moreover, in this case, β +(k) (c, r) is also k-positivefor any k-addable position (c, r). To see this, let γ = β +(k) (c, r) and suppose,for contradiction, there exist q < s ≤ k < t with γs < γq ≤ γt. Since k-additionlengthens row r and shortens rows above k, leaving all others the same, in orderfor β to be k-positive, we must have q = r and βr < βs < c, where t′ is themaximum row from which cells were dropped. However, this contradicts that (c, r)is k-addable. Thus we may k-add cells to β uniquely based on the added column,making intersections in (5.4) empty and the expansion nonnegative. �
The code of a permutation w is L(w)i equals the number of indices j > i forwhich wi > wj . When v is a grassmannian permutation, meaning it has atmost one descent, say at position k, the code L(v) is a partition of length k, andLascoux and Schutzenberger showed Sv = sL(v). Generalizing this, Lascoux andSchutzenberger define a vexillary permutation u as one where there do not existindices 1 ≤ a < b < c < d for which ub < ua < ud < uc, and they showedSu = κL(u). Macdonald characterized codes for vexillary permutations [18, (1.32)].
Theorem 5.2.3 ([18]). A permutation u is vexillary if and only if L(u) satisfies
(1) if L(u)r > L(u)t, then #{r < s < t | L(u)s < L(u)t} ≤ L(u)r − L(u)t;(2) if L(u)r ≤ L(u)t, then L(u)s ≥ L(u)r whenever r < s < t.
In particular, (2) implies L(u) is k-positive for all k whenever u is vexillary.
Corollary 5.2.4. For u vexillary and v grassmannian, we have
(5.6) SuSv = κβsλ =∑
γ
cγβ,λκγ ,
where L(u) = β, L(v) = λ, and cγβ,λ, the cardinality of LKT(γ/β, λ), is nonnegative.
While the right hand side of (5.6) is not the Schubert expansion, and so we are notcomputing Schubert structure constants, this formula is explicit, combinatorial andmanifestly nonnegative. Even without a rule for lifting this to a Schubert expansion,this formula may well yield interesting consequences, such as non-positivity, forcertain classes of Schubert structure constants.
MULTIPLICATION BY SCHUR POLYNOMIALS 21
5.3. Schubert times Schur polynomials. To generalize (5.6), we consider theRothe diagram of a permutation w given by
(5.7) D(w) = {(i, wj) | i < j and wi > wj}.
Notewt(D(w)) = L(w); see Fig. 14. We compress notation by KD(w) = KD(D(w)).
κ(013012) κ(023002) κ(033001) κ(014011) κ(024001)
Figure 14. Rothe diagram (left) and Yamanouchi Kohnert dia-grams for 13625847 giving the Demazure character expansion.
Kohnert conjectured the following analog of Definition 2.1.2 for Schubert polyno-mials for which Assaf [2] gave a short bijective proof, and Armon, Assaf, Bowlingand Ehrhard [1] gave representation theoretic proof. Earlier, Winkel [30] gave aproof not widely accepted given the opaque arguments.
Theorem 5.3.1 ([2, 1]). For w a permutation, we have
(5.8) Sw =∑
T∈KD(w)
xwt(T )n1 · · ·xwt(T )n
n .
where wt(T )r is the number of cells in row r of T .
Lascoux and Schutzenberger [16] gave a formula, proved by Reiner and Shimo-zono [23, Thm 4], for the nonnegative expansion of a Schubert polynomial intoDemazure characters using a generalization of RSK developed by Edelman andGreene [9]. Assaf [6, Thm 6.1.3] gave an alternative formulation using Kohnertdiagrams. For this, we omit the rather technical definition of Yamanouchi Kohnertdiagrams [6, Def 6.1.2] but give an example in Fig. 14.
Theorem 5.3.2 ([6]). For w a permutation, there exists a subset YKD(w) ⊆KD(w) of Yamanouchi Kohnert diagrams such that
(5.9) Sw =∑
T∈YKD(w)
κwt(T ).
Moreover, for T ∈ YKD(w), we have wt(T )r = 0 whenever L(w)r = 0.
Lemma 5.3.3. For T ∈ YKD(w), if wt(T ) has two k-addable cells (c, r), (c, s)with r < s, then there exists U ∈ YKD(w) ∩KD(T ) such that
Proof. Let β = wt(T ). By definition of k-addable, c > βr > βs and there exists alowest row t ≥ k for which βt = c−1. We claim there exists U ∈ YKD(T ) obtainedby moving rightmost c − βr cells in row t of T down to row r. In D(w), if somerow has no cell in a column strictly left of its rightmost cell, then there is no cellabove this in all of D(w) either, and this holds for T ∈ YKD(w) as well. Thus the
22 S. ASSAF
rightmost βt− βs cells of row t must lie strictly to the right of the rightmost cell inrow s. Furthermore, there are βr − βs > 0 occupied columns of row r with a cellbut the corresponding position in row s has no cell. Therefore these same columnsare empty in row t as well. In particular, there are at least βt − βs > βt − βr cellsin row t strictly to the right of all cells in row r, thereby proving we can applythe desired Kohnert moves. By Theorem 5.3.2, we may take U to be Yamanouchi.Moreover, wt(U)r = c, wt(U)s = βs and wt(U)t = βt − (c − βr) = βr − 1. Inparticular, (βr, s) is k-addable for wt(U). �
4
∩4
=4
Figure 15. An illustration of Lemma 5.3.3 for u = 13625847.
Theorem 5.3.4. For u any permutation and v any grassmannian permutation,
(5.10) SuSv =∑
T∈YKD(w)
κwt(T )sλ =∑
γ
cγu,λκγ ,
where the coefficients cγu,λ are each nonnegative.
Proof. By Lemma 5.1.3, in order for the Demazure character expansion of κwt(T )sλto have a term κγ occurring with negative sign, there must be multiple k-addable
cells for the same column of T . Suppose β ⊆k γ(1), . . . , γ(m) have the same multisetof added columns taken in the same order for β = wt(T ) for some T ∈ YKD(w).By Lemma 5.3.3, there is another diagram U ∈ YKD(w) with U ∈ KD(T ) andwt(U) ⊆k γ such that KD(γ(1))∩· · ·∩KD(γ(m)) = KD(γ). Thus by Theorem 5.1.5,each negative term cancels with a later (as determined by Kohnert moves) term,ultimately resulting in a positive expansion. �
As noted in the introduction, we can deduce the nonnegativity of the coefficientscγu,λ in (5.10) from Theorem 5.3.2 and the geometric fact that cwu,λ is nonnegative.However, our proof of Theorem 5.3.4 is purely combinatorial, giving strong evidencethat the insertion algorithm introduced in this paper might well be used to derive acombinatorial proof of the nonnegativity of the Schubert structure constants cwu,λ.
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