aa r X i v : . [ m a t h . C O ] F e b THE “GROTHENDIECK TO LASCOUX” CONJECTURE
VICTOR REINER AND ALEXANDER YONGA
BSTRACT . This report formulates a conjectural combinatorial rule that positively expandsGrothendieck polynomials into Lascoux polynomials. It generalizes one such formula ex-panding Schubert polynomials into key polynomials, and refines another one expandingstable Grothendieck polynomials.
1. T
HE OPEN PROBLEM
We set up the notation needed to state the problem, Conjecture 1.13 below.1.1.
Grothendieck and Lascoux polynomials.
Define operators ∂ i , π i on polynomials f ∈ Z [ β ][ x , . . . , x n ] ∂ i ( f ) = 1 − s i fx i − x i +1 and π i ( f ) = ∂ i ((1 + βx i +1 ) f ) where s i = ( i ↔ i + 1) is a simple transposition in the symmetric group S n . The transpo-sition s i acts on f ∈ Z [ β ][ x , . . . , x n ] by permuting x i and x i +1 . Definition . The β - Grothendieck polynomial G ( β ) w is recursively defined by the initial condition G ( β ) w = x n − x n − · · · x n − where w is the longest permutation that swaps i ↔ n + 1 − i , and then setting G ( β ) w = π i ( G ( β ) ws i ) if w ( i ) < w ( i + 1) .The β parameter was introduced by S. Fomin–A. N. Kirillov [FK94]; [LS82] uses β = − .Define a further family of operators e π i on polynomials via e π i ( f ) = ∂ i ( x i (1 + βx i +1 ) f ) . Also, let
Comp be the set of (weak) compositions , that is, α = ( α , α , . . . ) ∈ N ∞ havingfinitely many nonzero entries, where N = { , , , . . . } . Definition . The β - Lascoux polynomials Ω ( β ) α are again defined recur-sively. For α ∈ Comp , define Ω ( β ) α = ( x α x α x α · · · if α ≥ α ≥ α ≥ . . ., e π i (Ω ( β ) αs i ) if α i < α i +1 .The nomenclature “Lascoux polynomial” first appears in C. Monical’s [M17]. We alsorefer to A. Lascoux’s [L13], A. N. Kirillov’s [K16], C. Monical-O. Pechenik-D. Searles’[MPS21], O. Pechenik-D. Searles’ [PS19], and the references therein for more about bothfamilies of polynomials. Date : February 18, 2021. .2. Increasing tableaux and K -jeu de taquin. We need some notions from [TY09].
Definition . An increasing tableaux of shape ν/λ is a filling of ν/λ using { , , . . . , | ν/λ |} such that the labels of T strictly increase along rows and columns.Let INC ( ν/λ ) denote the set of all increasing tableaux of shape ν/λ . Definition . A short ribbon R is a skew shape without a × subshape, where each rowand column has at most two boxes, and each box is filled with one of two symbols, butadjacent boxes are filled differently. Two boxes lie in the same component of R if there is apath between them passing through boxes that are adjacent vertically or horizontally. Definition . Define switch ( R ) to be the same short ribbon as R but where, in each non-singleton component, each box is filled with the other symbol.For example: R = ◦◦ •◦ • switch ( R ) = ◦• ◦• ◦ .In what follows, we assume ν/λ is contained in an ambient rectangle Λ . Definition . An outer corner of a skew shape ν/λ is a maximally northwest box of Λ /ν .Given T ∈ INC ( ν/λ ) , consider a set of outer corners { x i } filled with • . Let m be themaximum value label appearing in T . Define revKjdt { x i } ( T ) as follows: let R m be theshort ribbon consisting of • and m . Apply switch ( R m ) . Now let R m − be the short ribbonconsisting of • and m − and apply switch ( R m − ) . Repeat until one applies switch ( R ) ,and then erase the • entries. For example, if ν/λ = (3 , , / (2 , is contained in Λ =(3 , , , and we might have T = 22 • •• 7→ ••
21 2 • 7→ •• •
21 = 221 = revKjdt { x i } ( T ) . Definition . A reverse K -rectification of T is any sequence of revKjdt -slides giving areverse straight tableau revKrect ( T ) .Continuing the previous example, one can perform the following revKjdt -slides: • ••
21 21 • • • • to conclude revKrect ( T ) = 1 2 . Unlike textbook jeu de taquin, revKrect might dependon the choice of revKjdt -slides used (see [TY09, Example 1.6], [BS16, Example 3.4]).Given P ∈ INC ( ν ) we define the left key K − ( P ) to be a tableau, using revKjdt , as follows.By definition, the first columns of P and K − ( P ) agree. Assume that the first ℓ columns of K − ( P ) have been determined. Apply reverse rectification of the increasing tableau P ( ℓ +1) comprised of the first ℓ +1 columns of P , inside the smallest rectangle Λ ( ℓ +1) that P ( ℓ +1) fitsinside. For specificity, we define revKrect by using the leftmost outer corner for each interme-diate revKjdt -slide.
Let C ( ℓ +1) be the leftmost column in the reverse rectification of P ( ℓ +1) .Then C ( ℓ +1) (after upward-justification) is the ( ℓ + 1) -st column of K − ( P ) . Repeating this,the end result is K − ( P ) . xample . The reader can check that if P = 1 2 3 5 72 4 5 64 6 then K − ( P ) = 1 1 1 1 22 2 2 24 4 . The first two columns are easily seen from the definition, as reverse rectification doesnothing. To compute the third column one works out this revKrect : • → • → • • → • • → • . The first column C (3) = 12 of the rightmost tableau is the third column of K − ( P ) . (cid:3) Notice in our example, K − ( P ) has the same (straight) shape as P . Also, K − ( P ) is a key :the set of labels in column i are contained in the set of labels in column i − for i ≥ .These properties always hold, and are proven in Section 3.3. Let content ( K − ( P )) be theusual content of a semistandard tableau; here content ( K − ( P )) = (4 , , , .1.3. Reduced and Hecke words.
Let ℓ ( w ) be the Coxeter length of w ∈ S n , that is, ℓ ( w ) = { ≤ i < j ≤ n : w ( i ) > w ( j ) } . Definition . A sequence ( i , i , . . . , i ℓ ( w ) ) is a reduced word for w if s i s i · · · s i ℓ ( w ) = w .Let Red ( w ) denote the set of reduced words for w . Definition . The
Hecke monoid H n is generated by u , u , . . . , u n − , subject to: u i ≡ u i u i u j ≡ u j u i if | i − j | > u i u i +1 u i ≡ u i +1 u i u i +1 Definition . A sequence ( i , i , . . . , i N ) ∈ N N is a Hecke word for w ∈ S n if u i u i · · · u i N ≡ u a u a · · · u a ℓ ( w ) , for some ( a , . . . , a ℓ ( w ) ) ∈ Red ( w ) . Definition . For any tableau P , we will read off a word denoted word ( P ) , concatenat-ing its rightmost column read top-to-bottom, then its next-to-rightmost column, etc. Forexample, the tableau P in Example 1.8 has word ( P ) = (7 , , , , , , , , , , .1.4. The “Grothendieck to Lascoux” conjecture.
This is the open problem of this report:
Conjecture 1.13. G ( β ) w = X P β boxes ( P ) − ℓ ( w ) Ω ( β ) content ( K − ( P )) where P is any straight-shape increasing tableau such that word ( P ) is a Hecke word for w .Example . If w = 31524 the increasing tableaux and the left keys are respectively(1) P = 1 2 43 , , K − ( P ) = 1 1 13 , , . For instance, if P is the rightmost increasing tableaux, then word ( P ) = (4 , , , , . Now u u u u u ≡ u u u u ≡ u u u u , and s s s s = 31524 . Conjecture 1.13 predicts G ( β )31524 = Ω ( β )301 + Ω ( β )202 + β Ω ( β )302 . (cid:3) onjecture 1.13 generalizes one formula [RS95] and refines another [BKSTY08]. It hasbeen exhaustively checked (with computer assistance) for n ≤ (and spot-checked for n = 8 , ). It says the G ( β ) w to Ω ( β ) α expansion is positive; this is also open.The rest of this report surveys known and related results that motivate Conjecture 1.13.The proof that the shapes of P and K − ( P ) agree, and that K − ( P ) is a key, is in Section 3.3.2. H ISTORY OF THE PROBLEM
During the preparation of [BKSTY08], M. Shimozono privately conjectured to the sec-ond author that G ( β ) w expands positively in the Ω ( β ) α ’s; he also suggested ideas towards arule. Conjecture 1.13 was formulated in September 2011 during a visit of the first authorto UIUC. There are two limiting cases of Conjecture 1.13, as explained now.2.1. The β = 0 specialization and stable-limit symmetry. Definition . The key polynomial (or type A Demazure character ) is κ α := Ω (0) α .References about key polynomials include [LS90, RS95, L13]. A tableau formula for κ α is in [LS90] (see also [RS95]). From the definitions,(2) Ω ( β ) α = κ α + X k> β k p k , where p k is a homogeneous polynomial in x , x , . . . of degree | α | + k , where | α | = P i ≥ α i . Definition . The
Schubert polynomial is S w := G (0) w .A combinatorial rule for G ( β ) w as a sum of S v ’s is given by C. Lenart’s [Le99].All the aforementioned polynomial families (Lascoux, Key, Grothendieck, Schubert) are Z [ β ] -linear bases of Z [ β ][ x , x , . . . ] . There are symmetric versions of these polynomials. Definition . The β - stable Grothendieck polynomial is G ( β ) w ( x , x , . . . ) = lim n →∞ G ( β )1 n × w ( X ) , where (1 n × w )( i ) = i if ≤ i ≤ n and (1 n × w )( i ) = w ( i ) + n if i > n . Definition . The stable Schubert polynomial is F w := G (0) w . This is also known as the Stanleysymmetric polynomial . Definition . A permutation w is Grassmannian at position k if w ( i ) < w ( i + 1) for i = k .To such w , define a partition λ = λ ( w ) by λ i = w ( k − i + 1) − ( k − i + 1) for ≤ i ≤ k . Definition . A set-valued semistandard Young tableaux T of shape λ is a filling of the boxesof λ with nonempty sets such that if one chooses a singleton from each set, the result is asemistandard Young tableaux (row weakly increasing and column strict). It represents the class of a Schubert variety X w in the flag variety GL n /B under Borel’s isomorphism(see [Fu97]). The Grothendieck polynomial G w := G ( − w similarly represents the Schubert structure sheaf O X w in the Grothendieck ring K ( GL n /B ) of algebraic vector bundles on GL n /B . This explains the “combina-torial K -theory” nomenclature [B05]. heorem 2.7 ([B02]) . Let w be a Grassmannian permutation of shape λ . Then G ( β ) λ := G ( β ) w = X T β labels ( T ) −| λ | x T , where the sum is over set-valued semistandard Young tableaux T of shape λ , and x T := Q i ≥ x i ∈ Ti .Definition . The
Schur function is s λ := G (0) λ .Summarizing, one has a commutative diagram(3) Ω ( β ) α −→ κ α ↓ ↓ G ( β ) λ ( α ) −→ s λ ( α ) with horizontal arrows indicating β = 0 specialization, and vertical arrows called stabi-lization : for the right vertical arrow, let λ ( α ) be the sorting of α , then if N > n , s λ ( α ) ( x , . . . , x n ) = κ (0 N ,α ) ( x , . . . , x n , , , . . . ) . Monomial expansion formulas.Theorem 2.9 ([FK94]) . G ( β ) w ( X ) = X ( a , i ) β N − ℓ ( w ) x i where the sum is over all pairs of sequences ( a , i ) (called compatible sequences) such that (a) a = ( a , a , · · · , a N ) is a Hecke word for w ; (b) i = ( i , i , · · · , i N ) has ≤ i ≤ i ≤ . . . ≤ i N (c) i j ≤ a j ; and (d) a j ≤ a j +1 = ⇒ i j < i j +1 . Theorem 2.10 ([FK94]) . G ( β ) w = X ( a , i ) β N − ℓ ( w ) x i where ( a , i ) satisfies (a), (b) and (d) above. Therefore, one has a commutative diagram, with the same arrows(4) G ( β ) w Thm . P ( a , i ) β N − ℓ ( w ) x i −→ S w = P ( a , i ) x i ↓ ↓ G ( β ) w Thm . P ( a , i ) β N − ℓ ( w ) x i −→ F w = P ( a , i ) x i . In view of Definitions 2.2 and 2.4, the expressions in the right column sum over ( a , i ) with a ∈ Red ( w ) , versus sums over Hecke words a for w in their left column counterparts.The diagram (4) specializes when w = w ( λ ) is Grassmannian, giving(5) G ( β ) λ ( x , . . . , x n ) −→ s λ ( x , . . . , x n ) ↓ ↓ G ( β ) λ −→ s λ . .3. Prior expansion formulas.
Conjecture 1.13 generalizes a relationship between theSchubert and key polynomials.
Theorem 2.11 ([LS89, RS95]) . S w = X P κ content ( K − ( P )) where the sum is over all increasing tableaux P such that word ( P ) ∈ Red ( w ) . To be precise, in the formulation given in [LS89, Theorem 4], the description of the “leftnil key” K − ( P ) differs from our definition. We are asserting (proof omitted) that in thecase of the P in Theorem 2.11, the two definitions agree. This is because revKjdt can beused to compute the insertion tableau of Hecke insertion [BKSTY08] which specializes to
Edelman-Greene insertion [EG87]; see [TY11].Theorem 2.11 is the non-symmetric version of the following result:
Theorem 2.12 ([FG94]) . Let a w,λ = { P ∈ INC ( λ ) : word ( P ) ∈ Red ( w ) } . Then F w = X λ a w,λ s λ . The next result generalizes Theorem 2.12. Conjecture 1.13 is its non-symmetric version:
Theorem 2.13 ([BKSTY08]) . Let b w,λ = { P ∈ INC ( λ ) : word ( P ) is a Hecke word for w } . Then G ( β ) w = X λ β | λ |− ℓ ( w ) b w,λ G ( β ) λ . Example . G ( β )31524 = G ( β )31 + G ( β )22 + βG ( β )32 . This is witnessed by the P tableaux of (1).Notice that Conjecture 1.13 subdivides the witnessing tableaux P for b w,λ according to content ( K − ( P )) . It is in this sense that Conjecture 1.13 is a refinement of Theorem 2.13. (cid:3) In conclusion, Conjecture 1.13 captures some known facts, as expressed in this diagram(6) G ( β ) w Conj . P P β | shape ( P ) |− ℓ ( w ) Ω ( β ) content ( K − ( P )) −→ S w Thm . P P κ content ( K − ( P )) ↓ ↓ G ( β ) w Thm . P P β | shape ( P ) |− ℓ ( w ) G ( β ) shape ( P ) −→ F w Thm . P P s shape ( P ) . Here shape ( P ) is the partition of P . The horizontal, vertical maps are as before.3. F URTHER DISCUSSION
Formulas for Lascoux polynomials.
Combinatorial rules for the Lascoux polynomi-als are in V. Buciumas–T. Scrimshaw–K. Weber [BSW20]. Another rule, generalizing theKohnert moves of [K90], was conjectured in [RY15]. Also, see the skyline conjecturalrule of C. Monical [M17]. Finally, [RS95, Theorem 5] gives an alternative formula for κ α interms of compatible sequences; we do not know a generalization of this formula to Ω ( β ) α . The conjecture is accidentally misstated there. See the corrected version https://faculty.math.illinois.edu/~ayong/polynomials.Seminaire.revision.2017.pdf which isconsistent with the 2011 report by C. Ross https://faculty.math.illinois.edu/~ayong/student_projects/Ross.pdf . .2. Warning about stable-limits.
The results of Section 2 suggest combinatorial proper-ties for stable-limit polynomials will hold for their non-symmetric versions. This is notalways true. S. Fomin-C. Greene proved the following result ( cf.
C. Lenart’s [Le00]):
Theorem 3.1 ([FG94]) . (7) G ( β ) w = X λ β | λ |− ℓ ( w ) d w,λ s λ where d w,λ counts tableaux P of shape λ that are row strict and column weakly increasing, suchthat word ( P ) is a Hecke word for w . Thus, using the Grassmannian permutation w = s s s s = 23514 , G ( β )2 , , = s , , + β (3 s , , , + s , , ) + · · · . If one expands Ω ( β ) α in the keys (the non-symmetric analogue (7); see (3)), positivity fails: Ω ( β )1 , , , = κ , , , + β (2 κ , , , + κ , , , + κ , , − κ , , ) + · · · . Proof that K − ( P ) is a key of the same shape as P . We first show shape ( K − ( P )) = shape ( P ) . In the notation of K − ( P ) ’s description, it suffices to argue that the length b of C ( ℓ +1) equals the length t of the ℓ + 1 ( i.e. , rightmost) column of P ( ℓ +1) .To see this, consider the general situation of an increasing tableau T contained in an r × s dimension rectangle Λ , and a second increasing tableau in Λ that is complementaryto T . Recall the notion of Kinfusion defined in [TY09, Section 3]. In fact,(8)
Kinfusion ( T, U ) = (
A, B ) where A = Krect ( U ) and B = revKrect ( A ) .The rectification A uses the inner corners defined by T . Similarly the reverse rectification B uses the outer corners defined by U .Given V ∈ INC ( ν/λ ) define LDS ( V ) to be the length of the longest decreasing subse-quence of the left to right, bottom to top, row reading word of V . This is true: Theorem 3.2 ([TY09, Theorem 6.1], cf. [BS16, Corollary 6.8]) . LDS is invariant under K -theoretic (reverse) jeu de taquin slides. Now, suppose t and u are the lengths of the s -th (possibly empty) column of T and U ,respectively. Similarly, let a and b be the length of the first (leftmost) columns of A and B ,respectively. Thus t + u = a + b = r . By Theorem 3.2, u = LDS ( U ) = LDS ( A ) = a . Thus b = t . The result follows from (8) and setting T = P ( ℓ +1) and U being any complementaryincreasing tableau to T inside the smallest rectangle Λ ( ℓ +1) that P ( ℓ +1) sits inside.To see that K − ( P ) is a key we use an argument of G. Orelowitz: Since we choose theleftmost outer corner at each slide of revKrect , when computing C ( ℓ +1) we begin by com-puting revKrect ( P ( ℓ ) ) as a partial revKrect of P ( ℓ +1) . At this point, C ( ℓ ) is the leftmostcolumn of this partial revKrect . Thus, when completing the revKrect ( P ( ℓ +1) ) , by thedefinition of revKjdt , the entries of C ( ℓ +1) are contained in those of C ( ℓ ) , as desired. (cid:3) The above shape argument does not depend on the specific choice of revKrect used ateach stage of the definition of K − ( P ) . We suspect this choice does not affect K − ( P ) beinga key, however, the choice we use (suggested by G. Orelowitz) makes the proof easy. CKNOWLEDGEMENTS
AY thanks Mark Shimozono for initiating his interest in the expansion problem of thisreport. We are grateful to Shiliang Gao, Cara Monical, and Gidon Orelowitz who eachgave very helpful analysis. We thank Oliver Pechenik, Colleen Robichaux, Colleen Ross,Travis Scrimshaw, and Dominic Searles for discussions. AY is supported by a SimonsCollaboration Grant, NSF RTG grant DMS-1937241, and the UIUC Center for AdvancedStudy. VR is supported by NSF grant DMS-1601961. John Stembridge’s Maple package
Coxeter was used in our experiments. We used computing at Brown’s Center for Compu-tation and Visualization during AY’s virtual residence at ICERM in Spring 2021.R
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