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A proposed bijection for the K-Kohnert rule for Grothendieck polynomials

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A proposed bijection for the K-Kohnert rule for Grothendieck polynomials

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Content A PROPOSED BIJECTION FOR THE K-KOHNERT RULE FOR GROTHEDIECK POLYNOMIALS AN ANALYSIS OF COMBINATORIAL MODELS FOR SCHUBERT AND GROTHENDIECK POLYNOMIALS by Reena Somani A Thesis Presented to the FACULTY OF THE USC DANA AND DAVID DORNSIFE COLLEGE OF LETTERS, ARTS AND SCIENCES In Partial Fulfillment of the Requirements for the Degree MASTER OF ARTS (MATHEMATICS) August 2023 Copyright 2023 Reena Somani Table of Contents List of Figures ......................................................................iii Abstract ............................................................................iv Introduction .........................................................................1 Chapter 1: Pipe Dreams .............................................................3 Chapter 2: Kohnert Diagrams .......................................................5 Chapter 3: Bijection .................................................................7 Chapter 4: K-Pipe Dreams ..........................................................8 Chapter 5: K-Kohnert Diagrams .....................................................9 Chapter 6: Proposed K-Bijection ...................................................10 References ..........................................................................11 ii List of Figures A reduced pipe dream for w = 152869347 ............................................3 The set of reduced pipe dreams for w = 1432 starting with D bot (w) .................. 4 A diagram (left) and a Kohnert diagram for it (right) ................................5 The set of Kohnert diagrams for w = 1432 starting with the Rothe diagram ..........6 A non-reduced pipe dream for w = 152869347 ........................................8 The set of non-reduced pipe dreams for w = 1432 other than those in Figure 2 .......8 The Rothe diagram (left) for w = 25134 and a K-Kohnert diagram for it (right) where the gray circles are ghost bubbles and the labelled circles form a Kohnert family .....9 The set of K-Kohnert diagrams for w = 1432 other than those in Figure 4 ............9 iii Abstract In1990,KohnertproposedacombinatorialmodelforSchubertpolynomials,whichAs- safrecentlyprovedisinbijectionwithBergeronandBilley’sreducedpipedreams. Build- ing on this, Ross and Yong offered a conjecture for a K-Kohnert’s rule for Grothendieck polynomials, which are the K-theoretic analog of Schubert polynomials and can be rep- resented by non-reduced pipe dreams. In this paper, we review each of these models and offer an adaptation of Assaf’s bijection for non-reduced pipe dreams and K-Kohnert diagrams. iv Introduction In his celebrated treatise, Schubert began investigating the possible number of inter- sections between subspaces of certain dimensions. Years later, when Hilbert compiled his list of twenty three unsolved and significant problems, his fiftienth problem was to put Schubert’smethodsonarigorousmathematicalfoundation,whichledtothedevelopment of intersection theory. The modern iteration of this problem studies the intersections of Schubert varieties, whose points consist of flags in the complete flag manifold Fl(n). For a fixed flag F ∗ , we take a Schubert variety X u (F ∗ ) where u is a permutation and definetheSchubertclasses[ X u (F ∗ )]astheequivalenceclassesinH ∗ (Fl(n)),thecohomol- ogy ring of the complete flag manifold. H ∗ (Fl(n)) is isomorphic to Z[x 1 ,...,xn] /<e 1 ,...,en > where e 1 ,...,e n are the elementary sign polynomials. The cohomology class is independent of the choice of flag, so we define [ X u (F ∗ )] = σ u ∈ Z[x 1 ,...,xn] /<e 1 ,...,en >, which form a basis for the ring. To count the number of points in the intersection of Schubert varieties, we need a formula for the Schubert structure constants c w uv , defined by defined by σ u σ v = P w =c w uv σ w . Lascoux and Schutenberger [LS82a] defined Schubert polynomials S w ∈Z[x 1 ,...,x n ] such thatS u S v = P w c w uv S w . These polynomials allow us to understand the Schubert structure constants without working in the cohomology ring. Since the Schubert structure constants c w uv count points, we know they are positive integer coefficients. This inspired the search for a combinatorial model for Schubert polynomials. Pipedreams,introducedbyBergeronandBilley[BB93],gaveagraphicalmodelofthe reducedcompatiblesequencesthatBilley,Jockush,andStanleyprovedgenerateSchubert polynomials [BJS93]. With this model, Bergeron and Billey provided an algorithm to determine each of the terms in a Schubert polynomial and provided a new proof of Monk’s rule, which gives an explicit formula for multiplying a Schubert polynomial and a special Schubert polynomial. 1 Kohnert’s rule [Koh91] was the first conjectured combinatorial model for Schubert polynomials, but it took many years and several dubious attempts to prove. Assaf [Ass22a] recently gave a simple bijection between Kohnert diagrams and reduced com- patible sequences. Since the combinatorics of Kohnert diagrams is more amenable to RSK-like insertion, this model has led to a new understanding of Schubert structure con- stants by Assaf and Bergeron [AB] via insertion on Kohnert diagrams, which extends beyond the known cases of Monk’s rule and Pieri’s rule. Grothendieck polynomials [LS82b] are the K-theoretic analog of Schubert polyno- mials. Like Schubert polynomials, Grothedieck polynomials can be modeled with non- reduced pipe dreams. Lenart, Robinson, and Sottile [LRS06] gave an algorithm similar to Bergeron and Billey’s to derive the set of non-reduced pipe dreams that represent the monomial terms of a Grothendieck polynomial. Ross and Yong [RY15] proposed an analog to Kohnert’s rule for Grothedieck polyno- mials. Considering the ways Kohnert’s rule has advanced Schubert calculus, this model could offer an interesting new perspective for Grothedieck polynomials. In this paper, we define pipe dreams and Kohnert diagrams and explain Assaf’s bijec- tion. WethendiscusssimilarmodelsforGrothedieckpolynomialsandproposeabijection between non-reduced pipe dreams and K-Kohnert diagrams. 2 1. Chapter 1: Pipe Dreams A pipe dream is a tiling of the first quadrant of Z× Z with elbows and finitely many crosses . For a pipe dream D, the cell (i,j)∈ D if it is a cross tile and (i,j) / ∈ D if it is an elbow tile. The shape of a pipe dream is the permutation ofS ∞ obtained by following the pipes from the y-axis to the x-axis. A pipe dream is reduced if no two lines, or pipes, cross more than once. 7 4 3 9 6 8 2 5 1 1 2 3 4 5 6 7 8 9 Figure 1. A reduced pipe dream for w = 152869347 Definition 1.1 ([BB93]). For a permutation w∈S n , D bot (w) ={(n− i+1,c) :c≤ m i } where m i = #{j :j >i and w j <w i } Definition1.2 ([BB93]). Aladder move onD∈RPD(w)isD∪{(i− m,j+1)}\{(i,j)}, where D satisfies the following conditions: • (i,j)∈D, (i,j +1) / ∈D • (i− m,j),(i− m,j +1)∈D for some m such that 1≤ m≤ i− 1 • (i− k,j),(i− k,j +1)∈D for 1≤ k≤ m− 1 The following diagram shows us the ladder move on the affected rows and columns. We see that the ladder move does not affect the permutation represented by the diagram since the pipes enter and exit in the same way after the ladder move is applied. 3 L ij − − → Let RPD(w) be the reduced pipe dreams of shape w and L(D) be the set of pipe dreams that can be derived by applying ladder moves to some diagram D. Theorem 1.3 ([BB93]). The Schubert polynomialS w is given by S w = X P∈RPD(w) x wt(P) 1 1 ··· x wt(P)n n = X D∈L(D bot (w) x wt(D) 1 1 ··· x wt(D)n n , where wt(T) i is the number is crosses in row i of some pipe dream T. 2 3 4 1 1 2 3 4 2 3 4 1 1 2 3 4 2 3 4 1 1 2 3 4 2 3 4 1 1 2 3 4 2 3 4 1 1 2 3 4 Figure 2. The set of reduced pipe dreams for w = 1432 starting with D bot (w) 4 2. Chapter 2: Kohnert Diagrams A diagram is a finite collection of points, called cells, in the first quadrant of Z× Z. Definition 2.1. For u a permutation, the Rothe diagram D(u) is D(u) ={(u j ,i)|i<j and u i >u j }⊂ N× N. (2.1) For example, the Rothe diagram for the permutation 152869347 is the left diagram in Fig. 3. Definition 2.2. A Kohnert move on a diagram selects the rightmost cell of a given row andmovesthecelldownwithinitscolumntothefirstavailablepositionbelow,ifitexists, jumping over other cells in its way as needed. gg g gg gg gg ggg g gg gg g gg g ggg Figure 3. A diagram (left) and a Kohnert diagram for it (right). Definition2.3. Thesetof Kohnert diagrams forafixeddiagram D, denotedbyKD(D), is the set of all diagrams obtained by a sequence of Kohnert moves on D. Kohnert conjectured the set KD(w) = KD(D(w)) generates the Schubert polynomial S w . Winkel [Win99, Win02] gave two proofs, though neither the original nor revised proof is universally accepted. Assaf [Ass22a] gave an explicit bijection between Kohnert diagrams and the compatible sequences model of Billey, Jockusch, and Stanley [BJS93]. Armon, Assaf, Bowling and Ehrhard [AABE23] gave a representation theoretic proof using Kra´ skiewicz–Pragacz modules [KP87, KP04] whose characters are Schubert poly- nomials. 5 Theorem 2.4 ([Win99, Win02, Ass22a, AABE23]). The Schubert polynomial indexed by the permutation w is given by S w = X T∈KD(w) x wt(T) 1 1 ··· x wt(T)n n , (2.2) where wt(T) i is the number of cells in row i of T. g gg gg g g g g g gg g gg Figure 4. The set of Kohnert diagrams for w = 1432 starting with the Rothe diagram 6 3. Chapter 3: Bijection Assaf’s weight-preserving bijection [Ass22a] between pipe dreams and Kohnert dia- grams allows us to use KD(w) to understand Schubert polynomials. This result has led to the most recent progress in finding a combinatorial formula for Schubert’s structure constants. Definition 3.1. ([Ass22b]). Let c,c+1 be columns in a diagram. Suppose the highest occupied cell in column c+1 is in row i. Pair this cell with the nearest occupied cell in column c, row k where k≥ i. Moving from top to bottom, continue this pairing process for all cells occupied in column c+1. To rectify, move any unpaired cells in column c+1 to column c. To reverse rectify, move unpaired cells from column c to column c+1. Definition 3.2. ([Ass22a]). The map W : RPD(w) → KD(w) is defined as follows. Consider some RC-graph P(w) where w ∈ S n . For each row i such that 1 ≤ i ≤ n, shift the crosses i− 1 cells to the right. Place bubbles where there are crosses in row n. Suppose the leftmost cross in row n− 1 is in column c. Rectify any bubbles above this row and make the cross in (n− 1,c) a bubble. Iterate across the row and then repeat for the below row. Continue until all crosses have become bubbles. Definition 3.3. ([Ass22a]). The map D : KD(w) → RPD(w) is defined as follows. Consider some Kohnert diagram D(w) for w∈S n . Suppose the rightmost bubble in row 1 is in column c. Reverse rectify bubbles above this row, and replace the bubble with a cross in (1,c). Iterate across the row, and then repeat with each row, moving from bottom to top. Once all the bubbles are crosses, for each row i such that 1≤ i≤ n, shift the crosses i− 1 cells to the left. Theorem 3.4. ([Ass22a]) W : RPD(w) → KD(w) and D : KD(w) → RPD(w) are well-defined and inverses. To prove the maps are well-defined, Assaf inducts on the length of a permutation w, which is the minimal length of a reduced word of w. Given the maps are well-defined, it is clear that they are inverses. This proves Kohnert’s rule for Schubert polynomials. 7 4. Chapter 4: K-Pipe Dreams A non-reduced pipe dream is a pipe dream where two pipes may cross multiple times. The excess, denoted by ex(Q), of a non-reduced pipe dream is any additional crosses between two pipes [PS19]. A non-reduced pipe dream Q sends a pipe from w i on the y-axis to i on the x-axis once ex(Q) has been removed. 7 4 3 9 6 8 2 5 1 1 2 3 4 5 6 7 8 9 Figure 5. A non-reduced pipe dream for w = 152869347 Definition 4.1. [LRS06] A K-ladder move M ij (D) on a non-reduced pipe dream is M ij =D∪{(i− m,j+1)} where the conditions outlined in [2.2] are satisfied. The cross at (i,j) is a ghost cross. Theorem 4.2. ([LS82b, LRS06]) The Grothendieck polynomialG w is given by G w = X Q∈PD(w) (− 1) ex(Q) x wt(Q) 1 1 ··· x wt(Q)n n = X E∈M(D bot ) (− 1) g x wt(E) 1 1 ··· x wt(E)n n (4.1) where the wt(Q) i is the number of crosses in the reduced pipe dream that corresponds to Q and wt(E) i is the number of non-ghost crosses in row i. 2 3 4 1 1 2 3 4 2 3 4 1 1 2 3 4 2 3 4 1 1 2 3 4 2 3 4 1 1 2 3 4 2 3 4 1 1 2 3 4 2 3 4 1 1 2 3 4 Figure 6. The set of non-reduced pipe dreams for w = 1432 other than those in Figure 2 8 5. Chapter 5: K-Kohnert Diagrams We will now examine Ross and Yong’s [RY15] K-Kohnert diagrams. Definition 5.1. A ghost Kohnert move on a diagram moves the rightmost cell in a row down to the first available cell in its column and leaves behind a ghost bubble. A ghost bubble cannot be moved by a Kohnert or ghost Kohnert move. Definition 5.2. An offspring bubble is a bubble generated by a ghost Kohnert move. A Kohnert family is the set of bubbles generated by a series of ghost Kohnert moves. 1 1 1 Figure 7. The Rothe diagram (left) for w = 25134 and a K-Kohnert diagram for it (right) where the gray circles are ghost bubbles and the labelled circles form a Kohnert family Definition 5.3. The set of K-Kohnert diagrams for a fixed diagram D, denoted by KKD(D),isthesetofalldiagramsobtainedbyasequenceofKohnertandghostKohnert moves on D. For a given permutation w, KKD(w) = KKD(D(w)), where D(w) is the Rothe diagram for w. Conjecture 5.4. ([RY15]) The Grothendieck polynomial indexed by w is given by G w = X G∈KKD(w) (− 1) g x wt(G) 1 1 ··· x wt(G)n n (5.1) whereg is the number of ghost bubbles inG and wt(G) i is the number of non-ghost bubbles in row i. Figure 8. The set of K-Kohnert diagrams forw = 1432 other than those in Figure 4 9 6. Chapter 6: Proposed K-Bijection After computing many examples in search of a proof for Conjecture 6.4, I propose the following maps. Conjecture 6.1. W as given in [4.2] is a well-defined map from KPD(w) to KKD(w). Theplacementofthecrossesinanon-reducedpipedreamdetermineswhichareghost crosses. Thus, as exhibited in Figure 6, K-pipe dreams have unique structures, so Assaf’s weight-preserving bijection should serve as a map from KPD(w) to KKD(w). On the other hand, K-Kohnert diagrams have a more ambiguous structure. The rightmost diagrams in Figure 8 is an example of two KKD(w) with the same shape but different placement of ghost bubbles. As a result, the map from KKD(w) to KPD(w) must consider which bubbles are ghosts when reverse rectifying. Definition 6.2. Let c,c+1 be columns in a K-Kohnert diagram where some collection of bubbles have positive integers placed in them. Pair cells as described in [4.1]. To i-reverse rectify, leave any unpaired bubbles labelledi fixed and move any other unpaired bubbles in column c to column c+1. Definition 6.3. The map ˜ D : KKD → KPD is defined as follows. Consider some K-Kohnert diagram D(w) with n Kohnert families. For each Kohnert family, choose an integer i such that 1≤ i≤ n and place an i in every bubble of that family. ApplyD as defined in [4.3] when a bubble is unlabelled. If a bubble has an integer i in it, i-reverse rectify before turning the bubble into a cross. Conjecture 6.4. ˜ D :KKD→KPD is well-defined. Given that these maps are well-defined, it is clear they are inverses. This would prove the K-Kohnert rule. 10 References [AABE23] Sam Armon, Sami Assaf, Grant Bowling, and Henry Ehrhard, Kohnert’s rule for flagged Schur modules , J. Algebra 617 (2023), 352–381. [AB] SamiAssafandNantelBergeron,AninsertionalgorithmformultiplyingSchu- bert polynomials by Schur polynomials, arXiv:coming soon. [Ass22a] Sami H. Assaf, A bijective proof of Kohnert’s rule for Schubert polynomials, Comb. Theory 2 (2022), no. 1, Paper No. 5, 9. [Ass22b] ,Demazure crystals for Kohnert polynomials,Trans.Amer.Math.Soc. 375 (2022), no. 3, 2147–2186. [BB93] Nantel Bergeron and Sara Billey, RC-graphs and Schubert polynomials, Ex- periment. Math. 2 (1993), no. 4, 257–269. [BJS93] SaraC.Billey,WilliamJockusch,andRichardP.Stanley, Some combinatorial properties of Schubert polynomials, J.AlgebraicCombin.2(1993), no.4, 345– 374. [Koh91] Axel Kohnert, Weintrauben, Polynome, Tableaux, Bayreuth. Math. Schr. (1991), no. 38, 1–97, Dissertation, Universit¨ at Bayreuth, Bayreuth, 1990. [KP87] Witold Kra´ skiewicz and Piotr Pragacz, Foncteurs de Schubert, C. R. Acad. Sci. Paris S´ er. I Math. 304 (1987), no. 9, 209–211. [KP04] , Schubert functors and Schubert polynomials, European J. Combin. 25 (2004), no. 8, 1327–1344. [LRS06] Cristian Lenart, Shawn Robinson, and Frank Sottile, Grothendieck polynomi- als via permutation patters and chains in the bruhat order, Am. J. Math.128 (2006), no. 4, 805–846. [LS82a] Alain Lascoux and Marcel-Paul Sch¨ utzenberger, Polynˆ omes de Schubert, C. R. Acad. Sci. Paris S´ er. I Math. 294 (1982), no. 13, 447–450. 11 [LS82b] , Structure de hopf de l’anneau de cohomologie et de l’anneau de grothendieck d’une vari´et´e de drapeaux, C. R. Acad. Sci. Paris S´er. I Math 295 (1982), no. 11, 629–633. [PS19] Oliver Pchenik and Dominic Searles, Decompositions of grothendieck polyno- mials, Int. Math. Res. 2019 (2019), no. 10, 3214–3241. [RY15] ColleenRossandAlexanderYong, Combinatorial rules for three bases of poly- nomials, S´ em. Lothar. Combin. 74 (2015), Art. B74a, 11. [Win99] Rudolf Winkel, Diagram rules for the generation of Schubert polynomials, J. Combin. Theory Ser. A 86 (1999), no. 1, 14–48. [Win02] , A derivation of Kohnert’s algorithm from Monk’s rule, S´ em. Lothar. Combin. 48 (2002), Art. B48f, 14. 12

Abstract (if available)

Abstract In 1990, Kohnert proposed a combinatorial model for Schubert polynomials, which Assaf recently proved is in bijection with Bergeron and Billey's reduced pipe dreams. Building on this, Ross and Yong offered a conjecture for a K-Kohnert's rule for Grothendieck polynomials, which are the K-theoretic analog of Schubert polynomials and can be represented by non-reduced pipe dreams. In this paper, we review each of these models and offer an adaptation of Assaf's bijection for non-reduced pipe dreams and K-Kohnert diagrams.

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Creator Somani, Reena Manish (author)

Core Title A proposed bijection for the K-Kohnert rule for Grothendieck polynomials

School College of Letters, Arts and Sciences

Degree Master of Arts

Degree Program Mathematics

Degree Conferral Date 2023-08

Publication Date 06/01/2023

Defense Date 06/01/2023

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag combinatorics,Grothendieck polynomials,Kohnert diagrams,K-theory,OAI-PMH Harvest,pipe dreams,Schubert calculus,Schubert polynomials

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Language English

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Advisor Assaf, Sami (committee chair), Leslie, Trevor (committee member), Williams, Harold (committee member)

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