Introduction to the Cohomology of the Flag Variety

Brendan Pawlowski; Sara C. Billey; Yibo Gao

REVIEW 1 major objections 6 minor 2 cited by

Introduction to the Cohomology of the Flag Variety

T0 review · 1 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Every Schubert counting problem reduces to multiplying Schubert polynomials and reading off one coefficient.

desk verdict A solid, well-crafted survey chapter with no new theorem; the revisionist Monk-style route is a real pedagogical contribution, and the chapter deserves a serious referee. read the letter →

arxiv 2506.21064 v1 pith:3X2L7S47 submitted 2025-06-26 math.CO math.ACmath.AG

Sara C. Billey , Yibo Gao , Brendan Pawlowski This is my paper · ORCID

classification math.COmath.ACmath.AG MSC 14M1514N1505E14

keywords SchubertcalculusflagvarietypolynomialsChowringHilbert's15thproblemstructureconstantsGrassmannianMonk'sformula

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This introductory handbook chapter makes a complete, rigorous case that Hilbert's 15th problem is solved for flag varieties: every finite enumerative question about intersecting Schubert varieties becomes a computation in a polynomial ring. The chain is explicit: Schubert varieties have cohomology classes represented by Schubert polynomials, products of these polynomials expand in the Schubert basis with nonnegative integer coefficients, and those coefficients are exactly the generic intersection numbers. The chapter follows Monk's constructive path rather than the standard divided-difference route, so that each step is tied to counting flags in triple intersections. The payoff is that any Schubert problem can be answered by polynomial arithmetic and linear algebra, with the remaining obstacle purely computational.

What carries the argument

The central objects are the Schubert varieties $X_w$, their classes $[X_w]$ in the Chow ring $A^*(\mathrm{Fl}(n)) \cong H^*(\mathrm{Fl}(n))$, and the Schubert polynomials $S_w$ defined by the transition equation via the lex-largest inversion recurrence, which the chapter derives from Monk's formula. The mechanism doing the work is the basis property: the Schubert polynomials form a basis of the coinvariant algebra with leading monomial given by the Lehmer code, and the induced ring isomorphism sends the product of Schubert classes to the ordinary product of the corresponding polynomials, so that structure constants are read off as expansion coefficients.

What would settle it

Compute a Schubert structure constant $c^w_{uv}$ in a small case such as $\mathrm{Fl}(4)$ or $\mathrm{Fl}(5)$ by two independent routes: expand the product of the corresponding Schubert polynomials in the Schubert basis, and directly solve the generic polynomial system of rank conditions that defines the triple intersection; any mismatch between the expansion coefficient and the number of solutions would refute the claimed reduction.

Watch

Extended reading notes

Core claim

The chapter establishes that for the complete flag variety $\mathrm{Fl}(n)$, the Chow ring $A^*(\mathrm{Fl}(n))$ is isomorphic to the cohomology ring and to the coinvariant algebra $\mathbb{Z}[x_1,\dots,x_n]/I_n^+$, with the Schubert classes $[X_w]$ forming a basis. Multiplication of these classes corresponds to generic intersection of the corresponding Schubert varieties, so the structure constants $c^w_{uv}$ in $[X_u][X_v] = \sum c^w_{uv}[X_w]$ are nonnegative integers counting points in 0-dimensional triple intersections. The chapter proves that the Schubert polynomials $S_w$ represent the Schubert classes $[X_{w_0 w}]$ and form a basis of the coinvariant algebra, so that products of Schubert classes become ordinary products of polynomials expanded in the Schubert basis. Thus the answer to any Schubert problem is a specific coefficient in a product of Schubert polynomials, and the authors state in Section 3.10 that "the generic solutions can be found by polynomial arithmetic and some linear algebra. The theory is completely rigorous."

Load-bearing premise

The reduction rests on the unproved theory of rational equivalence and the moving lemma, taken as given: that multiplying Chow classes of Schubert varieties genuinely counts their points of intersection when the varieties are moved into generic position.

Editorial extensions

If this is right

Any 0-dimensional Schubert problem in the complete flag variety can be solved by expanding a product of Schubert polynomials in the Schubert basis and extracting one coefficient, which the chapter demonstrates on the classic problem of counting lines meeting four given lines.
The Schubert structure constants are always nonnegative integers, because by the Geometry Implies Positivity theorem each one counts the flags in a generic triple intersection of Schubert varieties.
The ring structure of the flag variety is generated by the special classes $[X_{w_0 s_i}]$, so Monk's formula for multiplication by these classes determines all structure constants recursively.
The cohomology rings of all partial flag varieties and Grassmannians are governed by the same structure constants, extending the polynomial-arithmetic reduction beyond complete flags.
The remaining barrier is computational: Narayanan's result that Schubert problems are at least as hard as $\#P$ bounds the feasibility of carrying out the polynomial expansions for large $n$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

This reduction implies that progress on computing Schubert structure constants directly translates into progress on enumerative geometry, while the $\#P$-hardness result suggests that no universal polynomial-time combinatorial rule for these constants can exist unless the polynomial hierarchy collapses.
The same polynomial-arithmetic framework should extend, with positivity lost, to equivariant and quantum cohomology and to double Schubert polynomials, where the chapter's degeneracy-locus discussion already points toward how the moving-flag language adapts.
The realizability question for permutation arrays -- which higher-dimensional tables of intersection dimensions actually arise from configurations of flags -- is the natural combinatorial testbed for generalizing the reduction from pairs of flags to arbitrary numbers of reference flags.
Because the chapter deliberately bypasses the divided-difference route, its transition-equation definition of Schubert polynomials is itself a testable claim: if the transition recurrence ever produced a polynomial not representing the corresponding Schubert class, the entire computational pipeline would need repair.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is an expository book chapter, not a research paper. It proves no new theorem, but it does something useful: it rewrites the standard path to Schubert polynomials through Monk's formula and transition equations, and it makes Proposition 3.70 explicit as an algorithm. The exposition is careful, the history is fun, and the examples are well chosen.

The Monk-centric route is genuinely illuminating. The authors show you can define Schubert polynomials via transition equations and get positivity and the divided-difference recurrence as consequences rather than starting points. The worked computations, especially the n=4 structure constants via shoebox diagrams, really help. The chapter covers a large amount of ground: pipe dreams, bumpless pipe dreams, puzzles, Stanley symmetric functions, singular loci, and open problems. The bibliographic pointers are extensive and appropriate. Self-citations such as Billey-Vakil are used for known results and are legitimate.

Soft spots: the 'completely rigorous' claim in Section 3.10, that Schubert problems reduce to polynomial arithmetic and linear algebra, rests on the Chow/cohomology isomorphism and the moving lemma. The chapter explicitly says in Section 3.7 that defining rational equivalence is out of scope, citing Fulton for the foundation. For a survey that is legitimate rather than a flaw. Still, the phrase 'completely rigorous' is a bit strong, because the rigorous part is delegated to books. The chapter itself hedges Hilbert's 15th problem in Section 2.1, which is appropriate. The permutation arrays section (Section 3.6) is more technical than the rest and feels like a detour; it is interesting but only loosely connected to the main path. No load-bearing mathematical error found.

This is for graduate students and researchers wanting a computational, combinatorial entry to Schubert calculus, and for teachers. It deserves a serious referee, mainly for accuracy and organization. My recommendation: engage with it. It is a well-made survey; send it for review, and ask the authors to soften the 'completely rigorous' claim or to point more explicitly to where the rigorous foundation is proved.

Referee Report

1 major / 6 minor

Summary. This paper is a survey chapter intended for the 'Handbook of Combinatorial Algebraic Geometry: Subvarieties of the Flag Variety.' Its central claim is that Schubert problems on complete flag varieties can be reduced, rigorously and algorithmically, to expanding products of Schubert polynomials in the Schubert basis and extracting coefficients. The chapter develops the necessary background: flag varieties and Schubert varieties, Bruhat order, the Chow/cohomology ring, Monk's formula, the transition equation definition of Schubert polynomials, and their combinatorial models (pipe dreams, bumpless pipe dreams, etc.). It also surveys related geometry (matrix Schubert varieties, singular loci, degeneracy loci) and lists open problems. The exposition is constructive and intentionally follows a 'revisionist' path, starting from Monk's approach rather than the BGG/Demazure divided-difference formalism.

Significance. If the chapter's central assertion is accepted, it provides a clear and useful survey of a foundational area at the interface of algebraic geometry and combinatorics. The text is careful to credit original sources and gives a large number of exercises with citations, making it a valuable pedagogical resource. The explicit worked computation in §3.10, showing that (S_{1324})^4 expands with coefficient 2 on S_{3412} and thereby recovers the answer to the classical 'lines meeting four lines' problem, convincingly demonstrates the reduction. The chapter also serves as a broad introduction to modern tools (pipe dreams, puzzles, Stanley symmetric functions, etc.) and openly discusses computational complexity and open problems. For a handbook survey, the delegation of the deepest intersection-theoretic foundations to Fulton's books is appropriate rather than a flaw.

major comments (1)

[§3.7 and §3.10] The chapter's main claim that 'the theory is completely rigorous' depends on Theorem 3.82 and Theorem 3.115(e), but the exposition does not state the precise transversality hypothesis (e.g., Kleiman's transversality theorem) underlying the equality between Chow ring multiplication and generic intersection numbers. I recommend adding a remark in §3.7 that explicitly names the external results on which Theorem 3.82 rests and notes their validity for the complete flag variety, with a specific citation to [133] or [134]. This would make the rigor claim in §3.10 fully supportable within the chapter's own text rather than leaving the reader to infer the missing hypothesis.

minor comments (6)

[§3.9, Eq. (3.50)] The monomial for the longest permutation is typeset incompletely; it should read x_1^{n-1} x_2^{n-2} ... x_{n-1}^1 x_n^0, with the final exponent 0 explicitly written.
[§3.6] After stating that the Realizability Conjecture is true for d = 1, 2, 3, the text says 'Nonetheless, the conjecture is false' without specifying the dimension; the counterexamples in [40] occur for higher d, and the sentence would be clearer if it said so explicitly.
[§3.10, Eq. (3.56)] The expansion (x1+x2)^4 = 2 S_3412 + 3 S_25134 + S_162345 is asserted without indicating that S_25134 and S_162345 are permutations in S5 and S6, respectively; adding the one-line notation or a citation would help the reader verify the computation.
[§3.5, Example 3.51] The statement 'z42 is not a flag minor' may confuse readers because z42 is a matrix entry; consider rephrasing to 'the entry z42 is not a flag minor' to emphasize the distinction.
[§3.2, after Eq. (3.6)] The phrase 'tij are called transpositions' uses an untypeset t_{ij}; fix the indexing to t_{ij} throughout for consistency.
[Note to Readers] The note states 'We hope to have it polished up soon for publication in 2025'; the final published version should remove this preliminary-status language or update it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the chapter's reduction of Schubert problems to Schubert-polynomial arithmetic is grounded in standard intersection theory and proved from Monk's formula; self-citations are historical or auxiliary, not load-bearing.

full rationale

The chapter is an expository survey whose central reduction is: Schubert problems → products of Schubert classes → cohomology/coinvariant algebra → Schubert polynomials. The bridge is Theorem 3.82, which is not derived from Schubert polynomials; it is the standard Chow-ring statement that intersection numbers are coefficients in products of Chow classes, and the paper explicitly delegates the foundational moving-lemma/rational-equivalence machinery to Fulton's books, saying in §3.7: "Defining rational equivalence explicitly is beyond the scope of this chapter." Theorem 3.115, relating Schubert polynomials to Schubert classes, is proved in the text: Schubert polynomials are first defined by the transition equation (3.42), and the proof of Theorem 3.115 uses Lemma 3.114, whose proof is geometric (pullback under Fl(n) → Fl(n+1)), together with the independently known fact that Schubert classes form a basis of A*(Fl(n)). The structure-constant equality (e) is a consequence of the ring isomorphism f, not an input to it. Monk's formula is proved geometrically via triple intersections, and the polynomial results are then derived from it; no step fits a parameter and calls the result a prediction, and no object is defined in terms of its own conclusion. Self-citations such as Billey–Vakil in §3.6 and Billey–Jockusch–Stanley in §4.1 concern auxiliary algorithms and historical attributions; the main combinatorial identities used later are either proved or sketched in the chapter, and the load-bearing geometry is cited to standard references. The paper also hedges the Hilbert-15 claim in §2.1. I find no circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or new entities. The chapter's claims rest on standard, cited theorems of intersection theory and Schubert polynomials.

assumptions (4)

standard math The Chow ring A*(Fl(n)) is free with basis the Schubert classes [X_w] for w in S_n.
Invoked throughout Sections 3.7 and 3.8 to define Schubert structure constants; cited to Fulton [133].
standard math A*(Z) is isomorphic to singular cohomology H*(Z;Z) when Z is smooth and has a cellular decomposition.
Used to transfer Schubert calculus to cohomology; cited to [133, Ch. 1 and 19].
standard math Schubert polynomials S_w represent the Schubert class [X_{w0 w}] in the coinvariant algebra R_n.
Central identification used to turn intersection problems into polynomial multiplication; cited to Lascoux-Schutzenberger [250] and Macdonald [265].
domain assumption The moving lemma and rational equivalence for flag varieties: generic translates of Schubert varieties intersect as their Chow classes predict.
The paper states that the moving is straightforward for flag varieties but does not prove it; it is essential for Theorem 3.82.

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Pith. "Pith review of Introduction to the Cohomology of the Flag Variety." pith.science (2026). https://pith.science/paper/3X2L7S47

@misc{pith2026250621064,
  author       = {Pith},
  title        = {Pith review of: Introduction to the Cohomology of the Flag Variety},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3X2L7S47}},
  note         = {Machine review of arXiv:2506.21064}
}

read the original abstract

One hundred years ago, Hilbert gave a list of important open problems in mathematics. His 15th problem asked for the development of a rigorous calculus explaining Schubert's enumerative results for intersecting varieties defined by rank conditions on vector spaces. Today by way of many contributions in algebraic topology, geometry, and combinatorics, we consider this solved. Yet, deep questions remain about the subtleties of actually carrying out the process. In this chapter, we hope to summarize the rigorous development of what has become known as Schubert calculus, with an eye toward computation. We discuss Grassmannians and flag varieties and their cohomology rings, following Monk's constructive algebraic approach. We derive formulas for Schur and Schubert polynomials, which represent cohomology classes of Schubert varieties. We hint at the vast literature in this area and point to the other references in the Handbook for more information. Finally, we identify open problems that remain a challenge even with modern tools at our fingertips in hopes of inspiring further contributions in this fascinating field. This is intended as the first chapter of a book entitled "Handbook of Combinatorial Algebraic Geometry: Subvarieties of the Flag Variety", a compendium of topics in the area. The book is being edited by Erik Insko, Martha Precup, and Ed Richmond. In addition to this introductory chapter, others will cover more advanced topics such as Kazhdan-Lusztig varieties, generalized smooth Schubert varieties, Richardson varieties and positroid varieties, spherical and torus orbit closures, spanning line configurations, different types of Hessenberg varieties, and generalizations to Kac-Moody flag varieties, each written by experts in those areas. We hope you enjoy this chapter enough to seek out the others, and that you send us any comments or corrections you find as you read this article!

Figures

Figures reproduced from arXiv: 2506.21064 by the authors.

**Figure 1.** Projective representation of a flag in R 4 as a point on a line in a plane, which is spanned by one wall of a shoebox. R 4 meets the hyperplane in a line. Every 3-dimensional subspace of R 4 meets the hyperplane in a plane. Therefore, drawn projectively, a flag in Fl(4) is a point, on a line, in a plane, contained in one side of a shoebox, which represents C 4 projected on the fixed hyperplane. See [PITH_FULL_IMAGE… view at source ↗

**Figure 2.** A flag on a flag pole. Go Schubert Team! For each pair of flags, we can consider how their subspaces relate to each other. We can classify such pairs according to the intersection table of dimensions of the i th subspace in the first flag intersected with the j th subspace of the second flag. For example, again in n = 4, consider a flag B• drawn in black and R• drawn in red as in [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗

**Figure 3.** Pairs of flags in 3 different relative positions along with their intersection tables. The rows and columns are labeled the same way in each case. The zero-dimensional intersections are represented by empty cells in these tables for ease of reading. Continuing with Example 3.2, the flag F• = (6e1 + 3e2, 4e1 + 2e3, 9e1 + e3 + e4, e2) can be represented by the matrices (3.1)     6 4 9 0 3 0 0 1 0 2 1 0 0 0 1 0  … view at source ↗

Figures from the paper (58 more)

**Figure 4.** Figure 4: The diagram of w = 43152 is the set of outlined cells, so D(w) = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (4, 2)}. One more notation for a permutation w ∈ Sn is its Lehmer code, or simply its code, which is the n-tuple (3.12) c(w) = (c(w)1, c(w)2, . . . , c(w)n) where …

**Figure 5.** Figure 5: A shoebox diagram of two flags in Fl(4). The black flag represents the standard flag E•. The red flag R• is an element of C2341(E•). From the intersection conditions for Cw(E•) in Definition 3.21, one can observe that there is a close connection between the diagram D(w…

**Figure 6.** Figure 6: The Hasse diagram of S4 Exercise 3.33. Prove that the Hasse diagram of Sn is self-dual, rank-symmetric and rankunimodal. Testing if v ≤ w in Bruhat order via the transposition relations can be cumbersome since one would need to consider many possible paths from v to w…

**Figure 7.** Figure 7: The lower interval in the Bruhat order below 3412 One can see that the Schubert variety X3412 is not a smooth manifold since its Poincar´e polynomial is not symmetric (palindromic) which implies that Poincar´e duality does not hold for H∗ (X3412). There are many additi…

**Figure 8.** Figure 8: Slices of a permutation array. For each y ∈ [n] d , let P[y] = {x ∈ P | x ⪯ y} be the principal subarray of P containing all points of P which are dominated by y. Define rkjP = #{1 ≤ k ≤ n | there exists x ∈ P with xj = k}. P is rankable of rank r if rkjP = r for all 1…

**Figure 9.** Figure 9: Rank table for P = {(3, 4, 1), (4, 2, 2), (1, 4, 3), (3, 3, 3), (2, 3, 4), (3, 2, 4), (4, 1, 4)}, where the empty boxes mean the rank is 0 for readability. know G1 ⊂ G3 so the green point in the shoebox diagram corresponding to G1 is contained in the line at the inters…

**Figure 10.** Figure 10: Flags in position determined by the permutation array P from (3.21). Example 3.59. A subset of a totally rankable array may or may not be rankable. Consider the subset of P above given by the array A = {(3, 4, 1), (4, 2, 2), (2, 3, 4)}; then one can check A is totally…

**Figure 11.** Figure 11: The Pappus line configuration on 9 lines. The arc represents an impossible “hop” by the *blue line* over an intersection. This leads to an unrealizable permutation array. For j > 0 and k > 1, assume dj (s1), dj (s2, . . . , sk), and dj−1(s1, s2, . . . , sk) are known …

**Figure 12.** Figure 12: C46287351(E•) ∩ C53712648( E •) contains only the flag corresponding with the permutation matrix of 46287351. We claim the cells of Xw(E•) of dimension strictly smaller than Cw do not intersect Xw0w( E •). To prove the claim, recall from Exercise 3.39 that the boundar…

**Figure 13.** Figure 13: Projective representation of three transverse flags in C 4 looking into a shoebox bounded by the three planes.     0 0 0 1 0 0 1 2 1 1 2 3 1 2 3 4         0 0 0 1 0 1 1 2 0 1 2 3 1 2 3 4         0 1 1 1 0 1 2 2 1 2 3 3 1 2 3 4     dim(Bi ∩ Fj )…

**Figure 14.** Figure 14: Tables of intersection dimensions for F• ∈ X3421(B•) on the left, F• ∈ X4231(R•) in the middle, and F• ∈ X3124(G•) on the right from Example 3.89. Example 3.89. Let u = 3421, v = 4231, and w = 2431. We want to prove c w uv = 1 in this case. Observe that 2 = coinv(3421…

**Figure 15.** Figure 15: The purple flag represents the unique flag X3421(B•)∩X4231(R•)∩ X3124(G•). Example 3.90. Let u = 3421, v = 4231, and w = 4132. We want to prove c w uv = 0 in this case. Observe that 2 = coinv(3421) + coinv(4231) = coinv(4132), so it is possible c w uv is not zero by T…

**Figure 16.** Figure 16: The purple flag represents the unique flag in position 3421 with respect to the black flag, in position 4231 with respect to the red flag, and in position 2314 with respect to the green flag. Example 3.94. Let n = 4, w = 3421, and i = 2. Monk’s formula implies [X4231]…

**Figure 17.** Figure 17: Monk’s representatives of Schubert classes in Z[γ1, . . . , γn]. He writes permutations in Sn as bijections on the set {0, 1, 2, . . . , n − 1}. The “order” column gives the degree of the corresponding Schubert variety as a projective variety when one embeds Fl(n) int…

**Figure 18.** Figure 18: The diagrams of v = 1437256 and w = 1437265 with the difference highlighted in gray [PITH_FULL_IMAGE:figures/full_fig_p054_18.png]

**Figure 19.** Figure 19: Schubert polynomials for permutations in S4. and for ℓ(w) > 1, we have Sw = f([Xw0w]) by comparing (3.41) and Definition 3.104. Therefore, the Schubert polynomials multiply as in Monk’s formula Theorem 3.92. Therefore, (3.49) follows by applying the map w → w0w. □ Exa…

**Figure 20.** Figure 20: The wiring diagram for the reduced word (4, 3, 5, 6, 4, 3, 5) ∈ R(1265734) notated in three different ways: with the intermediate permutations w (t) shown on the left, the left-labelling in the middle, and the right-labelling on the right. The crossings in columns 2 a…

**Figure 21.** Figure 21: Left: a reduced pipe dream D for w = [3, 1, 4, 6, 5, 2]. Middle: the reading order for the crossings, with numbers indicating position in the order. The associated reduced word is rD = (5, 2, 1, 3, 4, 5) ∈ R(w). Reading the sequence of row numbers and column numbers i…

**Figure 22.** Figure 22: Chute moves on pipe dreams move one cross to the left and preserve the permutation. · · + + + + + · · + + + + + · · [PITH_FULL_IMAGE:figures/full_fig_p072_22.png]

**Figure 23.** Figure 23: Ladder moves on pipe dreams move one cross to the right and preserve the permutation. Theorem 4.8. [24, Thm 3.7] For any w ∈ Sn, every reduced pipe dream for w can be obtained from a sequence of ladder moves on Dbot(w), and every reduced pipe dream for w can be obtain…

**Figure 24.** Figure 24: Construction of RP(w) by ladder moves on Dbot(1432) [PITH_FULL_IMAGE:figures/full_fig_p072_24.png]

**Figure 25.** Figure 25: An example of the sequence of wiring diagrams for the words a ′ which appear when running the bounded bump algorithm on input a = (4, 3, 5, 6, 4, 3, 5), b = (2, 2, 2, 2, 2, 2, 2), t0 = 4, and ϵ = −. The arrows indicate which crossing will move in the next step. After …

**Figure 26.** Figure 26: Computing B − 6 (rD, jD) for D starting in position (2, 7) with rD = (7, 6, 5, 4, 3, 1, 8, 7, 6, 5, 4) and iD = (1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3) via 3 stack pushes. ϵ = −1 bounded bump applied to the crossing of the wr-wire and the ws-wire in D always terminates in a…

**Figure 27.** Figure 27: If D is the pipe dream on the left with reduced word rD = (4, 3, 5, 6, 4, 3, 5), then Tw(D) is the pipe dream on the right. In between we show the stack pushes in the bounded bump algorithm. The crossing initiating a stack push is highlighted for each step. Here, w = …

**Figure 28.** Figure 28: A term in S(w) and its corresponding pipe dream the two pictures due to the order that we read the crossings in a pipe dream as shown in [PITH_FULL_IMAGE:figures/full_fig_p083_28.png]

**Figure 29.** Figure 29: Example of the mitosis operator The monomial weight of the pipe dream D is x D = x1x 4 2x3x4, and we can calculate that ∂2x D = x1x 3 2x3x4 + x1x 2 2x 2 3x4 + x1x2x 3 3x4 = x D2,1 + x D2,3 + x D2,4 . This calculation may be a little bit misleading in general, but it i…

**Figure 30.** Figure 30: Balanced labellings (top) and standard Young tableaux (bottom) for λ = (3, 2) One intuition towards the balanced condition on permutation diagrams is in connection with reflection orders. Recall that R(w) denotes the set of reduced words for a permutation w ∈ Sn. For …

**Figure 31.** Figure 31: Construction of the tableau Ta for a = (2, 3, 2, 1, 4, 2) ∈ R(43152) Schubert polynomials can also be expanded using the idea of balanced labellings. To be precise, Schubert polynomials are generating functions for column-injective flagged balanced labellings of permu…

**Figure 32.** Figure 32: Column-injective flagged balanced labellings of shape D(1432) 4.5. Bumpless Pipe Dreams. Lam, Lee, and Shimozono [237] introduced bumpless pipe dreams (BPDs) in their work on the infinite flag variety and back-stable Schubert calculus and used them to give a formula f…

**Figure 33.** Figure 33: for example. Exercise 4.65. Show that in a bumpless pipe dream D, we have |blank(D)| = |cross(D)|. Exercise 4.66. Show that a bumpless pipe dream D is determined by blank(D) and cross(D) together. Remarkably, Lam-Lee-Shimozono proved that the reduced bumpless pipe dre…

**Figure 34.** Figure 34: Reduced pipe dreams for 2143. Remark 4.69. It is shown in [237] that bumpless pipe dreams can also compute double Schubert polynomials. After adding variables y1, y2, . . ., the weight of a bumpless pipe dream D becomes (x − y) D := Y (i,j)∈blank(D) (xi − yj ). Lam-Le…

**Figure 35.** Figure 35: The bijection between BPDs and ASMs ASM whether its corresponding BPD is reduced. Not necessarily reduced BPDs (and also not necessarily reduced PDs) are utilized to compute Grothendieck polynomials [367], the K-theoretic analogs of Schubert polynomials. It is worth n…

**Figure 36.** Figure 36: Generalized chute moves any pipe dream of w can be obtained from the bottom pipe dream of w using (generalized) chute moves (Theorem 4.8). A parallel story exists for bumpless pipe dreams. Definition 4.72. [237] For a (reduced) bumpless pipe dream D, a droop move from…

**Figure 37.** Figure 37: A droop move (but not a min-droop) on bumpless pipe dreams Definition 4.73. [237] The Rothe bumpless pipe dream (or the bottom bumpless pipe dream) for a permutation w is the bumpless pipe dream where pipe i only turns once at the -tile at (i,w(i)), for all i. Lemma 4…

**Figure 38.** Figure 38: The posets of pipe dreams and of bumpless pipe dreams for 1432 Of course, one immediate question is to describe an explicit and “natural” weight-preserving bijection between reduced pipe dreams and reduced bumpless pipe dreams. Since both objects compute Schubert poly…

**Figure 39.** Figure 39: Step (2) of Definition 4.77 × × . . . . . . . . . × × . . . . . . . . [PITH_FULL_IMAGE:figures/full_fig_p095_39.png]

**Figure 40.** Figure 40: Step (4) of Definition 4.77 Definition 4.78. [146] For D ∈ BPD(w) with ℓ(w) = ℓ, define φ(D) = r = (r1, . . . , rℓ), i = (i1, . . . ,iℓ) where pop(∇k−1D) = (rk,ik) for k = 1, . . . , ℓ. Example 4.79. Consider D ∈ BPD(w) with w = 2157346 in [PITH_FULL_IMAGE:figures/…

**Figure 41.** Figure 41: Steps for obtaining ∇D from D by the algorithm in Definition 4.77. Theorem 4.80. [146] The map φ in Definition 4.78 is a weight-preserving bijection from reduced bumpless pipe dreams to reduced compatible pairs of a fixed permutation [PITH_FULL_IMAGE:figures/full_fig…

**Figure 42.** Figure 42: The bijection between BPDs and PDs To prove the theorem given D ∈ BPD(w), it is straightforward to check φ(D) has the same monomial weight. To prove φ(D) is a reduced compatible sequence for the same permutation is not so hard. Given that Theorem 4.67 has already been…

**Figure 43.** Figure 43: An example of the map xα⇝ on BPD with α = 1 and w = 21534 We can define analogous maps xα⇝ and mk,α on pipe dreams to prove Theorem 4.81 bijectively. We will not spell out the details on these maps to avoid confusion with earlier material in Section 4.1 on little bump…

**Figure 44.** Figure 44: A hybrid pipe dream of w = 13542 and type τ = OBOBO with weight x1x3x4x5. Theorem 4.87. [217] For all w ∈ Sn and all τ ∈ {O, B} n , the Schubert polynomial satisfies Sw(x1, x2, . . . , xn) = X D∈HPD(w,τ) x D. In fact, the notion of hybrid pipe dreams gives us 2n diffe…

**Figure 45.** Figure 45: Left: a diagram D; middle: construction for wordj,I for I = {1, 2}; right: construction for wordj,I for I = {3}. Example 4.92. Consider the diagram D ∈ [4]2 in [PITH_FULL_IMAGE:figures/full_fig_p100_45.png]

**Figure 46.** Figure 46: Perfect tableaux of shape D(1432) We can represent the data of a perfect tableau using a matrix {aij} n i,j=1 where aij denotes the number of i’s in column j. Rewriting the conditions from Definition 4.97, we arrive at the following system of linear inequalities: (4.1…

**Figure 47.** Figure 47: The seven labeled triangular puzzle pieces used in the BuchKresch-Purbhoo-Tamvakis proof [74] of the 2-step puzzle rule for Schubert calculus. There is also a 3-step puzzle rule conjectured by Knutson-Buch and proved by Knutson and Zinn-Justin [218]. See also the spe…

**Figure 48.** Figure 48: All three completed puzzles used to compute c w u,v using the 3-step puzzle rule for Schubert calculus [72] where u = 2314, v = 2143 with the resulting permutation w = 4213, 3412 and 3241 from left to right [PITH_FULL_IMAGE:figures/full_fig_p111_48.png]

**Figure 49.** Figure 49: The transition tree for 35124786 terminating with vexillary leaves. The same expansion can also be obtained via the Edelman-Greene map ( [PITH_FULL_IMAGE:figures/full_fig_p119_49.png]

**Figure 50.** Figure 50: The P-tableaux using the Edelman-Greene correspondence for the permutation 35124786 Theorem 4.122 shows that the cohomology class of any subvariety Z ⊆ Gr(k, n) can be represented as a nonnegative linear combination of Schur functions. This suggests an inverse questio…

**Figure 18.** Figure 18: Compare those (northwest) diagrams with the southwest diagrams shown in [PITH_FULL_IMAGE:figures/full_fig_p128_18.png]

**Figure 51.** Figure 51: The essential sets (shaded) of v = 1532674 = (1437256)−1 and w = 1532764 = (1437265)−1 0 1 1 1 0 1 1 2 0 1 2 3 1 2 3 4 • • • • [PITH_FULL_IMAGE:figures/full_fig_p129_51.png]

**Figure 52.** Figure 52: The southwest rank table and essential set for 3124. Theorem 5.19 (Fulton’s Essential Set Theorem). The ideal I(MXw) is generated by the size rkSW(w)[i, j] + 1 minors of [zpq]i≤p≤n,1≤q≤j for all (i, j) ∈ Ess(w). Furthermore, no subset of these essential set rank equat…

**Figure 53.** Figure 53: A projection of the Horn polyhedron for n = 2. To visualize this polyhedron, divide through by α and plot γ/α against β/α to get a rectangular polyhedron as show in in [PITH_FULL_IMAGE:figures/full_fig_p136_53.png]

**Figure 54.** Figure 54: A trivial bundle and the M¨obius bundle over S 1 . To be precise, we have drawn some of the fibers Ez, and E is the surface formed by all Ez. The projection π sends each point in the fiber Ez (in blue) to the point z where that fiber intersects the base space S 1 (in …

**Figure 55.** Figure 55: The bundles O(1)R and O(−2)R over S 1 . Example 5.38. Let us work out what it means to have a section ϕ of the line bundle O(r) over CPn for a positive integer r. By definition, this means ϕ(Cv) is a linear functional on the line (Cv) ⊗r ⊆ (C n+1) ⊗r for each v ∈ C n+…

**Figure 56.** Figure 56: The northwest rank table and northwest essential set for 4123. Example 5.49. Fix a nondegenerate symmetric bilinear form ⟨ , ⟩ on C 4 , like the standard dot product ⟨u, v⟩ = P i uivi . We use Fulton’s formula to calculate the cohomology class of the subvariety Z = {F…

**Figure 57.** Figure 57: The Hasse diagram of the interval [id, 4231] Therefore, the singular locus of Xw is the union of Schubert varieties Sing(Xw) = [ Xv over all v ≤ w in Sn such that #{(i < j) | vtij ≤ w} > ℓ(w). Remark 5.66. We will return to the question of finding the singular locus o…

**Figure 58.** Figure 58: The Bruhat graph of w = 4213 [PITH_FULL_IMAGE:figures/full_fig_p151_58.png]

**Figure 59.** Figure 59: The matrix for the permutation w = 319827546 with w ′ = 31872654 is shown with a dot in each position (w(i), i) and regions A, B, C, D noted. For example, D = {(8, 4), (7, 6)}. as in [PITH_FULL_IMAGE:figures/full_fig_p153_59.png]

**Figure 60.** Figure 60: Patterns for the singular locus of Xw in the 4231, 3412, and 45312 cases respectively. Here ◦’s denote 1’s in w only, •’s denote 1’s in v only, and the circle around a dot denotes a 1 in both v,w. Multiplication of w by the cycle (α1, . . . , αm, β1, . . . , βm) rotat…

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The Schubert cycle expansion of a Springer fiber component equal to a Richardson variety, indexed by a Richardson tableau, is computed combinatorially using translation-equivalent Bruhat intervals and Sottile's Pieri rule.
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