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Set-valued Rothe Tableaux and Grothendieck Polynomials

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Pith's one-line read The double Grothendieck polynomial has a Rothe-tableau formula exactly for 1432-avoiding permutations.

desk verdict A genuinely new tableau model and an iff result for 1432-avoiding permutations, with one load-bearing lemma left as a sketch and a couple of notational slips that need cleaning up. read the letter →

arxiv 1908.04164 v1 pith:3EHRZDMI submitted 2019-08-12 math.CO

Neil J.Y. Fan , Peter L. Guo This is my paper · ORCID
classification math.CO MSC 05E1005E0514M15
keywords set-valuedRothetableauxdoubleGrothendieckpolynomials1432-avoidingpermutationsdiagramsisobaricdivideddifferenceoperatorstableaucomplexesSchubertpatternavoidance
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 paper introduces set-valued Rothe tableaux, which fill the Rothe diagram of a permutation with finite nonempty sets of positive integers so that rows are weakly decreasing and columns are strictly increasing. It argues that the double Grothendieck polynomial of a permutation, a polynomial representative for the K-theory class of a Schubert variety, is generated by these tableaux exactly when the permutation avoids the pattern 1432. For 1432-avoiding permutations the polynomial is the signed sum specified in Theorem 1.1; for every other permutation the paper proves that the same sum fails. The result matters because explicit tableau formulas for Grothendieck polynomials are known only for special permutation families, and this paper adds a new family while specializing to the known 321-avoiding tableau formula. Two further tableau formulas are derived from the tableau-complex viewpoint.

What carries the argument

The load-bearing object is the set-valued Rothe tableau: a filling of the Rothe diagram $D(w)$ with finite nonempty subsets of positive integers in which rows are weakly decreasing and columns strictly increasing under the set order $A<B$ when $\max A<\min B$ and $A\leq B$ when $\max A\leq\min B$. The proof mechanism is an induction on length using the isobaric divided difference operator $\pi_r$, where the two tableau sets $\mathrm{SVRT}(ws_r,f_0)$ and $\mathrm{SVRT}(w,f_0)$ are partitioned into equivalence classes supported on the squares containing $r$ or $r+1$; a bijection between class sets and a factorization of the class contributions show $\pi_r G_{ws_r}(C;x,y)=G_w(\Phi(C);x,y)$. The reverse direction uses the balanced-labeling model of Schubert polynomials to produce a labeling of $D(w)$ that is not a single-valued Rothe tableau. A secondary mechanism, the tableau complex of the cited reference [17], converts the main formula into two alternative tableau formulas.

What would settle it

Expanding both sides of (1.2) for the 1432-avoiding, non-321-avoiding permutation $w=35142$ and comparing coefficients would test the forward claim: equality on every monomial is exactly what Theorem 2.1 predicts, while any mismatch disproves it. For the reverse claim, the same coefficient comparison for $w=1432$ should exhibit at least one monomial where the two sides differ, since Theorem 2.2 asserts that this obstruction always exists.

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Extended reading notes

Core claim

The central discovery is an exact characterization. A permutation $w$ is 1432-avoiding if and only if $$G_w(x,y)=\sum_{T\in \mathrm{SVRT}(w,f_0)}(-1)^{|T|-\ell(w)}\prod_{(i,j)\in D(w)}\prod_{t\in T(i,j)}\bigl(x_t\oplus y_{m_{ij}(w)+i-t}\bigr),$$ where $D(w)$ is the Rothe diagram of $w$, $\mathrm{SVRT}(w,f_0)$ is the set of set-valued Rothe tableaux of shape $D(w)$ flagged by $f_0=(1,2,\ldots,n)$, $\ell(w)$ is the inversion length, $m_{ij}(w)$ is the number of diagram squares in row $i$ at or to the left of column $j$, and $a\oplus b=a+b-ab$. The forward direction is proved by induction along the first ascent using the isobaric divided difference operator, and the reverse direction is proved by showing that a 1432 pattern forces a failure already in the single-variable Schubert polynomial obtained from the lowest-degree part.

Load-bearing premise

The load-bearing premise is that the equivalence-class case analysis for tableaux of shape $D(w)$ (Theorem 2.10) is complete exactly as sketched, even though the paper gives only a sketch of the proof and relies on the operator identity stated in Lemma 2.12; a missing configuration there would break the induction connecting the two sides.

Editorial extensions

If this is right

  • Setting all $y_i$ to zero turns the main formula into a signed tableau sum for the single Grothendieck polynomial of every 1432-avoiding permutation.
  • Taking the lowest-degree homogeneous part and replacing $y_i$ by $-y_i$ yields the corresponding formulas for double and single Schubert polynomials of 1432-avoiding permutations.
  • Restricted to 321-avoiding permutations, the same formula agrees with the known flagged set-valued Young tableau formula for that family, so the Rothe-tableau model contains the older model as a special case.
  • The converse direction is an obstruction: if $w$ contains a 1432 pattern, the double Grothendieck polynomial cannot be represented by this particular Rothe-tableau sum, and the failure is visible already at the level of single Schubert polynomials.
  • Via tableau complexes, two equivalent formulas hold for 1432-avoiding permutations, one using limit set-valued Rothe tableaux and one using single-valued Rothe tableaux with correction factors.

Reading between the lines

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

  • An outside reader may test whether the same equivalence-class induction, which uses only the first-ascent condition and local configurations of squares, extends to other single-pattern avoidance classes beyond 1432; the paper does not make this claim.
  • Because the paper notes the equal enumeration of 1432-avoiding and 2143-avoiding permutations, one may look for a direct statistic-preserving bijection between set-valued Rothe tableaux of 1432-avoiding permutations and flagged set-valued Young tableaux of 2143-avoiding permutations; the paper does not construct one.
  • The tableau-complex formulas suggest a purely topological check: proving that the Rothe tableau complex is shellable would give an independent, noncomputational confirmation of the K-polynomial identities, a question not addressed here.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces set-valued Rothe tableaux, a filling model for the Rothe diagram of a permutation, and proposes that for a 1432-avoiding permutation w the double Grothendieck polynomial G_w(x,y) equals the sign-weighted sum over flagged set-valued Rothe tableaux given in (1.2). The converse is also claimed: if w contains a 1432 pattern, the formula fails. The proof of the forward direction proceeds by induction on length using the isobaric divided difference operator π_r, where r is the first ascent of w; the proof is organized around an equivalence relation on tableaux and a bijection between equivalence classes of SVRT(w s_r, f0) and SVRT(w, f0). The converse is proved by passing to single Schubert polynomials and using the balanced-labeling model of Fomin, Greene, Reiner, and Shimozono. A final section uses Knutson–Miller–Yong tableau complexes to derive two alternative formulas for G_w(x,y).

Significance. If the main theorem is correct, the paper provides the first tableau formula for double Grothendieck polynomials of 1432-avoiding permutations, a class equinumerous with vexillary permutations, and it specializes to Matsumura's formula for 321-avoiding permutations. The strategy of using equivalence classes to make the tableau model compatible with divided differences is natural and, in outline, sound; the paper also gives a useful connection to tableau complexes and to the balanced-labeling model for the converse statement. However, the current version of the manuscript does not contain a complete proof of a central lemma (Theorem 2.10), and several displayed formulas have indexing/domain problems that are load-bearing for the induction step. The result is plausible and significant, but the proof needs substantial repair before the claims are fully supported.

major comments (3)
  1. [Section 2.2, Theorem 2.10] Theorem 2.10, the formula for G_w(C';x,y) for an equivalence class C' in SVRT(w,f0), is only sketched: the text says 'The proof is nearly the same as the arguments for Theorem 2.7' and that the only difference is that P(T',r) is empty and P(T',r+1) contributes h(C',r+1;x,y). This is load-bearing because Theorem 2.11, specifically equation (2.32), uses this exact factor to identify π_r of the r-row contribution with -h(C',r+1;x,y), and Theorem 2.11 is the induction step establishing Theorem 2.5 and hence Theorem 2.1. The omitted row r+1 case is precisely where D(w) and D(w s_r) differ: the square (r,w_r) is deleted and the r-row squares to its right are shifted down to row r+1. That is exactly the configuration that cannot be assumed to follow from the analysis for i>r+1. A complete proof of Theorem 2.10, including verification of the y-exponents in h(C',r+1;x,y) and of the first factor in (2.24), is required for the induction to go through.
  2. [Section 2.2, equations (2.9) and (2.29)] Equations (2.9) and (2.29) are not well-defined as written. In (2.9), G_{w s_r}(C;x,y) is defined with a product over (i,j)∈D(w), but the tableaux T∈C have shape D(w s_r), so T(i,j) is undefined for squares of D(w) that are not in D(w s_r). Similarly, the left-hand side of (2.29) multiplies over (i,j)∈D(w) with T∈SVRT(w s_r,f0), and the right-hand side multiplies over D(w) with T'∈SVRT(w,f0); neither product is meaningful without an explicit identification of the square sets. This is not a cosmetic issue: (2.29) is substituted into (2.28), and the comparison between the two sides is the step that transfers the non-r/non-r+1 contribution from w s_r to w after the geometric shift. The intended index sets (presumably D(w s_r) on the left, together with the explicit matching of shifted squares) must be stated, and the equality proved for each type of square, especially the shifted row-r squares.
  3. [Section 2.2, equation (2.10)] The definition of ℓ_i(T) in (2.10) uses m_{i p_i}(w s_r) for every tableau T, but T may belong to SVRT(w,f0) as well as SVRT(w s_r,f0). For T∈SVRT(w,f0), the leftmost square (r+1,p_{r+1}) of P(T,r+1) is typically not a square of D(w s_r), so m_{i p_i}(w s_r) is undefined. This matters because the polynomial h(C',r+1;x,y) in Theorem 2.10 and the equality ℓ_r(T)=ℓ_{r+1}(T')-1 in (2.32) depend on this quantity. The authors should either define ℓ_i(T) separately for the two Rothe diagrams, with m computed in the corresponding diagram, or prove that the two definitions agree on the relevant squares.
minor comments (4)
  1. [Abstract and Introduction] There are several typographical slips, including 'Bu ch' in the abstract, 'shew diagram' after Figure 2.1, and 'there exits' in the proof of Theorem 2.13; these should be corrected.
  2. [Section 2.2, Lemma 2.9] The proof of Lemma 2.9 refers to 'the following two claims' and proves them, but the final sentence of the proof relies on the flag condition T(t,k)={t} for 1≤t≤i; this implication could be made more explicit for the reader.
  3. [Section 2.2, proof of (2.17)] The proof of (2.17) invokes Figure 2.5 and says 'we see that (2.17) holds'; adding a short explanation of how the definition of m_{i,j}(w s_r) in (1.1) yields the consecutive exponents ℓ_r(T)+1, ..., ℓ_r(T)+b_r(T) would improve readability.
  4. [Section 3, Theorem 3.1] The substitution in the proof of Theorem 1.4, replacing t_{x/arrownot↦a} by x_a y_{m_{ij}(w)+i-a} and then replacing x_t by 1-x_t and y_t by 1/(1-y_t), is stated very briefly; a few words on how the substitution acts on the K-polynomial and on the expression (x_t ⊕ y_{m}) would help the reader verify the formulas (1.3) and (1.4).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 1432-avoiding tableau formula is proved by a divided-difference induction from the standard definition of Grothendieck polynomials; no fitted parameter or load-bearing self-citation is present.

full rationale

The proposed tableau formula (1.2) is not used to define the objects: SVRT(w,f0) is a purely combinatorial set of flagged fillings, and the flag f0 and the exponents m_ij(w)+i-t are fixed a priori from w. The proof follows the standard inductive definition of double Grothendieck polynomials: the base case w0 is checked directly against the product formula, and for w != w0 with first ascent r the paper proves Gbar_w = pi_r Gbar_{wsr} (Theorem 2.5) through a bijection of equivalence classes and the class formulas in Theorems 2.7 and 2.10. Since the actual Grothendieck polynomial satisfies G_w = pi_r G_{wsr} by (2.7), induction yields equality. No parameter is fitted to data and then renamed as a prediction, and no load-bearing result is imported from the authors' own prior work. The external ingredients, namely Matsumura's Lemma 2.12, the balanced-labeling formula (2.34) of Fomin-Greene-Reiner-Shimozono, and the tableau-complex theorem 3.1 of Knutson-Miller-Yong, are independent theorems by other authors; they are cited as tools, not as substitutes for the main derivation. The noted weaknesses are correctness and completeness concerns rather than circularity: Theorem 2.10 is presented only as a sketch ('the proof is nearly the same as the arguments for Theorem 2.7'), and equations (2.9) and (2.29) display D(w) where D(wsr) would be expected. These issues would need to be repaired for a fully rigorous proof, but they do not make the target formula equivalent to its input by construction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No data-derived free parameters appear in the paper; the flag f0 = (1,2,...,n) is fixed by the theorem statement, not fitted. The listed axioms are external theorems used as black boxes. No ad hoc entities are introduced to force the result; the new tableaux are the object of study, not hidden postulates.

assumptions (5)
  • standard math The isobaric divided difference operators π_i satisfy the Coxeter relations and the recursion (2.7) uniquely defines double Grothendieck polynomials.
    Used in Section 2.1 to define G_w(x,y) and to justify the induction in Theorem 2.5.
  • standard math Matsumura's Lemma 2.12 (π_r applied to a product of (x_r ⊕ y_{a_j}) factors) is correct.
    Cited from [24, Lemma 4.1] and applied in the proof of Theorem 2.11 to evaluate the factor h(C,i;x,y).
  • standard math Knutson-Miller-Yong Theorem 3.1 gives the K-polynomial formulas (3.1)-(3.3) for tableau complexes.
    Used in Section 3 to prove the alternative formulas of Theorem 1.4.
  • standard math Fomin-Greene-Reiner-Shimozono balanced labeling formula (2.34) computes S_w(x).
    Used in the proof of Theorem 2.13 to show that the tableau formula fails for 1432-containing permutations.
  • standard math The lowest-degree homogeneous component of G_w(x,y), after y substitution, is S_w(x,y).
    Used to derive Corollary 1.3 and the contradiction in Theorem 2.2.

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Pith. "Pith review of Set-valued Rothe Tableaux and Grothendieck Polynomials." pith.science (2026). https://pith.science/paper/3EHRZDMI

@misc{pith2026190804164,
  author       = {Pith},
  title        = {Pith review of: Set-valued Rothe Tableaux and Grothendieck Polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3EHRZDMI}},
  note         = {Machine review of arXiv:1908.04164}
}
abstract

The notion of set-valued Young tableaux was introduced by Buch in his study of the Littlewood-Richardson rule for stable Grothendieck polynomials. Knutson, Miller and Yong showed that the double Grothendieck polynomials of 2143-avoiding permutations can be generated by set-valued Young tableaux. In this paper, we introduce the structure of set-valued Rothe tableaux of permutations. Given the Rothe diagram $D(w)$ of a permutation $w$, a set-valued Rothe tableau of shape $D(w)$ is a filling of finite nonempty subsets of positive integers into the squares of $D(w)$ such that the rows are weakly decreasing and the columns are strictly increasing. We show that the double Grothendieck polynomials of 1432-avoiding permutations can be generated by set-valued Rothe tableaux. When restricted to 321-avoiding permutations, our formula specializes to the tableau formula for double Grothendieck polynomials due to Matsumura. Employing the properties of tableau complexes given by Knutson, Miller and Yong, we obtain two alternative tableau formulas for the double Grothendieck polynomials of 1432-avoiding permutations.

Figures

Figures reproduced from arXiv: 1908.04164 by the authors.

Figure 1.1
Figure 1.1. (a) The Rothe diagram D(w), (b) a set-valued Rothe tableau, (c) a limit set-valued Rothe tableau for w = 426315. As aforementioned, a set-valued Rothe tableau of shape D(w) is a filling of finite nonempty subsets of positive integers into the squares of D(w) such that the rows are weakly decreasing and the columns are strictly increasing. For example, [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. D(w) and the corresponding skew shape σ(w) for w = 312465. Therefore, each set-valued Rothe tableau T ∈ SVRT(w,f0) can be viewed as a set￾valued (skew) Young tableau of shape σ(w) flagged by f(w). For a square (i, j) ∈ D(w), assume that α is the corresponding square of σ(w). Then we need to show that λr(α) + fr(α) − c(α) + 1 = mij (w) + i. (2.4) 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. An illustration for the proof of Lemma 2.8. [PITH_FULL_IMAGE:figures/full_fig_p012_2_2.png] view at source ↗
Figures from the paper (7 more)
Figure 2.3
Figure 2.3. Figure 2.3: An illustration for the proof of Lemma 2.9. [PITH_FULL_IMAGE:figures/full_fig_p012_2_3.png]
Figure 2.4
Figure 2.4. Figure 2.4: The first r + 1 rows of D(w) and D(wsr). Obviously, the first wr − 1 squares in the r-th row (respectively, (r + 1)-th row) of T are filled with the set {r} (respectively, {r + 1}). This implies that each square in the (r + 1)-th row of D(wsr) belongs to Q(T) and the…
Figure 2.5
Figure 2.5. Figure 2.5: An illustration of the squares in P(T, r). We next prove (2.18). For i > r, by Lemma 2.8 and Lemma 2.9, the configuration of the squares of P(T) and Q(T) must be as illustrated as in [PITH_FULL_IMAGE:figures/full_fig_p015_2_5.png]
Figure 2.6
Figure 2.6. Figure 2.6: The configuration of the squares in P(T, i) with i > r + 1. in mind that P(T ′ , i) = P(T, i). Then we have the following two cases. Case 1: In T ′ , the first k (0 ≤ k ≤ bi(T)) squares in P(T, i) contain r + 1, and the remaining bi(T) − k squares in P(T, i) contain …
Figure 2.7
Figure 2.7. Figure 2.7: Two balanced labelings for w = 25143. A balanced labeling of D(w) is said to be column strict if no column contains two equal labels. Let CSBL(w,f0) denote the set of column strict balanced labelings of D(w) such that L(i, j) ≤ i for each square (i, j) ∈ D(w). Fomin,…
Figure 2.8
Figure 2.8. Figure 2.8: An example for the proof of Theorem 2.13. [PITH_FULL_IMAGE:figures/full_fig_p021_2_8.png]
Figure 2.9
Figure 2.9. Figure 2.9: A balanced labeling in CSBL(w,f0), but not in SRT(w,f0). By the construction of L, it is not hard to check that L is a column strict balanced labeling in CSBL(w,f0). Moreover, the entries in the j-th column of L are not increasing. So L does not belong to SRT(w,f0). …

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    1999

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.