A Pieri rule for Demazure characters of the general linear group

Danjoseph Quijada; Sami Assaf

A Pieri rule for Demazure characters of the general linear group

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Pith's one-line read A signed Pieri rule for key polynomials, proved by box insertion into Kohnert diagrams.

desk verdict A substantial new bijective Pieri rule for key polynomials, with the main caveat being a load-bearing imported lemma that the authors do not reprove. read the letter →

arxiv 1908.08502 v1 pith:XO7TJTWA submitted 2019-08-22 math.CO

Sami Assaf , Danjoseph Quijada This is my paper · ORCID

classification math.CO MSC 05E0505E1014N1014N15

keywords DemazurecharacterskeypolynomialsPieriruleKohnertdiagramsRobinson-Schensted-KnuthinsertionleftswaporderSchubertmultiplicity-freeexpansion

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

The paper establishes a nonsymmetric Pieri rule for key polynomials, the characters of Demazure modules for the general linear group. It proves that multiplying any key polynomial $\kappa_a$ by a one-row Schur polynomial $s_{(1)}(x_1,\ldots,x_k)$ expands in the key basis with no coefficients other than $+1$ and $-1$, and it gives an explicit indexing of the terms by $k$-addable cells and subsets of rows. The proof is bijective: a single-box version of Robinson--Schensted--Knuth insertion on Kohnert diagrams gives a weight-preserving bijection between the diagram set of $a$ times a single box and the union of diagram sets $KD(b+e_j)$ for $b$ below $a$ in left swap order and $j\le k$, with signs arising only from overlaps in that union. If correct, the formula supplies the missing nonsymmetric analog of the classical Pieri rule and yields explicit positivity criteria, including a nonnegative expansion for products of Schubert polynomials with Grassmannian Schubert polynomials in the vexillary case.

What carries the argument

The machinery is Kohnert diagrams together with the left swap order and thread decomposition. A Kohnert diagram is a finite set of unit cells obtained from the key diagram of a weak composition by repeatedly moving the rightmost cell of a row down to the first available position in its column, and key polynomials are the generating polynomials of these diagrams. The left swap order $b\preceq a$ is the transitive closure of swapping a smaller part of $a$ with a larger part to its right; it controls containment, since $\mathrm{key}_b\in KD(a)$ exactly when $b\preceq a$, and more generally a diagram $T$ lies in $KD(a)$ exactly when its thread weight $\theta(T)$ lies below $a$ in this order. The paper's new insertion maps---bottom insertion, rectification, top insertion, and stratum maps---play the role of RSK insertion on tableaux and prove the bijection of Theorem 3.1.4. The drop composition $\mathrm{drop}(c,R)_a$ is the weak composition obtained by lowering supporting cells from above the added cell down to the rows of $R$; it is the object that records the inclusion-exclusion data in the final formula.

What would settle it

Search for a generic Kohnert diagram $T$ and a weak composition $a$ with $\theta(T)\preceq a$ but $T\notin KD(a)$; one such pair refutes the imported thread-weight criterion and, with it, Theorem 3.3.6. Because all objects are finite and algorithmically enumerable, this is a concrete brute-force check.

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Extended reading notes

Core claim

The central claim is Theorem 3.3.6: for a weak composition $a$ and positive integer $k$, $\kappa_a s_{(1)}(x_1,\ldots,x_k)$ equals the signed sum over $k$-addable columns $c$ of $a$ and nonempty subsets $R$ of the $k$-addable row set $\mathrm{Row}^c_{a,k}$ of $(-1)^{|R|-1}\kappa_{\mathrm{drop}(c,R)_a+e_{\min(R)}}$, and the terms are pairwise distinct. The indexing data are read directly from the key diagram of $a$: a $k$-addable cell is a position in a row at most $k$ to which a box can be added after supporting cells above row $k$ are dropped down through left swaps, and the drop composition records how far those supporting cells fall. The theorem is proved through the weight-preserving bijection of Theorem 3.1.4, $KD(a)\times KD(e_k)\leftrightarrow\bigcup_{b\preceq a,\,1\le j\le k} KD(b+e_j)$, constructed by an insertion algorithm on Kohnert diagrams. The negative signs are not cancellations inside a single disjoint union; they come from genuine overlaps of the sets $KD(b+e_j)$, and the inclusion-exclusion in the formula removes exactly that redundancy.

Load-bearing premise

The load-bearing premise is the imported criterion, used without proof, that a generic Kohnert diagram lies in $KD(a)$ exactly when its thread weight lies below $a$ in the left swap order; if that criterion has hidden restrictions, the bijection and the signed formula collapse.

Editorial extensions

If this is right

The product of any key polynomial with $x_1+\cdots+x_k$ has a cancellation-free expansion in the key basis, with at most one positive term per $k$-addable column and signs recorded by subsets of rows.
For weakly increasing $a$ and $k=n$, the formula specializes to the classical Schur Pieri rule $s_\lambda s_{(1)}=\sum_{\mu\supset\lambda,\,|\mu/\lambda|=1} s_\mu$.
For $k\ge \ell(a)$, the expansion is nonnegative and is indexed by choosing columns $c_1<\cdots<c_m$ with each $c_i-1$ among the parts of the current composition, as stated in Corollary 6.3.3.
For weak compositions satisfying condition (2) of Macdonald's vexillary definition, the key expansion of $\kappa_a s_{(m)}(x_1,\ldots,x_k)$ is nonnegative for every $k$, giving an explicit formula for the Schubert polynomial product $S_wS_{v((m),k)}$ in Theorem 6.3.8.
Iterating the insertion gives a horizontal-strip bijection $KD(a)\times KD(m e_k)\leftrightarrow D^{(m)}(a,k)$, expanding the product $\kappa_a\kappa_{m e_k}$ by $k$-addable horizontal $m$-strips, as in Theorem 6.1.5 and Corollary 6.1.8.

Reading between the lines

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

The signs, which arise from overlaps of diagram sets rather than from a sign-reversing involution, suggest that the formula could be lifted to a K-theoretic or Euler-characteristic Pieri rule for Demazure characters; this is a testable extension, not a claim of the paper.
The same stratum machinery might yield a direct rule for products of a key polynomial with an arbitrary Schur polynomial in a fixed number of variables, with explicit sign data refining the earlier formula; the paper only treats the single-row case.
A brute-force verification on all weak compositions of a small fixed length would test both the formula and the imported thread-weight criterion, since the two are logically tied.

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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

You should know about this paper if you care about key polynomials, Demazure characters, or bijective combinatorics. It proves a nonsymmetric Pieri rule for the product of an arbitrary key polynomial with a single-part key polynomial, with no restriction on the number of variables. The central bijection (Theorem 3.1.4) between KD(a) × KD(e_k) and a union of Kohnert diagram spaces is genuinely new, and the signed inclusion-exclusion formula (Theorem 3.3.6) with pairwise distinct terms is a clean statement that reduces to the classical Pieri rule when the composition is weakly increasing. I did not find a concrete error.

What the paper does well: it builds the bijection from scratch, treating the k=1 case with bottom insertion, the k ≥ ℓ(a) case with top insertion via rectification, and the middle cases with stratification maps. The reduction to k-addable cells is natural, and the worked examples are extensive and consistent. The final applications to vexillary compositions and Schubert polynomials give real payoff, and the connection to Haglund–Luoto–Mason–van Willigenburg is handled honestly.

Where I am cautious: nearly every containment or membership step in the proof relies on Lemma 2.3.8, imported from Assaf–Searles and stated as implicit in their Theorem 3.7. That lemma says a generic Kohnert diagram T lies in KD(a) iff its thread weight is below a in left swap order. The stress-test note is right to identify this as load-bearing: if the imported criterion has a hidden genericity restriction, the bijection and formula could collapse. The paper does not reprove the lemma, and a few later arguments are compressed enough that I would not want to referee this without that lemma in front of me. But I have no evidence of an actual counterexample, and the authors clearly state the dependence.

Bottom line: this is a serious combinatorial advance, aimed at algebraic combinatorics and Schubert calculus readers. It deserves a careful peer review, not a desk reject. My recommendation: send it out, and ask the authors to either prove Lemma 2.3.8 in a self-contained appendix or give a precise statement with a more explicit reference. I would cite it if I worked in this area.

Referee Report

0 major / 4 minor

Summary. The paper proves a nonsymmetric generalization of the Pieri rule for key polynomials (Demazure characters of GL_n). The main bijective statement, Theorem 3.1.4, asserts a weight-preserving bijection between KD(a) × KD(e_k) and the union of KD(b + e_j) over b weakly below a in left swap order and 1 ≤ j ≤ k. From this, Theorem 3.3.6 derives a cancellation-free, multiplicity-free signed formula for κ_a · s_(1)(x_1, ..., x_k) as a sum over k-addable columns and nonempty subsets R of the corresponding row sets. Sections 4 and 5 prove the bijection through bottom insertion, top insertion via rectification, and a stratification by stratum maps; Section 6 extends the construction to horizontal strips and characterizes when the key expansion is nonnegative, with applications to Schubert polynomials.

Significance. If the result stands, it is a substantial contribution: it gives the first explicit nonsymmetric Pieri rule for key polynomials, generalizes the classical RSK-based proof of Pieri's rule, and provides a signed but cancellation-free expansion whose signs are fully understood. The paper is also notable for its detailed, example-rich combinatorial proof and for the clean reduction to classical Schur Pieri in the weakly increasing case. The positivity characterization in Section 6.3, including the vexillary case, is a further valuable dividend. The main caveat is that the proof leans on the imported thread-decomposition criterion of Lemma 2.3.8; this is a cited result rather than an internal error, but its exact statement should be made fully explicit given how many later statements depend on it.

minor comments (4)

[Section 2.3, Lemma 2.3.8] Lemma 2.3.8, stated as implicit in [5, Theorem 3.7], is the load-bearing criterion that T ∈ KD(a) if and only if θ(T) ≼ a. It is used repeatedly—for example in Proposition 2.3.9, Corollary 3.1.5, Lemma 3.2.5, Theorem 3.2.10, Lemma 3.3.4, and Theorem 4.1.8—and therefore underpins Theorem 3.3.6. Since the paper does not reproduce or prove this criterion, I ask the authors to provide either a self-contained proof or a precise quotation of [5, Theorem 3.7] together with an explanation of why the stated 'if and only if' follows for generic Kohnert diagrams. This is a completeness issue rather than an apparent error, but it should be resolved before publication.
[Section 6.3, Corollary 6.3.1] In the displayed formula, the condition 'c_i−1 ∈ {a_1,...,a_n, c_i−1}' is tautological as written. Based on Example 6.3.2 and the surrounding text, it should read 'c_i−1 ∈ {a_1,...,a_n, c_{i−1}}' (with the second term being the previously chosen value, not the current value). Please correct this notation.
[Section 4.3, Theorem 4.3.15 and text before it] The notation 'ℓ = max_i{a_i > 0}' is ambiguous; it should be 'ℓ = max{i : a_i > 0}' or 'ℓ(a) = max{i : a_i > 0}', matching the use earlier in the paper. As written, the expression resembles a maximum over a set of inequalities rather than the largest index with a positive part.
[Section 5.2, proof of Theorem 5.1.13] In the paragraph after Lemma 5.2.8, the sentence 'By Lemma 5.1.8, we have wt(M_{b+e_k}|_{U^-}) = wt(M_{b+e_k}|_{U^-})' is tautological and appears to contain a typo. Please check whether the intended equality involves θ(U^-) or the restriction of the Kohnert labeling, and revise accordingly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Pieri formula is derived from an independently constructed Kohnert-diagram bijection, with only a non-load-bearing external thread-criterion citation.

full rationale

The derivation of Theorem 3.3.6 does not assume the Pieri-type expansion it proves. The target bijection in Theorem 3.1.4 is constructed independently: bottom insertion is proved directly in Theorem 4.1.13, top insertion via rectification in Theorem 4.3.15, and the general case through the injective stratum maps of Theorems 5.1.13 and 5.1.14. None of these arguments presuppose the coefficient formula or reduce it to a definition. The addable-cell reductions and intersection computations, such as Lemmas 3.2.5, 3.2.8, 3.3.4, and Theorem 3.2.10, are containment statements in the left-swap order, and these rely on the imported Lemma 2.3.8 from Assaf-Searles [5]. That lemma is a parameter-free criterion characterizing membership in Kohnert diagram sets in terms of thread weights; its statement does not involve products of key polynomials, the Pieri coefficients, or the desired expansion. Although the present paper cites prior work with overlapping authorship and does not reprove the lemma, the cited result is independent evidence rather than a restatement of the goal, so it does not make the argument circular. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported to forbid alternatives, and the final signed expansion is obtained by generating functions and inclusion-exclusion from a genuinely constructed bijection. Hence the derivation chain is self-contained in the relevant sense: the theorem is not equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data; the formulas are parameter-free combinatorial rules. No new physical or mathematical objects are postulated beyond the combinatorial constructions already defined in the proof. The proof does depend on several prior results, most notably the Assaf-Searles thread decomposition criterion, which is the weakest imported premise.

assumptions (6)

domain assumption Key polynomials are the generating polynomials of Kohnert diagrams (Kohnert's theorem, taken as Definition 2.2.3).
The paper defines kappa_a via the Kohnert diagram generating function. This is standard but is an external characterization.
domain assumption A diagram T is a generic Kohnert diagram iff the column inequality (2.3.1) holds (Proposition 2.3.2, from Assaf-Searles).
Used throughout to decide when diagrams are generic, including rectification and matching arguments.
domain assumption For T generic, T is in KD(a) iff theta(T) is at most a in the left swap order (Lemma 2.3.8, cited to Assaf-Searles).
This is the load-bearing membership criterion; the paper relies on it in Proposition 2.3.9 and all containment arguments.
domain assumption The injective map from Kohnert diagrams to semistandard Young tableaux of sorted shape is a bijection exactly for weakly increasing compositions (Assaf-Searles, Proposition 2.2.4).
Used in Corollary 2.2.6 and in the RSK-to-rectification identification in Theorem 4.2.7.
standard math RSK insertion yields the Pieri bijection for semistandard Young tableaux (Theorem 3.1.1 and Theorem 6.1.1).
Imported classical result used in the extremal k=n case and in the horizontal-strip iteration.
standard math Hall's Marriage Theorem implies the matching sequence characterization of Kohnert diagrams.
Invoked in Section 4.1 to connect the column inequality to the existence of matchings.

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Pith. "Pith review of A Pieri rule for Demazure characters of the general linear group." pith.science (2026). https://pith.science/paper/XO7TJTWA

@misc{pith2026190808502,
  author       = {Pith},
  title        = {Pith review of: A Pieri rule for Demazure characters of the general linear group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XO7TJTWA}},
  note         = {Machine review of arXiv:1908.08502}
}

read the original abstract

The Pieri rule is a nonnegative, multiplicity-free formula for the Schur function expansion of the product of an arbitrary Schur function with a single row Schur function. Key polynomials are characters of Demazure modules for the general linear group that generalize the Schur function basis of symmetric functions to a basis of the full polynomial ring. We prove a nonsymmetric generalization of the Pieri rule by giving a cancellation-free, multiplicity-free formula for the key polynomial expansion of the product of an arbitrary key polynomial with a single part key polynomial. Our proof is combinatorial, generalizing the Robinson--Schensted--Knuth insertion algorithm on tableaux to an insertion algorithm on Kohnert diagrams.

Figures

Figures reproduced from arXiv: 1908.08502 by the authors.

**Figure 1.** The Young diagram for the partition (5, 4, 4, 1). A semistandard Young tableau of shape λ is a filling of the cells of the Young diagram of λ with positive integers such that entries weakly increase left to right along rows and strictly increase top to bottom down columns. Let SSYTn(λ) denote the set of semistandard Young tableaux of shape λ with image in {1, 2, . . . , n}. Here we emphasize that entries may not exc… view at source ↗

**Figure 2.** The set SSYT3(3, 2) of semistandard Young tableaux of shape (3, 2) with largest entry 3. whose ith component is equal to the number of occurrences of i in T . For example, the weights of the first column of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** shows the key diagram for the weak composition (4, 1, 5, 0, 4). ❣❣❣❣ ❣❣❣❣❣ ❣ ❣❣❣❣ [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (25 more)

**Figure 4.** Figure 4: Construction of Kohnert diagrams for (0, 3, 2), where southeast edges indicate Kohnert moves on the second row, and south or southwest edges indicate Kohnert moves on the third row. To each diagram T of cells in the first quadrant we associate the weak composition wt(…

**Figure 5.** Figure 5: An example of the injective map from Kohnert diagrams to semistandard Young tableaux. In particular, this immediately gives a combinatorial proof of the following. Corollary 2.2.6 ([16]). For a weakly increasing of length n, we have (2.2.2) κa = srev(a)(x1, . . . , xn…

**Figure 6.** Figure 6: The left swap order on weak compositions of length 4 that sort to the partition (3, 2, 2). Example 2.3.5. Setting n = 4 and taking λ = (3, 2, 2), the Hasse diagram for the left swap order on weak compositions of length n that sort to λ is shown in [PITH_FULL_IMAGE:fig…

**Figure 7.** Figure 7: An example of thread decomposition of a generic Kohnert diagram, where the cells of a given thread are labeled the same. Implicit in [5, Theorem 3.7], we have the following useful fact. Lemma 2.3.8 ([5]). For a generic Kohnert diagram T , we have T ∈ KD(a) if and only…

**Figure 8.** Figure 8: The four addable cells (•) for the partition (5, 4, 4, 1). Example 3.1.3. The partition λ = (5, 4, 4, 1) has four addable cells illustrated in [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: The four 3-addable cells (•) for the weak composition (4, 1, 5, 0, 4), where for the right diagram we use (4, 4, 5, 0, 1) ≺ (4, 1, 5, 0, 4) to support the addable cell. When the cell in row r and column c is an addable cell for a, we also require a way to construct the…

**Figure 10.** Figure 10: The five addable cells (•) for (4, 6, 4, 3, 0, 1, 1, 2, 5, 4) in column 5. Here the marked cells (⊗) drop to positions (+) in row r in creating the maximal support composition. The first, third and fourth are 6-addable but the second and fifth are not. We can now re-c…

**Figure 11.** Figure 11: The four nonempty, non-singleton subsets of rows (•) of the 6-addable cells for (4, 6, 4, 3, 0, 1, 1, 2, 5, 4) in column 5. Here marked cells (⊗) will drop down to the indicated position (+) below in creating the maximal drop composition. Lemma 3.3.4. Let a be a weak …

**Figure 12.** Figure 12: The Kohnert diagrams for (0, 3, 2) along with the appended cell (⊕) under the bottom insertion map ∆1. In order to show ∆1(T ) is a generic Kohnert diagram, we reformulate the criterion given in Proposition 2.3.2 by generalizing the thread decomposition given in Def…

**Figure 13.** Figure 13: The four possible matching sequences of a generic Kohnert diagram along with their anchor weights (below), where the matched cells in adjacent columns are labeled the same. Example 4.1.5. Consider the generic Kohnert diagram from Example 2.3.7. Considering columns 4 …

**Figure 14.** Figure 14: A generic Kohnert diagram U in D((4, 1, 5, 0, 4), n) for n ≥ 3 with added column 2. Example 4.1.12. Let a = (4, 1, 5, 0, 4), and consider the generic Kohnert diagram U in [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]

**Figure 15.** Figure 15: A diagram T (left) with each position (c, r) with c > 1 labeled by mT (c, r), and its image under ρ (right). the diagram on the right side of [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]

**Figure 16.** Figure 16: Rectification of a weak but not generic Kohnert diagram, where ̺ acts by moving the colored cell left. Example 4.2.6. Consider the diagram on the left side of [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]

**Figure 17.** Figure 17: An example of RSK insertion on an element of SSYT5(5, 4, 4, 1). Example 4.2.8. Consider the tableau T ∈ SSYT5(5, 4, 4, 1) on the left side of [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]

**Figure 18.** Figure 18: Three copies of the Kohnert diagrams for (0, 2, 1) along with the appended cell (⊕) under the top insertion map ∆∞(−, j) for j = 3, 2, 1, from left to right. Remark 4.3.3. Observe the maps ∆1 and ∆∞(−, 1) differ in general. For instance, we have ∆1(key(0,2,1)) = key(1…

**Figure 19.** Figure 19: An illustration of the relation between the matching sequences Mθ(T \ {x}) (solid) and Mθ(T \ {y}) (dashed). and the result holds with t = 1. We now proceed by induction on the number of threads of Mθ(T ), noting that by Lemma 4.3.8, Mθ(T ) is simply M together with a…

**Figure 20.** Figure 20: An illustration of the infinite descending sequence of cells constructed from the matchings M = Mθ(U) (thick) and M∗ = Mθ(U \ {x}) (dotted). Continuing, we may set y2 = M(z1) 6= M(z) = y1. Since threading selects the lowest available cell, we must have M(z1) = y2 belo…

**Figure 21.** Figure 21: A generic Kohnert diagram U (left) in D(a, k) for a = (1, 5, 2, 1, 2, 6, 3) and k = 3 with excised column c = 4, its thread decomposition (middle), and the key diagram keyθ(U) (right) with the cell in position (c, k) indicated. Example 5.1.7. Consider the weak composi…

**Figure 22.** Figure 22: The partitioning of the generic Kohnert diagram U (left) in D(a, k) for a = (1, 5, 2, 1, 2, 6, 3) and k = 3 using the Kohnert labeling to obtain U + (middle) and U − (right). Example 5.1.10. Continuing with Ex. 5.1.7, we take the Kohnert labeling the diagram U with re…

**Figure 23.** Figure 23: The rectification of the diagram U + ∗ obtained from U + by removing the cell in position (1, 3). Example 5.1.12. Continuing with Ex. 5.1.10, we remove the cell in position (1, k) of U + and rectify to obtain the diagram on the right side of [PITH_FULL_IMAGE:figures/…

**Figure 24.** Figure 24: An illustration of the Pieri rule for computing the Schur expansion of the product s(3,2,2)s(2), where the two marked cells (•) denote the added horizontal 2-strip. The target space of the full key-Pieri bijection may be stated as the union of the Kohnert spaces of al…

**Figure 25.** Figure 25: An illustration of the Pieri rule computing the key expansion of the product κ(2,0,3,2)κ(0,0,2), where the two marked cells (•) denote the added horizontal 2-strip and crossed cells (×) drop to the marked positions (+). such that the columns indices c1, c2, . . . , cm…

**Figure 26.** Figure 26: An example of the nonnegative Pieri rule for bottom insertion, computing the key expansion of the product κ(1,4,0,3)κ(2), where the two marked cells (•) denote the added horizontal 2-strip and crossed cells (×) drop to the marked positions (+). Second, consider the su…

**Figure 27.** Figure 27: An example of the nonnegative Pieri rule for top insertion, computing the key expansion of κ(1,4,0,3)κ(0,0,0,2), where the two marked cells (•) denote the added horizontal 2-strip and crossed cells (×) drop to the marked positions (+). Our third subcase is more invol…

**Figure 28.** Figure 28: An example of the nonnegative Pieri rule in the vexillary case, computing the key expansion of κ(0,1,4,3)κ(0,0,2), where the two marked cells (•) denote the added horizontal 2-strip and crossed cells (×) drop to the marked positions (+). Lascoux and Sch¨utzenberger […

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