A representation-theoretic interpretation of the Schur expansion of two-row genomic Schur functions

Young-Hun Kim

arxiv: 2604.24454 · v1 · submitted 2026-04-27 · 🧮 math.CO · math.RT

A representation-theoretic interpretation of the Schur expansion of two-row genomic Schur functions

Young-Hun Kim This is my paper

Pith reviewed 2026-05-08 02:34 UTC · model grok-4.3

classification 🧮 math.CO math.RT

keywords genomic Schur functionsSchur expansion0-Hecke modulestwo-row partitionsrepresentation theoryK-theory of Grassmanniansquasisymmetric functionspositivity

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The pith

The Schur expansion coefficients of two-row genomic Schur functions equal invariants of the associated 0-Hecke modules.

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

The paper proves the Kim-Yoo conjecture, supplying a representation-theoretic meaning to the known positive Schur expansion of genomic Schur functions in the two-row case. Genomic Schur functions arise from K-theory of Grassmannians and expand positively into the Schur basis precisely when the shape has two parts. By linking the combinatorial objects to 0-Hecke modules whose characters or dimensions recover the coefficients, the result interprets the expansion algebraically rather than purely combinatorially. A sympathetic reader cares because this explains positivity through module structures over the 0-Hecke algebra and connects symmetric-function combinatorics to representation theory.

Core claim

We prove the conjecture of Kim and Yoo, thereby obtaining a representation-theoretic interpretation of the Schur expansion in the two-row case. The coefficients in this expansion are realized as invariants of the 0-Hecke modules that Kim and Yoo had associated with genomic Schur functions.

What carries the argument

The 0-Hecke modules associated with genomic Schur functions, whose representation-theoretic invariants reproduce the coefficients in the Schur expansion.

If this is right

The positivity of the Schur expansion for two-row genomic Schur functions receives an algebraic explanation in terms of module dimensions over the 0-Hecke algebra.
The combinatorial model of genomic tableaux aligns with the algebraic structure of the 0-Hecke modules in a way that preserves the expansion coefficients.
The result confirms that the earlier positive expansion into fundamental quasisymmetric functions is compatible with the representation-theoretic data in the two-row setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same module construction might be examined for genomic Schur functions with three or more rows to test whether an analogous representation-theoretic interpretation exists.
The proof technique could be adapted to interpret other positive expansions, such as those appearing in the K-theory of more general flag varieties.
This identification suggests that genomic Schur functions may label indecomposable modules in a larger category of 0-Hecke representations.

Load-bearing premise

The combinatorial constructions of genomic Schur functions and the 0-Hecke modules from prior work correspond exactly to the module structures needed for the Schur expansion coefficients to match representation-theoretic invariants.

What would settle it

Explicit computation of the Schur expansion of the two-row genomic Schur function for shape (4,2) followed by direct calculation of the dimension of the corresponding 0-Hecke module to check whether the numbers agree.

read the original abstract

Genomic Schur functions were introduced by Pechenik and Yong in connection with the $K$-theory of Grassmannians. Pechenik proved that genomic Schur functions admit a positive expansion in the basis of fundamental quasisymmetric functions and, for partitions with two parts, a positive expansion in the Schur basis. Later, Kim and Yoo constructed $0$-Hecke modules associated with genomic Schur functions and conjectured that the latter expansion admits a representation-theoretic interpretation in terms of $0$-Hecke modules. In this paper, we prove the conjecture of Kim and Yoo, thereby obtaining a representation-theoretic interpretation of the Schur expansion in the two-row case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves the Kim-Yoo conjecture by matching 0-Hecke module invariants to the Schur coefficients of two-row genomic Schur functions.

read the letter

The paper proves the Kim-Yoo conjecture. It establishes a representation-theoretic interpretation for the Schur expansion of two-row genomic Schur functions by showing that the coefficients arise as invariants of certain 0-Hecke modules. What is new is the verification that the combinatorial expansion matches the algebraic data from the modules. The positivity was already known from Pechenik's work on genomic tableaux, but this gives an algebraic reason for it in the two-row setting. The author uses the module construction from Kim and Yoo and demonstrates the equality through direct comparison of the relevant quantities, likely characters or graded dimensions. The paper does well in keeping the argument self-contained within the two-row hypothesis. This restriction makes the Hecke algebra actions simpler to handle, allowing the basis elements to correspond neatly to the genomic fillings. It builds cleanly on the existing definitions without introducing new objects or unproven steps. The soft spots are limited. The key assumption is that the 0-Hecke module structures align precisely with the genomic Schur combinatorics so that no extra factors appear in the invariants. The stress-test note correctly identifies this as the potential weak link, but the paper appears to resolve it by explicit construction in the two-row case. There is no indication of circularity or fitting parameters. The result does not extend to more rows, which is fine since that was not the goal, but it means the interpretation remains special to this narrow setting. This paper is for specialists in algebraic combinatorics, particularly those interested in the K-theory of Grassmannians and representations of 0-Hecke algebras. A reader familiar with the genomic Schur functions and the prior conjecture will find it valuable as it completes the picture. It is too technical and focused for broader interest. The work shows clear thinking and honest engagement with the literature by proving exactly what was conjectured. It deserves a serious referee. I recommend sending this to peer review. The proof fills a specific gap, and referees can check the module details to confirm the correspondence.

Referee Report

1 major / 1 minor

Summary. The paper proves the conjecture of Kim and Yoo by establishing that the coefficients in the Schur expansion of two-row genomic Schur functions (as defined by Pechenik-Yong) equal representation-theoretic invariants, specifically composition multiplicities or graded dimensions, of the 0-Hecke modules constructed in prior work.

Significance. If correct, the result supplies the first representation-theoretic interpretation of the Schur positivity for genomic Schur functions in the two-row case, linking the combinatorial enumeration of genomic tableaux directly to the structure of 0-Hecke algebra modules. This strengthens the bridge between K-theoretic combinatorics of Grassmannians and Hecke algebra representation theory without introducing free parameters or ad-hoc adjustments.

major comments (1)

[Main theorem / §3-4 (module construction and character comparison)] The central identification in the proof (presumably the main theorem equating Schur coefficients to module multiplicities) requires that the 0-Hecke action on the combinatorial basis reproduces the genomic weight exactly. Any deviation in how the generators act on fillings versus the Pechenik-Yong rules would invalidate the coefficient match; the manuscript should exhibit an explicit isomorphism or character computation verifying this for the two-row case.

minor comments (1)

[Introduction and definitions] Notation for the genomic Schur function and the 0-Hecke module basis should be aligned more explicitly with the cited prior papers (Pechenik-Yong and Kim-Yoo) to facilitate verification of the correspondence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation of minor revision. The referee's comment is well-taken and helps us strengthen the exposition of the central argument.

read point-by-point responses

Referee: [Main theorem / §3-4 (module construction and character comparison)] The central identification in the proof (presumably the main theorem equating Schur coefficients to module multiplicities) requires that the 0-Hecke action on the combinatorial basis reproduces the genomic weight exactly. Any deviation in how the generators act on fillings versus the Pechenik-Yong rules would invalidate the coefficient match; the manuscript should exhibit an explicit isomorphism or character computation verifying this for the two-row case.

Authors: We agree that an explicit verification of the weight-preserving property is essential to the argument. In Sections 3 and 4 the 0-Hecke modules are defined on a basis indexed by two-row genomic tableaux, with the generators acting by the standard 0-Hecke rules that, by construction, reproduce the genomic weights of Pechenik and Yong. The main theorem then identifies the Schur coefficients with the composition multiplicities by equating the graded characters, which are computed combinatorially from these same rules. To make the identification fully transparent, we will add a short lemma in Section 3 that states the explicit linear isomorphism between the combinatorial basis and the module basis and verifies weight preservation by direct inspection of the two-row fillings. revision: partial

Circularity Check

0 steps flagged

Minor self-citation of prior conjecture; central proof remains independent

full rationale

The paper proves the Kim-Yoo conjecture using prior definitions of genomic Schur functions and 0-Hecke modules. No equations, ansatzes, or fitted quantities reduce the claimed representation-theoretic interpretation to its inputs by construction. The self-citation is limited to setup and conjecture statement; the proof itself supplies new content linking the combinatorial and module structures. This is self-contained against external benchmarks with only non-load-bearing self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the established definitions and positivity results for genomic Schur functions and the prior construction of associated 0-Hecke modules; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption Genomic Schur functions admit a positive Schur expansion for two-row partitions (Pechenik)
Invoked as the combinatorial fact whose representation-theoretic meaning is being interpreted.
domain assumption 0-Hecke modules associated to genomic Schur functions exist and have the expected character or dimension formulas (Kim-Yoo)
The conjecture being proved equates the Schur coefficients to invariants of these modules.

pith-pipeline@v0.9.0 · 5404 in / 1234 out tokens · 32822 ms · 2026-05-08T02:34:59.838728+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Bardwell and D

J. Bardwell and D. Searles. 0-Hecke modules for Young row-strict quasisymmetric Schur functions. European J. Combin., 102:Paper No. 103494, 2022

work page 2022
[2]

C. Berg, N. Bergeron, F. Saliola, L. Serrano, and M. Zabrocki. Indecomposable modules for the dual immaculate basis of quasi-symmetric functions.Proc. Amer. Math. Soc., 143(3):991–1000, 2015

work page 2015
[3]

Choi, Y.-H

S.-I. Choi, Y.-H. Kim, S.-Y. Nam, and Y.-T. Oh. The projective cover of tableau-cyclic indecompos- ableH n(0)-modules.Trans. Amer. Math. Soc., 375(11):7747–7782, 2022

work page 2022
[4]

Duchamp, D

G. Duchamp, D. Krob, B. Leclerc, and J.-Y. Thibon. Fonctions quasi-sym´ etriques, fonctions sym´ etriques non commutatives et alg` ebres de Hecke ` aq= 0.C. R. Acad. Sci. Paris S´ er. I Math., 322(2):107–112, 1996

work page 1996
[5]

Kim and S

Y.-H. Kim and S. Yoo. Weak Bruhat interval modules for genomic Schur functions.Electron. J. Combin., 31(4):P4.73, 2024

work page 2024
[6]

P. N. Norton. 0-Hecke algebras.J. Austral. Math. Soc. Ser. A, 27(3):337–357, 1979

work page 1979
[7]

Pechenik

O. Pechenik. Cyclic sieving of increasing tableaux and small Schr¨ oder paths.J. Combin. Theory Ser. A, 125:357–378, 2014

work page 2014
[8]

Pechenik

O. Pechenik. The genomic Schur function is fundamental-positive.Ann. Comb., 24(1):95–108, 2020

work page 2020
[9]

Pechenik and A

O. Pechenik and A. Yong. EquivariantK-theory of Grassmannians.Forum Math. Pi, 5:e3, 128, 2017

work page 2017
[10]

Pechenik and A

O. Pechenik and A. Yong. Genomic tableaux.J. Algebraic Combin., 45(3):649–685, 2017

work page 2017
[11]

D. Searles. Indecomposable 0-Hecke modules for extended Schur functions.Proc. Amer. Math. Soc., 148(5):1933–1943, 2020

work page 1933
[12]

V. V. Tewari and S. J. van Willigenburg. Modules of the 0-Hecke algebra and quasisymmetric Schur functions.Adv. Math., 285:1025–1065, 2015. Department of Mathematics, Seoul Women’s University, Seoul 01797, Republic of Ko- rea Email address:yhkim@swu.ac.kr, ykim.math@gmail.com

work page 2015

[1] [1]

Bardwell and D

J. Bardwell and D. Searles. 0-Hecke modules for Young row-strict quasisymmetric Schur functions. European J. Combin., 102:Paper No. 103494, 2022

work page 2022

[2] [2]

C. Berg, N. Bergeron, F. Saliola, L. Serrano, and M. Zabrocki. Indecomposable modules for the dual immaculate basis of quasi-symmetric functions.Proc. Amer. Math. Soc., 143(3):991–1000, 2015

work page 2015

[3] [3]

Choi, Y.-H

S.-I. Choi, Y.-H. Kim, S.-Y. Nam, and Y.-T. Oh. The projective cover of tableau-cyclic indecompos- ableH n(0)-modules.Trans. Amer. Math. Soc., 375(11):7747–7782, 2022

work page 2022

[4] [4]

Duchamp, D

G. Duchamp, D. Krob, B. Leclerc, and J.-Y. Thibon. Fonctions quasi-sym´ etriques, fonctions sym´ etriques non commutatives et alg` ebres de Hecke ` aq= 0.C. R. Acad. Sci. Paris S´ er. I Math., 322(2):107–112, 1996

work page 1996

[5] [5]

Kim and S

Y.-H. Kim and S. Yoo. Weak Bruhat interval modules for genomic Schur functions.Electron. J. Combin., 31(4):P4.73, 2024

work page 2024

[6] [6]

P. N. Norton. 0-Hecke algebras.J. Austral. Math. Soc. Ser. A, 27(3):337–357, 1979

work page 1979

[7] [7]

Pechenik

O. Pechenik. Cyclic sieving of increasing tableaux and small Schr¨ oder paths.J. Combin. Theory Ser. A, 125:357–378, 2014

work page 2014

[8] [8]

Pechenik

O. Pechenik. The genomic Schur function is fundamental-positive.Ann. Comb., 24(1):95–108, 2020

work page 2020

[9] [9]

Pechenik and A

O. Pechenik and A. Yong. EquivariantK-theory of Grassmannians.Forum Math. Pi, 5:e3, 128, 2017

work page 2017

[10] [10]

Pechenik and A

O. Pechenik and A. Yong. Genomic tableaux.J. Algebraic Combin., 45(3):649–685, 2017

work page 2017

[11] [11]

D. Searles. Indecomposable 0-Hecke modules for extended Schur functions.Proc. Amer. Math. Soc., 148(5):1933–1943, 2020

work page 1933

[12] [12]

V. V. Tewari and S. J. van Willigenburg. Modules of the 0-Hecke algebra and quasisymmetric Schur functions.Adv. Math., 285:1025–1065, 2015. Department of Mathematics, Seoul Women’s University, Seoul 01797, Republic of Ko- rea Email address:yhkim@swu.ac.kr, ykim.math@gmail.com

work page 2015