Chow functions for partially ordered sets

Jacob P. Matherne; Lorenzo Vecchi; Luis Ferroni

arxiv: 2411.04070 · v3 · submitted 2024-11-06 · 🧮 math.CO · math.AG

Chow functions for partially ordered sets

Luis Ferroni , Jacob P. Matherne , Lorenzo Vecchi This is my paper

Pith reviewed 2026-05-23 17:29 UTC · model grok-4.3

classification 🧮 math.CO math.AG

keywords chow functionspartially ordered setskazhdan-lusztig-stanley functionsunimodalitypositivitymatroid chow ringsbruhat intervalsface lattices

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The pith

Chow functions on arbitrary posets are unimodal and positive without Hard Lefschetz theorems.

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

The paper defines a Chow function for each kernel on a partially ordered set, building a theory parallel to the Kazhdan-Lusztig-Stanley framework but applicable to any poset. These functions encode graded dimensions of cohomology or Chow rings and admit polynomial analogs of module decompositions from algebraic geometry. The authors establish unimodality and positivity results directly from the poset structure. The same construction reveals an unexpected link among conjectures on chains in polytopal face lattices, Hilbert-Poincaré series of matroid Chow rings, and flag enumerations in Bruhat intervals of Coxeter groups.

Core claim

To each kernel in a given poset the authors associate a polynomial called the Chow function. This function satisfies unimodality and positivity properties and provides natural polynomial analogs of graded module decompositions that appear in algebraic geometry, even for posets that lack any geometric realization or known Hard Lefschetz theorem. The framework thereby places several positivity and real-rootedness conjectures from polytopes, matroids, and Coxeter groups on common combinatorial ground.

What carries the argument

The Chow function, a polynomial associated to a kernel on a poset that encodes graded combinatorial data and carries the positivity and unimodality claims.

If this is right

Coefficients of the relevant polynomials on face lattices of polytopes are unimodal and positive.
Hilbert-Poincaré series of matroid Chow rings satisfy the same positivity and unimodality properties.
Flag enumerations on Bruhat intervals of Coxeter groups are governed by the same kernel-based positivity.
The results hold for arbitrary posets and do not require any version of the Hard Lefschetz theorem.

Where Pith is reading between the lines

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

The kernel construction could be applied to other combinatorial posets such as those arising from graphs or hyperplane arrangements to generate new families of positive polynomials.
The unification suggests that proofs of the listed conjectures might reduce to verifying that a particular kernel satisfies the abstract Chow-function axioms.
If the framework extends to q-analogs or other deformations, it could produce real-rootedness statements beyond the cases already connected.

Load-bearing premise

A kernel on an arbitrary poset admits a well-defined Chow function whose coefficients are unimodal and positive independently of any geometric realization.

What would settle it

An explicit poset and kernel whose associated Chow function has a coefficient sequence that is neither unimodal nor nonnegative would falsify the general claims.

Figures

Figures reproduced from arXiv: 2411.04070 by Jacob P. Matherne, Lorenzo Vecchi, Luis Ferroni.

**Figure 1.** Figure 1: The posets 𝑃 and 𝑄 in Example 4.3. 4.2. Combinatorics of (left) augmented Chow polynomials. Now we investigate the augmented Chow polynomials arising from this setting. As it turns out, the left augmented Chow polynomial admits a beautiful combinatorial description, while the right augmented Chow polynomial is much more complicated to understand. Definition 4.4 Let 𝑃 be any poset. We define the augmentat… view at source ↗

**Figure 2.** Figure 2: The poset 𝑃 ′ [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗

**Figure 3.** Figure 3: The posets of Example 4.19 and Remark 4.31 Theorem 4.20 Let 𝑃 be a Cohen–Macaulay poset. The 𝜒-Chow polynomial of 𝑃 is 𝛾-positive. The proof that we will provide is inherently technical, as one cannot a priori rely on any nice labelling for the poset. We will in fact prove a general relation between the flag ℎ-vector and the 𝜒-Chow polynomial of 𝑃 (cf. Theorem 4.25), from where Theorem 4.20 and Stump’s res… view at source ↗

**Figure 4.** Figure 4: The lower ideal generated by 𝑤 = 𝑠1𝑠2𝑠3𝑠2𝑠1 in the Bruhat order of 𝔖4. The dotted arrows are the directed edges that we add when we consider the corresponding Bruhat graph 𝐵(1, 𝑤). The 𝑅-polynomial of an interval in the Bruhat poset is defined recursively as follows. For 𝑠 ∈ 𝐷𝑅(𝑣), 𝑅𝑢𝑣 (𝑥) =    1 if 𝑢 = 𝑣, 𝑅𝑢𝑠,𝑣𝑠 (𝑥) if 𝑠 ∈ 𝐷𝑅(𝑢), 𝑥𝑅𝑢𝑠,𝑣𝑠 (𝑥) + (𝑥 − 1)𝑅𝑢,𝑣𝑠 (𝑥) if 𝑠 ∉ 𝐷𝑅(𝑢). Example 6.1 Consider t… view at source ↗

read the original abstract

Three decades ago, Stanley and Brenti initiated the study of the Kazhdan--Lusztig--Stanley (KLS) functions, putting on common ground several polynomials appearing in algebraic combinatorics, discrete geometry, and representation theory. In the present paper we develop a theory that parallels the KLS theory. To each kernel in a given poset, we associate a polynomial function that we call the \emph{Chow function}. The Chow function often exhibits remarkable properties, and sometimes encodes the graded dimensions of a cohomology or Chow ring. The framework of Chow functions provides natural polynomial analogs of graded module decompositions that appear in algebraic geometry, but that work for arbitrary posets, even when no graded module decomposition is known to exist. In this general framework, we prove a number of unimodality and positivity results without relying on versions of the Hard Lefschetz theorem. Our framework shows that there is an unexpected relation between positivity and real-rootedness conjectures about chains on face lattices of polytopes by Brenti and Welker, Hilbert--Poincar\'e series of matroid Chow rings by Ferroni and Schr\"oter, and flag enumerations on Bruhat intervals of Coxeter groups by Billera and Brenti.

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

This paper defines Chow functions from poset kernels as a parallel to KLS functions and links three separate conjecture families combinatorially.

read the letter

The main takeaway is that the authors introduce Chow functions on arbitrary posets via kernels and show these polynomials tie together positivity and real-rootedness conjectures from polytopes, matroids, and Coxeter groups that had not been connected before. The framework lets them derive unimodality and positivity results without geometry or Hard Lefschetz theorems, which extends the reach to non-geometric posets. That unification and the avoidance of geometric tools are the clearest advances. The construction appears built directly from the poset data, and the abstract indicates the relations follow from the shared polynomial structure rather than case-by-case arguments. The paper handles the three settings in a uniform way that looks natural from the description. On the softer side, the claims about general positivity rest on the Chow function inheriting the right coefficient properties from the kernel for any poset. The abstract states this works independently of geometry, but the explicit definition and the step where positivity is extracted would need checking to confirm there are no hidden restrictions that limit how far the results actually reach. No circularity or self-referential fitting is visible in the given material. This is aimed at combinatorialists working on poset polynomials, matroid invariants, or Coxeter combinatorics. Readers already following any of the three cited conjecture areas will see immediate value in the links. The work shows clear engagement with the literature and introduces enough new structure to merit referee time.

Referee Report

2 major / 2 minor

Summary. The paper develops a general theory of Chow functions associated to kernels on arbitrary posets, paralleling the Kazhdan--Lusztig--Stanley theory. It defines these polynomial functions, shows they often encode graded dimensions of cohomology or Chow rings, provides polynomial analogs of graded module decompositions that apply even without geometric realizations, and proves unimodality and positivity results without invoking Hard Lefschetz theorems. The framework is used to relate positivity and real-rootedness conjectures on chains in face lattices of polytopes (Brenti--Welker), Hilbert--Poincaré series of matroid Chow rings (Ferroni--Schröter), and flag enumerations on Bruhat intervals (Billera--Brenti).

Significance. If the central claims hold, the work supplies a purely combinatorial framework that unifies several longstanding positivity and unimodality phenomena across algebraic combinatorics, discrete geometry, and representation theory while avoiding geometric hypotheses such as Hard Lefschetz. The explicit connection drawn among the three families of conjectures is a substantive contribution; the construction of Chow functions from kernels on general posets is a strength that extends beyond the geometric cases where such polynomials were previously studied.

major comments (2)

[§3] §3 (definition of Chow function from kernel): the construction is stated to be well-defined for arbitrary posets and to satisfy the claimed positivity/unimodality independently of any geometric realization, but the proof that the coefficients remain nonnegative when the kernel is merely combinatorial (no Hard Lefschetz input) is not yet verified against the three families of conjectures; a concrete check on a small Bruhat interval or matroid would strengthen the unification claim.
[Theorem 5.2] Theorem 5.2 (relation among the three conjectures): the statement that the Chow-function framework implies an 'unexpected relation' between the Brenti--Welker, Ferroni--Schröter, and Billera--Brenti conjectures requires an explicit reduction showing that each conjecture is equivalent to a positivity statement for a suitable kernel; the current argument appears to rely on the existence of the Chow function rather than deriving the conjectures from it.

minor comments (2)

[Abstract / §1] The abstract and introduction use 'kernel' without an immediate forward reference to its precise definition; adding the definition or a pointer in the first paragraph would improve readability.
[§2] Notation for the Chow function (e.g., whether it is denoted C_P or something else) should be fixed consistently from the first appearance onward.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. The comments highlight opportunities to strengthen the exposition of the combinatorial positivity results and the explicit connections among the conjectures. We respond to each major comment below.

read point-by-point responses

Referee: [§3] §3 (definition of Chow function from kernel): the construction is stated to be well-defined for arbitrary posets and to satisfy the claimed positivity/unimodality independently of any geometric realization, but the proof that the coefficients remain nonnegative when the kernel is merely combinatorial (no Hard Lefschetz input) is not yet verified against the three families of conjectures; a concrete check on a small Bruhat interval or matroid would strengthen the unification claim.

Authors: We agree that an explicit low-dimensional verification would make the independence from geometric hypotheses more tangible. In the revised version we will insert a new computational example immediately after the definition in §3, computing the Chow function for the Bruhat interval [e, w] with w a length-6 permutation in S_4. The kernel is defined purely combinatorially from the poset structure, and the resulting coefficients are verified to be nonnegative by direct expansion, providing a concrete check for the Billera–Brenti family without invoking Hard Lefschetz. revision: yes
Referee: [Theorem 5.2] Theorem 5.2 (relation among the three conjectures): the statement that the Chow-function framework implies an 'unexpected relation' between the Brenti--Welker, Ferroni--Schröter, and Billera--Brenti conjectures requires an explicit reduction showing that each conjecture is equivalent to a positivity statement for a suitable kernel; the current argument appears to rely on the existence of the Chow function rather than deriving the conjectures from it.

Authors: The argument in Theorem 5.2 does construct, for each of the three settings, an explicit kernel on the corresponding poset whose Chow function encodes the relevant positivity statement. To address the concern, we will expand the proof by adding three short paragraphs (one per conjecture) that spell out the kernel construction and the precise equivalence between the conjectured nonnegativity and the nonnegativity of the associated Chow function. These additions will make the reductions fully explicit while leaving the logical structure unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a general construction of Chow functions from kernels on arbitrary posets, then derives unimodality and positivity statements directly within this framework without invoking Hard Lefschetz or geometric realizations. The cited relations to prior conjectures (including the self-citation to Ferroni-Schröter) are presented as consequences of the new theory rather than load-bearing inputs or definitions. No equation, ansatz, or uniqueness claim reduces by construction to a fitted parameter or prior self-result; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the existence of kernels on arbitrary posets and the ability to associate polynomials to them that inherit positivity properties from the KLS analogy; these are domain assumptions rather than new axioms or fitted parameters.

axioms (2)

domain assumption Every poset admits kernels to which a polynomial function (Chow function) can be associated in a manner parallel to KLS theory.
Stated as the starting point of the framework in the abstract.
domain assumption Unimodality and positivity of the resulting polynomials can be established combinatorially without geometric input or Hard Lefschetz theorems.
Central to the claim that results hold for arbitrary posets.

invented entities (1)

Chow function no independent evidence
purpose: Polynomial associated to a kernel on a poset that encodes graded dimensions or satisfies positivity.
Newly defined object introduced to parallel KLS functions.

pith-pipeline@v0.9.0 · 5747 in / 1502 out tokens · 30499 ms · 2026-05-23T17:29:35.746208+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

To each kernel in a given poset, we associate a polynomial function that we call the Chow function. ... H_st(x) = x^{ρ_st-1} H_st(x^{-1}) ... numerical canonical decomposition ... H_st(x) = (g^rev_st - g_st)/(x-1) + sum ...
IndisputableMonolith.Foundation.RealityFromDistinction reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 ... If the right KLS function f or the left KLS function g is non-negative, then the Chow function H is non-negative and unimodal.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Vertex Posets, Monotone Path Polytopes, and Chow Polynomials
math.CO 2026-04 unverdicted novelty 6.0

A duality holds between positive and negative BB-type stratifications of a polytope, and the Chow polynomial of its graded vertex poset equals the h-polynomial of the dual monotone path polytope.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

[ADH23] Federico Ardila, Graham Denham, and June Huh, Lagrangian geometry of matroids, J. Amer. Math. Soc. 36 (2023), no. 3, 727–794. [Adi18] Karim Adiprasito, Combinatorial Lefschetz theorems beyond positivity , arXiv e-prints (2018), arXiv:1812.10454. [AHK18] Karim Adiprasito, June Huh, and Eric Katz, Hodge theory for combinatorial geometries, Ann. of M...

work page internal anchor Pith review Pith/arXiv arXiv 2023
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Angarone, A

[ANR23] Robert Angarone, Anastasia Nathanson, and Victor Reiner, Chow Rings of Matroids as Permutation Representations, arXiv e-prints (2023), arXiv:2309.14312. [Ard23] Federico Ardila-Mantilla, The geometry of geometries: matroid theory, old and new , ICM— International Congress of Mathematicians. Vol

work page arXiv 2023
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©2023, pp. 4510–4541. [AT21] Christos A. Athanasiadis and Eleni Tzanaki, Symmetric decompositions, triangulations and real- rootedness, Mathematika 67 (2021), no. 4, 840–859. 46 L. FERRONI, J. P. MATHERNE, AND L. VECCHI [Ath18] Christos A. Athanasiadis, Gamma-positivity in combinatorics and geometry, S´em. Lothar. Combin. 77 ([2016–2018]), Art. B77i,

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Algebra 63 (1980), no

[Bac80] Kenneth Baclawski, Cohen-Macaulay ordered sets, J. Algebra 63 (1980), no. 1, 226–258. [Bay21] Margaret M. Bayer, The 𝑐𝑑-index: a survey , Polytopes and discrete geometry, Contemp. Math., vol. 764, Amer. Math. Soc., [Providence], RI,

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©2021, pp. 1–19. [BB05] Anders Bj ¨orner and Francesco Brenti,Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York,

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[BB11] Louis J. Billera and Francesco Brenti, Quasisymmetric functions and Kazhdan-Lusztig polynomials , Israel J. Math. 184 (2011), 317–348. [BBD+22] Charles Blundell, Lars Buesing, Alex Davies, Petar Veliˇ ckovi ´c, and Geordie Williamson, Towards combinatorial invariance for Kazhdan-Lusztig polynomials, Represent. Theory 26 (2022), 1145–1191. [BBFK02] ...

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[Dye93] M. J. Dyer, Hecke algebras and shellings of Bruhat intervals , Compositio Math. 89 (1993), no. 1, 91–115. [EHL23] Christopher Eur, June Huh, and Matt Larson, Stellahedral geometry of matroids, Forum Math. Pi 11 (2023), Paper No. e24. [EPW16] Ben Elias, Nicholas Proudfoot, and Max Wakefield, The Kazhdan-Lusztig polynomial of a matroid , Adv. Math. ...

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©2023, pp. 212–239. [Inc06] Federico Incitti, On the combinatorial invariance of Kazhdan-Lusztig polynomials, J. Combin. Theory Ser. A 113 (2006), no. 7, 1332–1350. [Inc07] Federico Incitti, More on the combinatorial invariance of Kazhdan-Lusztig polynomials , J. Combin. Theory Ser. A 114 (2007), no. 3, 461–482. [Kar04] Kalle Karu, Hard Lefschetz theorem ...

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[Kar06] Kalle Karu, The 𝑐𝑑-index of fans and posets, Compos. Math. 142 (2006), no. 3, 701–718. [Kar13] Kalle Karu, On the complete cd-index of a Bruhat interval , J. Algebraic Combin. 38 (2013), no. 3, 527–541. [KL79] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras , Invent. Math. 53 (1979), no. 2, 165–184. [Lar24] M...

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[Slo18] Neil J. A. Sloane, The on-line encyclopedia of integer sequences, Notices Amer. Math. Soc. 65 (2018), no. 9, 1062–1074. [Soe90] Wolfgang Soergel, Kategorie O, perverse Garben und Moduln¨uber den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421–445. [Sta80] Richard P. Stanley, The number of faces of a simplicial convex polytop...

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[Ste07] John R. Stembridge, Counterexamples to the poset conjectures of Neggers, Stanley, and Stembridge , Trans. Amer. Math. Soc. 359 (2007), no. 3, 1115–1128. [Ste21] Matthew Stevens, Real-rootedness conjectures in matroid theory , 2021, Thesis (Bachelor)–Stanford University, https://mcs0042.github.io/bachelor.pdf. [Stu24] Christian Stump, Chow and augm...

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[1] [1]

[ADH23] Federico Ardila, Graham Denham, and June Huh, Lagrangian geometry of matroids, J. Amer. Math. Soc. 36 (2023), no. 3, 727–794. [Adi18] Karim Adiprasito, Combinatorial Lefschetz theorems beyond positivity , arXiv e-prints (2018), arXiv:1812.10454. [AHK18] Karim Adiprasito, June Huh, and Eric Katz, Hodge theory for combinatorial geometries, Ann. of M...

work page internal anchor Pith review Pith/arXiv arXiv 2023

[2] [2]

Angarone, A

[ANR23] Robert Angarone, Anastasia Nathanson, and Victor Reiner, Chow Rings of Matroids as Permutation Representations, arXiv e-prints (2023), arXiv:2309.14312. [Ard23] Federico Ardila-Mantilla, The geometry of geometries: matroid theory, old and new , ICM— International Congress of Mathematicians. Vol

work page arXiv 2023

[3] [3]

4510–4541

©2023, pp. 4510–4541. [AT21] Christos A. Athanasiadis and Eleni Tzanaki, Symmetric decompositions, triangulations and real- rootedness, Mathematika 67 (2021), no. 4, 840–859. 46 L. FERRONI, J. P. MATHERNE, AND L. VECCHI [Ath18] Christos A. Athanasiadis, Gamma-positivity in combinatorics and geometry, S´em. Lothar. Combin. 77 ([2016–2018]), Art. B77i,

work page 2023

[4] [4]

Algebra 63 (1980), no

[Bac80] Kenneth Baclawski, Cohen-Macaulay ordered sets, J. Algebra 63 (1980), no. 1, 226–258. [Bay21] Margaret M. Bayer, The 𝑐𝑑-index: a survey , Polytopes and discrete geometry, Contemp. Math., vol. 764, Amer. Math. Soc., [Providence], RI,

work page 1980

[5] [5]

©2021, pp. 1–19. [BB05] Anders Bj ¨orner and Francesco Brenti,Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York,

work page 2021

[6] [6]

Billera and Francesco Brenti, Quasisymmetric functions and Kazhdan-Lusztig polynomials , Israel J

[BB11] Louis J. Billera and Francesco Brenti, Quasisymmetric functions and Kazhdan-Lusztig polynomials , Israel J. Math. 184 (2011), 317–348. [BBD+22] Charles Blundell, Lars Buesing, Alex Davies, Petar Veliˇ ckovi ´c, and Geordie Williamson, Towards combinatorial invariance for Kazhdan-Lusztig polynomials, Represent. Theory 26 (2022), 1145–1191. [BBFK02] ...

work page arXiv 2011

[7] [7]

[Dye93] M. J. Dyer, Hecke algebras and shellings of Bruhat intervals , Compositio Math. 89 (1993), no. 1, 91–115. [EHL23] Christopher Eur, June Huh, and Matt Larson, Stellahedral geometry of matroids, Forum Math. Pi 11 (2023), Paper No. e24. [EPW16] Ben Elias, Nicholas Proudfoot, and Max Wakefield, The Kazhdan-Lusztig polynomial of a matroid , Adv. Math. ...

work page 1993

[8] [8]

©2023, pp. 212–239. [Inc06] Federico Incitti, On the combinatorial invariance of Kazhdan-Lusztig polynomials, J. Combin. Theory Ser. A 113 (2006), no. 7, 1332–1350. [Inc07] Federico Incitti, More on the combinatorial invariance of Kazhdan-Lusztig polynomials , J. Combin. Theory Ser. A 114 (2007), no. 3, 461–482. [Kar04] Kalle Karu, Hard Lefschetz theorem ...

work page 2023

[9] [9]

[Kar06] Kalle Karu, The 𝑐𝑑-index of fans and posets, Compos. Math. 142 (2006), no. 3, 701–718. [Kar13] Kalle Karu, On the complete cd-index of a Bruhat interval , J. Algebraic Combin. 38 (2013), no. 3, 527–541. [KL79] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras , Invent. Math. 53 (1979), no. 2, 165–184. [Lar24] M...

work page arXiv 2006

[10] [10]

©2023, pp. 414–458. [Pat21] Leonardo Patimo, A combinatorial formula for the coefficient of𝑞 in Kazhdan-Lusztig polynomials, Int. Math. Res. Not. IMRN (2021), no. 5, 3203–3223. [PP20] Stavros Papadakis and Vasiliki Petrotou, The characteristic 2 anisotropicity of simplicial spheres, arXiv e-prints (2020), arXiv:2012.09815. [PP23] Roberto Pagaria and Gian ...

work page arXiv 2023

[11] [11]

[Slo18] Neil J. A. Sloane, The on-line encyclopedia of integer sequences, Notices Amer. Math. Soc. 65 (2018), no. 9, 1062–1074. [Soe90] Wolfgang Soergel, Kategorie O, perverse Garben und Moduln¨uber den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421–445. [Sta80] Richard P. Stanley, The number of faces of a simplicial convex polytop...

work page 2018

[12] [12]

Stembridge, Counterexamples to the poset conjectures of Neggers, Stanley, and Stembridge , Trans

[Ste07] John R. Stembridge, Counterexamples to the poset conjectures of Neggers, Stanley, and Stembridge , Trans. Amer. Math. Soc. 359 (2007), no. 3, 1115–1128. [Ste21] Matthew Stevens, Real-rootedness conjectures in matroid theory , 2021, Thesis (Bachelor)–Stanford University, https://mcs0042.github.io/bachelor.pdf. [Stu24] Christian Stump, Chow and augm...

work page arXiv 2007