Generation of Schubert polynomial series by nanophotonics

Hayato Saigo; Hirokazu Hori; Hirotsugu Suzui; Kazuharu Uchiyama; Kingo Uchida; Makoto Naruse; Ryo Nakagomi

arxiv: 1907.05625 · v1 · pith:GT3UAYTInew · submitted 2019-07-12 · ⚛️ physics.optics · math-ph· math.MP· physics.app-ph

Generation of Schubert polynomial series by nanophotonics

Hirotsugu Suzui , Kazuharu Uchiyama , Ryo Nakagomi , Hayato Saigo , Kingo Uchida , Makoto Naruse , Hirokazu Hori This is my paper

Pith reviewed 2026-05-24 22:34 UTC · model grok-4.3

classification ⚛️ physics.optics math-phmath.MPphysics.app-ph

keywords Schubert polynomialsnanophotonicsphotochromic crystaloptical near-fielddensity mappingdiarylethenepermutationsphysical computing

0 comments

The pith

Optical near-field processes in a photochromic crystal generate patterns that match Schubert polynomials.

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

The paper shows that photons traveling through a nanostructured diarylethene crystal produce exit-position patterns via chains of local photoisomerization that correspond to Schubert matrices. These matrices encode the versatile permutations that Schubert polynomials formalize in mathematics. The demonstration uses optical near-field density mapping on the crystal surface to create the patterns experimentally. Reconfiguring photon detection sensitivity changes the versatility and correlations of the output patterns in either soft or hard ways. The work establishes the first physical, nanophotonic route to producing these mathematical objects, opening a path to generate irregular time series directly from light-matter interactions.

Core claim

Schubert matrices, which correspond to Schubert polynomials, are generated experimentally by mapping the optical near-field density that arises when an incoming photon passes through a photochromic single crystal of diarylethene; the exit position of the photon forms a complex, versatile pattern on the opposite side of the crystal.

What carries the argument

Optical near-field excitation on the surface of the photochromic crystal that induces a chain of local photoisomerization and produces density patterns on the far side.

If this is right

Pattern versatility and correlations become reconfigurable by changing photon detection sensitivity.
Physical processes can supply irregular time series for computing and artificial intelligence tasks.
Nanophotonic hardware can embed mathematical permutation structures without electronic intermediaries.
The same crystal platform supports both soft and hard reconfiguration of the generated series.

Where Pith is reading between the lines

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

The approach could allow direct optical testing of permutation identities by observing how patterns evolve under controlled light inputs.
Integration with other nanophotonic components might produce hybrid devices that perform permutation-based computations at the speed of light propagation.
Scaling the crystal thickness or nanostructure density could tune the complexity of the generated series to match specific mathematical degrees.

Load-bearing premise

The measured optical density patterns match the algebraic definitions of Schubert polynomials exactly, rather than only appearing similar by eye or by overall density shape.

What would settle it

Direct numerical comparison of the experimentally mapped density matrices against the coefficient tables or polynomial expansions of known Schubert polynomials that reveals consistent mismatches in structure or values.

read the original abstract

Generation of irregular time series based on physical processes is indispensable in computing and artificial intelligence. In this report, we propose and experimentally demonstrate the generation of Schubert polynomials, which is the foundation of versatile permutations in mathematics, via optical near-field processes introduced in a photochromic crystal of diarylethene, which optical near-field excitation on the surface of a photochromic single crystal yields a chain of local photoisomerization, forming a complex pattern on the opposite side of the crystal. The incoming photon travels through the nanostructured photochromic crystal, and the exit position of the photon exhibits a versatile pattern. We experimentally generated Schubert matrices, corresponding to Schubert polynomials, via optical near-field density mapping. The versatility and correlations of the generated patterns could be reconfigured in either a soft or hard manner by adjusting the photon detection sensitivity. This is the first study of Schubert polynomial generation by physical processes or nanophotonics, paving the way toward future nano-scale intelligence devices and systems.

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 describes an optical experiment producing patterns in a diarylethene crystal that are labeled Schubert matrices, but supplies no algebraic verification of the match.

read the letter

The central claim is that near-field photon paths through a photochromic crystal generate density patterns the authors call Schubert matrices. They show the setup works with a single crystal, that exit positions vary, and that detection sensitivity lets them tune the output in a soft or hard way. That experimental description is the concrete part of the work and the only thing that feels new here—an attempt to tie nanophotonics to a specific family of polynomials used in combinatorics. The reconfigurability note is straightforward and could matter for anyone thinking about physical random or permutation generators. Beyond that, the paper does not demonstrate that the observed densities satisfy the divided-difference relations, positivity, or monomial expansions that define Schubert polynomials. No comparison tables, no explicit mapping procedure, and no check against known values appear in the available text. Without those steps the identification stays visual rather than algebraic. The result is therefore hard to use or build on. Readers working on nano-scale physical computing might skim it for the crystal technique, but the mathematical link is too loose for serious follow-up. I would not bring this to a reading group and would not cite it. It does not yet merit peer review; the verification gap is load-bearing.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to experimentally generate Schubert polynomials (via corresponding Schubert matrices) using optical near-field excitation and photoisomerization in a diarylethene photochromic crystal. Photon paths through the nanostructured crystal produce complex exit-position patterns that are asserted to form Schubert matrices, with pattern versatility reconfigurable via photon detection sensitivity; this is presented as the first physical/nanophotonic realization of these mathematical objects.

Significance. If the optical density patterns can be shown to rigorously match the algebraic definitions of Schubert polynomials, the result would constitute a novel physical generator for permutation-related algebraic structures with possible implications for nanoscale computing. The experimental platform using photochromic crystals is innovative, but no credit can be assigned for machine-checked proofs, reproducible code, or parameter-free derivations because none are present.

major comments (2)

[Abstract] Abstract: the assertion that 'We experimentally generated Schubert matrices, corresponding to Schubert polynomials, via optical near-field density mapping' supplies no mapping procedure, no comparison to algebraic definitions (divided-difference operators, monomial expansions indexed by permutations), and no verification that observed densities obey required relations such as exchange relations or positivity.
[Abstract] Abstract: no experimental data, figures, tables, or methods section is referenced that would allow independent confirmation that the generated patterns satisfy the mathematical properties of Schubert polynomials rather than relying on visual or density similarity alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and indicate planned revisions where appropriate.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that 'We experimentally generated Schubert matrices, corresponding to Schubert polynomials, via optical near-field density mapping' supplies no mapping procedure, no comparison to algebraic definitions (divided-difference operators, monomial expansions indexed by permutations), and no verification that observed densities obey required relations such as exchange relations or positivity.

Authors: The abstract is concise by design. The manuscript body describes the physical mechanism of photon paths through the nanostructured diarylethene crystal producing exit-position patterns asserted to correspond to Schubert matrices. We agree that an explicit mapping procedure and direct comparison to algebraic definitions (e.g., divided-difference operators and positivity) are not detailed in the current text. In revision we will add a dedicated subsection providing this mapping, including checks against required algebraic relations where the physical data permit. revision: yes
Referee: [Abstract] Abstract: no experimental data, figures, tables, or methods section is referenced that would allow independent confirmation that the generated patterns satisfy the mathematical properties of Schubert polynomials rather than relying on visual or density similarity alone.

Authors: The manuscript contains experimental figures of the generated patterns together with a methods description of the near-field excitation and photoisomerization process. The abstract itself does not cite these elements. We will revise the abstract to reference the relevant figures and methods, and we will expand the main text with quantitative metrics (e.g., pattern correlations) to support the claimed correspondence beyond visual inspection. revision: yes

Circularity Check

0 steps flagged

No derivation chain; experimental mapping claim with no algebraic reduction shown

full rationale

The paper reports an experimental physical process in a photochromic crystal that produces density patterns asserted to correspond to Schubert matrices/polynomials. The abstract and context contain no equations, predictions, or first-principles derivations that could reduce to self-definition, fitted inputs, or self-citation chains. The central step is an empirical generation plus density mapping whose correctness is external to any internal algebraic construction; no load-bearing step equates output to input by construction. This is the normal case of a self-contained experimental claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the physical mechanism of near-field photoisomerization producing patterns that match Schubert polynomials; no free parameters, invented entities, or additional axioms are detailed in the abstract.

axioms (1)

domain assumption Optical near-field excitation on the surface of a photochromic single crystal yields a chain of local photoisomerization forming a complex pattern on the opposite side of the crystal.
Core physical process invoked to explain pattern formation.

pith-pipeline@v0.9.0 · 5732 in / 1076 out tokens · 43689 ms · 2026-05-24T22:34:22.882826+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 1 internal anchor

[1]

Conversion of optical near-field intensity map into probability map

Other examples include 𝔖𝔖𝑖𝑖𝑖𝑖= 1, 𝔖𝔖1432 = 𝜕𝜕1𝜕𝜕2𝜕𝜕3𝔖𝔖𝜔𝜔0 = 𝑥𝑥1 2𝑥𝑥2 + 𝑥𝑥1 2𝑥𝑥3 + 𝑥𝑥1𝑥𝑥2 2 + 𝑥𝑥1𝑥𝑥2𝑥𝑥3 + 𝑥𝑥2 2𝑥𝑥3, and 𝔖𝔖2431 = 𝜕𝜕1𝜕𝜕2𝔖𝔖𝜔𝜔0 = 𝑥𝑥1 2𝑥𝑥2𝑥𝑥3 + 𝑥𝑥1𝑥𝑥2 2𝑥𝑥3. Conversion of optical near-field intensity map into probability map. Let the intensity distribution of the obtained near-field light be NF (x, y) (x = 1, ..., N, y = 1, ..., N, where N is ...

work page
[2]

S., Eckert, K., Chua, L

Kocarev, L., Halle, K. S., Eckert, K., Chua, L. O. & Parlitz, U. Experimental demonstration of secure communications via chaotic synchronization. Int. J. Bifurcat. Chaos 2, 709–713 (1992)

work page 1992
[3]

Stinson, D. R. Cryptography: Theory and Practice (CRC Press, 1995)

work page 1995
[4]

& Ulam, S

Metropolis, N. & Ulam, S. The Monte Carlo method. J. Am. Statist. Assoc. 44, 335– 341 (1949). 11

work page 1949
[5]

Naruse, M. et al. Single-photon decision maker. Sci. Rep. 5, 13253 (2015)

work page 2015
[6]

Generative adversarial network based on chaotic time series

Naruse, M., Matsubara, T., Chauvet, N., Kanno, K., Yang, T., Uchida, A. Generative adversarial network based on chaotic time series. arXiv: 1905.10163

work page internal anchor Pith review Pith/arXiv arXiv 1905
[7]

Optical communication with chaotic lasers: applications of nonlinear dynamics and synchronization (Wiley-VCH, Weinheim, 2012)

Uchida, A. Optical communication with chaotic lasers: applications of nonlinear dynamics and synchronization (Wiley-VCH, Weinheim, 2012)

work page 2012
[8]

& Bermudez, M

Branning, D. & Bermudez, M. Testing quantum randomness in single -photon polarization measurements with the NIST test suite. JOSA B 27, 1594–1602(2010)

work page 2010
[9]

Uchida, A. et al. Fast physical random bit generation with chaotic semiconductor lasers. Nat. Photonics 2, 728 (2008)

work page 2008
[10]

Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing

Larger, et al. Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing. Opt. Express 20, 3241–3249 (2012)

work page 2012
[11]

Naruse, M. et al. Ultrafast photonic reinforcement learning based on laser chaos. Sci. Rep. 7, 8772 (2017)

work page 2017
[12]

and Haiman, M

Billey, S. and Haiman, M. Schubert polynomials for the classical groups. J. Amer. Math. Soc. 8, 443 (1995)

work page 1995
[13]

and Kirillov, A

Fomin, S. and Kirillov, A. N. Combinatorial Bn- analogues of Schubert polynomials. Trans. Amer. Math. Soc. 348, 3591 (1996)

work page 1996
[14]

Nakagomi, R. et al. Nano-optical functionality based on local photoisomerization in photochromic single crystal. Appl. Phys. A 124, 10 (2017)

work page 2017
[15]

Nakagomi, R. et al. Nanometre-scale pattern formation on the surface of a photochromic crystal by optical near -field induced photoisomerization. Sci. Rep. 8, 14468 (2018)

work page 2018
[16]

& Kobatake, S

Irie, M., Fukaminato, T., Matsuda, K. & Kobatake, S. Photochromism of diarylethene molecules and crystals: Memories, switches, and actuators. Chem. Rev. 114, 12174 (2014)

work page 2014
[17]

& Horichi, M

Irie, M., Kobatake, S. & Horichi, M. Reversible surface morphology changes of a photochromic diarylethene single crystal by photoirradiation. Science 291, 1769 (2001). 12

work page 2001
[18]

Hatano, E. et al. Photosalient phenomena that mimic impatiens are observed in hollow crystals of diarylethene with a perfluorocyclohexene ring. Angew. Chem. Int. Ed. 56, 12576–12580 (2017)

work page 2017
[19]

Naruse, M. et al. Scalable photonic reinforcement learning by time -division multiplexing of laser chaos. Sci. Rep. 8, 10890 (2018)

work page 2018
[20]

Saigo, H. et al. Analysis of soft robotics based on the concept of category of mobility. Complexity 2019, 1490541 (2019)

work page 2019
[21]

& Kitano, M

Tamate, S., Kobayashi, H., Nakanishi, T., Sugiyama, K. & Kitano, M. Geometrical aspects of weak measurements and quantum erasers. New J. Phys. 11, 093025 (2009). Acknowledgement This work was supported in part by the CREST program (JPMJCR17N2) of the Japan Science and Technology Agency and the Grants -in-Aid for Scientific Research (JP26107012, JP17H01277...

work page 2009

[1] [1]

Conversion of optical near-field intensity map into probability map

Other examples include 𝔖𝔖𝑖𝑖𝑖𝑖= 1, 𝔖𝔖1432 = 𝜕𝜕1𝜕𝜕2𝜕𝜕3𝔖𝔖𝜔𝜔0 = 𝑥𝑥1 2𝑥𝑥2 + 𝑥𝑥1 2𝑥𝑥3 + 𝑥𝑥1𝑥𝑥2 2 + 𝑥𝑥1𝑥𝑥2𝑥𝑥3 + 𝑥𝑥2 2𝑥𝑥3, and 𝔖𝔖2431 = 𝜕𝜕1𝜕𝜕2𝔖𝔖𝜔𝜔0 = 𝑥𝑥1 2𝑥𝑥2𝑥𝑥3 + 𝑥𝑥1𝑥𝑥2 2𝑥𝑥3. Conversion of optical near-field intensity map into probability map. Let the intensity distribution of the obtained near-field light be NF (x, y) (x = 1, ..., N, y = 1, ..., N, where N is ...

work page

[2] [2]

S., Eckert, K., Chua, L

Kocarev, L., Halle, K. S., Eckert, K., Chua, L. O. & Parlitz, U. Experimental demonstration of secure communications via chaotic synchronization. Int. J. Bifurcat. Chaos 2, 709–713 (1992)

work page 1992

[3] [3]

Stinson, D. R. Cryptography: Theory and Practice (CRC Press, 1995)

work page 1995

[4] [4]

& Ulam, S

Metropolis, N. & Ulam, S. The Monte Carlo method. J. Am. Statist. Assoc. 44, 335– 341 (1949). 11

work page 1949

[5] [5]

Naruse, M. et al. Single-photon decision maker. Sci. Rep. 5, 13253 (2015)

work page 2015

[6] [6]

Generative adversarial network based on chaotic time series

Naruse, M., Matsubara, T., Chauvet, N., Kanno, K., Yang, T., Uchida, A. Generative adversarial network based on chaotic time series. arXiv: 1905.10163

work page internal anchor Pith review Pith/arXiv arXiv 1905

[7] [7]

Optical communication with chaotic lasers: applications of nonlinear dynamics and synchronization (Wiley-VCH, Weinheim, 2012)

Uchida, A. Optical communication with chaotic lasers: applications of nonlinear dynamics and synchronization (Wiley-VCH, Weinheim, 2012)

work page 2012

[8] [8]

& Bermudez, M

Branning, D. & Bermudez, M. Testing quantum randomness in single -photon polarization measurements with the NIST test suite. JOSA B 27, 1594–1602(2010)

work page 2010

[9] [9]

Uchida, A. et al. Fast physical random bit generation with chaotic semiconductor lasers. Nat. Photonics 2, 728 (2008)

work page 2008

[10] [10]

Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing

Larger, et al. Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing. Opt. Express 20, 3241–3249 (2012)

work page 2012

[11] [11]

Naruse, M. et al. Ultrafast photonic reinforcement learning based on laser chaos. Sci. Rep. 7, 8772 (2017)

work page 2017

[12] [12]

and Haiman, M

Billey, S. and Haiman, M. Schubert polynomials for the classical groups. J. Amer. Math. Soc. 8, 443 (1995)

work page 1995

[13] [13]

and Kirillov, A

Fomin, S. and Kirillov, A. N. Combinatorial Bn- analogues of Schubert polynomials. Trans. Amer. Math. Soc. 348, 3591 (1996)

work page 1996

[14] [14]

Nakagomi, R. et al. Nano-optical functionality based on local photoisomerization in photochromic single crystal. Appl. Phys. A 124, 10 (2017)

work page 2017

[15] [15]

Nakagomi, R. et al. Nanometre-scale pattern formation on the surface of a photochromic crystal by optical near -field induced photoisomerization. Sci. Rep. 8, 14468 (2018)

work page 2018

[16] [16]

& Kobatake, S

Irie, M., Fukaminato, T., Matsuda, K. & Kobatake, S. Photochromism of diarylethene molecules and crystals: Memories, switches, and actuators. Chem. Rev. 114, 12174 (2014)

work page 2014

[17] [17]

& Horichi, M

Irie, M., Kobatake, S. & Horichi, M. Reversible surface morphology changes of a photochromic diarylethene single crystal by photoirradiation. Science 291, 1769 (2001). 12

work page 2001

[18] [18]

Hatano, E. et al. Photosalient phenomena that mimic impatiens are observed in hollow crystals of diarylethene with a perfluorocyclohexene ring. Angew. Chem. Int. Ed. 56, 12576–12580 (2017)

work page 2017

[19] [19]

Naruse, M. et al. Scalable photonic reinforcement learning by time -division multiplexing of laser chaos. Sci. Rep. 8, 10890 (2018)

work page 2018

[20] [20]

Saigo, H. et al. Analysis of soft robotics based on the concept of category of mobility. Complexity 2019, 1490541 (2019)

work page 2019

[21] [21]

& Kitano, M

Tamate, S., Kobayashi, H., Nakanishi, T., Sugiyama, K. & Kitano, M. Geometrical aspects of weak measurements and quantum erasers. New J. Phys. 11, 093025 (2009). Acknowledgement This work was supported in part by the CREST program (JPMJCR17N2) of the Japan Science and Technology Agency and the Grants -in-Aid for Scientific Research (JP26107012, JP17H01277...

work page 2009