Papers - Iwao, Shinsuke

Tetrahedron equation and Schur functions

Shinsuke Iwao, Kohei Motegi, Ryo Ohkawa

Journal of Physics A: Mathematical and Theoretical  2024.11

Accepted

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Closed k-Schur Katalan functions as K-homology Schubert representatives of the affine Grassmannian

Takeshi Ikeda, Shinsuke Iwao, Satoshi Naito

Transactions of the American Mathematical Society, Series B  2024.03

Accepted,  ISSN  2330-0000

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<p>Recently, Blasiak–Morse–Seelinger introduced symmetric func- tions called Katalan functions, and proved that the <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-theoretic <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k">
<mml:semantics>
<mml:mi>k</mml:mi>
<mml:annotation encoding="application/x-tex">k</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-Schur functions due to Lam–Schilling–Shimozono form a subfamily of the Katalan functions. They conjectured that another subfamily of Katalan functions called closed <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k">
<mml:semantics>
<mml:mi>k</mml:mi>
<mml:annotation encoding="application/x-tex">k</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-Schur Katalan functions is identified with the Schubert structure sheaves in the <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-homology of the affine Grassmannian. Our main result is a proof of this conjecture.</p>

<p>We also study a <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-theoretic Peterson isomorphism that Ikeda, Iwao, and Maeno constructed, in a nongeometric manner, based on the unipotent solution of the relativistic Toda lattice of Ruijsenaars. We prove that the map sends a Schubert class of the quantum <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-theory ring of the flag variety to a closed <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k">
<mml:semantics>
<mml:mi>k</mml:mi>
<mml:annotation encoding="application/x-tex">k</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-Schur Katalan function up to an explicit factor related to a translation element with respect to an antidominant coroot. In fact, we prove this map coincides with a map whose existence was conjectured by Lam, Li, Mihalcea, Shimozono, and proved by Kato, and more recently by Chow and Leung.</p>

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Free fermions and canonical Grothendieck polynomials

Shinsuke Iwao, Kohei Motegi, Travis Scrimshaw

Algebraic Combinatorics 7 ( 1 ) 245 - 274 2024.02

Accepted,  ISSN  2589-5486

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We give a presentation of refined (dual) canonical Grothendieck polynomials and their skew versions using free fermions. Using this, we derive a number of identities, including the skew Cauchy identities, branching rules, expansion formulas, and integral formulas.

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Free fermions and Schur expansions of multi-Schur functions

Shinsuke Iwao

Journal of Combinatorial Theory. Series A 198 2023.08

Accepted,  ISSN  0097-3165

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Multi-Schur functions are symmetric functions that generalize the supersymmetric Schur functions, the flagged Schur functions, and the refined dual Grothendieck functions, which have been intensively studied by Lascoux. In this paper, we give a new free-fermionic presentation of them. The multi-Schur functions are indexed by a partition and two “tuples of tuples” of indeterminates. We construct a family of linear bases of the fermionic Fock space that are indexed by such data and prove that they correspond to the multi-Schur functions through the boson-fermion correspondence. By focusing on some special bases, which we call refined bases, we give a straightforward method of expanding a multi-Schur function in the refined dual Grothendieck polynomials. We also present a sufficient condition for a multi-Schur function to have its Hall-dual function in the completed ring of symmetric functions.

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Free-fermions and skew stable Grothendieck polynomials

Shinsuke Iwao

Journal of Algebraic Combinatorics (Springer Science and Business Media {LLC})  56 ( 2 ) 493 - 526 2022.04

Accepted,  ISSN  0925-9899

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We present a free-fermionic presentation of the skew (dual) stable Grothendieck polynomials. A direct proof of their determinantal formulas is given from this presentation. We also introduce a combinatorial method to describe the multiplication map and its adjoint over the space of skew (dual) stable Grothendieck polynomials. This calculation requires the use of noncommutative supersymmetric Schur functions.

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Neutral-fermionic presentation of the K-theoretic Q-function

Shinsuke Iwao

Journal of Algebraic Combinatorics (Springer Science and Business Media {LLC})  55 ( 2 ) 629 - 662 2022.03

Accepted,  ISSN  0925-9899

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We show a new neutral-fermionic presentation of Ikeda-Naruse’s K-theoretic Q-functions, which represent a Schubert class in the K-theory of coherent sheaves on the Lagrangian Grassmannian. Our presentation provides a simple description and yields a straightforward proof of two types of Pfaffian formulas for them. We present a dual space of GΓ, the vector space generated by all K-theoretic Q-functions, by constructing a non-degenerate bilinear form which is compatible with the neutral-fermionic presentation. We give a new family of dual K-theoretic Q-functions, their neutral-fermionic presentations, and Pfaffian formulas.

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Grothendieck polynomials and the boson-fermion correspondence

Shinsuke Iwao

Algebraic Combinatorics (Cellule MathDoc/CEDRAM)  3 ( 5 ) 1023 - 1040 2020.10

Accepted,  ISSN  2589-5486

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In this paper we study algebraic and combinatorial properties of symmetric Grothendieck polynomials and their dual polynomials by means of the boson-fermion correspondence. We show that these symmetric functions can be expressed as a vacuum expectation value of some operator that is written in terms of free-fermions. By using the free-fermionic expressions, we obtain alternative proofs of determinantal formulas and Pieri type formulas.

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Jeu de taquin, uniqueness of rectification and ultradiscrete KP

Shinsuke Iwao

Journal of Integrable Systems (Oxford University Press (OUP))  4 ( 1 )  2019.01

Accepted,  ISSN  2058-5985

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<title>Abstract</title>
In this article, we study tropical-theoretic aspects of the ‘rectification algorithm’ on skew Young tableaux. It is shown that the algorithm is interpreted as a time evolution of some tropical integrable system. By using this fact, we construct a new combinatorial map that is essentially equivalent to the rectification algorithm. Some of properties of the rectification can be seen more clearly via this map. For example, the uniqueness of a rectification boils down to an easy combinatorial problem. Our method is mainly based on the two previous researches: the theory of geometric tableaux by Noumi–Yamada, and the study on the relationship between jeu de taquin slides and the ultradiscrete KP equation by Mikami and Katayama–Kakei.

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Peterson Isomorphism in K-theory and Relativistic Toda Lattice

Takeshi Ikeda, Shinsuke Iwao, Toshiaki Maeno

International Mathematics Research Notices (Oxford University Press (OUP))  2020 ( 19 ) 6421 - 6462 2018.04

Accepted,  ISSN  1073-7928

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<title>Abstract</title>
The K-homology ring of the affine Grassmannian of $SL_{n}(\mathbb{C})$ was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On the other hand, for the quantum K-theory of the flag variety $F\,\! l_{n}$, Kirillov and Maeno provided a conjectural presentation based on the results obtained by Givental and Lee. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on Ruijsenaars’s relativistic Toda lattice with unipotent initial condition. From this result, we obtain a K-theory analogue of the so-called Peterson isomorphism for (co)homology. We provide a conjecture on the detailed relationship between the Schubert bases, and, in particular, we determine the image of Lenart–Maeno’s quantum Grothendieck polynomial associated with a Grassmannian permutation.

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Tropical integrable systems and Young tableaux: shape equivalence and Littlewood–Richardson correspondence

Shinsuke Iwao

Journal of Integrable Systems (Oxford University Press ({OUP}))  3 ( 1 )  2018.01

Accepted,  ISSN  2058-5985

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We present a new characterization of the shape equivalent class and the
Littlewood-Richardson correspondence of Young tableaux in terms of tropical
(ultradiscrete) integrable systems. As an application, an alternative proof of
the "shape change theorem" is given.


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The discrete Toda equation revisited: Dual β-Grothendieck polynomials, ultradiscretization, and static solitons

Shinsuke Iwao, Hidetomo Nagai

Journal of Physics A: Mathematical and Theoretical (Institute of Physics Publishing)  51 ( 13 )  2018

Accepted,  ISSN  1751-8121

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This paper presents a study of the discrete Toda equation that was introduced in 1977. In this paper, it is proved that the determinantal solution of the discrete Toda equation, obtained via the Lax formalism, is naturally related to the dual Grothendieck polynomials, a K-theoretic generalization of the Schur polynomials. A tropical permanent solution to the ultradiscrete Toda equation is also derived. The proposed method gives a tropical algebraic representation of the static solitons. Lastly, a new cellular automaton realization of the ultradiscrete Toda equation is proposed.

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Peterson Isomorphism in $K$-theory and Relativistic Toda Lattice

Takeshi Ikeda, Shinsuke Iwao, Toshiaki Maeno

 2017.03

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The $K$-homology ring of the affine Grassmannian of $SL_n(C)$ was studied by
Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf
subring of the ring of symmetric functions. On the other hand, for the quantum
$K$-theory of the flag variety $Fl_n$, Kirillov and Maeno provided a
conjectural presentation based on the results obtained by Givental and Lee. We
construct an explicit birational morphism between the spectrums of these two
rings. Our method relies on Ruijsenaars's relativistic Toda lattice with
unipotent initial condition. From this result, we obtain a $K$-theory analogue
of the so-called Peterson isomorphism for (co)homology. We provide a conjecture
on the detailed relationship between the Schubert bases, and, in particular, we
determine the image of Lenart--Maeno's quantum Grothendieck polynomial
associated with a Grassmannian permutation.


Tropical Krichever construction for the non-periodic box and ball system

Shinsuke Iwao, Hidetomo Nagai, Shin Isojima

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL (IOP PUBLISHING LTD)  45 ( 39 ) 395202 2012.09

Accepted,  ISSN  1751-8113

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A solution for an initial value problem of the box and ball system is constructed from a solution of the periodic box and ball system. The construction is done through a specific limiting process based on the theory of tropical geometry. This method gives a tropical analogue of the Krichever construction, which is an algebro-geometric method to construct exact solutions to integrable systems, for the non-periodic system.

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Tropical curves and integrable piecewise linear maps

Rei Inoue, Shinsuke Iwao

TROPICAL GEOMETRY AND INTEGRABLE SYSTEMS (AMER MATHEMATICAL SOC)  580   21 - 39 2012

Accepted,  ISSN  0271-4132

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We present applications of tropical geometry to some integrable piecewise-linear maps, based on the lecture given by one of the authors (R. I.) at the workshop "Tropical Geometry and Integrable Systems" (University of Glasgow, July 2011), and on some new results obtained afterward. After a brief review on tropical curve theory, we study the spectral curves and the isolevel sets of the tropical periodic Toda lattice and the periodic Box-ball system.

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The two-dimensional periodic box-ball system and its fundamental cycle

Shinsuke Iwao

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL (IOP PUBLISHING LTD)  45 ( 39 ) 395204 2012

Accepted,  ISSN  1751-8113

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We study a two-dimensional box-ball system which is an ultradiscrete analogue of the discrete KP equation. We construct an algorithm to calculate the fundamental cycle, which is an important conserved quantity of the two-dimensional box-ball system with periodic boundary condition, by using the tropical curve theory.

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TROPICAL SPECTRAL CURVES, FAY'S TRISECANT IDENTITY, AND GENERALIZED ULTRADISCRETE TODA LATTICE

Rei Inoue, Shinsuke Iwao

NEW TRENDS IN QUANTUM INTEGRABLE SYSTEMS (WORLD SCIENTIFIC PUBL CO PTE LTD)     101 - 116 2010.10

Accepted

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We study the generalized ultradiscrete periodic Toda lattice J(M, N) which has tropical spectral curve. We introduce a tropical analogue of Fay's trisecant identity, and apply it to construct a general solution to J(M, N).

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Solution of the generalized periodic discrete Toda equation II: theta function solution

Shinsuke Iwao

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL (IOP PUBLISHING LTD)  43 ( 15 ) 155208 2010.03

Accepted,  ISSN  1751-8113

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We construct the theta function solution to the initial value problem for the generalized periodic discrete Toda equation.

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Integration over Tropical Plane Curves and Ultradiscretization

Shinsuke Iwao

INTERNATIONAL MATHEMATICS RESEARCH NOTICES (OXFORD UNIV PRESS)  2010 ( 1 ) 112 - 148 2010

Accepted,  ISSN  1073-7928

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In this article we study holomorphic integrals on tropical plane curves in view of ultradiscretization. We prove that the lattice integrals over tropical curves can be obtained as a certain limit of complex integrals over Riemann surfaces.

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Solution to the initial value problem of the ultradiscrete periodic Toda equation

Makoto Idzumi, Shinsuke Iwao, Jun Mada, Tetsuji Tokihiro

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL (IOP PUBLISHING LTD)  42 ( 31 ) 315209 2009

Accepted,  ISSN  1751-8113

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We present an expression for the solution to the initial value problem for the ultradiscrete periodic Toda equation. The expression provides explicit forms of all dependent variables of the equation, while the previously known solutions give only half of the dependent variables while the others have to be determined implicitly using the conserved quantities.

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Solutions of the generalized periodic discrete Toda equation

Shinsuke Iwao

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL (IOP PUBLISHING LTD)  41 ( 11 ) 115201 2008

Accepted,  ISSN  1751-8113

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A box-ball system with more than one kind of ball is obtained by the generalized periodic discrete Toda equation (pd Toda equation). We present an algebraic geometric study of the periodic Toda equation. The time evolution of the pd Toda equation is linearized on the Picard group of an algebraic variety, and theta function solutions are obtained.

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