DSDaisuke Sagaki
Daisuke Sagaki
*This is an algorithmically generated author record based on publicly available information.
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
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Research Subjects
Daisuke Sagaki has a total of 24 co-authors. Here are the top co-authors by number of shared publications:
- Satoshi Naito
Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan
27 shared publications - Satoshi Naito
Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
18 shared publications - Cristian Lenart
Department of Mathematics and Statistics, State University of New York at Albany, Albany, USA
9 shared publications - Takafumi Kouno
Waseda University
5 shared publications - Anne Schilling
Department of Mathematics, UC Davis, Davis, USA
4 shared publications - Mark Shimozono
Department of Mathematics 460 McBryde Hall 225 Stanger St. 0151 24061 Blacksburg MC Virginia Tech VA U. S. A
4 shared publications
Latest Publications
- 19 March 2025
- Published by Wiley in Journal of the London Mathematical Society
- Vol. 111 (3)
A presentation of the torus‐equivariant quantum K‐theory ring of flag manifolds of type A, Part I: The defining ideal
We give a presentation of the torus‐equivariant (small) quantum ‐theory ring of flag manifolds of type , as the quotient of a polynomial ring by an explicit ideal. This result is the torus‐equivariant version of our previous one, which gives a presentation of the nonequivariant quantum ‐theory ring of flag manifolds of type . However, the method of proof for the torus‐equivariant one is entirely different from that for the nonequivariant one; our proof is based on the result in the limit, and uses Nakayama‐type arguments to upgrade it to the quantum situation. Also, in contrast to the nonequivariant case in which we used the Chevalley formula, we make use of the inverse Chevalley formula for the torus‐equivariant ‐group of semi‐infinite flag manifolds to obtain relations that yield our presentation.
- 19 March 2025
- Published by Wiley in Journal of the London Mathematical Society
- Vol. 111 (3)
A presentation of the torus‐equivariant quantum K‐theory ring of flag manifolds of type A, Part I: The defining ideal
We give a presentation of the torus‐equivariant (small) quantum ‐theory ring of flag manifolds of type , as the quotient of a polynomial ring by an explicit ideal. This result is the torus‐equivariant version of our previous one, which gives a presentation of the nonequivariant quantum ‐theory ring of flag manifolds of type . However, the method of proof for the torus‐equivariant one is entirely different from that for the nonequivariant one; our proof is based on the result in the limit, and uses Nakayama‐type arguments to upgrade it to the quantum situation. Also, in contrast to the nonequivariant case in which we used the Chevalley formula, we make use of the inverse Chevalley formula for the torus‐equivariant ‐group of semi‐infinite flag manifolds to obtain relations that yield our presentation.
- Open Access
- 30 January 2025
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 13
A presentation of the torus-equivariant quantum K-theory ring of flag manifolds of type A, Part II: quantum double Grothendieck polynomials
In our previous paper, we gave a presentation of the torus-equivariant quantum K-theory ring $QK_{H}(Fl_{n+1})$ of the (full) flag manifold $Fl_{n+1}$ of type $A_{n}$ as a quotient of a polynomial ring by an explicit ideal. In this paper, we prove that quantum double Grothendieck polynomials, introduced by Lenart-Maeno, represent the corresponding (opposite) Schubert classes in the quantum K-theory ring $QK_{H}(Fl_{n+1})$ under this presentation. The main ingredient in our proof is an explicit formula expressing the semi-infinite Schubert class associated to the longest element of the finite Weyl group, which is proved by making use of the general Chevalley formula for the torus-equivariant K-group of the semi-infinite flag manifold associated to $SL_{n+1}(\mathbb {C})$ .- 30 January 2025
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 13
Open AccessA presentation of the torus-equivariant quantum K-theory ring of flag manifolds of type A, Part II: quantum double Grothendieck polynomials
In our previous paper, we gave a presentation of the torus-equivariant quantum K-theory ring $QK_{H}(Fl_{n+1})$ of the (full) flag manifold $Fl_{n+1}$ of type $A_{n}$ as a quotient of a polynomial ring by an explicit ideal. In this paper, we prove that quantum double Grothendieck polynomials, introduced by Lenart-Maeno, represent the corresponding (opposite) Schubert classes in the quantum K-theory ring $QK_{H}(Fl_{n+1})$ under this presentation. The main ingredient in our proof is an explicit formula expressing the semi-infinite Schubert class associated to the longest element of the finite Weyl group, which is proved by making use of the general Chevalley formula for the torus-equivariant K-group of the semi-infinite flag manifold associated to $SL_{n+1}(\mathbb {C})$ . - 28 March 2024
- Published by Springer Nature in Selecta Mathematica
- Vol. 30 (3) , 1-44
A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory
- 28 March 2024
- Published by Springer Nature in Selecta Mathematica
- Vol. 30 (3) , 1-44
A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory
- Open Access
A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory
We give a Chevalley formula for an arbitrary weight for the torus-equivariant $K$-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum $K$-theory $QK_{T}(G/B)$ of an (ordinary) flag manifold $G/B$; this has been a longstanding conjecture about the multiplicative structure of $QK_{T}(G/B)$. In type $A_{n-1}$, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum $K$-theory $QK(SL_{n}/B)$; we also obtain very explicit information about the coefficients in the respective Chevalley formula.Open AccessA general Chevalley formula for semi-infinite flag manifolds and quantum K-theory
We give a Chevalley formula for an arbitrary weight for the torus-equivariant $K$-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum $K$-theory $QK_{T}(G/B)$ of an (ordinary) flag manifold $G/B$; this has been a longstanding conjecture about the multiplicative structure of $QK_{T}(G/B)$. In type $A_{n-1}$, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum $K$-theory $QK(SL_{n}/B)$; we also obtain very explicit information about the coefficients in the respective Chevalley formula. - Open Access
- 6 February 2024
- Published by Elsevier in Journal of Algebra
- Vol. 645, 1-53
Quantum K-theory Chevalley formulas in the parabolic case
- 6 February 2024
- Published by Elsevier in Journal of Algebra
- Vol. 645, 1-53
Open AccessQuantum K-theory Chevalley formulas in the parabolic case
- Open AccessA presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part I: the defining idealWe give a presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, as a quotient of a polynomial ring by an explicit ideal. This is the torus-equivariant version of our previous result, which gives a presentation of the non-equivariant quantum $K$-theory ring of flag manifolds of type $A$. However, the method of proof for the torus-equivariant one is completely different from that for the non-equivariant one; our proof is based on the result in the $Q = 0$ limit, and uses Nakayama-type arguments to upgrade it to the quantum situation. Also, in contrast to the non-equivariant case in which we used the Chevalley formula, we make use of the inverse Chevalley formula for the torus-equivariant $K$-group of semi-infinite flag manifolds to obtain a relation which yields our presentation.Open AccessA presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part I: the defining idealWe give a presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, as a quotient of a polynomial ring by an explicit ideal. This is the torus-equivariant version of our previous result, which gives a presentation of the non-equivariant quantum $K$-theory ring of flag manifolds of type $A$. However, the method of proof for the torus-equivariant one is completely different from that for the non-equivariant one; our proof is based on the result in the $Q = 0$ limit, and uses Nakayama-type arguments to upgrade it to the quantum situation. Also, in contrast to the non-equivariant case in which we used the Chevalley formula, we make use of the inverse Chevalley formula for the torus-equivariant $K$-group of semi-infinite flag manifolds to obtain a relation which yields our presentation.
- Open Access
Quantum K-theory Chevalley formulas in the parabolic case
We derive cancellation-free Chevalley-type multiplication formulas in the T-equivariant quantum K-theory of Grassmannians of type A and C, and also those of two-step flag manifolds of type A. They are obtained based on the uniform Chevalley formula in the T-equivariant quantum K-theory of arbitrary flag manifolds G/B, which was derived earlier in terms of the quantum alcove model, by the last three authors.
Open AccessQuantum K-theory Chevalley formulas in the parabolic case
We derive cancellation-free Chevalley-type multiplication formulas in the T-equivariant quantum K-theory of Grassmannians of type A and C, and also those of two-step flag manifolds of type A. They are obtained based on the uniform Chevalley formula in the T-equivariant quantum K-theory of arbitrary flag manifolds G/B, which was derived earlier in terms of the quantum alcove model, by the last three authors.
- Open Access
Pieri-type multiplication formula for quantum Grothendieck polynomials
The purpose of this paper is to prove a Pieri-type multiplication formula for quantum Grothendieck polynomials, which was conjectured by Lenart-Maeno. This formula would enable us to compute explicitly the quantum product of two arbitrary (opposite) Schubert classes in $QK(Fl_{n})$ on the basis of the fact that the quantum Grothendieck polynomials represent the corresponding (opposite) Schubert classes in the (small) quantum $K$-theory $QK(Fl_{n})$ of the full flag manifold $Fl_{n}$ of type $A_{n-1}$.Open AccessPieri-type multiplication formula for quantum Grothendieck polynomials
The purpose of this paper is to prove a Pieri-type multiplication formula for quantum Grothendieck polynomials, which was conjectured by Lenart-Maeno. This formula would enable us to compute explicitly the quantum product of two arbitrary (opposite) Schubert classes in $QK(Fl_{n})$ on the basis of the fact that the quantum Grothendieck polynomials represent the corresponding (opposite) Schubert classes in the (small) quantum $K$-theory $QK(Fl_{n})$ of the full flag manifold $Fl_{n}$ of type $A_{n-1}$. - Open AccessA presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part II: quantum double Grothendieck polynomialsIn our previous paper, we gave a presentation of the torus-equivariant quantum $K$-theory ring $QK_{H}(Fl_{n+1})$ of the (full) flag manifold $Fl_{n+1}$ of type $A_{n}$ as a quotient of a polynomial ring by an explicit ideal. In this paper, we prove that quantum double Grothendieck polynomials, introduced by Lenart-Maeno, represent the corresponding (opposite) Schubert classes in the quantum $K$-theory ring $QK_{H}(Fl_{n+1})$ under this presentation. The main ingredient in our proof is an explicit formula expressing the semi-infinite Schubert class associated to the longest element of the finite Weyl group, which is proved by making use of the general Chevalley formula for the torus-equivariant $K$-group of the semi-infinite flag manifold associated to $SL_{n+1}(\mathbb{C})$.Open AccessA presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part II: quantum double Grothendieck polynomialsIn our previous paper, we gave a presentation of the torus-equivariant quantum $K$-theory ring $QK_{H}(Fl_{n+1})$ of the (full) flag manifold $Fl_{n+1}$ of type $A_{n}$ as a quotient of a polynomial ring by an explicit ideal. In this paper, we prove that quantum double Grothendieck polynomials, introduced by Lenart-Maeno, represent the corresponding (opposite) Schubert classes in the quantum $K$-theory ring $QK_{H}(Fl_{n+1})$ under this presentation. The main ingredient in our proof is an explicit formula expressing the semi-infinite Schubert class associated to the longest element of the finite Weyl group, which is proved by making use of the general Chevalley formula for the torus-equivariant $K$-group of the semi-infinite flag manifold associated to $SL_{n+1}(\mathbb{C})$.
- Open Access
- 7 July 2021
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 9
InverseK-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type
We prove an explicit inverse Chevalley formula in the equivariantK-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb {Z}\left [q^{\pm 1}\right ]$ -linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars $e^{\lambda }$ , where $\lambda $ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type $E_8$ . The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetricq-Toda operators for minuscule weights in ADE type.- 7 July 2021
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 9
Open AccessInverseK-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type
We prove an explicit inverse Chevalley formula in the equivariantK-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb {Z}\left [q^{\pm 1}\right ]$ -linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars $e^{\lambda }$ , where $\lambda $ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type $E_8$ . The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetricq-Toda operators for minuscule weights in ADE type.