Daisuke Sagaki
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View article: Quantum $K$-theoretic divisor axiom for flag manifolds
Quantum $K$-theoretic divisor axiom for flag manifolds Open
We prove an identity for (torus-equivariant) 3-point, genus 0, $K$-theoretic Gromov-Witten invariants of flag manifolds $G/P$, which can be thought of as a replacement for the ``divisor axiom'' in their (torus-equivariant) quantum $K$-theo…
View article: A presentation of the torus-equivariant quantum <i>K</i>-theory ring of flag manifolds of type <i>A</i>, Part II: quantum double Grothendieck polynomials
A presentation of the torus-equivariant quantum <i>K</i>-theory ring of flag manifolds of type <i>A</i>, Part II: quantum double Grothendieck polynomials Open
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,…
View article: A presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part II: quantum double Grothendieck polynomials
A presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part II: quantum double Grothendieck polynomials Open
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…
View article: A presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part I: the defining ideal
A presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part I: the defining ideal Open
We 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 pre…
View article: Pieri-type multiplication formula for quantum Grothendieck polynomials
Pieri-type multiplication formula for quantum Grothendieck polynomials Open
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 …
View article: Symmetric and Nonsymmetric Macdonald Polynomials via a Path Model with a Pseudo-crystal Structure
Symmetric and Nonsymmetric Macdonald Polynomials via a Path Model with a Pseudo-crystal Structure Open
In this paper we derive a counterpart of the well-known Ram-Yip formula for symmetric and nonsymmetric Macdonald polynomials of arbitrary type. Our new formula is in terms of a generalization of the Lakshmibai-Seshadri paths (originating i…
View article: Inverse $K$-Chevalley formulas for semi-infinite flag manifolds, II: arbitrary weights in ADE type
Inverse $K$-Chevalley formulas for semi-infinite flag manifolds, II: arbitrary weights in ADE type Open
We continue the study, begun in [Kouno-Naito-Orr-Sagaki, 2021], of inverse Chevalley formulas for the equivariant $K$-group of semi-infinite flag manifolds. Using the language of alcove paths, we reformulate and extend our combinatorial in…
View article: Quantum K-theory Chevalley formulas in the parabolic case
Quantum K-theory Chevalley formulas in the parabolic case Open
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 Chevalle…
View article: Inverse<i>K</i>-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type
Inverse<i>K</i>-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type Open
We prove an explicit inverse Chevalley formula in the equivariant K -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 …
View article: A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory
A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory Open
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…
View article: Inverse $K$-Chevalley formulas for semi-infinite flag manifolds, I:\n minuscule weights in ADE type
Inverse $K$-Chevalley formulas for semi-infinite flag manifolds, I:\n minuscule weights in ADE type Open
We prove an explicit inverse Chevalley formula in the equivariant $K$-theory\nof semi-infinite flag manifolds of simply-laced type. By an inverse Chevalley\nformula, we mean a formula for the product of an equivariant scalar with a\nSchube…
View article: Inverse $K$-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type
Inverse $K$-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type Open
We prove an explicit inverse Chevalley formula in the equivariant $K$-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 …
View article: Chevalley formula for anti-dominant minuscule fundamental weights in the equivariant quantum $K$-group of partial flag manifolds
Chevalley formula for anti-dominant minuscule fundamental weights in the equivariant quantum $K$-group of partial flag manifolds Open
In this paper, we give an explicit formula of Chevalley type, in terms of the Bruhat graph, for the quantum multiplication with the class of the line bundle associated to the anti-dominant minuscule fundamental weight $- \varpi_{k}$ in the…
View article: A Chevalley formula for semi-infinite flag manifolds and quantum K-theory (Extended abstract)
A Chevalley formula for semi-infinite flag manifolds and quantum K-theory (Extended abstract) Open
We give a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant K-theory of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formul…
View article: A uniform model for kirillov-reshetikhin crystals
A uniform model for kirillov-reshetikhin crystals Open
We present a uniform construction of tensor products of one-column Kirillov–Reshetikhin (KR) crystals in all untwisted affine types, which uses a generalization of the Lakshmibai–Seshadri paths (in the theory of the Littelmann path model).…
View article: Chevalley formula for anti-dominant weights in the equivariant $K$-theory of semi-infinite flag manifolds
Chevalley formula for anti-dominant weights in the equivariant $K$-theory of semi-infinite flag manifolds Open
We prove a Pieri-Chevalley formula for anti-dominant weights and also a Monk formula in the torus-equivariant $K$-group of the formal power series model of semi-infinite flag manifolds, both of which are described explicitly in terms of se…
View article: Tensor product decomposition theorem for quantum Lakshmibai-Seshadri paths and standard monomial theory for semi-infinite Lakshmibai-Seshadri paths
Tensor product decomposition theorem for quantum Lakshmibai-Seshadri paths and standard monomial theory for semi-infinite Lakshmibai-Seshadri paths Open
Let $λ$ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let $\mathrm{QLS}(λ)$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $λ$. For an element $w$ of a finite Weyl group $W$, the specializ…
View article: Level-zero van der Kallen modules and specialization of nonsymmetric Macdonald polynomials at $t = \infty$
Level-zero van der Kallen modules and specialization of nonsymmetric Macdonald polynomials at $t = \infty$ Open
Let $λ\in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary coset representative of minimal length for the cosets in $W/W_λ$, where $W_λ$ is the stabilizer of $λ$ in a finite Weyl group $W$. In this paper, we give a mod…
View article: Path model for an extremal weight module over the quantized hyperbolic Kac-Moody algebra of rank 2
Path model for an extremal weight module over the quantized hyperbolic Kac-Moody algebra of rank 2 Open
Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank 2, and set $λ=Λ_{1} - Λ_{2}$, where $Λ_{1}$, $Λ_{2}$ are the fundamental weights. Denote by $V(λ)$ the extremal weight module of extremal weight $λ$ with $v_λ$ the extremal weigh…
View article: Pieri-Chevalley type formula for equivariant $K$-theory of semi-infinite flag manifolds
Pieri-Chevalley type formula for equivariant $K$-theory of semi-infinite flag manifolds Open
We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model $\mathbf{Q}_{G}$ of semi-infinite flag manifold and prove the Pieri-Chevalley formula, which describes the product, in…
View article: A Uniform Model for Kirillov–Reshetikhin Crystals II. Alcove Model, Path Model, and $P=X$
A Uniform Model for Kirillov–Reshetikhin Crystals II. Alcove Model, Path Model, and $P=X$ Open
We establish the equality of the specialization $P_\\lambda(x;q,0)$ of the Macdonald\n polynomial at $t=0$ with the graded character $X_\\lambda(x;q)$ of a tensor product of\n "single-column" Kirillov-Reshetikhin (KR) modules for untwisted…
View article: An explicit formula for the specialization of nonsymmetric Macdonald polynomials at $t = \infty$
An explicit formula for the specialization of nonsymmetric Macdonald polynomials at $t = \infty$ Open
In this paper, we give an explicit description of the specialization $E_{\mu}(q, \infty)$ of the nonsymmetric Macdonald polynomial $E_{\mu}(q, t)$ at $t = \infty$ for an arbitrary untwisted affine root system in terms of the quantum Bruhat…
View article: Specialization of nonsymmetric Macdonald polynomials at $t=\infty$ and Demazure submodules of level-zero extremal weight modules
Specialization of nonsymmetric Macdonald polynomials at $t=\infty$ and Demazure submodules of level-zero extremal weight modules Open
In this paper, we give a representation-theoretic interpretation of the specialization $E_{w_{\circ} λ} (q,\infty)$ of the nonsymmetric Macdonald polynomial $E_{w_{\circ} λ}(q,t)$ at $t=\infty$ in terms of the Demazure submodule $V_{w_\cir…
View article: Application of a ℤ₃-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices
Application of a ℤ₃-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices Open
By applying Miyamoto’s -orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices and their automorphisms of order , we construct holomorphic vertex operator algebras of central charge whose Lie algebra…
View article: Explicit description of the degree function in terms of quantum Lakshmibai-Seshadri paths
Explicit description of the degree function in terms of quantum Lakshmibai-Seshadri paths Open
We give an explicit and computable description, in terms of the parabolic quantum Bruhat graph, of the degree function defined for quantum Lakshmibai-Seshadri paths, or equivalently, for "projected" (affine) level-zero Lakshmibai-Seshadri …
View article: Explicit description of the degree function in terms of quantum Lakshmibai-Seshadri paths
Explicit description of the degree function in terms of quantum Lakshmibai-Seshadri paths Open
We give an explicit and computable description, in terms of the parabolic\nquantum Bruhat graph, of the degree function defined for quantum\nLakshmibai-Seshadri paths, or equivalently, for "projected" (affine) level-zero\nLakshmibai-Seshad…