arXiv subjects

Jun'ichi Shiraishi

Publications and source records attributed to Jun'ichi Shiraishi.

At least 19 recordsLinked to original sources

arXiv2026

Non-stationary difference equation and affine Laumon space III : Generalization to $\widehat{\mathfrak{gl}}_N$

In a series of papers we have considered a non-stationary difference equation which was originally discovered for the deformed Virasoro conformal block. The equation involves mass parameters and, when they are tuned appropriately, the equation is regarded as a quantum KZ equation for $U_q(A_{1}^{(1)})$. We introduce a $\widehat{\mathfrak{gl}}_N$ generalization of the non-stationary difference equation. The Hamiltonian is expressed in terms of $q$-commuting variables and allows both factorized forms and a normal ordered form. By specializing the mass parameters appropriately, the Hamiltonian can be identified with the $R$-matrix of the symmetric tensor representation of $U_q(A_{N-1}^{(1)})$, which in turn comes from the 3D (tetrahedron) $R$-matrix. We conjecture that the affine Laumon partition function of type $A_{N-1}^{(1)}$ gives a solution to our $\widehat{\mathfrak{gl}}_N$ non-stationary difference equation. As a check of our conjecture, we work out the four dimensional limit and find that the non-stationary difference equation reduces to the Fuji-Suzuki-Tsuda system.

math.QA↗

arXiv2026

Shifted quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ and the Pieri rule of the super Macdonald polynomials

The super Macdonald polynomials indexed by the super partitions form a basis of the level zero super Fock module (combinatorial representation) of the quantum toroidal algebra $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_{1|1})$. The action of the super charges of $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_{1|1})$ implies the Pieri rule of the super Macdonald polynomials. We can express the Pieri rule in terms of differential operators in the power sums $p_k$ and the fermionic power sums $π_k$, which leads to the operators on the Fock space of a free boson and a free fermion. From the Pieri rule we compute the supersymmetric Hamiltonians given by the anti-commutator of the super charges and recover the results previously obtained in the literature. It is remarkable that we have to deal with a shifted quantum toroidal algebra.

math.QA↗

arXiv2026

Quantum Corner Polynomials: A Generalization of Super Macdonald Polynomials and Their VOA Correspondence

In this paper, we introduce a family of partially symmetric polynomials, which we call quantum corner polynomials, as a generalization of the Sergeev-Veselov super Macdonald polynomials. We show that these quantum corner polynomials are precisely the partially symmetric polynomials corresponding to the quantum corner VOAs. Furthermore, we provide a detailed proof of the partial symmetricity of these polynomials.

hep-th↗

arXiv2025

Quantum Corner VOA and the Super Macdonald Polynomials

In this paper, we establish a relation between the quantum corner VOA $q\widetilde{Y}_{L,0,N}[Ψ]$, which can be regarded as a generalization of quantum $W_N$ algebra, and Sergeev-Veselov super Macdonald polynomials. We demonstrate precisely that, under a specific map, the correlation functions of the currents of $q\widetilde{Y}_{L,0,N}[Ψ]$, coincide with the Sergeev-Veselov super Macdonald polynomials.

hep-th↗

arXiv2025

Super Macdonald polynomials and BPS state counting on the blow-up

We explore the relation of the super Macdonald polynomials and the BPS state counting on the blow-up of $\mathbb{P}^2$, which is mathematically described by framed stable perverse coherent sheaves. Fixed points of the torus action on the moduli space of BPS states are labeled by super partitions. From the equivariant character of the tangent space at the fixed points we can define the Nekrasov factor for a pair of super partitions, which is used for the localization computation of the partition function. The Nekrasov factor also allows us to compute matrix elements of the action of the quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ on the $K$ group of the moduli space. We confirm that these matrix elements are consistent with the Pieri rule of the super Macdonald polynomials.

hep-th↗

arXiv2025

$K$-theoretic wall-crossing formulas and multiple basic hypergeometric series

We study $K$-theoretic integrals over famed quiver moduli via wall-crossing phenomena. We study the chainsaw quiver varieties, and consider generating functions defined by two types of $K$-theoretic classes. In particular, we focus on integrals over the handsaw quiver varieties of type $A_{1}$, and get functional equations for each of them. We also give explicit formula for these partition functions. In particular, we obtain geometric interpretation of transformation formulas for multiple basic hypergeometric series including the Kajihara transformation formula, and the one studied by Langer-Schlosser-Warnaar and Hallnäs-Langman-Noumi-Rosengren.

math.AG↗

arXiv2024

Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation

We show that Shakirov's non-stationary difference equation, when it is truncated, implies the quantum Knizhnik-Zamolodchikov ($q$-KZ) equation for $U_{\mathsf v}\bigl(A_1^{(1)}\bigr)$ with generic spins. Namely, we can tune mass parameters so that the Hamiltonian acts on the space of finite Laurent polynomials. Then the representation matrix of the Hamiltonian agrees with the $R$-matrix, or the quantum $6j$ symbols. On the other hand, we prove that the $K$ theoretic Nekrasov partition function from the affine Laumon space is identified with the well-studied Jackson integral solution to the $q$-KZ equation. Combining these results, we establish that the affine Laumon partition function gives a solution to Shakirov's equation, which was a conjecture in our previous paper. We also work out the base-fiber duality and four-dimensional limit in relation with the $q$-KZ equation.

math.QA↗

arXiv2024

Elliptic Deformation of the Gaiotto-Rapčák Corner VOA and the Associated Partially Symmetric Polynomials

We construct the elliptic Miura transformation and use it to obtain the expression of the currents of elliptic corner VOA. We subsequently prove a novel combinatorial formula that is essential for deriving the quadratic relations of the currents. In addition, we give a conjecture that relates the correlation function of the currents of elliptic corner VOA to a certain family of partially symmetric polynomials. The elliptic Macdonald polynomials, constructed recently by Awata-Kanno-Mironov-Morozov-Zenkevich, and Fukuda-Ohkubo-Shiraishi, can be obtained as a particular case of this family.

hep-th↗

arXiv2023

Non-Stationary Difference Equation and Affine Laumon Space: Quantization of Discrete Painlevé Equation

We show the relation of the non-stationary difference equation proposed by one of the authors and the quantized discrete Painlevé VI equation. The five-dimensional Seiberg-Witten curve associated with the difference equation has a consistent four-dimensional limit. We also show that the original equation can be factorized as a coupled system for a pair of functions $\bigl(\mathcal{F}^{(1)},\mathcal{F}^{(2)}\bigr)$, which is a consequence of the identification of the Hamiltonian as a translation element in the extended affine Weyl group. We conjecture that the instanton partition function coming from the affine Laumon space provides a solution to the coupled system.

nlin.SI↗

arXiv2021

Branching Formula for $q$-Toda Function of Type B

We present a proof of the explicit formula for the asymptotically free eigenfunctions of the $B_N$ $q$-Toda operator which was conjectured by the first and third authors. This formula can be regarded as a branching formula from the $B_N$ $q$-Toda eigenfunction restricted to the $A_{N-1}$ $q$-Toda eigenfunctions. The proof is given by a contigulation relation of the $A_{N-1}$ Toda eigenfunctions and a recursion relation of the branching coefficients.

math.QA↗

arXiv2020

Non-Stationary Ruijsenaars Functions for $κ=t^{-1/N}$ and Intertwining Operators of Ding-Iohara-Miki Algebra

We construct the non-stationary Ruijsenaars functions (affine analogue of the Macdonald functions) in the special case $κ=t^{-1/N}$, using the intertwining operators of the Ding-Iohara-Miki algebra (DIM algebra) associated with $N$-fold Fock tensor spaces. By the $S$-duality of the intertwiners, another expression is obtained for the non-stationary Ruijsenaars functions with $κ=t^{-1/N}$, which can be regarded as a natural elliptic lift of the asymptotic Macdonald functions to the multivariate elliptic hypergeometric series. We also investigate some properties of the vertex operator of the DIM algebra appearing in the present algebraic framework; an integral operator which commutes with the elliptic Ruijsenaars operator, and the degeneration of the vertex operators to the Virasoro primary fields in the conformal limit $q \rightarrow 1$.

math.QA↗

arXiv2020

Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions

We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $BC_n$ Koornwinder polynomials $P_{(1^r)}(x|a,b,c,d|q,t)$ with one column diagrams, to the type $BC_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the matrix $\mathcal{C}$ enjoy a set of four terms recursion relations. These recursions provide us with the branching rules for the Koornwinder polynomials with one column diagrams, namely the restriction rules from $BC_n$ to $BC_{n-1}$. To have a good description of the transition matrices involved, we introduce the following degeneration scheme of the Koornwinder polynomials: $P_{(1^r)}(x|a,b,c,d|q,t) \longleftrightarrow P_{(1^r)}(x|a,-a,c,d|q,t)\longleftrightarrow P_{(1^r)}(x|a,-a,c,-c|q,t) \longleftrightarrow P_{(1^r)}\big(x|t^{1/2}c,-t^{1/2}c,c,-c|q,t\big) \longleftrightarrow P_{(1^r)}\big(x|t^{1/2},-t^{1/2},1,-1|q,t\big)$. We prove that the transition matrices associated with each of these degeneration steps are given in terms of the matrix inversion formula of Bressoud. As an application, we give an explicit formula for the Kostka polynomials of type $B_n$, namely the transition matrix from the Schur polynomials $P^{(B_n,B_n)}_{(1^r)}(x|q;q,q)$ to the Hall-Littlewood polynomials $P^{(B_n,B_n)}_{(1^r)}(x|t;0,t)$. We also present a conjecture for the asymptotically free eigenfunctions of the $B_n$ $q$-Toda operator, which can be regarded as a branching formula from the $B_n$ $q$-Toda eigenfunction restricted to the $A_{n-1}$ $q$-Toda eigenfunctions.

math.QA↗

arXiv2020

Lattice models, deformed Virasoro algebra and reduction equation

We study the fused currents of the deformed Virasoro algebra (DVA). By constructing a homotopy operator we show that for special values of the parameter of the algebra fused currents pairwise coincide on the cohomologies of the Felder resolution. Within the algebraic approach to lattice models these currents are known to describe neutral excitations of the solid-on-solid (SOS) models in the transfer-matrix picture. It allows us to prove the closeness of the system of excitations for a special nonunitary series of restricted SOS (RSOS) models. Though the results of the algebraic approach to lattice models were consistent with the results of other methods, the lack of such proof had been an essential gap in its construction.

hep-th↗

arXiv2019

Affine Screening Operators, Affine Laumon Spaces, and Conjectures Concerning Non-Stationary Ruijsenaars Functions

Based on the screened vertex operators associated with the affine screening operators, we introduce the formal power series f^{hat{gl}_N}(x,p|s,kappa|q,t) which we call the non-stationary Ruijsenaars function. We identify it with the generating function for the Euler characteristics of the affine Laumon spaces. When the parameters s and kappa are suitably chosen, the limit t rightarrow q of f^{hat{gl}_N}(x,p|s,kappa|q,q/t) gives us the dominant integrable characters of hat{sl}_N multiplied by 1/(p^N;p^N)_infty (i.e. the hat{gl}_1 character). Several conjectures are presented for f^{hat{gl}_N}(x,p|s,kappa|q,t), including the bispectral and the Poincare dualities, and the evaluation formula. Main Conjecture asserts that (i) one can normalize f^{hat{gl}_N}(x,p|s,kappa|q,t) in such a way that the limit kappa rightarrow 1 exists, and (ii) the limit f^{st.hat{gl}_N}(x,p|s|q,t) gives us the eigenfunction of the elliptic Ruijsenaars operator. The non-stationary affine q-difference Toda operator T^{hat{gl}_N}(kappa) is introduced, which comes as an outcome of the study of the Poincare duality conjecture in the affine Toda limit t rightarrow 0. Main Conjecture is examined also in the limiting cases of the affine q-difference Toda (t rightarrow 0), and the elliptic Calogero-Sutherland (q,t rightarrow 1) equations.

math.QA↗

arXiv2019

Generalized Macdonald Functions on Fock Tensor Spaces and Duality Formula for Changing Preferred Direction

An explicit formula is obtained for the generalized Macdonald functions on the $N$-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property of the arbitrary matrix elements of the multi-valent intertwining operator (or refined topological vertex operator) associated with the Ding--Iohara--Miki algebra (DIM algebra) with respect to the generalized Macdonald functions, which was conjectured by Awata, Feigin, Hoshino, Kanai, Yanagida and one of the authors. Our proof is based on the combinatorial and analytic properties of the asymptotic eigenfunctions of the ordinary Macdonald operator of $A$-type, and the Euler transformation formula for Kajihara and Noumi's multiple basic hypergeometric series. That factorization formula provides us with a reasonable algebraic description of the 5D (K-theoretic) Alday-Gaiotto-Tachikawa (AGT) correspondence, and the interpretation of the invariance under the preferred direction from the point of view of the $SL(2,\mathbb{Z})$ duality of the DIM algebra.

math.QA↗

arXiv2018

Macdonald Polynomials of Type $C_n$ with One-Column Diagrams and Deformed Catalan Numbers

We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $C_n$ degeneration of the Koornwinder polynomials $P_{(1^r)}(x\,|\,a,-a,c,-c\,|\,q,t)$ with one column diagrams, to the type $C_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the matrix $\mathcal{C}$ enjoy a set of three term recursion relations, which can be regarded as a $(a,c,t)$-deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type $(C_n,C_n)$ Macdonald polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,b;q,t)= P_{(1^r)}\big(x\,|\,b^{1/2},-b^{1/2},q^{1/2}b^{1/2},-q^{1/2}b^{1/2}\,|\,q,t\big)$. It is also shown that the $q$-ballot numbers appear as the Kostka polynomials, namely in the transition matrix from the Schur polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,q;q,q)$ to the Hall-Littlewood polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,t;0,t)$.

math.QA↗

arXiv2015

Tableau Formulas for One-Row Macdonald Polynomials of Types $C_n$ and $D_n$

We present explicit formulas for the Macdonald polynomials of types $C_n$ and $D_n$ in the one-row case. In view of the combinatorial structure, we call them "tableau formulas". For the construction of the tableau formulas, we apply some transformation formulas for the basic hypergeometric series involving very well-poised balanced ${}_{12}W_{11}$ series. We remark that the correlation functions of the deformed $\mathcal{W}$ algebra generators automatically give rise to the tableau formulas when we principally specialize the coordinate variables.

math.CO↗

arXiv2013

Macdonald polynomials, Laumon spaces and perverse coherent sheaves

Let $G$ be an almost simple simply connected complex Lie group, and let $G/U_-$ be its base affine space. In this paper we formulate a conjecture, which provides a new geometric interpretation of the Macdonald polynomials associated to $G$ via perverse coherent sheaves on the scheme of formal arcs in the affinization of $G/U_-$. We prove our conjecture for $G=SL(N)$ using the so called Laumon resolution of the space of quasi-maps (using this resolution one can reformulate the statement so that only "usual" (not perverse) coherent sheaves are used). In the course of the proof we also give a $K$-theoretic version of the main result of arXiv/0811.4454.

math.AG↗