Kernel function and quantum algebras

Abstract

We introduce an analogue $K_n(x,z;q,t)$ of the Cauchy-type kernel function for the Macdonald polynomials, being constructed in the tensor product of the ring of symmetric functions and the commutative algebra $\mathcal{A}$ over the degenerate $\mathbb{C}{\mathbb{P}}^1$. We show that a certain restriction of $K_n(x,z;q,t)$ with respect to the variable $z$ is neatly described by the tableau sum formula of Macdonald polynomials. Next, we demonstrate that the integer level representation of the Ding-Iohara quantum algebra naturally produces the currents of the deformed $\mathcal{W}$ algebra. Then we remark that the $K_n(x,z;q,t)$ emerges in the highest-to-highest correlation function of the deformed $\mathcal{W}$ algebra.

20 pages,

References in corpus (2)

• Quantum W-algebras and Elliptic Algebras
• Quantum $W_N$ Algebras and Macdonald Polynomials

Cited by in corpus (31)

• Ding-Iohara-Miki symmetry of network matrix models
• Explicit examples of DIM constraints for network matrix models
• Instanton moduli spaces and bases in coset conformal field theory
• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• Plane partitions with a "pit": generating functions and representation theory
• Notes on Ding-Iohara algebra and AGT conjecture
• Cluster integrable systems, q-Painleve equations and their quantization
• Anomaly in RTT relation for DIM algebra and network matrix models
• A slow review of the AGT correspondence
• An M-Theoretic Derivation of a 5d and 6d AGT Correspondence, and Relativistic and Elliptized Integrable Systems
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• AGT, Burge pairs and minimal models
• (q,t)-KZ equation for Ding-Iohara-Miki algebra
• A Note on Quiver Quantum Toroidal Algebra
• Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence
• Quantum toroidal gl(1) and Bethe ansatz
• Quantum toroidal algebras and solvable structures in gauge/string theory
• $q$-Deformation of Corner Vertex Operator Algebras by Miura Transformation
• A Macdonald refined topological vertex
• Bootstrapping the $S^5$ partition function
• Commutative Families of the Elliptic Macdonald Operator
• Kac determinant and singular vector of the level N representation of Ding-Iohara-Miki algebra
• Twisted Representations of Algebra of $q$-Difference Operators, Twisted $q$-$W$ Algebras and Conformal Blocks
• Non-Stationary Ruijsenaars Functions for $\kappa =t^{-1/N}$ and Intertwining Operators of Ding-Iohara-Miki Algebra
• Quasi-Hopf twist and elliptic Nekrasov factor
• Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator
• Tableau Formulas for One-Row Macdonald Polynomials of Types $C_n$ and $D_n$
• Free field approach to the Macdonald process
• Intertwining operator and integrable hierarchies from topological strings
• Singular Vector of Ding-Iohara-Miki Algebra and Hall-Littlewood Limit of 5D AGT Conjecture
• Twisted Fock module of toroidal algebra via DAHA and vertex operators