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.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2010
- DOI:
- 10.48550/arXiv.1002.2485
- arXiv:
- arXiv:1002.2485
- Bibcode:
- 2010arXiv1002.2485F
- Keywords:
-
- Mathematics - Quantum Algebra;
- Mathematics - Combinatorics
- E-Print:
- 20 pages,