核函数与量子代数
Kernel function and quantum algebras
摘要：
我们介绍了一个类似的柯西型核函数的Macdonald多项式$K_n(x,z;q,t)$，被构造在张量积的对称函数环和交换代数$\mathcal{A}$退化$\mathbb{C} \mathbb{P}^1$。我们表明，一定的限制$K_n(x,z;q,t)$相对于变量$z$是整齐地描述的表和公式的麦克唐纳多项式。接下来，我们证明了丁Iohara量子代数的整数水平表示自然产生的变形$\mathcal{W}$代数的电流。然后，我们注意到，$K_n(x,z;q,t)$出现在变形的$\mathcal{W}$代数的最高到最高的相关函数。
英文摘要：
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.
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