KERNEL FUNCTION AND QUANTUM ALGEBRAS (Representation Theory and Combinatorics)
Boris Lvovich Feigin, Ayumu Hoshino, Jun Shibahara, Jun’ichi Shiraishi, Shintarou Yanagida · Institutional Repositories DataBase (IRDB) · 2010
We introduce an aiialogue $ K_{t1}(x, z;q, t)$ of the Cauchy-type kernel function for the MMac- donald polynomials, being constructed in the tensor product of the ring $\Lambda_{F}$ of symmetric functions and the commutative algebra $\mathcal{A}$ over the degenerate $\mathbb{C}\mathbb{P}^{1}$ .We show that a certain restriction of $K_{n}(x,\tilde{k};q, t)$ with respect to the variable $z$ ls neatly described by the tableau sum formula of Macdon- ald polynomials.Next, we demonstrate that the level $rn$ representation of the Ding-Iohara quantum algebra $\mathcal{U}(q, t)$ naturally produces the currents of the deformed $V\backslash \prime_{q,p}(\epsilon 1_{r\iota})$ .Then we remark that the $I\backslash ^{r}n(x, z;q, t)$ emerges in the highest-to-higliest correlation function of the deformed $w_{J,p}^{\backslash }(\mathfrak{s}1_{\eta})$ algebra. KERNEL FUNCTION1.1.The algebra $\mathcal{A}$ .We briefly recall the definition and the basic facts about the commutative algebra $\mathcal{A}$ introduced in [FHHSY].Let $q_{1},$ $q_{2}$ be two independent indeterminates and set $q_{3}:=1q_{1}q_{2}$ .We also use the symbols $\mathbb{F}:=\mathbb{Q}(q_{1}, q_{2}),$ $\mathbb{N}:=\{0,1,2, \ldots\}$ and $\mathbb{N}_{+}:=\{1,2, \ldots\}$ .For $n,$ $k\in \mathbb{N}+$ , we define two operators $\partial^{(0,k)},$ $\partial^{(\infty,k)}$ acting on the space of symmetric rational functions inwhenever the limit exists.We also set $\partial^{(0k)}$ ) $c=0,$ $\partial^{(\infty,k)}c=0$ for $c\in \mathbb{F}$ .Finally we define $\partial^{(0,0)}$ and $\partial^{(\infty,0)}$ to be the identity operator.Definition 1.1.For $n\in \mathbb{N}$ , the vector space $A_{n}=\mathcal{A}_{n}(q_{1}, q_{2}, q_{3})$ is defined by the following conclitions (i), (ii), (iii) and (iv).(i) $\mathcal{A}_{0};=\mathbb{F}$ .For $n\in \mathbb{N}+,$ $f(x_{1}, \ldots, x_{n})\in \mathcal{A}_{n}$ is a rational function with coefficients in $\mathbb{F}$ , and symmetric with respect to the $x_{i}' s$ .(ii) For $n\in \mathbb{N},$ $0\leq k\leq n$ and $f\in \mathcal{A}_{n}$ , the limits $\partial^{(\infty,k)}f$ and $\partial^{(0,k)}f$ both exist and coincide: $\partial^{(\infty,k)}f=\partial^{(0_{2}k)}f$ (degenerate $\mathbb{C}\mathbb{P}^{1}$ condition).(iii) The poles of $f\in \mathcal{A}_{n}$ are located only on the diagonal $\{(x_{1}, \ldots, x_{n})|\exists(i,j), i eq j, x_{i}=x_{j}\}$ , and the orders of the poles are at most two.(iv) For $n\geq 3,$ $!\in \mathcal{A}_{n}$ satisfies the wheel conditions $f(x_{1}, q_{1}x_{1}, q_{1}q_{2}x_{1}, x_{4}, \ldots)=0$ , $f(x_{1}, q_{2}x_{1}, q_{1}q_{2}x_{1}, x_{4}, \ldots)=0$ .Then we set the graded vector space $\mathcal{A}=\mathcal{A}(q_{1}, q_{2}, q_{3}):=\oplus_{r\downarrow\geq 0}\mathcal{A}_{n}$.Definition 1.2.For an m-variable symmetric rational function $f$ and an n-variable symmetric rational function $g$ , we define an $(m+n)$ -variable symmetric rational function $f*g$ by $(f*g)(x_{1}, \ldots, x_{m+n}):=$ Sym $[\alpha'\beta m+1\leq\beta\leq m+n$ .