Kac determinant and singular vector of the level N representation of Ding–Iohara–Miki algebra

Abstract

In this paper, we obtain the formula for the Kac determinant of the algebra arising from the level N representation of the Ding–Iohara–Miki algebra. It is also discovered that its singular vectors correspond to generalized Macdonald functions (the q-deformed version of the AFLT basis).

Appendices

Appendix

Appendix A: Macdonald functions

In this appendix, we give the definition of the ordinary Macdonald functions [26, Chap. VI].

Let $\Lambda$ be the ring of symmetric functions, and $p_{\lambda } = \prod _{k\ge 1} p_{\lambda _k}$ ($p_n =\sum _{i\ge 1} x_i^n$) is the power sum symmetric functions. Define the inner product $\langle -,- \rangle _{q,t}$ over $\Lambda$ by the condition that

$$\begin{aligned} \langle p_{\lambda }, p_{\mu } \rangle _{q,t} = z_{\lambda } \prod _{k=1}^{\ell (\lambda )} \frac{1-q^{\lambda _k}}{1-t^{\lambda _k}} \delta _{\lambda , \mu }, \quad z_{\lambda } \mathbin {:=}\prod _{i \ge 1} i^{m_i} m_i !, \end{aligned}$$

(A.1)

where $m_i=m_i (\lambda )$ is the number of entries in $\lambda$ equal i. For a partition $\lambda$, Macdonald functions $P_{\lambda } \in \Lambda$ are defined to be the unique functions in $\Lambda$ satisfying the following two conditions:

$$\begin{aligned}&\lambda \ne \mu \quad \Rightarrow \quad \langle P_{\lambda }, P_{\mu } \rangle _{q,t} = 0; \end{aligned}$$

(A.2)

$$\begin{aligned}&P_{\lambda } = m_{\lambda } + \sum _{\mu < \lambda } c_{\lambda \mu } m_{\mu }. \end{aligned}$$

(A.3)

Here $m_{\lambda }$ is a monomial symmetric function and < is the ordinary dominance partial ordering. In this paper, the power sum symmetric functions $p_n$ ($n \in {\mathbb {N}}$) are regarded as the variables of Macdonald functions. That is to say, $P_{\lambda }= P_{\lambda }(p_n; q,t)$. Here $P_{\lambda }(p_n; q,t)$ is an abbreviation for $P_{\lambda }(p_1, p_2, \ldots ; q,t)$.

Appendix B: Definition of DIM algebra and level N representation

In this section, we recall the definition of the DIM algebra and the level N representation. For the notations, we follow [19]. The DIM algebra has two parameters q and t. Let g(z) be the formal series

$$\begin{aligned} g(z)\mathbin {:=}\frac{G^+(z)}{G^-(z)}, \quad G^{\pm }(z)\mathbin {:=}(1-q^{\pm 1}z)(1-t^{\mp 1}z)(1-q^{\mp 1}t^{\pm 1}z). \end{aligned}$$

(B.1)

Then this series satisfies $g(z)=g(z^{-1})^{-1}$.

Definition B.1

Define the algebra ${\mathcal {U}}$ to be the unital associative algebra over ${\mathbb {Q}}(q,t)$ generated by the currents $x^\pm (z)=\sum _{n\in {\mathbb {Z}}}x^\pm _n z^{-n}$, $\psi ^\pm (z)=\sum _{\pm n\in {\mathbb {Z}}_{\ge 0}}\psi ^\pm _n z^{-n}$ and the central element $\gamma ^{\pm 1/2}$ satisfying the defining relations

$$\begin{aligned}&\psi ^\pm (z) \psi ^\pm (w) = \psi ^\pm (w) \psi ^\pm (z), \quad \psi ^+(z)\psi ^-(w) = \dfrac{g(\gamma ^{+1} w/z)}{g(\gamma ^{-1}w/z)}\psi ^-(w)\psi ^+(z), \end{aligned}$$

(B.2)

$$\begin{aligned}&\psi ^+(z)x^\pm (w) = g(\gamma ^{\mp 1/2}w/z)^{\mp 1} x^\pm (w)\psi ^+(z), \end{aligned}$$

(B.3)

$$\begin{aligned}&\psi ^-(z)x^\pm (w) = g(\gamma ^{\mp 1/2}z/w)^{\pm 1} x^\pm (w)\psi ^-(z), \end{aligned}$$

(B.4)

$$\begin{aligned}&\,[x^+(z),x^-(w)] =\dfrac{(1-q)(1-1/t)}{1-q/t} \big ( \delta (\gamma ^{-1}z/w) \psi ^+(\gamma ^{1/2}w)\nonumber \\&\quad \quad \quad \quad \quad \quad \quad \quad \quad - \delta (\gamma z/w) \psi ^-(\gamma ^{-1/2}w) \big ), \end{aligned}$$

(B.5)

$$\begin{aligned}&G^{\mp }(z/w)x^\pm (z)x^\pm (w)=G^{\pm }(z/w)x^\pm (w)x^\pm (z) . \end{aligned}$$

(B.6)

This algebra ${\mathcal {U}}$ is an example of the family of topological Hopf algebras introduced by Ding and Iohara [15]. This family is a sort of generalization of the Drinfeld realization of the quantum affine algebras. However, Miki introduce a deformation of the $W_{1+\infty }$ algebra in [27], which is the quotient of the algebra ${\mathcal {U}}$ by the Serre-type relation. Hence we call the algebra ${\mathcal {U}}$ the Ding-Iohara-Miki algebra (DIM algebra). Since the algebra ${\mathcal {U}}$ has a lot of background, there are a lot of other names such as quantum toroidal ${\mathfrak {g}}{\mathfrak {l}}_1$ algebra [21, 22], elliptic Hall algebra [14] and so on. This algebra has a Hopf algebra structure. The formulas for its coproduct $\Delta$ are

$$\begin{aligned}&\Delta (\psi ^\pm (z))= \psi ^\pm (\gamma _{(2)}^{\pm 1/2}z)\otimes \psi ^\pm (\gamma _{(1)}^{\mp 1/2}z), \end{aligned}$$

(B.7)

$$\begin{aligned}&\Delta (x^+(z))= x^+(z)\otimes 1+ \psi ^-(\gamma _{(1)}^{1/2}z)\otimes x^+(\gamma _{(1)}z), \end{aligned}$$

(B.8)

$$\begin{aligned}&\Delta (x^-(z))= x^-(\gamma _{(2)}z)\otimes \psi ^+(\gamma _{(2)}^{1/2}z)+1 \otimes x^-(z), \end{aligned}$$

(B.9)

and $\Delta (\gamma ^{\pm 1/2})=\gamma ^{\pm 1/2} \otimes \gamma ^{\pm 1/2}$, where $\gamma _{(1)}^{\pm 1/2} \mathbin {:=}\gamma ^{\pm 1/2}\otimes 1$ and $\gamma _{(2)}^{\pm 1/2} \mathbin {:=}1\otimes \gamma ^{\pm 1/2}$. Since we do not use the antipode and the counit in this paper, we omit them. The DIM algebra ${\mathcal {U}}$ can be represented by the Heisenberg algebra $a_{n}$ ($n\in {\mathbb {Z}}$) with the relation

$$\begin{aligned} \,[a_n, a_m] = n\frac{1-q^{|n|}}{1-t^{|n|}} \delta _{n+m,0}. \end{aligned}$$

(B.10)

Fact B.2

([19, 20]) Let u be an complex parameter or indeterminate. The morphism $\rho _u$ defined as follows is a representation of the DIM algebra:

$$\begin{aligned}&\rho _u(x^+(z))=u\, \eta (z),\quad \rho _u(x^-(z))=u^{-1} \xi (z), \end{aligned}$$

(B.11)

$$\begin{aligned}&\rho _u(\psi ^\pm (z))=\varphi ^\pm (z), \quad \rho _u(\gamma ^{\pm 1/2})=(t/q)^{\pm 1/4}, \end{aligned}$$

(B.12)

where

$$\begin{aligned}&\eta (z)\mathbin {:=}\exp \left( \sum _{n=1}^{\infty } \dfrac{1-t^{-n}}{n} z^{n} a_{-n} \right) \exp \left( -\sum _{n=1}^{\infty } \dfrac{1-t^{n} }{n} z^{-n} a_n \right) , \end{aligned}$$

(B.13)

$$\begin{aligned}&\xi (z)\mathbin {:=}\exp \left( -\sum _{n=1}^{\infty } \dfrac{1-t^{-n}}{n}(t/q)^{n/2} z^{n}a_{-n}\right) \exp \left( \sum _{n=1}^{\infty } \dfrac{1-t^{n}}{n} (t/q)^{n/2} z^{-n}a_n \right) , \end{aligned}$$

(B.14)

$$\begin{aligned}&\varphi _{+}(z)\mathbin {:=}\exp \left( -\sum _{n=1}^{\infty } \dfrac{1-t^{n}}{n} (1-t^n q^{-n})(t/q)^{-n/4} z^{-n}a_n \right) , \end{aligned}$$

(B.15)

$$\begin{aligned}&\varphi _{-}(z)\mathbin {:=}\exp \left( \sum _{n=1}^{\infty } \dfrac{1-t^{-n}}{n} (1-t^n q^{-n})(t/q)^{-n/4} z^{n}a_{-n} \right) . \end{aligned}$$

(B.16)

Not that the zero mode $\eta _0$ of $\eta (z)=\sum _n \eta _n z^{-n}$ can be essentially identified with the Macdonald difference operator [26, 32]. By using the coproduct of ${\mathcal {U}}$, we can consider the tensor representation of $\rho _u$. For an N-tuple of parameters $\vec {u}=(u_1,u_2,\ldots ,u_N)$, define the morphism $\rho _{\vec {u}}^{(N)}$ by

$$\begin{aligned} \rho _{\vec {u}}^{(N)}\mathbin {:=}(\rho _{u_1}\otimes \rho _{u_2}\otimes \cdots \otimes \rho _{u_N}) \circ \Delta ^{(N)}, \end{aligned}$$

(B.17)

where $\Delta ^{(N)}$ is inductively defined by $\Delta ^{(1)}\mathbin {:=}\mathrm {id}$, $\Delta ^{(2)}\mathbin {:=}\Delta$ and $\Delta ^{(N)}\mathbin {:=}( \mathrm {id} \otimes \cdots \otimes \mathrm{id}\otimes \Delta )\circ \Delta ^{(N-1)}$. The representation $\rho _{\vec {u}}^{(N)}$ is called the level N representation after the property $\rho _{\vec {u}}^{(N)}(\gamma )=(t/q)^{\frac{N}{2}}$. $\rho _{\vec {u}}^{(N)}$ is also called the level (N, 0) representation or the horizontal representation to distinguish another one called the level (0, N) representation or the vertical representation [4, 17, 23]. These representations can be regarded as a sort of duality through an automorphism of DIM algebra. In the representation $\rho _{\vec {u}}^{(N)}$, we write the i-th bosons as

$$\begin{aligned} a^{(i)}_n \mathbin {:=}\underbrace{1 \otimes \cdots \otimes 1 \otimes a_n}_i \otimes 1 \otimes \cdots \otimes 1 \end{aligned}$$

(B.18)

for simplicity. The generator $X^{(1)}(z)$ in Definition 2.2 is obtained by

$$\begin{aligned} X^{(1)}(z) = \rho ^{(N)}_{\vec {u}} (x^{+}(z)). \end{aligned}$$

(B.19)

Note that, in this paper, the parameters $u_i$ are realized by the operators $U_i$ and the highest weight vector in order to use screening currents.