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

TL;DR

This paper derives the Kac determinant formula for the level N representation of the Ding-Iohara-Miki algebra and links its singular vectors to generalized Macdonald functions, advancing understanding of algebraic structures in quantum algebra.

Contribution

It provides the first explicit formula for the Kac determinant in this algebra and identifies singular vectors with generalized Macdonald functions.

Findings

Kac determinant formula derived for level N representation

Singular vectors correspond to generalized Macdonald functions

Enhances understanding of algebraic structures in quantum 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).

Equations

$${\rho }_{\vec{u}}^{(N)}$$

$${\Delta }^{(N)}$$

$$X^{(1)}(z):=\sum _{n\in \mathbb{Z}}{X}_n^{(1)}{z}^{-n}:={\rho }^{(N)}(x^{+}(z)).$$

$$|X_{\vec{\lambda }}⟩:=X_{-{\lambda }_1^{(1)}}^{(1)}{X}_{-{\lambda }_2^{(1)}}^{(1)}\cdots X_{-{\lambda }_1^{(2)}}^{(2)}{X}_{-{\lambda }_2^{(2)}}^{(2)}\cdots X_{-{\lambda }_1^{(N)}}^{(N)}{X}_{-{\lambda }_2^{(N)}}^{(N)}\cdots |\vec{u}⟩,$$

$$\det {\left(⟨X_{\vec{\lambda }}⟩X_{\vec{\mu }}\right)}_{|\vec{\lambda }|=|\vec{\mu }|=n}=\prod _{\vec{\lambda }⊢n}\prod _{k=1}^N{b}_{{\lambda }^{(k)}}(q)b_{{\lambda }^{(k)}}^{\prime }(t^{-1})$$

$$\times \prod _{\begin{array}{c}1\le r,s\\ rs\le n\end{array}}{\left((u_1{u}_2\cdots u_N{)}^2\prod _{1\le i<j\le N}(u_i-q^s{t}^{-r}{u}_j)(u_i-q^{-r}{t}^s{u}_j)\right)}^{P^{(N)}(n-rs)},$$

$$|{\chi }_{\vec{r},\vec{s}}⟩:=\oint \prod _{k=1}^{N-1}\prod _{i=1}^{r_k}d{z}_i^{(k)}$$

$$\cdots S^{(1)}(z_1^{(1)})\cdots S^{(1)}(z_{r_1}^{(1)})|\vec{v}⟩$$

$$[a_n^{(i)},a_m^{(j)}]=n\frac{1-q^{|n|}}{1-t^{|n|}}{\delta }_{n+m,0}{\delta }_{i,j},[a_n^{(i)},Q^{(j)}]={\delta }_{n,0}{\delta }_{i,j}.$$

$${\eta }^{(i)}(z)$$

$${\phi }^{(i)}(z)$$

$$X^{(i)}(z):=\sum _{1\le j_1<\cdots <j_i\le N}to0.0pt\text{raise}1.72218pt\bullet \text{hss}\text{lower}1.72218pt\bullet {\Lambda }_{j_1}(z)\cdots {\Lambda }_{j_i}(p^{i-1}z)to0.0pt\text{raise}1.72218pt\bullet \text{hss}\text{lower}1.72218pt\bullet ,$$

$${\Lambda }^i(z):={\phi }^{(1)}(z){\phi }^{(2)}(z{p}^{-\frac{1}{2}})\cdots {\phi }^{(i-1)}(z{p}^{-\frac{i-2}{2}}){\eta }^{(i)}(z{p}^{-\frac{i-1}{2}})U_i,$$

$$U_i:=q^{\sqrt{\beta }{a}_0^{(i)}}{p}^{-\frac{N+1}{2}+i}.$$

$$f^{(1)}\left(\frac{w}{z}\right)X^{(1)}(z)X^{(1)}(w)-X^{(1)}(w)X^{(1)}(z)f^{(1)}\left(\frac{z}{w}\right)$$

$$=\frac{(1-q)(1-t^{-1})}{1-p}\left\{\delta \left(\frac{w}{pz}\right)X^{(2)}(z)-\delta \left(\frac{pw}{z}\right)X^{(2)}(w)\right\},$$

$$f^{(2)}\left(\frac{w}{z}\right)X^{(2)}(z)X^{(2)}(w)-X^{(2)}(w)X^{(2)}(z)f^{(2)}\left(\frac{z}{w}\right)=0,$$

$$f^{(1)}\left(\frac{pw}{z}\right)X^{(1)}(z)X^{(2)}(w)-X^{(2)}(w)X^{(1)}(z)f^{(1)}\left(\frac{z}{w}\right)=0,$$

$$f^{(1)}(z):=\exp \left\{\sum _{n>0}\frac{(1-q^n)(1-t^{-n})}{n}{z}^n\right\},$$

$$f^{(2)}(z):=\exp \left\{\sum _{n>0}\frac{(1-q^n)(1-t^{-n})(1+p^n)}{n}{z}^n\right\}.$$

$$[X_n^{(1)},X_m^{(1)}]=-\sum _{l=1}^{\infty }{f}_l^{(1)}(X_{n-l}^{(1)}{X}_{m+l}^{(1)}-X_{m-l}^{(1)}{X}_{n+l}^{(1)})$$

$$+\frac{(1-q)(1-t^{-1})}{1-p}(p^m-p^n)X_{n+m}^{(2)},$$

$$[X_n^{(2)},X_m^{(2)}]=-\sum _{l=1}^{\infty }{f}_l^{(2)}(X_{n-l}^{(2)}{X}_{m+l}^{(2)}-X_{m-l}^{(2)}{X}_{n+l}^{(2)}),$$

$$[X_n^{(1)},X_m^{(2)}]=-\sum _{l=1}^{\infty }{f}_l^{(1)}(p^l{X}_{n-l}^{(1)}{X}_{m+l}^{(2)}-X_{m-l}^{(2)}{X}_{n+l}^{(1)}).$$

$$|X_{\vec{\lambda }}⟩:=X_{-{\lambda }_1^{(1)}}^{(1)}{X}_{-{\lambda }_2^{(1)}}^{(1)}\cdots X_{-{\lambda }_1^{(2)}}^{(2)}{X}_{-{\lambda }_2^{(2)}}^{(2)}\cdots X_{-{\lambda }_1^{(N)}}^{(N)}{X}_{-{\lambda }_2^{(N)}}^{(N)}\cdots |\vec{u}⟩,$$

$$⟨X_{\vec{\lambda }}|:=⟨\vec{u}|\cdots X_{{\lambda }_2^{(N)}}^{(N)}{X}_{{\lambda }_1^{(N)}}^{(N)}\cdots X_{{\lambda }_2^{(2)}}^{(2)}{X}_{{\lambda }_1^{(2)}}^{(2)}\cdots X_{{\lambda }_2^{(1)}}^{(1)}{X}_{{\lambda }_1^{(1)}}^{(1)}.$$

$${\det }_n$$

$$\times \prod _{\begin{array}{c}1\le r,s\\ rs\le n\end{array}}{\left((u_1{u}_2\cdots u_N{)}^2\prod _{1\le i<j\le N}(u_i-q^s{t}^{-r}{u}_j)(u_i-q^{-r}{t}^s{u}_j)\right)}^{P^{(N)}(n-rs)},$$

$${\det }_n=\prod _{\lambda ⊢n}{b}_{\lambda }(q)b_{\lambda }^{\prime }(t^{-1})\times u_1^{2{\sum }_{\lambda ⊢n}\ell (\lambda )}.$$

$$X_n^{(i)}|\chi ⟩=0$$

$$b_{-n}^{\prime }:=\frac{(1-t^{-n})(1-p^n)}{n(1-p^{Nn})}{p}^{(N-1)n}\sum _{k=1}^N{p}^{(\frac{-k+1}{2})n}{a}_{-n}^{(k)},$$

$$b_n^{\prime }:=-\frac{(1-t^n)(1-p^n)}{n(1-p^{Nn})}{p}^{(N-1)n}\sum _{k=1}^N{p}^{\left(\frac{-k+1}{2}\right)n}{a}_n^{(k)}(n>0),$$

$$b_0^{\prime }:=a_0^{(1)}+\cdots +a_0^{(N)},Q^{\prime }:=\frac{Q^{(1)}+\cdots +Q^{(N)}}{N}.$$

$$b_{-n}^{(i)}:=\frac{1-t^{-n}}{n}{a}_{-n}^{(i)}-p^{(\frac{-i+1}{2})n}{b}_{-n}^{\prime },b_n^{(i)}:=\frac{1-t^n}{n}{a}_n^{(i)}+p^{(\frac{-i+1}{2})n}{b}_n^{\prime }(n>0).$$

$$h_{-n}^{(i)}:=(1-p^{-n})\left(\sum _{k=1}^{i-1}{p}^{\frac{-k+1}{2}n}{b}_{-n}^{(k)}\right)+p^{\frac{-i+1}{2}n}{b}_{-n}^{(i)},h_n^{(i)}:=-p^{\frac{i-1}{2}n}{b}_n^{(i)}(n>0),$$

$$h_0^{(i)}:=a_0^{(i)}-\frac{b_0^{\prime }}{N},Q_h^{(i)}:=Q^{(i)}-Q^{\prime }.$$

$$[b_n^{\prime },b_m^{\prime }]=-\frac{(1-q^{|n|})(1-t^{-|n|})(1-p^{|n|})}{n(1-p^{N|n|})}{\delta }_{n+m,0},$$

Full text

1. Introduction

In 1995, Mimachi and Yamada discovered the surprising relation that the Jack functions with rectangular Young diagrams have a one-to-one correspondence with singular vectors of the Virasoro algebra [26]. As a $q$-difference deformation of the Jack polynomials, there is a system of orthogonal functions called the Macdonald functions [24]. Awata, Kubo, Odake and Shiraishi introduced in [31] a $q$-deformation of the Virasoro algebra, which is named the deformed Virasoro algebra. This deformed algebra is designed so that singular vectors of Verma modules correspond to Macdonald symmetric functions associated with rectangular Young diagrams. It is also possible to obtain the Jack and Macodnald functions with general partitions from the singular vectors of the $W_{N}$-algebra and the deformed $W_{N}$-algebra, which is the (deformed) Virasoro algebra when $N=2$ [27, 8, 7]. To be exact, singular vectors of the (deformed) $W_{N}$-algebra can be realized by $N-1$ families of bosons under the free field representation. By a certain projection to one of these bosons, we can obtain the Jack (or Macdonald) functions associated with Young diagrams with $N-1$ edges (see Figure 1). As we will see later in Introduction, we discover a sort of generalization of these correspondence using generalized Macdonald functions arising from AGT conjecture.

AGT (Alday-Gaiotto-Tachikawa) conjecture states that the Virasoro conformal blocks of two-dimensional conformal field theories correspond to the instanton partition functions of four-dimensional $\mathcal{N}=2$ $SU(2)$ supersymmetric gauge theories [2]. It is also expected that the four-dimensional gauge theories with the higher gauge group $SU(N)$ correspond with the $W_{N}$-algebra [32]. $q$-deformed version of the AGT conjecture is also provided. That is, the deformed Virasoro/$W$-algebra is related to five-dimensional gauge theories (5D AGT conjecture) [9, 10]. It is known that a good basis called AFLT basis (Alba-Fateev-Litvinov-Tarnopolski basis) [1, 15] exists in the representation space of the tensor product of Virasoro/$W_{N}$ algebra and $U(1)$ Heisenberg algebra, and by this basis the conformal block can be combinatorially expanded like the Nekrasov partition functions. Since the AFLT basis corresponds to the torus fixed points in the instanton moduli space, it is also called the fixed point basis. The AFLT basis can be regarded as a sort of generalization of Jack functions under the free field representation. Also Dotsenko-Fateev integral are expanded in the form of the Nekrasov funciton with the help of that generalized Jack functions [28]. $q$-deformed version of the AFLT basis [3] can be constructed in the representation space of the Ding-Iohara-Miki algebra $\mathcal{U}$ (DIM algebra), and regarded as generalized Macdonald functions. By using, the generalized Macdonald fucntions, $q$-Dotsenko-Fateev integral are also combinatorially expanded [33].

The DIM algebra is the associative algebra generated by four currents $x^{\pm}(z)=\sum_{n\in\mathbb{Z}}x^{\pm}_{n}z^{-n}$, $\psi^{\pm}(z)=\sum_{\pm n\in\mathbb{Z}_{\geq 0}}\psi^{\pm}_{n}z^{-n}$ and the central element $\gamma^{\pm 1/2}$ satisfying the relations (B.2)-(B.6) as explained in Appendix B. The DIM algebra $\mathcal{U}$ has the face of a $q$-deformation of the $W_{1+\infty}$ algebra as introduced by Miki in [25], and the deformed Virasoro/$W$-algebra appear in its representation [18]. Since the DIM algebra has a lot of background, there are a lot of other names such as quantum toroidal $\mathfrak{gl}_{1}$ algebra [19, 20], elliptic Hall algebra [13] and so on. Furthermore, the DIM algebra $\mathcal{U}$ is associated with the Macdonald functions. The generator $x^{+}_{0}$ can be essentially identified with Macdonald’s difference operator under the free field representation $\rho_{u}:\mathcal{U}\rightarrow\mathrm{End}(\mathcal{F}_{u})$, where $\mathcal{F}_{u}$ is the Fock module and $u$ expresses its highest wight [17]. (See also Fact B.2.) The DIM algebra $\mathcal{U}$ has a Hopf algebra structure which does not exist in the deformed Virasoro/$W$-algebra. Using the coproduct $\Delta$ of the DIM algebra, we can consider the tensor representation of $\rho_{u}$:

[TABLE]

where $\vec{u}=(u_{1},\ldots,u_{N})$. This representation $\rho^{(N)}_{\vec{u}}$ is called the level $N$ representation. The generalized Macdonald functions are defined to be eigenfunctions of the generator $X^{(1)}_{0}$ given by

[TABLE]

Whereas the ordinary Macdonald functions are indexed by partitions, the generalized Macdonald ones are indexed by $N$-tuples of partitions $\vec{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(N)})$ and denoted by $\Ket{P_{\vec{\lambda}}}$. They correspond to the ordinary Macdonald function in the case of $N=1$. Introducing the generator $X^{(i)}_{n}$ which appearing in the commutation relations of $X^{(j)}_{n}$ ($j=1,\ldots,i-1$), we can define an algebra $\mathcal{A}(N):=\langle X^{(i)}_{n}|i=1,\ldots,N,n\in\mathbb{Z}\rangle$. (See Definition 2.2.) With the help of the integral form of generalized Macdonald functions, which are defined by using PBW type vectors

[TABLE]

it was conjectured that the conformal block of a certain vertex operator of DIM algebra corresponds to 5D (K-theoretic) partition functions [3]. The geometric interpretation of this kind of correspondence is given in [29], and the algebra $\mathcal{A}(N)$ coincides with the $qW$ algebra of type $\mathfrak{gl}_{N}$ (not $\mathfrak{sl}_{N}$) which is important in that interpretation. Let us also mention that by using another representation of DIM algebra, it is proved that the correlation function of an intertwining operator of DIM algebra coincides with the 5D Nekrasov formula [4]. Furthermore, it is conjectured that R-matrix of DIM algebra can be written in terms of the integral form of generalized Macdonald functions [5, 6, 22]

In this paper, we focus on the representation of the algebra $\mathcal{A}(N)$. One of the results in this paper (Theorem 2.6) is the formula for the Kac determinant with respect to the PBW type vector $\Ket{X_{\vec{\lambda}}}$.

Theorem.

[TABLE]

where $b_{\lambda}(q)\mathbin{:=}\prod_{i\geq 1}\prod_{k=1}^{m_{i}}(1-q^{k})$, $b^{\prime}_{\lambda}(q)\mathbin{:=}\prod_{i\geq 1}\prod_{k=1}^{m_{i}}(-1+q^{k})$, and $P^{(N)}(n)$ denotes the number of the $N$-tuples of Young diagrams of size $n$. Further, $m_{i}=m_{i}(\lambda)$ is the number of entries in $\lambda$ equal to $i$.

This determinant can be proved by using the fact that the generators $X^{(i)}_{n}$ can be decomposed into the deformed $W$-algebra part and the $U(1)$ part by a linear transformation of the bosons, and using the screening currents of the deformed $W$-algebra. By this formula, we can solve the conjecture [3, Conjecture 3.4] that the PBW type vectors $\Ket{X_{\vec{\lambda}}}$ of the algebra $\mathcal{A}(N)$ are a basis.

We also discover that singular vectors of the algebra $\mathcal{A}(N)$ correspond to the generalized Macdonald functions (Theorem 4.4). By this result, we can get singular vectors from generalized Macdonald functions. The singular vectors are intrinsically the same as those of the deformed $W$-algebra. However, as the projection of bosons is required for coincidence with the ordinary Macdonald functions [7], the result of this paper that does not need projections can be regarded as an extension of correspondence with ordinary Macdonald functions. As a corollary of this fact, we can find a new relation of the ordinary Macdonald functions and the generalized Macdonald functions by the projection of bosons.

Concretely, the vector $\Ket{\chi_{\vec{r},\vec{s}}}$ defined to be

[TABLE]

is a singular vector. Here $S^{(i)}(z)$ denotes the screening operator defined in (3.22), and $\vec{v}=(v_{1},\ldots,v_{N})$ is an $N$-tuple of parameters depending on the number of screening currents $r_{k}$ and positive integers $s_{k}$ ($k=1,\ldots,N-1$). For more details, see Section 4. In the case that $r_{k}$ satisfy $0\leq r_{k}\leq r_{k+1}$ for all $k$, the singular vector $\Ket{\chi_{\vec{r},\vec{s}}}$ coincides with the generalized Macdonald function $\Ket{P_{\vec{\lambda}}}$ with the $N$-tuple of Young diagrams in Figure 2.

In fact, Figure 1 means the same Young diagram being on the rightmost side in Figure 2. Hence the projection of this generalized Macdonald function corresponds to the ordinary Macdonald functions associated with the rightmost Young diagram with $N-1$ edges in Figure 2 (Corollary 4.7). When the condition $r_{k+1}\geq r_{k}$ is removed, the above figure is not a Young diagram. However it turns out that the vector $\Ket{\chi_{\vec{r},\vec{s}}}$ coincides with the generalized Macdonald function obtained by cutting off the protruding part and moving boxes to the Young diagrams on the left side. For details, see Theorem 4.4 and Example 4.6.

This paper is organized as follows. In Section 2, we define the algebra $\mathcal{A}(N)=\langle X^{(i)}_{n}\rangle$ and give a factorized formula for the Kac determinant. Section 3 is devoted to the proof of that formula. Along [18], we provide the linear transformations of bosons to decompose the algebra $\mathcal{A}(N)$ into the deformed $W$-algebra and $U(1)$ Heisenberg algebra. By using some results of the deformed $W$-algebra, we prove the formula for the Kac determinant. In Section 4, we give the definition of the generalized Macdonald functions and show that the singular vectors $\ket{\chi_{\vec{r},\vec{s}}}$ coincide with generalized Macdonald functions. In Appendix A, the definition of ordinary Macdonald functions are explained. At last, in Appendix B, we describe the definition of the DIM algebra and the level $N$ representation.

Notation of partitions

Let us explain the notation of partitions and Young diagrams. A partition $\lambda=(\lambda_{1},\ldots,\lambda_{n})$ is a non-increasing sequence of integers $\lambda_{1}\geq\ldots\geq\lambda_{n}\geq 0$. We write $\left|\lambda\right|\mathbin{:=}\sum_{i}\lambda_{i}$. The length of $\lambda$, denoted by $\ell(\lambda)$, is the number of elements $\lambda_{i}$ with $\lambda_{i}\neq 0$. $m_{i}=m_{i}(\lambda)$ denotes the number of elements that are equal to $i$ in $\lambda$, and we occasionally write partitions as $\lambda=(1^{m_{1}},2^{m_{2}},3^{m_{3}},\ldots)$. For example, $\lambda=(6,6,6,2,2,1)=(6^{3},2^{2},1)$. The partitions are identified with the Young diagrams, which are the figures written by putting $\lambda_{i}$ boxes on the $i$-th row and aligning the left side. We use $(i,j)\in\mathbb{N}^{2}$ for a coordinate of a box in the $i$-th row and the $j$-th column in a Young diagram. For example, if $\lambda=(6,4,3,3,1)$, its Young diagram is

$s$

.

The coordinate of the box $s$ is $(2,3)$. For an $N$-tuple of partition $\vec{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(N)})$, we write $|\vec{\lambda}|:=|\lambda^{(1)}|+\cdots+|\lambda^{(N)}|$. If $|\vec{\lambda}|=m$, we occasionally use the symbol $\vdash$ as $\vec{\lambda}\vdash m$.

2. Kac determinant of level $N$ representation of DIM algebra

Firstly, we define the algebra $\mathcal{A}(N)=\langle X^{(i)}_{n}\rangle$ obtained by the level $N$ representation of the DIM algebra. This algebra $\mathcal{A}(N)$ and the level $N$ representation play an important role in 5D AGT conjecture corresponding to $SU(N)$ gauge theory. Let $q$ and $t=q^{\beta}$ be generic complex parameters and $p\mathbin{:=}q/t$. $a_{n}^{(i)}$ and $Q^{(i)}$ ($n\in\mathbb{Z}$, $i=1,\ldots N$) are the generators of the Heisenberg algebra such that

[TABLE]

Let us define the vertex operators $\eta^{(i)}(z)$ and $\varphi^{(i)}(z)$.

Definition 2.1**.**

Set

[TABLE]

Definition 2.2**.**

Define generators $X^{(i)}(z)=\sum_{n}X^{(i)}_{n}z^{-n}$ by

[TABLE]

where \scriptscriptstyle\bullet$$\scriptscriptstyle\bullet   \scriptscriptstyle\bullet$$\scriptscriptstyle\bullet denotes the usual normal ordering product and

[TABLE]

The generator $X^{(1)}(z)$ arises from the level $N$ representation of Ding-Iohara-Miki algebra [17, 18] and is obtained by acting the coproduct $N$ times to a generator of the DIM algebra corresponding to $\eta(z)$ (see Appendix B). The other generators $X^{(i)}_{n}$ appear in the commutation relations of generators $X^{(i-k)}_{n}$ ($k=1,\ldots i-1$). Let $\Ket{0}$ be the highest weight vector in the Fock module of the Heisenberg algebra such that $a^{(i)}_{n}\Ket{0}=0$ for $n\geq 0$. For an $N$-tuple of complex parameters $\vec{u}=(u_{1},\ldots,u_{N})$ with $u_{i}=q^{\sqrt{\beta}w_{i}}p^{-\frac{N+1}{2}+i}$, set $\Ket{\vec{u}}:=e^{\sum_{i=1}^{N}w_{i}Q^{(i)}}\Ket{0}$. Then they satisfy the relation $U_{i}\Ket{\vec{u}}=u_{i}\Ket{\vec{u}}$. Similarly, let $\Bra{0}$ be the dual highest weight vector, and $\Bra{\vec{u}}:=\Bra{0}e^{-\sum_{i=1}^{N}w_{i}Q^{(i)}}$. $\mathcal{F}_{\vec{u}}$ is the highest weight module generated by $\Ket{\vec{u}}$, and $\mathcal{F}_{\vec{u}}^{*}$ is the dual module generated by $\Bra{\vec{u}}$. The bilinear form (Shapovalov form) $\mathcal{F}_{\vec{u}}^{*}\otimes\mathcal{F}_{\vec{u}}\rightarrow\mathbb{C}$ is uniquely determined by the condition $\Braket{\vec{u}}{\vec{u}}=1$.

Definition 2.3**.**

Define the algebra $\mathcal{A}(N)$ to be the subalgebra $\langle X^{(i)}_{n}|i=1,\ldots,N,n\in\mathbb{Z}\rangle$ in some completion of the endomorphism algebra of the Fock module $\mathcal{F}_{\vec{u}}$ for our Heisenberg algebra.

Example 2.4**.**

If $N=2$, the commutation relations of the generators are

[TABLE]

where $\delta(x)=\sum_{n\in\mathbb{Z}}x^{n}$ is the multiplicative delta function and the structure constant $f^{(i)}(z)=\sum_{l=0}^{\infty}f^{(i)}_{l}z^{l}$ is defined by

[TABLE]

[TABLE]

These relations are equivalent to

[TABLE]

The calculation of Example 2.4 is similar to the ordinary deformed Virasoro and W algebra [7, 31]. The algebra $\mathcal{A}(N)$ corresponds to the $qW$ algebra of type $\mathfrak{gl}_{N}$ (not $\mathfrak{sl}_{N}$) defined in [29]. The relations of generators $X^{(i)}(z)$ for the general $N$ case are also shown in [29]. By the following PBW type vectors $\Ket{X_{\vec{\lambda}}}$, we can define the integral form of the generalized Macdonald functions, which have significant properties in the AGT correspondence [3].

Definition 2.5**.**

For an $N$-tuple of partitions $\vec{\lambda}=(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(N)})$, set

[TABLE]

The PBW theorem cannot be used because the algebra $\mathcal{A}(N)$ is not a Lie algebra. In [3] it was conjectured that the PBW type vectors $\Ket{X_{\vec{\lambda}}}$ and $\Bra{X_{\vec{\lambda}}}$ are a basis over $\mathcal{F}_{\vec{u}}$ and $\mathcal{F}_{\vec{u}}^{*}$, respectively. This conjecture can be solved by the following Kac determinant of the algebra $\mathcal{A}(N)$.

Theorem 2.6**.**

Let $\mathrm{det}_{n}\mathbin{:=}\det\left(\Braket{X_{\vec{\lambda}}}{X_{\vec{\mu}}}\right)_{\vec{\lambda},\vec{\mu}\vdash n}$. Then

[TABLE]

where $b_{\lambda}(q)\mathbin{:=}\prod_{i\geq 1}\prod_{k=1}^{m_{i}}(1-q^{k})$, $b^{\prime}_{\lambda}(q)\mathbin{:=}\prod_{i\geq 1}\prod_{k=1}^{m_{i}}(-1+q^{k})$. $P^{(N)}(n)$ denotes the number of $N$-tuples of Young diagrams of size $n$, i.e., \#\big{\{}\vec{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(N)})\big{|}\vec{\lambda}\vdash n\big{\}}. In particular, if $N=1$,

[TABLE]

Corollary 2.7**.**

If $u_{i}\neq 0$ and $u_{i}\neq q^{r}t^{-s}u_{j}$ for any numbers $i$, $j$ and integers $r$, $s$, then the PBW type vectors $\Ket{X_{\vec{\lambda}}}$ (resp. $\Bra{X_{\vec{\lambda}}}$) are a basis over $\mathcal{F}_{\vec{u}}$ (resp. $\mathcal{F}^{*}_{\vec{u}}$).

Also it can be seen that the representation of the algebra $\mathcal{A}(N)$ on the Fock Module $\mathcal{F}_{\vec{u}}$ is irreducible if and only if the parameters $\vec{u}$ satisfy the condition that $u_{i}\neq 0$ and $u_{i}\neq q^{r}t^{-s}u_{j}$. The proof of Theorem 2.6 is given in the next section.

3. Proof of Theorem 2.6

It is known that the algebra $\mathcal{A}(N)$ obtained by the level $N$ representation of the DIM algebra can be regarded as the tensor product of the deformed $W_{N}$-algebra and the Heisenberg algebra associated with the $U(1)$ factor [18]. This fact is obtained by a linear transformation of bosons. The point of proof of Theorem 2.6 is to construct singular vectors by using screening currents of the deformed $W_{N}$-algebra under the decomposition of the generators $X^{(i)}(z)$ into the deformed $W_{N}$-algebra part and the $U(1)$ part. In general, a vector $\Ket{\chi}$ in the Fock module $\mathcal{F}_{\vec{u}}$ is called the singular vector of the algebra $\mathcal{A}(N)$ if it satisfies

[TABLE]

for all $i$ and $n>0$. The singular vectors obtained by the screening currents are intrinsically the same one of the deformed $W$-algebra. From this singular vector, we can get the vanishing line of the Kac determinant in the similar way of the deformed $W_{N}$-algebra. The formulas for the Kac determinant of the deformed Virasoro algebra and the deformed $W_{N}$-algebra are proved in [11, 12].

First, in the $N\geq 2$ case, we introduce the following bosons.

U(1) part boson

[TABLE]

[TABLE]

Orthogonal component of $a^{(i)}_{n}$ for $b^{\prime}$

[TABLE]

Fundamental boson of the deformed $W_{N}$-algebra part

[TABLE]

[TABLE]

Then they satisfy the following relations

[TABLE]

where $\theta(P)$ is $1$ or [math] if the proposition $P$ is true or false, respectively.111Note that $h^{(i)}_{n}$ and $Q_{h}^{(i)}$ correspond to the fundamental bosons $h^{N-i+1}_{n}$ and $Q_{h}^{N-i+1}$ in [7], respectively. Using these bosons, we can decompose the generator $X^{(i)}(z)$ into the U(1) part and the deformed $W$-algebra part. That is to say,

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

$W^{(i)}(z)$ is the generator of the deformed $W_{N}$-algebra. Let us introduce the new parameters $u^{\prime}_{i}$ and $u^{\prime\prime}$ defined by

[TABLE]

Then the inner product of PBW type vectors can be written as

[TABLE]

Hence, its determinant is also in the form

[TABLE]

where $F(u^{\prime}_{1},\ldots,u^{\prime}_{N})$ is a polynomial in $u^{\prime}_{i}$ ($i=1,\ldots,N$) which is independent of $u^{\prime\prime}$. Now in [7], the screening currents of the deformed $W_{N}$-algebra are introduced:

[TABLE]

where $\alpha^{(i)}_{n}$ is the root boson defined by $\alpha^{(i)}_{n}\mathbin{:=}h^{(i+1)}_{n}-h^{(i)}_{n}$ and $Q^{(i)}_{\alpha}\mathbin{:=}Q_{h}^{(i+1)}-Q_{h}^{(i)}$. The bosons $\alpha^{(i)}_{n}$ and $Q^{(i)}_{\alpha}$ commute with $b^{\prime}_{n}$, and it is known that the screening charge $\oint dzS^{(i)}(z)$ commutes with the generators $W^{(j)}(z)$. Therefore, $\oint dzS^{(i)}(z)$ commutes with any generator $X^{(j)}_{n}$, and it can be regarded as the screening charges of the algebra $\mathcal{A}(N)$. Define parameters $h^{(i)}$ and $\alpha^{\prime}$ by $h^{(i)}+\frac{\alpha^{\prime}}{N}=w_{i}$ and $\sum_{i=1}^{N}h^{(i)}=0$. We set $\alpha^{(i)}\mathbin{:=}h^{(i+1)}-h^{(i)}$. For $i=1,\ldots,N-1$, we define $Q^{(i)}_{\Lambda}\mathbin{:=}\sum_{k=i+1}^{N}Q^{(k)}_{h}$. Then $\Ket{\vec{u}}=e^{\sum_{i=1}^{N-1}\alpha^{(i)}Q^{(i)}_{\Lambda}+\alpha^{\prime}Q^{\prime}}\Ket{0}$ and we can see the relations $[\alpha^{(i)}_{n},Q^{(j)}_{\Lambda}]=\delta_{i,j}$ and $\alpha^{(i)}_{0}\ket{\vec{u}}=\alpha^{(i)}\ket{\vec{u}}$. For any number $i=1,\dots,N$, the vector arising from the screening current $S^{(i)}(z)$,

[TABLE]

is a singular vector. $\ket{\chi^{(i)}_{r,s}}$ is in the Fock module $\mathcal{F}_{\vec{u}}$ with the parameter $\vec{u}$ satisfying $u_{k}=v_{k}$ for $k\neq i,i+1$ and $u_{i+1}=t^{r}v_{i+1}$, $u_{i}=t^{-r}v_{i}$. The obtained relation $u_{i}=q^{s}t^{-r}u_{i+1}$ is equivalent to the condition that the parameter $\alpha^{(i)}$ defined above satisfies $\alpha^{(i)}=\sqrt{\beta}(1+r)-\frac{1}{\sqrt{\beta}}(1+s)$. The $P^{(N)}(n-rs)$ vectors obtained by this singular vector

[TABLE]

contribute the vanishing point $(u^{\prime}_{i}-q^{s}t^{-r}u^{\prime}_{i+1})^{P^{(N)}(n-rs)}$ in the polynomial $F$. Similarly to the case of the deformed $W_{N}$-algebra (see [12]), by the $\mathfrak{sl}_{N}$ Weyl group invariance of the eigenvalues of $W^{(i)}_{0}$, the polynomial $F$ has the factor $(u^{\prime}_{i}-q^{s}t^{-r}u^{\prime}_{j})^{P^{(N)}(n-rs)}$ ($\forall i,j$). Considering the degree of polynomials $F(u^{\prime}_{1},\ldots,u^{\prime}_{N})$, we can see that when $N\geq 2$, the Kac determinant is

[TABLE]

where $g_{N,n}(q,t)$ is a rational function in parameters $q$ and $t$ and independent of the parameters $u_{i}$. In (3.25), we used the following Lemma.

Lemma 3.1**.**

Let $f$ be a (complex valued) function on $\mathbb{N}$. Then for any $n\in\mathbb{N}$ and $k=1,\ldots,N$,

[TABLE]

This Lemma is a fairly straightforward generalization of [11, Lemma 3.2]. The proof is similar. If $N=1$, the Kac determinant $\mathrm{det}_{n}$ is clearly in the form

[TABLE]

Next, the prefactor $g_{N,n}(q,t)$ can be computed in general $N$ case by introducing another boson

[TABLE]

The commutation relation of the boson $\mathsf{a}^{(i)}_{n}$ is

[TABLE]

Define the matrix $H^{(n,\pm)}_{\vec{\lambda},\vec{\mu}}$ by the expansions

[TABLE]

where $\mathsf{a}_{-\vec{\mu}}$ and $\mathsf{a}_{\vec{\mu}}$ are

[TABLE]

We write their determinants as $H^{(n,\pm)}\mathbin{:=}\det(H^{(n,\pm)}_{\vec{\lambda},\vec{\mu}})$. By using these determinants, the Kac determinant can be written as

[TABLE]

Here $G_{n}(q,t)$ is the determinant of the diagonal matrix $(\Bra{\vec{u}}\mathsf{a}_{\vec{\lambda}}\mathsf{a}_{-\vec{\mu}}\Ket{\vec{u}})_{\vec{\lambda},\vec{\mu}\vdash n}$. This factor is independent of the parameters $u_{i}$, and we have $G_{n}(q,t)=\prod_{\vec{\lambda}\vdash n}\prod_{k=1}^{N}b_{\lambda^{(k)}}(q)b^{\prime}_{\lambda^{(k)}}(t^{-1})$. In (3.25), the factor depending on $u_{i}$ in $\mathrm{det}_{n}$ was already clarified. Hence, we can determine the prefactor $g_{N,n}(q,t)$ by computing the leading term in $H^{(n,+)}\times H^{(n,-)}$. That is, the prefactor $g_{N,n}(q,t)$ can be written as

[TABLE]

where we introduce the function $\mathsf{lt}(f;u)$ which gives the leading term of $f$ as the polynomial in $u$, and $\mathsf{lt}(f;u_{1},\ldots,u_{N})\mathbin{:=}\mathsf{lt}(\cdots\mathsf{lt}(\mathsf{lt}(f;u_{1});u_{2})\cdots;u_{N})$. To calculate this leading term, define the operators $A^{(k)}(z)=\sum_{n\in\mathbb{Z}}A^{(k)}_{n}z^{-n}$, $B^{(k)}(z)=\sum_{n\in\mathbb{Z}}B^{(k)}_{n}z^{-n}$ by

[TABLE]

$\mathsf{lt}(H^{(n,-)};u_{1},\ldots,u_{N})$ arises from only the operator

[TABLE]

in $X^{(k)}(z)$. Then

[TABLE]

Let $L^{(n,-)}_{\vec{\lambda},\vec{\mu}}$ and $C^{(n,-)}_{\vec{\lambda},\vec{\mu}}$ be the matrices given by

[TABLE]

where $\mathcal{L}_{-\vec{\lambda}}$ and $A_{-\vec{\lambda}}$ are defined in the usual way:

[TABLE]

Then $\mathsf{lt}(H^{(n,-)};u_{1},\ldots,u_{N})$ is expressed as

[TABLE]

Since the matrix $L^{(n,-)}_{\vec{\lambda},\vec{\mu}}$ is lower triangular with respect to the partial ordering $\overset{**}{>}^{\mathrm{R}}$ 222 Here the partial orderings $\overset{**}{>}^{\mathrm{R}}$ and $\overset{**}{>}^{\mathrm{L}}$ are defined as follows:

$\displaystyle\vec{\lambda}\overset{**}{\geq}^{\mathrm{R}}\vec{\mu}\quad\overset{\mathrm{def}}{\Leftrightarrow}\quad$ $\displaystyle|\vec{\lambda}|=|\vec{\mu}|,\quad\sum_{i=1}^{k}|\lambda^{(i)}|\geq\sum_{i=1}^{k}|\mu^{(i)}|\quad(\forall k)$

(3.43)

$\displaystyle\mathrm{or}\quad"(|\lambda^{(1)}|,\ldots,|\lambda^{(N)}|)=(|\mu^{(1)}|,\ldots,|\mu^{(N)}|)\quad\mathrm{and}\quad\lambda^{(i)}\geq\mu^{(i)}\quad(\forall i)",$

(3.44)

$\displaystyle\vec{\lambda}\overset{**}{\geq}^{\mathrm{L}}\vec{\mu}\quad\overset{\mathrm{def}}{\Leftrightarrow}\quad$ $\displaystyle|\vec{\lambda}|=|\vec{\mu}|,\quad\sum_{i=K}^{N}|\lambda^{(i)}|\geq\sum_{i=k}^{N}|\mu^{(i)}|\quad(\forall k)$

(3.45)

$\displaystyle\mathrm{or}\quad"(|\lambda^{(1)}|,\ldots,|\lambda^{(N)}|)=(|\mu^{(1)}|,\ldots,|\mu^{(N)}|)\quad\mathrm{and}\quad\lambda^{(i)}\geq\mu^{(i)}\quad(\forall i)".$

(3.46)

Then we have $L^{(n,-)}_{\vec{\lambda},\vec{\mu}}=0$ unless $\vec{\lambda}\overset{**}{<}^{\mathrm{R}}\vec{\mu}$.

and its diagonal elements are

[TABLE]

we have

[TABLE]

The transition matrix $C^{(-)}_{\vec{\lambda},\vec{\mu}}$ is upper triangular with respect to the partial ordering $\overset{**}{>}^{\mathrm{L}}$, and all diagonal elements are $1$. Hence, $\det(C^{(-)}_{\vec{\lambda},\vec{\mu}})_{\vec{\lambda},\vec{\mu}\vdash n}=1$. Similarly, with the help of the base transformation to $\Bra{\vec{u}}B_{\vec{\lambda}}$, it can be seen that

[TABLE]

Therefore the prefactor $g_{N,n}(q,t)$ is

[TABLE]

This completes the proof of Theorem 2.6.

4. Singular vectors and generalized Macdonald functions

In this subsection, the singular vectors of the algebra $\mathcal{A}(N)$ are discussed. Trivially, when $u_{i}=0$, the Kac determinant (2.17) degenerates, and it can be easily seen that the vectors $a^{(i)}_{-\lambda}\Ket{\vec{u}}$ are singular vectors. Since the screening operator $S^{(i)}(z)$ is the same one of the deformed $W_{N}$-algebra, the situation of the singular vectors of $\mathcal{A}(N)$ except contribution arising when $u_{i}=0$ is the same as the deformed $W_{N}$-algebra. We discover that singular vectors obtained by the screening currents $S^{(i)}(z)$ coincides with generalized Macdonald functions which is also called the AFLT basis.333 The AFLT basis in 4D AGT conjecture can be constructed by the spherical double affine Hecke algebra with central charges ($SH^{c}$ algebra). The relation between singular vectors of the $SH^{c}$ algebra and the AFLT basis is investigated in [23].

Firstly, let us introduce the generalized Macdonald functions. To state their existence theorem, we prepare the following ordering.

Definition 4.1**.**

For $N$-tuple of partitions $\vec{\lambda}$ and $\vec{\mu}$,

[TABLE]

Note that the second condition can be replaced with

[TABLE]

The generalized Macdonald functions are eigenfunctions of the operator $X^{(1)}_{0}$. The operator $X^{(1)}_{0}$ is triangulated in the basis of the product of the ordinary Macdonald functions $\prod_{i=1}^{N}P_{\lambda^{(i)}}(a^{(i)}_{-n};q,t)\Ket{\vec{u}}$ arranged in the above ordering, where $P_{\lambda}(a_{-n}^{(i)};q,t)$ are Macdonald symmetric functions defined in Appendix A with substituting the bosons $a^{(i)}_{-n}$ for the power sum symmetric functions $p_{n}$. By the triangulation of $X^{(1)}_{0}$, the following existence theorem of the generalized Macdonald functions holds.

Fact 4.2** ([30]).**

For each $N$-tuple of partitions $\vec{\lambda}$, there exists a unique vector $\Ket{P_{\vec{\lambda}}}\in\mathcal{F}_{\vec{u}}$ such that

[TABLE]

where $c_{\vec{\lambda},\vec{\mu}}=c_{\vec{\lambda},\vec{\mu}}(u_{1},\ldots,u_{N};q,t)$ is a constant and $\epsilon_{\vec{\lambda}}=\epsilon_{\vec{\lambda}}(u_{1},\ldots,u_{N};q,t)$ is the eigenvalue of $X^{(1)}_{0}$. Then the eigenvalues are

[TABLE]

The unique vectors $\Ket{P_{\vec{\lambda}}}$ are called the generalized Macdonald functions or the q-deformed version of the AFLT basis. The vectors $\Ket{P_{\vec{\lambda}}}$ is first introduced in [3] and play an important role in the 5D AGT coorespondence. Although the ordering of Definition 4.1 is different from the one in [3], the eigenfunctions $\Ket{P_{\vec{\lambda}}}$ are quite the same. By this proposition, it can be seen that $\Ket{P_{\vec{\lambda}}}$ is a basis over $\mathcal{F}_{\vec{u}}$, and the eigenvalues of $X^{(1)}_{0}$ are non-degenerate when $u_{i}$ is generic.

In this paper, we discover the relation between the vectors $\Ket{P_{\vec{\lambda}}}$ and the singular vectors. First, we have the following simple theorem. Hereafter, we consider the $N\geq 2$ case.

Theorem 4.3**.**

For a number $i\in\{1,\ldots,N-1\}$, if $u_{i}=q^{s}t^{-r}u_{i+1}$ and the other $u_{j}$ are generic, there exists a unique singular vector $\ket{\chi^{(i)}_{r,s}}$ in $\mathcal{F}_{\vec{u}}$ up to scalar multiple, and it corresponds to the generalized Macdonald function $\Ket{P_{\vec{\lambda}}}$ with

[TABLE]

That is,

[TABLE]

Proof.

Existence and uniqueness are understood by the formula for the Kac determinant (2.17) in the usual way. In fact, the unique singular vector $\ket{\chi^{(i)}_{r,s}}$ is the one of (3.23). Since the screening charges commute with $X^{(1)}_{0}$, the singular vector is an eigenfunction of $X^{(1)}_{0}$ of the eigenvalue $\sum_{i=1}^{N}v_{i}$. Using the relations $u_{k}=v_{k}$ for $k\neq i,i+1$ and $u_{i+1}=t^{r}v_{i+1}$, $u_{i}=t^{-r}v_{i}$, we have

[TABLE]

where $\epsilon_{\vec{\lambda}}=\epsilon_{\vec{\lambda}}(u_{1},\ldots,u_{N})$ is the eigenvalue of the generalized Macdonald functions introduced in (4.5). Thus, the singular vector $\ket{\chi^{(i)}_{r,s}}$ and the generalized Macdonald function $\Ket{P_{(\emptyset,\ldots,\emptyset,(s^{r}),\emptyset,\ldots,\emptyset)}}$ are in the same eigenspace of $X^{(1)}_{0}$. Moreover, by comparing the eigenvalues $\epsilon_{\vec{\lambda}}$, it can be shown that the dimension of the eigenspace of the eigenvalue $\epsilon_{(\emptyset,\ldots,\emptyset,(s^{r}),\emptyset,\ldots,\emptyset)}$ is $1$ even when $u_{i}=q^{s}t^{-r}u_{i+1}$. Therefore, this theorem follows. ∎

Let us consider more complicated cases. For variables $\alpha^{(k)}$ ($k=1,\ldots,N-1$), define the function $\check{h}^{(i)}$ by

[TABLE]

Then it satisfies $\alpha^{(i)}=\check{h}^{(i+1)}(\alpha^{(k)})-\check{h}^{(i)}(\alpha^{(k)})$. We focus on the following singular vectors

[TABLE]

where the parameter $\vec{v}=(v_{1},\ldots,v_{N})$ is $v_{i}=v^{\prime\prime}v^{\prime}_{i}$, $v^{\prime}_{i}=q^{\sqrt{\beta}\check{h}^{(i)}(\widetilde{\alpha}^{k}_{\vec{r},\vec{s}})}p^{-\frac{N+1}{2}+i}$, and for non-negative integers $s_{k}>0$ and $r_{k}$ ($k=1,\ldots,N-1$),

[TABLE]

The singular vector $\Ket{\chi_{\vec{r},\vec{s}}}$ is in the Fock module $\mathcal{F}_{\vec{u}}$ of the highest weight $\vec{u}=(u_{1},\ldots,u_{N})$ defined by $u_{i}=u^{\prime\prime}u^{\prime}_{i}$, $u^{\prime}_{i}=q^{\sqrt{\beta}\check{h}^{(i)}(\alpha^{k}_{\vec{r},\vec{s}})}p^{-\frac{N+1}{2}+i}$, $u^{\prime\prime}=v^{\prime\prime}$ and

[TABLE]

The equation (4.12) is equivalent to the condition $u_{i}=q^{s_{i}}t^{-r_{i}+r_{i+1}}u_{i+1}$ ($\forall i$) of parameters $u_{i}$. We can also see that $\ket{\vec{u}}=e^{\sum_{i=1}^{N-1}\alpha_{\vec{r},\vec{s}}^{(i)}Q^{(i)}_{\Lambda}+\alpha^{\prime}Q^{\prime}}\ket{0}$ under the parametrization $u^{\prime\prime}=q^{\sqrt{\beta}\frac{\alpha^{\prime}}{N}}$. Here, $Q^{(i)}_{\Lambda}$ and $Q^{\prime}$ are defined in Section 3.444 The screening currents $S^{(i)}(z)$, the parameters $\alpha^{(k)}_{\vec{r},\vec{s}}$ and integers $r_{i}$, $s_{i}$ in this paper correspond to $S^{N-i}_{+}(z)$, $\alpha^{N-k}_{r,s}$, $r_{N-i}$ and $s_{N-i}$ in [7], respectively.

To introduce the partition of a generalized Macdonald function corresponding to this singular vector, we prepare following notations. For a partition $\lambda=(\lambda_{1},\lambda_{2},\ldots)$ and a non-negative integer $n<\ell(\lambda)$, we define

[TABLE]

When $n\geq\ell(\lambda)$, we put $\mathsf{T}(\lambda;s):=\lambda$, $\mathsf{R}(\lambda;s):=\emptyset$. For partitions $\lambda$ and $\mu$ satisfying $\lambda_{\ell(\lambda)}\geq\mu_{1}$, define

[TABLE]

In this paper, for an $n$-tuple of integers $\vec{k}=(k_{1},\ldots,k_{n})$, let $\vec{k}^{*}$ denote the $(n-1)$-tuple of integers except the last integer, i.e., $\vec{k}^{*}:=(k^{*}_{1},\ldots,k^{*}_{n-1}):=(k_{1},\ldots,k_{n-1})$, and we put $k^{*}_{n}=0$.

For $(N-1)$-tuples of non-negative integers $\vec{r}=(r_{1},\ldots,r_{N-1})$ and $\vec{s}=(s_{1},\ldots,s_{N-1})$, the $N$-tuple of partitions $\Theta_{\vec{r},\vec{s}}=(\theta_{\vec{r},\vec{s}}^{(1)},\ldots,\theta^{(N)}_{\vec{r},\vec{s}})$ is inductively defined by $\Theta_{(r_{1}),(s_{1})}:=(\emptyset,(s_{1}^{r_{1}}))$ and

[TABLE]

The meaning of this $N$-tuple of partitions will be described in Example 4.6.

We obtain the following theorem which states the coincidence of the singular vectors and generalized Macdonald functions with the $N$-tuple of partitions $\Theta_{\vec{r},\vec{s}}$. This theorem is an extension of the result in [7].

Theorem 4.4**.**

Let $\vec{s}$ be an $(N-1)$-tuple of positive integers and $\vec{r}$ be an $(N-1)$-tuple of non-negative integers. If parameters $u_{i}$ satisfy $u_{i}=q^{s_{i}}t^{-r_{i}+r_{i+1}}u_{i+1}$ for all $i$ ($r_{N}:=0$), then the singular vector $\Ket{\chi_{\vec{r},\vec{s}}}$ coincides with the generalized Macdonald function $\Ket{P_{\Theta_{\vec{r},\vec{s}}}}$, i.e.,

[TABLE]

Before the proof of this theorem, we note that the eigenvalues $e_{\lambda}$ of ordinary Macdonald functions can be written by contribution from the edges of the Young diagram:

[TABLE]

where $A(\lambda)$ and $R(\lambda)$ are the sets of coordinates of the boxes which can be added to or removed from Young diagram of $\lambda$, respectively. We prepare a simple Lemma with respect to the eigenvalues $e_{\lambda}$.

Lemma 4.5**.**

For a partition $\lambda$ with $\ell(\lambda)\leq r$

[TABLE]

For a partition $\lambda$ and a non-negative integer $n$,

[TABLE]

Using partitions $\lambda$ and $\mu$ satisfying $\lambda_{\ell(\lambda)}\leq\mu_{1}$, the equation (4.19) can be also written as

[TABLE]

This lemma is easily proved by the expression (4.17) of eigenvalues $e_{\lambda}$. We now prove Theorem 4.4.

Proof of Theorem 4.4. The method of the proof is similar to Theorem 4.3. Using the fact that generalized Macdonald functions and the singular vector $\Ket{\chi_{\vec{r},\vec{s}}}$ are eigenfunctions of the operator $X^{(1)}_{0}$, we can prove this Theorem by comparison of each eigenvalue.

First, we prove that the eigenvalue $\epsilon_{\Theta_{\vec{r},\vec{s}}}$ of the generalized Macdonald function coincides with the eigenvalue $\sum_{i=1}^{N}v_{i}$ of the singular vector $\Ket{\chi_{\vec{r},\vec{s}}}$. It is useful to show the coincidence of each quotient by $u_{N}$, i.e.,

[TABLE]

and

[TABLE]

Here, $s_{i}+\cdots+s_{N-1}=0$ for $i=N$. We use the mathematical induction on $N$. If $N=2$, it is clear that

[TABLE]

Assuming the coincidence of (4.21) and (4.22) for $N=n-1$, we will prove it for $N=n$. By definition of $\Theta_{\vec{r},\vec{s}}$ and Lemma 4.5,

[TABLE]

By assumption for $N=n-1$, we can see that (4.24) is

[TABLE]

Therefore, by induction on $N$, the eigenvalue of the generalized Macdonald function $\Ket{P_{\Theta_{\vec{r},\vec{s}}}}$ coincides with the one of the singular vector $\Ket{\chi_{\vec{r},\vec{s}}}$ for general $N$. By this fact, $\Ket{P_{\Theta_{\vec{r},\vec{s}}}}$ and $\Ket{\chi_{\vec{r},\vec{s}}}$ belong to the same eigenspace of $X^{(1)}_{0}$.

The proof of this Theorem is completed by showing that the dimension of the eigenspace of the eigenvalue $\sum_{i=1}^{N}v_{i}$ is $1$. Since we know all eigenvalues of $X^{(1)}_{0}$ by Fact 4.2, it is sufficient to prove that if an $N$-tuple of partitions $\vec{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(N)})$ satisfies $\epsilon_{\vec{\lambda}}=\sum_{i=1}^{N}v_{i}$, then $\vec{\lambda}=\Theta_{\vec{r},\vec{s}}$. That is to say, we will show

[TABLE]

Actually, we consider the quotient by $u_{N}$ instead of the equation $\epsilon_{\vec{\lambda}}=\sum_{i=1}^{N}v_{i}$ in (4.26):

[TABLE]

Let us denote by $E_{i}$ each term of LHS in (4.27), i.e.,

[TABLE]

First, we deal with the case that $r_{N-1}\neq 0$. Let $\vec{\lambda}$ satisfy the equation (4.27) and $i_{0}\leq N-1$ be the maximum number such that $r_{N-1}>r_{i_{0}}$ ($r_{0}:=0$). Then we have

[TABLE]

The property (4.29) is shown as follows. We focus on the term

[TABLE]

in the RHS of (4.27). In order that the LHS reproduces the term $\mathcal{T}_{i_{0}}$, it is necessary that there is a number $j_{0}$ such that the expansion of $E_{j_{0}}$ contains the term $\mathcal{T}_{i_{0}}$. We can see $j_{0}=N$, because for a number $j$ such that $i_{0}\geq j$, the degree with respect to $q$ of $E_{j}$ is larger than the degree $s_{i_{0}+1}+\cdots+s_{N-1}$ of $\mathcal{T}_{i_{0}}$ and for a number $j$ such that $N-1\geq j>i_{0}$, the degree with respect to $t$ of $E_{j}$ is smaller than $-r_{i_{0}}$ owing to the maximality of $i_{0}$. Therefore, the only term $E_{N}=e_{\lambda^{(N)}}$ in the LHS of (4.27) contains $\mathcal{T}_{i_{0}}$. Since $e_{\lambda}$ can be written by contribution from the edges of Young diagram as (4.17), the partition $\lambda^{(N)}$ satisfies

[TABLE]

By (4.31),

[TABLE]

where $\check{s}:=s_{i_{0}+1}+\cdots+s_{N-1}$ and $m_{\check{s}}=m_{\check{s}}(\lambda^{(N)})$ is the number of entries in $\lambda^{(N)}$ that are equal to $\check{s}$. Hence, the term

[TABLE]

is contained in $e_{\lambda^{(N)}}$. Since there is no term with the minus sign in the RHS of (4.27), $m_{\check{s}}$ should be a value such that the term (4.33) is canceled by other terms arising from $E_{i}$ ($i\neq N$). The degree with respect to $q$ of $E_{i}$ for $i_{0}\geq i$ is larger than the one of (4.33). In addition, the maximum degree with respect to $t$ of $E_{i}$ ($N>i>i_{0}$) is $\max\{-r_{i_{0}+1},\ldots,-r_{N-1}\}=-r_{N-1}$ by the maximality of $i_{0}$. Therefore, we have

[TABLE]

From (4.32) and (4.34), it can be seen that $\lambda^{(N)}_{r_{N-1}}=s_{i_{0}+1}+\cdots s_{N-1}$. On the other hand, in order to reproduce the term $t^{-r_{N-1}}$ in the RHS of (4.27), it is necessary that $\lambda^{(N)}$ satisfy

[TABLE]

because the degree with respect to $q$ of $E_{i}$ for $i\neq N$ is too large to reproduce the term $t^{-r_{N-1}}$. By (4.35), we have $\ell(\lambda^{(N)})=r_{N-1}$ and $r_{i_{0}}+m_{\check{s}}(\lambda^{(N)})=r_{N-1}$. Thus (4.29) is proved.

Next, let us show that

[TABLE]

(4.36) is proved as follows. When there exists a number $i$ such that

[TABLE]

let $i_{e}$ be the maximum number satisfying the condition (4.37). There is a number $j_{e}$ such that $E_{j_{e}}$ contains the term

[TABLE]

appearing in RHS of (4.27). Then $j_{e}$ satisfies

[TABLE]

by the comparison of degrees of $q$. However, $j_{e}$ does not satisfy

[TABLE]

because the degree of $t$ of $E_{j}$ ($N-1>j\geq i_{e}+1$) is smaller than $-r_{i_{e}}=-r_{N-1}$ owing to the maximality of $i_{0}$ and $i_{e}$. Furthermore, $j_{e}\neq N$ by the property (4.29). Therefore $j_{e}=N-1$. Since the only term $E_{N-1}=t^{r_{N-1}}q^{s_{N-1}}e_{\lambda^{(N-1)}}$ contains $\mathcal{T}_{i_{e}}$, we can see that $\lambda^{(N-1)}_{1}=s_{i_{e}+1}+\cdots+s_{N-2}$. Here, if $i_{e}=N-2$, then $\lambda^{(N-1)}_{1}=0$, i.e., $\lambda^{(N-1)}=\emptyset$. Next, when there is no number such that (4.37), a number $j$ such that $E_{j}$ can cancel the term (4.33) with the minus sign is only $N-1$ by similar discussion using degrees of $q$ and $t$. Hence we get $\lambda^{(N-1)}_{1}=s_{i_{0}+1}+\cdots+s_{N-2}$. Thus (4.36) is proved.

By the above properties, we are ready to write the following equation:

[TABLE]

where $\mathsf{J}(\lambda^{(N)}-(s_{N-1}^{r_{N-1}}),\lambda^{(N-1)})$ is well-defined by the properties (4.29) and (4.36). Put $\mu^{(i)}:=\lambda^{(i)}$ ($i=1,\ldots,N-2$), $\mu^{(N-1)}:=\mathsf{J}(\lambda^{(N)}-(s_{N-1}^{r_{N-1}}),\lambda^{(N-1)})$. Combining (4) with (4.27) yields

[TABLE]

We dealt with the case that $r_{N-1}\neq 0$. Also in the case that $r_{N-1}=0$, it is clear that $\lambda^{(N)}=\emptyset$, and we can get the same equation as (4.42). Assuming (4.26) holds for the case obtained by replacing $N$ with $N-1$, we obtain

[TABLE]

Hence, $\lambda^{(i)}=\theta^{(i)}_{\vec{r}^{*},\vec{s}^{*}}$ for $i\leq N-2$, and

[TABLE]

Since $\ell(\lambda^{(N)})=r_{N-1}$,

[TABLE]

Therefore, we have $\vec{\lambda}=\Theta_{\vec{r},\vec{s}}$. By induction on $N$, (4.26) holds for general $N$. This completes the proof. ∎

Example 4.6**.**

If $r_{k+1}\geq r_{k}\geq 0$ for all $k$, then the singular vector $\Ket{\chi_{\vec{r},\vec{s}}}$ coincides with the generalized Macdonald function $\Ket{P_{(\emptyset,\ldots,\emptyset,\lambda_{\vec{r},\vec{s}})}}$ with

[TABLE]

$\lambda_{\vec{r},\vec{s}}$ corresponds to the Young diagram with $N-1$ edges in Figure 2 in Introduction. If $r_{k}$ does not satisfy the condition $r_{k+1}\geq r_{k}\geq 0$, Figure 2 is not a Young diagram. In this case, the Young diagram corresponding to the singular vector $\Ket{\chi_{\vec{r},\vec{s}}}$ is obtained by cutting off the protruding part and moving the boxes to Young diagrams on the left side. This is the meaning of the $N$-tuple of partitions $\Theta_{\vec{r},\vec{s}}$. For example, when $N=6$ and $r_{1}>r_{4}>r_{3}>r_{5}>r_{2}$, the $N$-tuple of Young diagrams $\Theta_{\vec{r},\vec{s}}$ is the one pictured in Figure 3.

It is known that if $r_{k+1}\geq r_{k}\geq 0$ for all $k$, the projection of the singular vectors $\Ket{\chi_{\vec{r},\vec{s}}}$ onto the diagonal components of the boson $h^{(N)}_{n}$ corresponds to ordinary Macdonald functions [7, (35)]. Hence, ordinary Macdonald functions are obtained by the projection of generalized Macdonald functions.

Corollary 4.7**.**

When $u_{i}=q^{s_{i}}t^{-r_{i}+r_{i+1}}u_{i+1}$ for all $i$,

[TABLE]

Here, $p_{n}$ denotes the ordinary power sum symmetric functions.

Acknowledgments

The author show his greatest appreciation to H. Awata, M. Bershtein, B. Feigin, P. Gavrylenko, K. Hosomichi, H. Itoyama, H. Kanno, A. Marshakov, T. Matsumoto, A. Mironov, S. Moriyama, Al. Morozov, An. Morozov, H. Nagoya, A. Negut, T. Okazaki, T. Shiromizu, T. Takebe, M. Taki, S. Yanagida and Y. Zenkevich for variable discussions and comments. The author is supported in part by Canon Foundation Research Fellowship.

Appendix

Appendix A Macdonald functions

In this Appendix, we give the definition of the ordinary Macdonald functions [24, Chap. VI].

Let $\Lambda$ be the ring of symmetric functions, and $p_{\lambda}=\prod_{k\geq 1}p_{\lambda_{k}}$ ($p_{n}=\sum_{i\geq 1}x_{i}^{n}$) is the power sum symmetric functions. Define the inner product $\langle-,-\rangle_{q,t}$ over $\Lambda$ by the condition that

[TABLE]

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:

[TABLE]

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 [17]. The DIM algebra has two parameters $q$ and $t$. Let $g(z)$ be the formal series

[TABLE]

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 $x^{\pm}(z)=\sum_{n\in\mathbb{Z}}x^{\pm}_{n}z^{-n}$, $\psi^{\pm}(z)=\sum_{\pm n\in\mathbb{Z}_{\geq 0}}\psi^{\pm}_{n}z^{-n}$ and the central element $\gamma^{\pm 1/2}$ satisfying the defining relations

[TABLE]

This algebra $\mathcal{U}$ is an example of the family of topological Hopf algebras introduced by Ding and Iohara [14]. 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 [25], 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 $\mathfrak{gl}_{1}$ algebra [19, 20], elliptic Hall algebra [13] and so on. This algebra has a Hopf algebra structure. The formulas for its coproduct $\Delta$ are

[TABLE]

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

[TABLE]

Fact B.2** ([17, 18]).**

Let $u$ be an complex parameter or indeterminate. The morphism $\rho_{u}$ defined as follows is a representation of the DIM algebra:

[TABLE]

where

[TABLE]

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 [24, 31]. 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

[TABLE]

where $\Delta^{(N)}$ is inductively defined by $\Delta^{(1)}\mathbin{:=}\mathrm{id}$, $\Delta^{(2)}\mathbin{:=}\Delta$ and $\Delta^{(N)}\mathbin{:=}(\mathrm{id}\otimes\cdots\otimes{\rm id}\otimes\Delta)\circ\Delta^{(N-1)}$. 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 [16, 21, 4]. 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

[TABLE]

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

[TABLE]

Note that, in this paper, the parameters $u_{i}$ are realized by the operators $U_{i}$ and the highest weight vector $\Ket{\vec{u}}$ in order to use screening currents.