Principal subspaces for the affine Lie algebras in types $D$, $E$ and $F$

TL;DR

This paper constructs quasi-particle bases for principal subspaces of certain affine Lie algebra modules in types D, E, and F, leading to new character formulas and combinatorial identities.

Contribution

It generalizes Georgiev's approach to types D, E, and F, providing explicit bases, presentations, and character formulas for these principal subspaces.

Findings

Constructed quasi-particle bases for principal subspaces

Derived presentations and character formulas

Discovered new combinatorial identities

Abstract

We consider the principal subspaces of certain level $k\geqslant 1$ integrable highest weight modules and generalized Verma modules for the untwisted affine Lie algebras in types $D$, $E$ and $F$. Generalizing the approach of G. Georgiev we construct their quasi-particle bases. We use the bases to derive presentations of the principal subspaces, calculate their character formulae and find some new combinatorial identities.

Equations

$$\mathfrak{g}={\mathfrak{n}}_{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}_{+},\text{where}{\mathfrak{n}}_{\pm }=\underset{\alpha \in R_{\pm }}{⨁}{\mathfrak{n}}_{\alpha }\text{and}{\mathfrak{n}}_{\alpha }=\mathbb{C}{x}_{\alpha }\text{ for all }\alpha \in R,$$

$$A_l$$

$$C_l$$

$$E_6$$

$$E_8$$

$$G_2$$

$$\tilde{\mathfrak{g}}=\mathfrak{g}\otimes \mathbb{C}[t,t^{-1}]\oplus \mathbb{C}c\oplus \mathbb{C}d,$$

$$\left[c,\tilde{\mathfrak{g}}\right]=0,\left[d,x(m)\right]=mx(m),$$

$$\left[x(m),y(n)\right]=\left[x,y\right](m+n)+⟨x,y⟩m{\delta }_{m+n0}c.$$

$$N(\Lambda )=U(\tilde{\mathfrak{g}}){\otimes }_{U({\tilde{\mathfrak{g}}}^{⩾0})}{U}_{\lambda },$$

$${\tilde{\mathfrak{g}}}^{⩾0}=\underset{n⩾0}{⨁}(\mathfrak{g}\otimes t^n)\oplus \mathbb{C}c\oplus \mathbb{C}d$$

$$\mathfrak{g}\otimes t^n\cdot u=0\text{ for all }n>0,c\cdot u=ku\text{and}d\cdot u=0\text{for all }u\in U_{\lambda }.$$

$$N(k{\Lambda }_0)\cong U({\tilde{\mathfrak{g}}}^{<0}),\text{where}{\tilde{\mathfrak{g}}}^{<0}=\underset{n<0}{⨁}(\mathfrak{g}\otimes t^n).$$

$${\tilde{\mathfrak{n}}}_{+}={\mathfrak{n}}_{+}\otimes \mathbb{C}[t,t^{-1}],{\tilde{\mathfrak{n}}}_{+}^{⩾0}={\mathfrak{n}}_{+}\otimes \mathbb{C}[t]\text{and}{\tilde{\mathfrak{n}}}_{+}^{<0}={\mathfrak{n}}_{+}\otimes t^{-1}\mathbb{C}[t^{-1}].$$

$$W_V=U({\tilde{\mathfrak{n}}}_{+})v_V.$$

$$x_{{\alpha }_i}(z)=\sum _{m\in \mathbb{Z}}{x}_{{\alpha }_i}(m)z^{-m-1}\in \mathrm{Hom}(V,V((z)))\subset (\mathrm{End}V)[[z^{\pm 1}]],i=1,\dots ,l.$$

$$x_{n{\alpha }_i}(z)=\sum _{m\in \mathbb{Z}}{x}_{n{\alpha }_i}(m)z^{-m-n}=\underset{n\text{ times}}{\underbrace{x_{{\alpha }_i}(z)\cdots x_{{\alpha }_i}(z)}}=x_{{\alpha }_i}(z{)}^n$$

$$b=\left(x_{n_{r_l^{(1)},l}{\alpha }_l}(m_{r_l^{(1)},l})\dots x_{n_{1,l}{\alpha }_l}(m_{1,l})\right)\dots \left(x_{n_{r_1^{(1)},1}{\alpha }_1}(m_{r_1^{(1)},1})\dots x_{n_{1,1}{\alpha }_1}(m_{1,1})\right)$$

$$b=b_{{\alpha }_l}\cdots b_{{\alpha }_2}{b}_{{\alpha }_1},\text{where}{b}_{{\alpha }_i}=x_{n_{r_i^{(1)},i}{\alpha }_i}(m_{r_i^{(1)},i})\dots x_{n_{1,i}{\alpha }_i}(m_{1,i})\text{ for }i=1,\dots ,l.$$

$$\text{ for all}i=1,\dots ,l\text{and}p=1,\dots ,r_i^{(1)}-1$$

$$\text{if}{n}_{p+1,i}=n_{p,i}\text{then}{m}_{p+1,i}⩽m_{p,i}-2n_{p,i}.$$

$${\nu }_i=\{\begin{array}{ll}1,& \text{if }⟨{\alpha }_i,{\alpha }_i⟩=2,\\ 2,& \text{if }⟨{\alpha }_i,{\alpha }_i⟩=1,\\ 3,& \text{if }⟨{\alpha }_i,{\alpha }_i⟩=\frac{2}{3}\end{array}\text{and}{i}^{\prime }=\{\begin{array}{ll}l-2,& \text{if }i=l\text{ and }\mathfrak{g}=D_l,\\ 3,& \text{if }i=l\text{ and }\mathfrak{g}=E_6,E_7,\\ 5,& \text{if }i=l\text{ and }\mathfrak{g}=E_8,\\ i-1,& \text{otherwise.}\end{array}$$

$$\text{for all}i=1,\dots ,l\text{and}p=1,\dots ,r_i^{(1)}$$

$$m_{p,i}⩽-n_{p,i}+\sum _{q=1}^{r_{i^{\prime }}^{(1)}}\min \left\{\frac{{\nu }_i}{{\nu }_{i^{\prime }}}{n}_{q,i^{\prime }},n_{p,i}\right\}-2(p-1)n_{p,i},$$

$$n_{p,i}⩽k{\nu }_i\text{for all}i=1,\dots ,l\text{and}p=1,\dots ,r_i^{(1)}.$$

$${\mathfrak{B}}_V=\left\{b{v}_V:b\in B_V\right\}\subset W_V\text{for}V=N(k{\Lambda }_0),L(k{\Lambda }_0).$$

$$\Lambda =k_0{\Lambda }_0+k_j{\Lambda }_j,$$

$$j_t=\{\begin{array}{ll}0,& \text{if }1⩽t⩽k_0,\\ j,& \text{if}{k}_0<t⩽k_0+k_j.\end{array}$$

$$\text{for all}i=1,\dots ,l\text{and}p=1,\dots ,r_i^{(1)}$$

$$m_{p,i}⩽-n_{p,i}+\sum _{q=1}^{r_{i^{\prime }}^{(1)}}\min \left\{n_{q,i^{\prime }},n_{p,i}\right\}-2(p-1)n_{p,i}-\sum _{t=1}^{n_{p,i}}{\delta }_{i{j}_t}.$$

$${\mathfrak{B}}_{L(\Lambda )}=\left\{b{v}_{L(\Lambda )}:b\in B_{L(\Lambda )}\right\}\subset W_{L(\Lambda )}.$$

$$f_{L(\Lambda )}:U({\tilde{\mathfrak{n}}}_{+})$$

$$a$$

$$R_{{\alpha }_i}(-m)=\sum _{\begin{array}{c}{m}_1,\dots ,m_{k{\nu }_i+1}⩽-1\\ m_1+\dots +m_{k{\nu }_i+1}=-m\end{array}}{x}_{{\alpha }_i}(m_1)\dots x_{{\alpha }_i}(m_{k{\nu }_i+1}).$$

$$I_{L(k{\Lambda }_0)}=U({\tilde{\mathfrak{n}}}_{+}){\tilde{\mathfrak{n}}}_{+}^{⩾0}+\sum _{i=1}^l\sum _{m⩾k{\nu }_i+1}U({\tilde{\mathfrak{n}}}_{+})R_{{\alpha }_i}(-m).$$

Full text

1. Introduction

Starting with J. Lepowsky and S. Milne [30], the fascinating connection between Rogers–Ramanujan-type identities and affine Kac–Moody Lie algebras was extensively studied; see, e.g., [31, 32, 33, 35] and references therein. The principal subspaces of standard modules, i.e. of integrable highest weight modules for the affine Lie algebras, introduced by B. L. Feigin and A. V. Stoyanovsky [16], present a remarkable example of this interplay between combinatorics and algebra. In particular, their so-called quasi-particle bases provide an interpretation of the sum sides of various Rogers–Ramanujan-type identities; see [4, 5, 6, 7, 16, 20, 34]. Aside from quasi-particle bases, numerous research directions are focused on other aspects of principal subspaces and related structures such as certain generalized principal subspaces [2], Feigin–Stoyanovsky’s type subspaces [3, 22, 38], realizations of Jack symmetric functions [8], presentations of principal subspaces [9, 10, 11, 12, 36, 37, 39, 40], Rogers–Ramanujan-type recursions [13, 14], Koszul complexes [24], principal subspaces for quantum affine algebras and double Yangians [26, 27, 28] etc. The key ingredient that all the aforementioned studies have in common is the application of vertex-operator theoretic methods.

Let $\Lambda_{0},\ldots,\Lambda_{l}$ be the fundamental weights of the untwisted affine Lie algebra $\widetilde{\mathfrak{g}}$ associated with the simple Lie algebra $\mathfrak{g}$ of rank $l$. In this paper, we consider the principal subspaces $W_{N(k\Lambda_{0})}$ of the generalized Verma modules $N(k\Lambda_{0})$ and the principal subspaces $W_{L(k\Lambda_{0})}$ of the standard modules $L(k\Lambda_{0})$ of highest weights $k\Lambda_{0}$ for $\widetilde{\mathfrak{g}}$ in types $D$, $E$ and $F$. The main result is a construction of the quasi-particle bases $\mathfrak{B}_{N(k\Lambda_{0})}$ and $\mathfrak{B}_{L(k\Lambda_{0})}$ of the corresponding principal subspaces:

Theorem** (3.1).**

For any positive integer $k$ the set $\mathfrak{B}_{V}$ forms a basis of the principal subspace $W_{V}$ of the $\widetilde{\mathfrak{g}}$-module $V=N(k\Lambda_{0}),L(k\Lambda_{0})$.**

The bases are expressed in terms of monomials of certain operators, called quasi-particles, applied on the highest weight vector, whose charges and energies satisfy certain difference conditions. Theorem 3.1 for $\mathfrak{g}$ of type $A_{1}$ goes back to Feigin and Stoyanovsky [16]. The $\mathfrak{g}=A_{l}$ case was proved by G. Georgiev [20] for all rectangular weights, i.e. for all integral dominant highest weights $\Lambda=k_{0}\Lambda_{0}+k_{j}\Lambda_{j}$. The bases $\mathfrak{B}_{V}$ for $\mathfrak{g}=B_{l},C_{l},G_{2}$ were obtained by the first author in [4, 5, 6]. The $\mathfrak{g}=A_{l}$ case for basic modules can be also recovered from the recent result of K. Kawasetsu [25]. Our proof of Theorem 3.1 in types $D$, $E$ and $F$ follows the approach in [20] and relies on [4, 5, 22]. In addition to Theorem 3.1, in Theorem 3.2 we construct quasi-particle bases of the principal subspaces $W_{L(\Lambda)}$ for all rectangular highest weights $\Lambda$ in types $D$ and $E$, thus generalizing [20].

Next, in Theorem 4.1, we derive presentations of the principal subspaces $W_{L(k\Lambda_{0})}$ for all types of $\mathfrak{g}$. The presentations of principal subspaces of standard $\widetilde{\mathfrak{g}}$-modules $L(\Lambda)$ for the level $k$ integral dominant highest weights $\Lambda$ were established by Feigin and Stoyanovsky [16] for $\mathfrak{g}=A_{1}$ and $k=1$. Furthermore, the presentations were proved by C. Calinescu, Lepowsky and A. Milas [9, 10, 11] for $\mathfrak{g}=A_{1}$ and $k\geqslant 1$ and for $\mathfrak{g}=A,D,E$ and $k=1$, and by C. Sadowski [39] for $\mathfrak{g}=A_{2}$ and $k\geqslant 1$. As explained in [9], these a priori proofs do not rely on the detailed underlying structure, such as bases of the standard modules or of the principal subspaces. Finally, Sadowski [40] proved the general case $\mathfrak{g}=A_{l}$ for all $k\geqslant 1$ using Georgiev’s quasi-particle bases [20]. In contrast with [9, 10, 11, 39], our proof employs the sets $\mathfrak{B}_{L(k\Lambda_{0})}$ from Theorem 3.1, thus solving a simpler problem. In addition, using the quasi-particle bases from Theorem 3.2 we obtain presentations of the principal subspaces $W_{L(\Lambda)}$ for all rectangular highest weights $\Lambda$ in types $D$ and $E$; see Theorem 4.2. It is worth noting that, aside from the aforementioned cases covered in [9, 10, 11, 39], the a priori proof of these presentations, which were originally conjectured in [40], is still lacking.

In the end, we use the bases from Theorems 3.1 and 3.2 to explicitly write the character formulae for the principal subspaces. In particular, by regarding two different bases for $W_{N(k\Lambda_{0})}$ in types $D$, $E$ and $F$, we obtain three new families of combinatorial identities.

2. Preliminaries

Let $\mathfrak{g}$ be a complex simple Lie algebra of rank $l$ equipped with a nondegenerate invariant symmetric bilinear form $\langle\cdot,\cdot\rangle$ and let $\mathfrak{h}$ be its Cartan subalgebra. As the restriction of the form $\langle\cdot,\cdot\rangle$ on $\mathfrak{h}$ is nondegenerate, it defines a symmetric bilinear form on the dual $\mathfrak{h}^{\ast}$. Let $\Pi=\left\{\alpha_{1},\ldots,\alpha_{l}\right\}\subset\mathfrak{h}^{*}$ be the basis of the root system $R$ of $\mathfrak{g}$ with respect to $\mathfrak{h}$ and let $x_{\alpha}\in\mathfrak{g}$ with $\alpha\in R$ be the root vectors. The simple roots $\alpha_{1},\ldots,\alpha_{l}$ are labelledaaa In contrast with [21] and [23, Table Fin], we reverse the labels in the Dynkin diagram of type $C_{l}$ in Figure 2, so that the root lengths in the sequence $\alpha_{1},\ldots,\alpha_{l}$ increase for all types of $\mathfrak{g}$, thus getting a simpler formulation of Theorem 3.1. as in Figure 2. We denote by $\alpha_{1}^{\vee},\ldots,\alpha_{l}^{\vee}$ the corresponding simple coroots. Let $\lambda_{1},\ldots,\lambda_{l}\in\mathfrak{h}^{*}$ be the fundamental weights, i.e. the weights such that $\left<\lambda_{i},\alpha_{j}\right>=\delta_{ij}$. Let $Q=\sum_{i=1}^{l}\mathbb{Z}\alpha_{i}$ and $P=\sum_{i=1}^{l}\mathbb{Z}\lambda_{i}$ be the root lattice and the weight lattice of $\mathfrak{g}$ respectively. We assume that the form $\langle\cdot,\cdot\rangle$ is normalized so that $\langle\alpha,\alpha\rangle=2$ for every long root $\alpha\in R$. Hence, in particular, we have $\langle\alpha_{i},\alpha_{i}\rangle\in\left\{2/3,1,2\right\}$ for all $i=1,\ldots,l$. Denote by $R_{+}$ and $R_{-}$ the sets of positive and negative roots. Let

[TABLE]

be the triangular decomposition of $\mathfrak{g}$; see [21] for more details on simple Lie algebras.

[TABLE]

Figure 1. Finite Dynkin diagrams

The affine Kac–Moody Lie algebra $\widetilde{\mathfrak{g}}$ associated to $\mathfrak{g}$ is defined by

[TABLE]

where the elements $x(m)=x\otimes t^{m}$ for $x\in\mathfrak{g}$ and $m\in\mathbb{Z}$ are subject to relations

[TABLE]

We denote by $\alpha_{0},\alpha_{1},\ldots,\alpha_{l}$ and $\alpha_{0}^{\vee},\alpha_{1}^{\vee},\ldots,\alpha_{l}^{\vee}$ the simple roots and the simple coroots of $\widetilde{\mathfrak{g}}$. Let $\Lambda_{i}$ be the fundamental weights of $\widetilde{\mathfrak{g}}$, i.e. the weights such that $\Lambda_{i}(d)=0$ and $\Lambda_{i}(\alpha_{j}^{\vee})=\delta_{ij}$ for all $i,j=0,\ldots,l$. For more details on affine Lie algebras see [23].

Let $k_{0},\ldots,k_{l}$ be nonnegative integers such that $k=k_{0}+\ldots+k_{l}$ is positive and let $\lambda=k_{1}\lambda_{1}+\ldots+k_{l}\lambda_{l}$. Denote by $U_{\lambda}$ the finite-dimensional irreducible $\mathfrak{g}$-module of highest weight $\lambda$. The generalized Verma $\widetilde{\mathfrak{g}}$-module $N(\Lambda)$ of highest weight $\Lambda=k_{0}\Lambda_{0}+\ldots+k_{l}\Lambda_{l}$ and of level $k$ is defined as the induced $\widetilde{\mathfrak{g}}$-module

[TABLE]

where the action of the Lie algebra

[TABLE]

on $U_{\lambda}$ is given by

[TABLE]

Denote by $L(\Lambda)$ the standard $\widetilde{\mathfrak{g}}$-module of highest weight $\Lambda$ and of level $k$, i.e. the integrable highest weight $\widetilde{\mathfrak{g}}$-module which equals the unique simple quotient of the generalized Verma module $N(\Lambda)$. In particular, for $\lambda=0$ we obtain the generalized Verma $\widetilde{\mathfrak{g}}$-module $N(k\Lambda_{0})$ of highest weight $k\Lambda_{0}$ and level $k=k_{0}$ which possesses a vertex operator algebra structure. Moreover, $L(k\Lambda_{0})$ is a simple vertex operator algebra and the level $k$ standard $\widetilde{\mathfrak{g}}$-modules are $L(k\Lambda_{0})$-modules; see, e.g., [29, 33]. Finally, recall that Poincaré–Birkhoff–Witt theorem for the universal enveloping algebra implies the vector space isomorphism

[TABLE]

For more details on the representation theory of affine Lie algebras see [23].

3. Quasi-particle bases of principal subspaces

In this section, we state our main results, Theorems 3.1 and 3.2.

3.1. Quasi-particles

Introduce the following subalgebras of $\widetilde{\mathfrak{g}}$:

[TABLE]

Let $\Lambda$ be an arbitrary integral dominant weight of $\widetilde{\mathfrak{g}}$. Denote by $V$ the generalized Verma module $N(\Lambda)$ or the standard module $L(\Lambda)$ with a highest weight vector $v_{V}$. Following Feigin and Stoyanovsky [16], we define the principal subspace $W_{V}$ of $V$ by

[TABLE]

Consider the vertex operators

[TABLE]

Note that (2.1) implies $[x_{\alpha_{i}}(z_{1}),x_{\alpha_{i}}(z_{2})]=0$ so that

[TABLE]

is a well-defined element of $\mathop{\mathrm{Hom}}(V,V((z)))$ for all $n\geqslant 1$. In fact, $x_{n\alpha_{i}}(z)$ is the vertex operator associated with the vector $x_{\alpha_{i}}(-1)^{n}v_{V}\in V$. As in [20], define the quasi-particle of color $i$, charge $n$ and energy $-m$ as the coefficient $x_{n\alpha_{i}}(m)\in\mathop{\mathrm{End}}V$ of (3.1).

Consider the quasi-particle monomial

[TABLE]

in $\mathop{\mathrm{End}}V$. Note that the quasi-particle colors in ($m$) are increasing from right to left and that the integers $r_{j}^{(1)}\geqslant 0$ with $j=1,\ldots,l$ denote the parts of the conjugate partition of $n_{j}=n_{r_{j}^{(1)},j}+\cdots+n_{1,j}$; see [20, 4, 5, 6] for more details. It is convenient to write quasi-particle monomial ($m$) more briefly as

[TABLE]

3.2. Quasi-particle bases for $\Lambda=k\Lambda_{0}$

Suppose that $\Lambda=k\Lambda_{0}$ for some positive integer $k$ so that $V$ denotes the generalized Verma module $N(k\Lambda_{0})$ or the standard module $L(k\Lambda_{0})$. We introduce certain difference conditions for energies and charges of quasi-particles in ($m$). First, for the adjacent quasi-particles of the same color we require that

[TABLE]

Next, we turn to the difference conditions which describe the interaction of two quasi-particles of adjacent colors. For all $i=1,\ldots,l$ define

[TABLE]

Introduce the following difference conditions:

[TABLE]

where we set $r_{0}^{(1)}=0$ so that the sum in ($c_{2}$) is zero for $i=1$. In the end, we impose the following restrictions on the quasi-particle charges:

[TABLE]

Let $B_{N(k\Lambda_{0})}$ be the set of all monomials ($m$), regarded as elements of $\mathop{\mathrm{End}}N(k\Lambda_{0})$, satisfying conditions ($c_{1}$) and ($c_{2}$). Moreover, let $B_{L(k\Lambda_{0})}$ be the set of all monomials ($m$), regarded as elements of $\mathop{\mathrm{End}}L(k\Lambda_{0})$, satisfying ($c_{1}$), ($c_{2}$) and ($c_{3}$). Finally, let

[TABLE]

Theorem 3.1**.**

For any positive integer $k$ the set $\mathfrak{B}_{V}$ forms a basis of the principal subspace $W_{V}$ of the $\widetilde{\mathfrak{g}}$-module $V=N(k\Lambda_{0}),L(k\Lambda_{0})$.

Even though Theorem 3.1 is formulated for an arbitrary untwisted affine Lie algebra $\widetilde{\mathfrak{g}}$, we only give proof for $\mathfrak{g}$ of type $D$, $E$ and $F$; see Sections 5 and 6. The proofs for the remaining types can be found in [4, 5, 6, 20].

3.3. Quasi-particle bases for rectangular weights in types $D$ and $E$

Suppose that the affine Lie algebra $\widetilde{\mathfrak{g}}$ is of type $D_{l}^{(1)}$, $E_{6}^{(1)}$ or $E_{7}^{(1)}$. Let $\Lambda$ be the rectangular weight, i.e. the weight of the form

[TABLE]

where $k_{0},k_{j}$ are positive integers and $\Lambda_{j}$ is the fundamental weight of level one; cf. [20]. Recall that $j=1,l-1,l$ for $\widetilde{\mathfrak{g}}=D_{l}^{(1)}$, $j=1,6$ for $\widetilde{\mathfrak{g}}=E_{6}^{(1)}$ and $j=1$ for $\widetilde{\mathfrak{g}}=E_{7}^{(1)}$; see [23]. Denote by $k=k_{0}+k_{j}$ the level of $\Lambda$. Define

[TABLE]

Introduce the following difference condition:

[TABLE]

Note that this condition differs from ($c_{2}$) by a new term $\sum_{t=1}^{n_{p,i}}\delta_{ij_{t}}$. For a given rectangular weight $\Lambda$ denote by $B_{L(\Lambda)}$ be the set of all monomials ($m$), regarded as elements of $\mathop{\mathrm{End}}L(\Lambda)$, satisfying ($c_{1}$), ($c^{\prime}_{2}$) and ($c_{3}$). Finally, let

[TABLE]

Theorem 3.2**.**

Let $\widetilde{\mathfrak{g}}$ be the affine Lie algebra of type $D_{l}^{(1)}$, $E_{6}^{(1)}$ or $E_{7}^{(1)}$. For any rectangular weight $\Lambda$ the set $\mathfrak{B}_{L(\Lambda)}$ forms a basis of the principal subspace $W_{L(\Lambda)}$.

The proof of Theorem 3.2 is given in Section 6.

4. Presentations of the principal subspaces $W_{L(k\Lambda_{0})}$

In this section, we give the presentations of the principal subspaces $W_{L(k\Lambda_{0})}$ for an arbitrary untwisted affine Lie algebra $\widetilde{\mathfrak{g}}$; see Theorem 4.1 below. Next, in Theorem 4.2, we give the presentations of $W_{L(\Lambda)}$ for all rectangular weights $\Lambda$ in types $D$ and $E$. As pointed out in Section 1, the presentations of the principal subspaces of certain standard $\widetilde{\mathfrak{g}}$-modules in types $A$, $D$ and $E$ were originally found and proved in [16, 9, 10, 11, 39, 40] while their general form was conjectured in [40].

Let $\Lambda$ be an integral dominant highest weight. Consider the natural surjective map

[TABLE]

For any $i=1,\ldots,l$ and integer $m\geqslant k\nu_{i}+1$ define the elements $R_{\alpha_{i}}(-m)\in U(\widetilde{\mathfrak{n}}_{+})$ by

[TABLE]

Let $I_{L(k\Lambda_{0})}$ be the left ideal in the universal enveloping algebra $U(\widetilde{\mathfrak{n}}_{+})$ defined by

[TABLE]

We have the following natural presentations of the principal subspaces:

Theorem 4.1**.**

For all positive integers $k$ we have

[TABLE]

In Section 5, we employ the sets $\mathfrak{B}_{L(k\Lambda_{0})}$ from Theorem 3.1 to prove Theorem 4.1 for the affine Lie algebra $\widetilde{\mathfrak{g}}=F_{4}^{(1)}$. We omit the proof for other types of $\widetilde{\mathfrak{g}}$ since it goes analogously, by using the corresponding quasi-particle bases.

Let $\widetilde{\mathfrak{g}}$ be the affine Lie algebra of type $D_{l}^{(1)}$, $E_{6}^{(1)}$ or $E_{7}^{(1)}$. As in [40], for a given rectangular weight $\Lambda=k_{0}\Lambda_{0}+k_{j}\Lambda_{j}$ define the left ideal in the universal enveloping algebra $U(\widetilde{\mathfrak{n}}_{+})$ by

[TABLE]

Theorem 4.2**.**

Let $\widetilde{\mathfrak{g}}$ be the affine Lie algebra of type $D_{l}^{(1)}$, $E_{6}^{(1)}$ or $E_{7}^{(1)}$. For a given rectangular weight $\Lambda$ we have

[TABLE]

The proof of Theorem 4.2 is given in Section 6.

Remark 4.3**.**

The form of the elements $R_{\alpha_{i}}(-m)$ is motivated by the integrability condition

[TABLE]

which is due to Lepowsky and Primc [31]. It implies quasi-particle charges constraint ($c_{3}$).

5. Proof of Theorems 3.1 and 4.1 in type $F$

In this section, we prove Theorems 3.1 and 4.1 in type $F$. The proof is divided into six steps, i.e. Sections 5.1–5.6. We consider the affine Lie algebra $\widetilde{\mathfrak{g}}$ of type $F_{4}^{(1)}$ so that $l=4$ and the basis $\Pi$ of the root system $R$ for the corresponding simple Lie algebra $\mathfrak{g}$ consists of the simple roots $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$; see [21, Chap. III]. The maximal root $\theta$ equals

[TABLE]

5.1. Linear order on quasi-particle monomials

In this section, we briefly cover some basic concepts originated in [20] which are typically used to handle quasi-particle monomials. In particular, we introduce a certain linear order among such monomials which will come in useful in Section 5.5. Let

[TABLE]

be an element of $\mathop{\mathrm{End}}V$, where $V=N(k\Lambda_{0})$ or $V=L(k\Lambda_{0})$, such that

[TABLE]

Define the charge-type $\mathcal{C}$ and the energy-type $\mathcal{E}$ of $b$ by

[TABLE]

Moreover, define the color-type of $b$ as the quadruple $(n_{4},n_{3},n_{2},n_{1})$ such that $n_{j}$ denotes the sum of charges of all color $j$ quasi-particles, i.e. such that $n_{j}=n_{r_{j}^{(1)},j}+\ldots+n_{1,j}$.

Let $b_{1},b_{2}$ be any two quasi-particle monomials of the same color-type, expressed as in ($m_{F_{4}}$), such that their charges and energies satisfy (5.2). Denote their charge-types and energy-types by $\mathcal{C}_{1},\mathcal{C}_{2}$ and $\mathcal{E}_{1},\mathcal{E}_{2}$ respectively. Define the strict linear order among quasi-particle monomials of the same color-type by

[TABLE]

where the order on (finite) sequences of integers is defined as follows:

[TABLE]

5.2. Projection of the principal subspace

As in [4], we now generalize Georgiev’s projection [20] to type $F$. Consider quasi-particle monomial ($m_{F_{4}}$) as an element of $\mathop{\mathrm{End}}L(k\Lambda_{0})$. Suppose that its charges and energies satisfy (5.2). Define its dual charge-type $\mathcal{D}$ as

[TABLE]

where $r_{i}^{(n)}$ denotes the number of color $i$ quasi-particles of charge greater than or equal to $n$ in the monomial. Observe that, due to (4.4), the monomial does not posses any quasi-particles of color $i$ whose charge is strictly greater than $k\nu_{i}$.

The standard module $L(k\Lambda_{0})$ can be regarded as a submodule of the tensor product module $L(\Lambda_{0})^{\otimes k}$ generated by the highest weight vector $v_{L(k\Lambda_{0})}=v_{L(\Lambda_{0})}^{\otimes k}$. Let $\pi_{\mathcal{D}}$ be the projection of the principal subspace $W_{L(k\Lambda_{0})}$ on the tensor product space

[TABLE]

where $W_{(\mu^{(t)}_{4};\mu^{(t)}_{3};r_{2}^{(t)};r_{1}^{(t)})}$ denote the $\mathfrak{h}$-weight subspaces of the level $1$ principal subspace $W_{L(\Lambda_{0})}$ of weight $\mu^{(t)}_{4}\alpha_{4}+\mu^{(t)}_{3}\alpha_{3}+r^{(t)}_{2}\alpha_{2}+r_{1}^{(t)}\alpha_{1}\in R$ with

[TABLE]

We denote by the same symbol $\pi_{\mathcal{D}}$ the generalization of the projection to the space of formal series with coefficients in $W_{L(k\Lambda_{0})}$. Applying the generating function corresponding to ($m_{F_{4}}$) on the highest weight vector $v_{L(k\Lambda_{0})}=v_{L(\Lambda_{0})}^{\otimes k}$ we obtain

[TABLE]

Relations (4.4) imply that by applying the projection $\pi_{\mathcal{D}}$ on (5.8) we get

[TABLE]

multiplied by some nonzero scalar, where we set $x_{0\alpha_{i}}(z)=1$. The integers $n_{p,i}^{(t)}$ in (5.9) are uniquely determined by

[TABLE]

and by the requirement that at most one $n_{p,i}^{(t)}$ equals $1$ when $i=3,4$. Therefore, for every variable $z_{r,i}$, where $i=1,2,3,4$ and $r=1,\ldots,r_{i}^{(1)}$, the projection $\pi_{\mathcal{D}}$ places at most one generating function $x_{\alpha_{i}}(z_{r,i})$ if $i=1,2$ and at most two generating functions $x_{\alpha_{i}}(z_{r,i})$ if $i=3,4$ on each tensor factor of $W(\Lambda_{0})^{\otimes k}$.

5.3. Operators $A_{\theta}$ and $e_{\alpha}$

Let $b\in B_{L(k\Lambda_{0})}$ be a quasi-particle monomial of charge-type $\mathcal{C}$ and dual charge-type $\mathcal{D}$. Denote the charges and the energies of its quasi-particles as in ($m_{F_{4}}$). In this section, generalizing the approach from [6], we demonstrate how to reduce $b$ to obtain a new monomial $b^{\prime}\in B_{L(k\Lambda_{0})}$ such that its charge-type $\mathcal{C}^{\prime}$ satisfies $\mathcal{C}^{\prime}<\mathcal{C}$ with respect to linear order (5.4). This will be a key step in the proof of linear independence of the set $\mathfrak{B}_{L(k\Lambda_{0})}$ in Section 5.4.

Let $A_{\theta}$ be the constant term of the operator

[TABLE]

i.e. $A_{\theta}=x_{\theta}(-1)$, where $\theta$ is the maximal root; recall (5.1). Consider the image of the vector $\pi_{\mathcal{D}}\,bv_{L(k\Lambda_{0})}\in W_{L(k\Lambda_{0})}\subset W_{L(\Lambda_{0})}^{\otimes k}$ with respect to the operator

[TABLE]

This image can be obtained as the coefficient of the variables

[TABLE]

in the expression

[TABLE]

Due to [17], the operator $A_{\theta}$ commutes with the action of quasi-particles. Hence, using (5.9), we find that the $s$-th tensor factor (from the right) in (5.11) equals

[TABLE]

Consider the Weyl group translation operator $e_{\alpha}\in\mathop{\mathrm{End}}L(\Lambda_{0})$ defined by

[TABLE]

for $\alpha\in R$; see [23, Chap. 3]. It possesses the following properties:

[TABLE]

Using (5.12) and (5.13) for $\alpha=\theta$ we rewrite the $s$-th tensor factor $F_{s}$ as

[TABLE]

Recall (5.1) and notation (3.2). Taking the coefficient of variables (5.10) in (5.14) we find

[TABLE]

where $(e_{\theta})_{s}$ denotes the action of $e_{\theta}$ on the $s$-th tensor factor (from the right) and

[TABLE]

Therefore, by applying the above procedure we increased the energies of all quasi-particles of color $1$ and charge $s=n_{1,1}$ in the monomial $b\in B_{L(k\Lambda_{0})}$ by $1$. We may continue to apply the same procedure, now starting with $b^{+}v_{L(k\Lambda_{0})}$, until we obtain the monomial

[TABLE]

Since $b$ is an element of $B_{L(k\Lambda_{0})}$, the quasi-particle monomial $\widetilde{b}$ belongs to $B_{L(k\Lambda_{0})}$ as well. Moreover, the charge-type and the dual charge-type of $\widetilde{b}$ equal $\mathcal{C}$ and $\mathcal{D}$ respectively.

By (5.12) we have $x_{\alpha_{1}}(-1)v_{L(\Lambda_{0})}=-e_{\alpha_{1}}v_{L(\Lambda_{0})}$. Hence, the vector $\pi_{\mathcal{D}}\,\widetilde{b}v_{L(k\Lambda_{0})}$, which belongs to $W_{L(k\Lambda_{0})}\subset W_{L(\Lambda_{0})}^{\otimes k}$, equals the coefficient of the variables

[TABLE]

in

[TABLE]

where $\overline{z}$ is given by (5.10). We now employ (5.13) to move $1^{\otimes(k-s)}\otimes e_{\alpha_{1}}^{\otimes s}$ all the way to the left in (5.16). Next, by dropping the invertible operator $(-1)^{s}(1^{\otimes(k-s)}\otimes e_{\alpha_{1}}^{\otimes s})$ and taking the coefficient of variables (5.15) we get $\pi_{\mathcal{D}^{\prime}}\,b^{\prime}v_{L(k\Lambda_{0})}$, where the quasi-particle monomial $b^{\prime}$ of charge-type $\mathcal{C}^{\prime}$ and dual charge-type $\mathcal{D}^{\prime}$ is given by

[TABLE]

Clearly, the energies of the quasi-particles in colors $3$ and $4$ did not change. Furthermore, if the dual charge-type $\mathcal{D}$ of $b$ equals

[TABLE]

then the dual charge-type $\mathcal{D}^{\prime}$ of $b^{\prime}$ equals

[TABLE]

In particular, we have $\mathcal{C}^{\prime}<\mathcal{C}$ with respect to linear order (5.4). Finally, by arguing as in [5, Proposition 3.3.1] one can check that $b^{\prime}$ belongs to $B_{L(k\Lambda_{0})}$.

5.4. Linear independence of the sets $\mathfrak{B}_{V}$

In this section, we prove linear independence of the set $\mathfrak{B}_{L(k\Lambda_{0})}$. Linear independence of $\mathfrak{B}_{N(k\Lambda_{0})}$ can be verified by arguing as in [4, Sect. 3]. Suppose there exists a linear dependence relation among some elements $b^{a}v_{L(k\Lambda_{0})}\in\mathfrak{B}_{L(k\Lambda_{0})}$,

[TABLE]

and $A$ is a finite nonempty set. As the principal subspace $W_{L(k\Lambda_{0})}$ is a direct sum of its $\mathfrak{h}$-weight subspaces, we can assume that all $b^{a}\in B_{L(k\Lambda_{0})}$ posses the same color-type.

Recall strict linear order (5.4) and choose $a_{0}\in A$ such that $b^{a_{0}}<b^{a}$ for all $a\in A$, $a\neq a_{0}$. Suppose that the charge-type $\mathcal{C}$ and the dual charge-type $\mathcal{D}$ of $b^{a_{0}}$ are given by (5.3) and (5.6) respectively. Applying the projection $\pi_{\mathcal{D}}$ on (5.17) we obtain a linear combination of elements in

[TABLE]

recall (5.7). The definition of the projection $\pi_{\mathcal{D}}$ implies that all $b^{a}v_{L(k\Lambda_{0})}$ such that the charge-type of $b^{a}$ is strictly greater than $\mathcal{C}$ with respect to (5.5) are annihilated by $\pi_{\mathcal{D}}$. Therefore, we can assume that all $b^{a}$ posses the same charge-type $\mathcal{C}$ and, consequently, the same dual-charge-type $\mathcal{D}$.

As in (3.2), write the monomials $b^{a}$ as $b^{a}=b^{a}_{\alpha_{4}}b^{a}_{\alpha_{3}}b^{a}_{\alpha_{2}}b^{a}_{\alpha_{1}}$, where $b^{a}_{\alpha_{j}}$ consist of quasi-particles of color $j$. We now apply the procedure described in Section 5.3 on the linear combination

[TABLE]

We repeat it until all quasi-particles of color $1$ are removed from the first summand $c_{a_{0}}\pi_{\mathcal{D}}\,b^{a_{0}}v_{L(k\Lambda_{0})}$. This also removes all quasi-particles of color $1$ from other summands, so that (5.18) becomes

[TABLE]

for some quasi-particle monomials $\widetilde{b}^{a}_{\alpha_{2}}$ of color $2$ and scalars $\widetilde{c}_{a}\neq 0$ such that $\widetilde{\mathcal{D}}$ is the dual charge-type of all quasi-particle monomials $b^{a}_{\alpha_{4}}b^{a}_{\alpha_{3}}\widetilde{b}^{a}_{\alpha_{2}}$ in (5.19). The summation in (5.19) goes over all $a\neq a_{0}$ such that $b^{a}_{\alpha_{1}}=b^{a_{0}}_{\alpha_{1}}$ because the summands $\pi_{\mathcal{D}}\,b^{a}v_{L(k\Lambda_{0})}$ such that $b^{a_{0}}_{\alpha_{1}}<b^{a}_{\alpha_{1}}$ get annihilated in the process.

The vectors $b^{a}_{\alpha_{4}}b^{a}_{\alpha_{3}}\widetilde{b}^{a}_{\alpha_{2}}v_{L(k\Lambda_{0})}$ in (5.19) belong to $\mathfrak{B}_{L(k\Lambda_{0})}$. Furthermore, they can be realized as elements of the principal subspace of the level $k$ standard module $L(k\Lambda_{0})$ with the highest weight vector $v_{L(k\Lambda_{0})}$ for the affine Lie algebra of type $C_{3}^{(1)}$. Moreover, their realizations belong to the corresponding basis in type $C_{3}^{(1)}$, as given by Theorem 3.1 (for a detailed proof in type $C_{l}^{(1)}$ see [5]). This implies $\widetilde{c}_{a_{0}}=0$ and, consequently, $c_{a_{0}}=0$, thus contradicting (5.17). Finally, we conclude that the set $\mathfrak{B}_{L(k\Lambda_{0})}$ is linearly independent.

5.5. Small spanning sets $\bar{\mathfrak{B}}_{V}$

In this section, we construct certain small spanning sets $\bar{\mathfrak{B}}_{N(k\Lambda_{0})}$ and $\bar{\mathfrak{B}}_{L(k\Lambda_{0})}$ for the quotients $U(\widetilde{\mathfrak{n}}_{+})/I_{N(k\Lambda_{0})}$ and $U(\widetilde{\mathfrak{n}}_{+})/I_{L(k\Lambda_{0})}$ of the algebra $U(\widetilde{\mathfrak{n}}_{+})$ over its left ideals $I_{N(k\Lambda_{0})}=U(\widetilde{\mathfrak{n}}_{+})\widetilde{\mathfrak{n}}_{+}^{\geqslant 0}$ and $I_{L(k\Lambda_{0})}$ defined by (4.2). We denote by $\bar{x}$ the image of the element $x\in U(\widetilde{\mathfrak{n}}_{+})$ in these quotients with respect to the corresponding canonical epimorphisms. First, we consider $U(\widetilde{\mathfrak{n}}_{+})/I_{N(k\Lambda_{0})}$. By Poincaré–Birkhoff–Witt theorem for the universal enveloping algebra we have

[TABLE]

By (2.1) quasi-particles of the same color commute, so all monomials

[TABLE]

such that their charges and energies satisfy (5.2) form a spanning set for $U(\widetilde{\mathfrak{n}}_{+})/I_{N(k\Lambda_{0})}$.

We now list two families of quasi-particle relations which can be used to strengthen the conditions in (5.2), i.e. to obtain a smaller spanning set. The first family is given for quasi-particles on $N(k\Lambda_{0})$ of color $i=1,2,3,4$ and charges $n_{1}$ and $n_{2}$ such that $n_{2}\leqslant n_{1}$:

[TABLE]

where $p=0,1,\ldots,2n_{2}-1$ and $A_{p}(z),B_{p}(z)$ are some formal series with coefficients in the set of quasi-particle polynomials; see [16, 20, 22]. As demonstrated in [22, Remark 4.6], see also [4, Lemma 2.2.1], relations ($r_{1}$) can be used to express $2n_{2}$ monomials of the form

[TABLE]

as a linear combination of monomials

[TABLE]

and monomials which contain a quasi-particle of color $i$ and charge $n_{1}+1$, thus possessing the greater charge-type. In particular, for $n_{2}=n_{1}$ one can express $2n_{2}$ monomials

[TABLE]

as a linear combination of monomials

[TABLE]

and monomials which contain a quasi-particle of color $i$ and charge $n_{2}+1$, thus possessing the greater charge-type; cf. [4, Corollary 2.2.2].

The second family of relations for quasi-particles on $N(k\Lambda_{0})$ is given by

[TABLE]

They follow by a direct computation employing the commutator formula for vertex operators; see, e.g., [29, Chap. 6.2].

Remark 5.1**.**

Due to ($r_{2}$), the quasi-particles of colors $1$ and $2$ and the quasi-particles of colors $3$ and $4$ interact as the quasi-particles of colors $1$ and $2$ for the affine Lie algebra $A_{2}^{(1)}$ while the quasi-particles of colors $2$ and $3$ interact as the quasi-particles of colors $1$ and $2$ for the affine Lie algebra $B_{2}^{(1)}$.

Let $\bar{\mathfrak{B}}_{N(k\Lambda_{0})}$ be the set of all monomials ($\bar{m}_{F_{4}}$) satisfying difference conditions ($c_{1}$) and ($c_{2}$) (with $l=4$ and $i^{\prime}=i-1$ for all $i=1,2,3,4$). Using relations ($r_{1}$) and ($r_{2}$) and arguing as in [4, 20] one can show that every monomial of the form ($\bar{m}_{F_{4}}$) satisfying (5.2) can be expressed as a linear combination of some monomials in $\bar{\mathfrak{B}}_{N(k\Lambda_{0})}$, so that $\bar{\mathfrak{B}}_{N(k\Lambda_{0})}$ spans the quotient $U(\widetilde{\mathfrak{n}}_{+})/I_{N(k\Lambda_{0})}$. The proof goes by induction on the charge-type and total energy of quasi-particle monomials and relies on the properties of strict linear order (5.4). Roughly speaking, difference condition ($c_{1}$) follows from relations ($r_{1}$) for $n_{2}=n_{1}$; the last summand $-2(p-1)n_{p,i}$ in ($c_{2}$) follows from relations ($r_{1}$) for $n_{2}<n_{1}$; the sum in ($c_{2}$) follows from ($r_{2}$). Finally, the first summand $-n_{p,i}$ in ($c_{2}$) is due to the fact that each summand on the right hand side of

[TABLE]

contains at least one quasi-particle $x_{\alpha_{i}}(n_{j})$ with $n_{j}\geqslant 0$, so that $x_{m\alpha_{i}}(n)$ belongs to $I_{N(k\Lambda_{0})}$ for $n>-m$.

We now consider $U(\widetilde{\mathfrak{n}}_{+})/I_{L(k\Lambda_{0})}$. It is clear that all monomials ($\bar{m}_{F_{4}}$), regarded as elements of $U(\widetilde{\mathfrak{n}}_{+})/I_{L(k\Lambda_{0})}$ and satisfying difference conditions ($c_{1}$) and ($c_{2}$), form a spanning set for the quotient $U(\widetilde{\mathfrak{n}}_{+})/I_{L(k\Lambda_{0})}$. However, the form of the ideal $I_{L(k\Lambda_{0})}$, as defined in (4.2), implies additional relations

[TABLE]

in $U(\widetilde{\mathfrak{n}}_{+})/I_{L(k\Lambda_{0})}$ which we now use to obtain a smaller spanning set; recall Remark 4.3.

Suppose that monomial ($\bar{m}_{F_{4}}$) satisfies difference conditions ($c_{1}$) and ($c_{2}$) and contains a quasi-particle $\bar{x}_{n_{j,i}\alpha_{i}}(m_{j,i})$ of charge $n_{j,i}\geqslant k\nu_{i}+1$ and color $1\leqslant i\leqslant 4$. Clearly, such monomial coincides with the coefficient of the variables

[TABLE]

in the generating function

[TABLE]

Introduce the Laurent polynomial

[TABLE]

By combining relations ($r_{2}$) and ($r_{3}$) we find $P\bar{X}=0$ as the operator $\bar{x}_{n_{j,i}\alpha_{i}}(z_{j,i})$ in $P\bar{X}$ can be moved all the way to the right, thus annihilating the expression. By taking the coefficient of the variables (5.20) in $P\bar{X}=0$ we express ($\bar{m}_{F_{4}}$) as a linear combination of some quasi-particle monomials of the same charge-type and of the same total energy $m_{r_{4}^{(1)},4}+\ldots+m_{1,1}$, which are greater than ($\bar{m}_{F_{4}}$) with respect to linear order (5.4). However, there exists only finitely many such quasi-particle monomials which are nonzero. Hence, by repeating the same procedure for an appropriate number of times, now starting with these new monomials, we find, after finitely many steps, that ($\bar{m}_{F_{4}}$) equals zero. Therefore, we conclude that the set $\bar{\mathfrak{B}}_{L(k\Lambda_{0})}$ of all monomials ($\bar{m}_{F_{4}}$) in $U(\widetilde{\mathfrak{n}}_{+})/I_{L(k\Lambda_{0})}$ which satisfy difference conditions ($c_{1}$), ($c_{2}$) and ($c_{3}$) forms a spanning set for $U(\widetilde{\mathfrak{n}}_{+})/I_{L(k\Lambda_{0})}$.

5.6. Proof of Theorems 3.1 and 4.1

In Section 5.4, we established the linear independence of the sets $\mathfrak{B}_{N(k\Lambda_{0})}$ and $\mathfrak{B}_{L(k\Lambda_{0})}$. We now prove that they span the principal subspaces $W_{L(k\Lambda_{0})}$ and $W_{N(k\Lambda_{0})}$, thus finishing the proof of Theorem 3.1. Moreover, as a consequence of the proof, we obtain the presentations of the principal subspace $W_{L(k\Lambda_{0})}$ given by Theorem 4.1. Introduce the natural surjective map

[TABLE]

so that we can consider the cases $V=L(k\Lambda_{0})$ and $V=N(k\Lambda_{0})$ simultaneously. Recall that the surjective map $f_{L(k\Lambda_{0})}$ is given by (4.1), the left ideal $I_{L(k\Lambda_{0})}$ is defined by (4.2) and $I_{N(k\Lambda_{0})}=U(\widetilde{\mathfrak{n}}_{+})\widetilde{\mathfrak{n}}_{+}^{\geqslant 0}$ .

Let $V$ be $N(k\Lambda_{0})$ or $L(k\Lambda_{0})$. It is clear that the left ideal $I_{V}$ belongs to the kernel of $f_{V}$. Hence, there exists a unique map

[TABLE]

where $\pi_{V}$ is the canonical epimorphism $U(\widetilde{\mathfrak{n}}_{+})\to U(\widetilde{\mathfrak{n}}_{+})/I_{V}$. The map $\bar{f}_{V}$ is surjective as $f_{V}$ is surjective and, furthermore, it maps bijectively $\bar{\mathfrak{B}}_{V}$ to $\mathfrak{B}_{V}$. Therefore, the linearly independent set $\mathfrak{B}_{V}$ spans the principal subspace $W_{V}$ and so it forms a basis of $W_{V}$, which proves Theorem 3.1. This implies that the map (5.21) is a vector space isomorphism, so, in particular, we conclude that $\ker f_{L(k\Lambda_{0})}=I_{L(k\Lambda_{0})}$, thus proving Theorem 4.1.

6. Proof of Theorems 3.1, 3.2 and 4.2 in types $D$ and $E$

In this section, unless stated otherwise, we denote by $\widetilde{\mathfrak{g}}$ the affine Lie algebra of type $D_{l}^{(1)}$, $E_{6}^{(1)}$, $E_{7}^{(1)}$, $E_{8}^{(1)}$. First, we give an outline of the proof of Theorem 3.1 for $\widetilde{\mathfrak{g}}$. As the generalization of the arguments from Section 5.5 is straightforward, we only discuss the proof of linear independence. It relies on the coefficients of certain level 1 intertwining operators and on the vertex operator algebra construction of basic modules, thus resembling the corresponding proofs in types $A_{l}^{(1)}$, $B_{l}^{(1)}$ and $C_{l}^{(1)}$; see [20, 4, 5]. In Section 6.1 we recall the aforementioned construction while in Section 6.2 we demonstrate how to use the corresponding operators to complete the proof of Theorem 3.1. Next, in Section 6.3 we add some details as compared to Sections 5 and 6.2 to take care of the modifications needed to carry out the argument for rectangular weights, i.e. to prove Theorems 3.2 and 4.2. Finally, in Section 6.4 we construct different quasi-particle bases in type $E$, such that their linear independence can be verified by employing the operator $A_{\theta}$ associated with the maximal root $\theta$, thus resembling the corresponding proof in type $F$ from Section 5.

6.1. Vertex operator algebra construction of basic modules

We follow [19, 29] to review the vertex operator algebra construction of the basic modules $L(\Lambda_{i})$ [18, 41]. Set

[TABLE]

Let $M(1)=S(\widehat{\mathfrak{h}}^{<0})$ be the Fock space for the Heisenberg algebra $\widehat{\mathfrak{h}}_{*}$ with $h(-m)$ acting as multiplication and $h(m)$ acting as differentiation on $M(1)$ for all $h\in\mathfrak{h}$ and $m\in\mathbb{N}$. Consider the tensor products

[TABLE]

where $\mathbb{C}\left[P\right]$ and $\mathbb{C}\left[Q\right]$ denote the group algebras of the weight lattice $P$ and of the root lattice $Q$ with respective bases $\{e^{\lambda}:\lambda\in P\}$ and $\{e^{\alpha}:\alpha\in Q\}$. We use the identification of group elements $e^{\lambda}=1\otimes e^{\lambda}\in V_{P}$.

Let $e_{\lambda}\colon V_{P}\to V_{P}$ be the linear isomorphism defined by

[TABLE]

where $\epsilon$ is a certain map $P\times P\to\mathbb{C}^{\times}$ satisfying $\epsilon(\lambda,0)=\epsilon(0,\lambda)=1$ for all $\lambda\in P$; see [19, 29] for more details. The space $V_{Q}$ is equipped with a structure of a vertex operator algebra, with $V_{P}$ being a $V_{Q}$-module, by

[TABLE]

and $z^{\lambda}=1\otimes z^{\lambda}$ acts by $z^{\lambda}e^{\mu}=z^{\langle\lambda,\mu\rangle}$ for all $\lambda,\mu\in P$. Moreover, the space $V_{P}$ acquires a structure of level one $\widetilde{\mathfrak{g}}$-module via

[TABLE]

With respect to this action, the space $V_{Q}$ is identified with the standard module $L(\Lambda_{0})$ while the irreducible $V_{Q}$-submodules $V_{Q}e^{\lambda_{i}}$ of $V_{P}$ are identified with the standard modules $L(\Lambda_{i})$ for all $i$ such that the weight $\Lambda_{i}$ is of level one. The corresponding highest weight vectors are $v_{L(\Lambda_{0})}=1$ and $v_{L(\Lambda_{i})}=e^{\lambda_{i}}$.

6.2. Operators $A_{\lambda_{i}}$ and proof of Theorem 3.1

Let $b\in B_{L(k\Lambda_{0})}$ be a quasi-particle monomial as in ($m$), of charge-type $\mathcal{C}$ and dual charge-type

[TABLE]

We now demonstrate how to carry out the procedure from Section 5.3, i.e. how to reduce $b$ to obtain a new monomial $b^{\prime}\in B_{L(k\Lambda_{0})}$ such that its charge-type $\mathcal{C}^{\prime}$ satisfies $\mathcal{C}^{\prime}<\mathcal{C}$ with respect to linear order (5.4). Denote by $I(\cdot,z)$ the intertwining operator of type $\binom{V_{P}}{V_{P}\,\,V_{Q}}$,

[TABLE]

see [17, Sect. 5.4]. For $i=1,\ldots,l$ let $A_{\lambda_{i}}$ be the constant term of $I(e^{\lambda_{i}},z)$, that is

[TABLE]

We have

[TABLE]

In contrast with Section 5.3, which relies on the application of the operators $A_{\theta}$ and $e_{\theta}$, we here make use of $A_{\lambda_{i}}$ and $e_{\lambda_{i}}$ in a similar fashion. In particular, we employ the following property of $e_{\lambda_{i}}$:

[TABLE]

see [11] for more details. Moreover, we use the fact that the operators $A_{\lambda_{i}}$ commute with the action of $x_{\alpha}(z)$ for all $\alpha\in R$, which comes as a consequence of the commutator formula for $x_{\alpha}(z)$ and $I(e^{\lambda_{i}},z)$; see [17, Sect. 5.4].

As in Section 5.2, denote by $\pi_{\mathcal{D}}$ the projection of the principal subspace $W_{L(k\Lambda_{0})}$ on

[TABLE]

where $W_{(r^{(t)}_{l};\ldots;r_{2}^{(t)};r_{1}^{(t)})}$ denote the $\mathfrak{h}$-weight subspaces of the level $1$ principal subspace $W_{L(\Lambda_{0})}$ of the weight $r^{(t)}_{l}\alpha_{l}+\cdots+r^{(t)}_{2}\alpha_{2}+r_{1}^{(t)}\alpha_{1}\in R$. Arguing as in Section 5.3, we conclude that the image of $\pi_{\mathcal{D}}\,bv_{L(k\Lambda_{0})}\in W_{L(k\Lambda_{0})}\subset W_{L(\Lambda_{0})}^{\otimes k}$ with respect to the operator

[TABLE]

equals the coefficient of the variables

[TABLE]

in the expression

[TABLE]

Moreover, the $s$-th tensor factor in (6.5) (from the right) equals

[TABLE]

where the integers $n_{p,i}^{(t)}$ are given by

[TABLE]

By combining (6.1) and (6.3) we get

[TABLE]

Recall the notation from (3.2). By taking the coefficient of variables (6.4) in (6.6) we have

[TABLE]

where $(e_{\lambda_{1}})_{s}$ denotes the action of $e_{\lambda_{1}}$ on the $s$-th tensor factor (from the right) and

[TABLE]

Note that the monomial $b^{+}$ belongs to $B_{L(k\Lambda_{0})}$.

As in Section 5.3, we can now continue to apply this procedure until we obtain a monomial $b^{\prime}\in B_{L(k\Lambda_{0})}$ of charge-type $\mathcal{C}^{\prime}<\mathcal{C}$. Finally, by repeating the arguments from Section 5.4 almost verbatim, we can prove the linear independence of the set $\mathfrak{B}_{L(k\Lambda_{0})}$. However, in contrast with Section 5.4, where the quasi-particle basis in type $F_{4}^{(1)}$ was reduced to a basis in type $C_{3}^{(1)}$, the quasi-particle basis in type $D_{l}^{(1)}$, $E_{6}^{(1)}$, $E_{7}^{(1)}$ or $E_{8}^{(1)}$ is reduced, after sufficient number of steps, to a basis in type $A_{1}^{(1)}$ from Theorem 3.1. Note that such a modification of the argument is possible because we have the operators $A_{\lambda_{i}}$ and $e^{\lambda_{i}}$ satisfying (6.2) and (6.3) at our disposal; cf. corresponding properties (5.12) and (5.13) for $\alpha=\theta$.

6.3. Proof of Theorems 3.2 and 4.2

Let $\widetilde{\mathfrak{g}}$ be the affine Lie algebra of type $D_{l}^{(1)}$, $E_{6}^{(1)}$ or $E_{7}^{(1)}$ and let $\Lambda=k_{0}\Lambda_{0}+k_{j}\Lambda_{j}$ be an arbitrary rectangular weight, as defined in Section 3.3. First, we prove that the set $\mathfrak{B}_{L(\Lambda)}$ is linearly independent. As in Section 5.2, we regard the standard module $L(\Lambda)$ as the submodule of $L(\Lambda_{0})^{\otimes k_{0}}\otimes L(\Lambda_{j})^{\otimes k_{j}}$ generated by the highest weight vector $v_{L(\Lambda)}=v_{L(\Lambda_{0})}^{\otimes k_{0}}\otimes v_{L(\Lambda_{j})}^{\otimes k_{j}}$. Suppose that

[TABLE]

$A$ is a finite nonempty set and all $b^{a}\in B_{L(\Lambda)}$ posses the same color-type. Let $b^{a_{0}}$ be a monomial of dual charge-type $\mathcal{D}$ such that $b^{a_{0}}<b^{a}$ for all $a\in A$, $a\neq a_{0}$, with respect to linear order (5.4). Applying the corresponding projection $\pi_{\mathcal{D}}$, which is defined in parallel with Section 6.2, on linear combination (6.7), we obtain

[TABLE]

By Section 6.1, the highest weight vector $v_{L(\Lambda)}$ is identified with $1^{\otimes k_{0}}\otimes(e^{\lambda_{j}})^{\otimes k_{j}}$, so that, due to (6.1), we have

[TABLE]

Therefore, linear combination (6.8) can be expressed as

[TABLE]

By employing (6.3) to move $1^{\otimes k_{0}}\otimes e_{\lambda_{j}}^{\otimes k_{j}}$ all the way to the left and then dropping the invertible operator, we get

[TABLE]

for some quasi-particle monomials $\tilde{b}^{a}$. Using the fact that the original monomials $b^{a}$ belong to $B_{L(\Lambda)}$ one can verify that all $\tilde{b}^{a}$ belong to $B_{L(k\Lambda_{0})}$. Therefore, due to the identification $v_{L(\Lambda_{0})}^{\otimes k}=v_{L(k\Lambda_{0})}$, the linear independence of the set $\mathfrak{B}_{L(\Lambda)}$ now follows from Theorem 3.1.

We now proceed as in Section 5.5 and construct a spanning set for $U(\widetilde{\mathfrak{n}}_{+})/I_{L(\Lambda)}$. We denote the image of the element $x\in U(\widetilde{\mathfrak{n}}_{+})$ in the quotient $U(\widetilde{\mathfrak{n}}_{+})/I_{V}$, where $V=L(k\Lambda_{0}),L(\Lambda)$, by $\bar{x}$. Let $\bar{\mathfrak{B}}_{L(\Lambda)}$ be the set of all monomials

[TABLE]

in $U(\widetilde{\mathfrak{n}}_{+})/I_{L(\Lambda)}$ such that their charges and energies satisfy

[TABLE]

and difference conditions ($c_{1}$), ($c^{\prime}_{2}$) and ($c_{3}$). It is clear from Theorem 3.1 that the set of all monomials $\bar{b}$ as in ($\bar{m}$) satisfying (6.9) and difference conditions ($c_{1}$), ($c_{2}$) and ($c_{3}$) spans the quotient $U(\widetilde{\mathfrak{n}}_{+})/I_{L(\Lambda)}$. Suppose that such a monomial $\bar{b}$ does not satisfy the more restrictive condition ($c^{\prime}_{2}$). Introduce the generating functions

[TABLE]

where the subscript $V$ indicates that the coefficients of $\bar{X}_{V}$ are regarded as elements of the quotient $U(\widetilde{\mathfrak{n}}_{+})/I_{V}$. Clearly, $\bar{b}$ equals the coefficient of the variables

[TABLE]

in $\bar{X}_{L(\Lambda)}$. By Theorem 4.1 we have $W_{L(k\Lambda_{0})}\cong U(\widetilde{\mathfrak{n}}_{+})/I_{L(k\Lambda_{0})}$. Therefore, due to commutation relations

[TABLE]

with $M_{i}=\min\left\{\textstyle n_{q,i^{\prime}},n_{p,i}\right\}$, the product $P\bar{X}_{L(k\Lambda_{0})}$, where $P$ is the Laurent polynomial

[TABLE]

belongs to

[TABLE]

for $V=L(k\Lambda_{0})$. This implies that the product $P\bar{X}_{L(\Lambda)}$ belongs to (6.10) for $V=L(\Lambda)$. However, every vertex operator $\bar{x}_{n\alpha_{i}}(z)$ in the product $P\bar{X}_{L(\Lambda)}$ can be moved all the way to the right. By (4.3) we have $x_{\alpha_{j}}(-1)^{k_{0}+1}\in I_{L(\Lambda)}$, so that each $\bar{x}_{n\alpha_{i}}(z)$ increases the power of its variable $z$ in (6.10) by $\sum_{t=1}^{n}\delta_{ij_{t}}$. Therefore, we have

[TABLE]

By employing (6.11) and repeating the corresponding part of the proof of [20, Thm. 5.1] the monomial $\bar{b}$ can be expressed as a linear combination of elements of $\bar{\mathfrak{B}}_{L(\Lambda)}$. Hence we conclude that the set $\bar{\mathfrak{B}}_{L(\Lambda)}$ spans the quotient $U(\widetilde{\mathfrak{n}}_{+})/I_{L(\Lambda)}$.

Since the ideal $I_{L(\Lambda)}$ belongs to the kernel of the map $f_{L(\Lambda)}$ defined by (4.1), Theorems 3.2 and 4.2 can be now verified by arguing as in Section 5.6.

6.4. Operator $A_{\theta}$ revisited

As with type $G$ in [6], the linear independence proof in type $F$ employs certain operator $A_{\theta}=x_{\theta}(-1)$; see Sections 5.3 and 5.4. In this section we show that the operator $A_{\theta}$ associated with the maximal root $\theta$ in type $E$ can be also used to verify the linear independence, but of different bases. First, for $\mathfrak{g}=E_{l}$ set

[TABLE]

Introduce the following families of difference conditions:

[TABLE]

Let $B_{L(k\Lambda_{0})}^{E_{l}}$ be the set all monomials ($m$) which satisfy (6.9) and the following difference conditions:

• $\circ$

($c_{1}$), ($c_{3}$), ($c_{2}^{0}$), ($c_{2}^{j}$) for $j=3,4,5,6,8$ and ($c_{2}^{j,k}$) for $(j,k)=(7,2)$ if $l=8$;

• $\circ$

($c_{1}$), ($c_{3}$), ($c_{2}^{0}$), ($c_{2}^{j}$) for $j=3,4,5,7$ and ($c_{2}^{j,k}$) for $(j,k)=(6,1)$ if $l=7$;

• $\circ$

($c_{1}$), ($c_{3}$), ($c_{2}^{0}$), ($c_{2}^{j}$) for $j=3,5,6$ and ($c_{2}^{j,k}$) for $(j,k)=(4,1)$ if $l=6$.

Proposition 6.1**.**

For any positive integer $k$ the set

[TABLE]

forms a basis of the principal subspace $W_{L(k\Lambda_{0})}$ of the standard module $L(k\Lambda_{0})$ for the affine Lie algebra in type $E_{l}^{(1)}$.

Proof. The maximal root $\theta$ in type $E$ satisfies

[TABLE]

Therefore, as described in Section 5.4, by applying the procedure from Section 5.3 on an arbitrary linear combination of elements of $\mathfrak{B}_{L(k\Lambda_{0})}^{E_{8}}$, one can remove all quasi-particles of color $1$ from the corresponding quasi-particle monomials. The resulting linear combination can be identified as a linear combination of elements of $\mathfrak{B}_{L(k\Lambda_{0})}^{E_{7}}$; see Figure 2. Due to (6.12), by applying the same procedure once again, one can remove all quasi-particles of color $1$bbb Note that the quasi-particles of color $1$ in type $E_{7}$ correspond, with respect to the aforementioned identification, to the quasi-particles of color $7$ in type $E_{8}$; see Figure 2. from the corresponding quasi-particle monomials, thus obtaining the expression which can be identified as a linear combination of elements of the basis $\mathfrak{B}_{L(k\Lambda_{0})}$ from Theorem 3.1 for $\mathfrak{g}=D_{6}$; see Figure 2. As for type $E_{6}$, due to (6.12), by applying the procedure from Section 5.3 on an arbitrary linear combination of elements of $\mathfrak{B}_{L(k\Lambda_{0})}^{E_{6}}$, one can remove all quasi-particles of color $6$ from the corresponding quasi-particle monomials. The resulting expression can be identified as a linear combination of elements of the basis $\mathfrak{B}_{L(k\Lambda_{0})}$ from Theorem 3.1 for $\mathfrak{g}=A_{5}$; see Figure 2. Therefore, the proposition follows from Theorem 3.1 and the fact that the characters of the corresponding bases coincide which is verified by arguing as in Section 7. ∎

7. Character formulae and combinatorial identities

Let $\delta=\sum_{i=0}^{l}a_{i}\alpha_{i}$ be the imaginary root as in [23, Chap. 5], where the integers $a_{i}$ denote the labels in the Dynkin diagram [23, Table Aff] for $\widetilde{\mathfrak{g}}$. As before, let $V$ denote a standard module or a generalized Verma module. Define the character $\mathop{\mathrm{ch}}W_{V}$ of the corresponding principal subspace $W_{V}$ by

[TABLE]

where $q,y_{1},\ldots,y_{l}$ are formal variables and $(W_{V})_{-m\delta+n_{1}\alpha_{1}+\ldots+n_{l}\alpha_{l}}$ denote the weight subspaces of $W_{V}$ of weight $-m\delta+n_{1}\alpha_{1}+\cdots+n_{l}\alpha_{l}$ with respect to

[TABLE]

In order to simplify our notation, we set $\mu_{i}=\nu_{i}/\nu_{i^{\prime}}$ for $i=2,\ldots,l$; recall (3.3). Also, we write

[TABLE]

Theorem 3.1 implies the following character formulae:

Theorem 7.1**.**

(a) Set $n_{i}=\sum_{t=1}^{\nu_{i}k}r_{i}^{(t)}$ for $i=1,\ldots,l$. For any integer $k\geqslant 1$ we have

[TABLE]

(b) Set $n_{i}=\sum_{t=1}^{\nu_{i}u_{i}}r_{i}^{(t)}$ for $i=1,\ldots,l$. For any integer $k\geqslant 1$ we have

[TABLE]

Proof.

We give the proof of this theorem for the case $F_{4}^{(1)}$, since the proof for the cases $D_{l}^{(1)}$, $E_{6}^{(1)}$, $E_{7}^{(1)}$ and $E_{8}^{(1)}$ goes analogously. The proof for other types can be found in [20, 4, 5, 6]. In order to determine the character of $W_{L(k\Lambda_{0})}$, we write conditions on energies of quasi-particles of the set $B_{W_{L(k\Lambda_{0})}}$ in terms of $r_{i}^{(s)}$. For a fixed color-type $(n_{4},n_{3},n_{2},n_{1})$, charge-type

[TABLE]

and dual-charge-type

[TABLE]

the following equalities can be verified by a direct calculation:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By combining (7.1)–(7.4), difference conditions ($c_{1}$)–($c_{3}$) and the formula

[TABLE]

where $p_{r}(j)$ denotes the number of partitions of $j$ with at most $r$ parts, we get

[TABLE]

where $n_{i}=\sum_{t=1}^{k}r_{i}^{(t)}$ for $i=1,2$ and $n_{i}=\sum_{t=1}^{2k}r_{i}^{(t)}$ for $i=3,4$, as required. The character formula for the generalized Verma module is verified analogously. ∎

Theorem 3.2 implies the following character formulae in types $D_{l}^{(1)}$, $E_{6}^{(1)}$ and $E_{7}^{(1)}$ while the case $A_{l}^{(1)}$ is due to [20].

Theorem 7.2**.**

Set $n_{i}=r_{i}^{(1)}+\cdots+r_{i}^{(k)}$ for $i=1,\ldots,l$. For any rectangular weight $\Lambda=k_{0}\Lambda_{0}+k_{j}\Lambda_{j}$ of level $k=k_{0}+k_{j}$ we have

[TABLE]

Note that from (5.21) we have an isomorphism of $\widetilde{\mathfrak{n}}_{+}$-modules $W_{N(k\Lambda_{0})}$ and $U(\widetilde{\mathfrak{n}}_{+}^{<0})$, so we can obtain character formula of $W_{N(k\Lambda_{0})}$ by using Poincaré–Birkhoff–Witt basis of $U(\widetilde{\mathfrak{n}}_{+}^{<0})$ as well. For example, in the case $F_{4}^{(1)}$, we get

[TABLE]

where

[TABLE]

For any positive root $\alpha=a_{1}\alpha_{1}+\cdots a_{l}\alpha_{l}\in R_{+}$ we introduce the following notation

[TABLE]

so that for an arbitrary affine Lie algebra $\widetilde{\mathfrak{g}}$ character formula (7.5) generalizes to

[TABLE]

Theorem 7.1 and (7.6) imply the following generalization of Euler–Cauchy theorem; cf. [1].

Theorem 7.3**.**

For any untwisted affine Lie algebra $\widetilde{\mathfrak{g}}$ we have

[TABLE]

where $n_{i}=\sum_{t\geqslant 1}r_{i}^{(t)}$ for $i=1,\ldots,l$ and the sum on the right hand side of goes over all descending infinite sequences of nonnegative integers with finite support.

In particular, the theorem produces three new families of combinatorial identities which correspond to types $D$, $E$ and $F$.

Acknowledgement

The authors would like to thank Mirko Primc for useful discussions and support. The first author is partially supported by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004).