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Principal subspaces of higher-level standard sl(3)^-modules

We use the theory of vertex operator algebras and intertwining operators to obtain systems of q-difference equations satisfied by the graded dimensions of the principal subspaces of certain level k standard modules for \hat{\goth{sl}(3)}. As a consequence we establish new formulas for the graded dimensions of the principal subspaces corresponding to the highest-weights i\Lambda_1+(k-i)\Lambda_2, where 1 \leq i \leq k and \Lambda_1 and \Lambda_2 are fundamental weights of \hat{\goth{sl}(3)}.


Introduction
The theory of vertex operator algebras ( [B], [FLM]; cf. [LL]) leads to classical and new combinatorial identities (see e.g. [LW2]- [LW4], [Ca], [MP1] and [MP2]) and q-difference equations (recursions) satisfied by the graded dimensions (characters) of certain substructures of the standard representations of affine Lie algebras (see [CLM1]- [CLM2] and [C1]). This paper is a continuation of [C1], to which we refer the reader for background and notation (see also [C2]). In [C1] we have derived a complete set of recursions that characterize the graded dimensions of all the principal subspaces of the level 1 standard representations of sl(n) with n ≥ 3. Here we extend this approach to the principal subspaces of the higherlevel standard modules for sl(3). This work and the work done in [C1] can be viewed as a continuation of a program to obtain Rogers-Ramanujan-type recursions, which was initiated by Capparelli,Lepowsky and Milas in [CLM1]- [CLM2].
In the present paper we continue the study of a relationship between intertwining operators, in the sense of [FHL] and [DL], associated to standard modules and the corresponding principal subspaces of these modules. We consider the definition of the principal subspace W (Λ) from [FS1] and [FS2], namely, W (Λ) = U (n) · v Λ , where Λ and v Λ are the highest weight and a highest weight vector of the standard sl(3)-representation L(Λ) of level k > 1. Byn we mean n ⊗ C[t, t −1 ], where n is the subalgebra of sl(3) consisting of the strictly uppertriangular matrices. As is well known, the highest weights of the level k standard modules have the form Λ = k 0 Λ 0 + k 1 Λ 1 + k 2 Λ 2 , where Λ 0 , Λ 1 , Λ 2 are the fundamental weights of sl(3) and k 0 , k 1 , k 2 are nonnegative integers whose sum is k.
The main result of our paper is the following theorem, which gives two families of exact sequences that yield linear systems of q-difference equations: Theorem For any integer i with 1 ≤ i ≤ k there are natural sequences and these sequences are exact. (See Theorem 4.1 for details.) As a consequence of this theorem we derive a system of q-difference equations that characterize the graded dimensions of the principal subspaces W (iΛ 0 +(k −i)Λ j ), where 0 ≤ i ≤ k and j = 1, 2 (see Theorem 4.2 below). The graded dimensions of these subspaces were previously obtained by Georgiev using a different method. Combining our recursions with the formulas for the graded dimensions of W (iΛ 0 + (k − i)Λ j ) we obtain the graded dimensions of the principal subspaces W (iΛ 1 + (k − i)Λ 2 ) (see Corollary 4.1 in Section 4). These are new results in the process of obtaining graded dimensions of principal subspaces; the method used in [G1] did not give an answer for highest weights of type iΛ 1 + (k − i)Λ 2 with 1 ≤ i ≤ k − 1. Perhaps the main point of our strategy and results, though, is understanding the role of the intertwining operators involved in the construction of the exact sequences.
In this paper we formulate as a conjecture a presentation of the principal subspaces of all the higher-level standard modules for sl(3). It appears that in fact one can use an idea developed in [C1] to prove this result, but we will instead prove this conjecture in a different way, in a future publication that is part of ongoing joint work with Lepowsky and Milas.
The paper is organized as follows. Section 2 gives background and notation. In Section 3 we discuss the principal subspaces of the standard sl(3)-modules. In Section 4 we construct exact sequences that give q-difference equations and we obtain the graded dimensions of the principal subspaces W This paper is part of the author's Ph.D. dissertation written under the direction of James Lepowsky at Rutgers University.
Acknowledgement. I would like to thank James Lepowsky for his generous and inspiring guidance and for his encouragement during this work. I am thankful to Antun Milas for many useful conversations.

Preliminaries
The aim of this section is to recall the vertex operator construction of the higher-level standard modules for the untwisted affine Lie algebra sl(3) by using the corresponding contructions of the level 1 standard representations. We use the setting and notation from [C1].
In this paper we work with the finite-dimensional complex Lie algebra sl(3) that has a standard basis We fix the Cartan subalgebra h = Ch α 1 ⊕ Ch α 2 of sl(3). The standard symmetric invariant nondegenerate bilinear form x, y = tr(xy) defined for any x and y in sl(3) allows us to identify h with h * . Take α 1 and α 2 to be the (positive) simple roots corresponding to the vectors x α 1 and x α 2 . Under our identification we have h α 1 = α 1 and h α 2 = α 2 . The fundamental weights of sl(3) are linear functionals λ 1 and λ 2 in the dual space h * (= h). They are determined by the conditions λ i , α j = δ i,j for i, j = 1, 2, so that We denote by n the positive nilpotent subalgebra of sl (3), which can be viewed as the subalgebra consisting of the strictly upper-triangular matrices. Now we consider the untwisted affine Lie algebra associated to sl(3), where c is a non-zero central element and for any x, y ∈ sl(3) and m, n ∈ Z. By adjoining the degree operator [d, c] = 0) to the Lie algebra sl(3) one obtains the affine Kac-Moody algebra sl(3) = sl(3) ⊕ Cd (cf. [K]). We introduce the following subalgebras of sl (3): The latter is a Heisenberg subalgebra of sl(3) in the sense that its commutator subalgebra is equal to its center, which is one-dimensional. The usual extension of the form ·, · to h ⊕ Cc ⊕ Cd will be denoted by the same symbol ( c, c = 0, d, d = 0 and c, d = 1). We will identify h ⊕ Cc ⊕ Cd with its dual (h ⊕ Cc ⊕ Cd) * via this form. The simple roots of sl(3) are α 0 , α 1 , α 2 . We denote the fundamental weights of sl(3) by Λ 0 , Λ 1 , Λ 2 . Then We say that a sl(3)-module has level k ∈ C if c acts as multiplication by k. It is well known that any standard module L(Λ) with Λ ∈ (h ⊕ Cc ⊕ Cd) * has nonnegative integral level, given by Λ, c (cf. [K]). We denote by L(Λ 0 ), L(Λ 1 ), L(Λ 2 ) the standard sl(3)-modules of level 1 with v Λ 0 , v Λ 1 and v Λ 2 highest weight vectors.
We form the induced h-module such that h ⊗ C[t] acts trivially and c acts as identity on the one-dimensional module C. Let Q = Zα 1 ⊕ Zα 2 be the root lattice and P = Zλ 1 ⊕ Zλ 2 be the weight lattice of sl(3). Denote by C[Q] and C [P ] the group algebras of the lattices Q and P with bases {e α |α ∈ Q} and {e λ |λ ∈ P }. Consider the following vector spaces: and It is well known ( [FK], [S]; cf. [FLM]) that the vector spaces V P , V Q , V Q e λ 1 and V Q e λ 2 admit a natural sl(3)-module structure via certain vertex operators. Moreover, V Q , V Q e λ 1 and V Q e λ 2 are the level 1 standard representations of sl(3) with highest weights Λ 0 , Λ 1 and Λ 2 and highest weight vectors v Λ 0 = 1 ⊗ 1, v Λ 1 = 1 ⊗ e λ 1 and v Λ 2 = 1 ⊗ e λ 2 . We shall identify and we shall write (2.8) For any weight λ ∈ P , by e λ we mean a vector of V P or an operator on V P , depending on the context. The space V Q has a natural structure of vertex operator algebra and V Q e λ i are V Q -modules for i = 1, 2 ( [B] and [FLM]). See Section 2 in [C1] for further details and background about the vertex operator construction of the level 1 standard sl(3)-modules. The highest weights of the level k standard sl(3)-modules are given by where k 0 , k 1 , k 2 are nonnegative integers such that k 0 + k 1 + k 2 = k. Throughout this paper k > 1 stands for the level of a representation. We consider L(Λ) a standard sl(3)-module of level k with highest weight Λ as in (2.9) and a highest weight vector v Λ . Set where exactly k 0 indices are equal to 0, k 1 indices are equal to 1 and k 2 indices are equal to 2. Then of course v i 1 ,...,i k is a highest weight vector for sl(3), and (2.12) (cf. [K]). Thus there is an embedding of L(Λ) into V ⊗k P : uniquely determined by the identification v Λ = v i 1 ,...,i k . The action of sl(3) on V ⊗k P is given by the usual comultiplication a · v = ∆(a)v = (a ⊗ 1 ⊗ · · · ⊗ 1 + · · · + 1 ⊗ · · · ⊗ 1 ⊗ a)v (2.14) for a ∈ sl(3) and v ∈ V ⊗k P and this action extends in the usual way to U ( sl(3)). Throughout this paper we will write x(m) for the action of x ⊗ t m on any sl(3)-module, where x ∈ sl(3) and m ∈ Z. In particular, for any root α and integer m we set x α (m) for the action of x α ⊗ t m ∈ sl(3) on L(Λ). In this paper we will also use the notation x(m) for the Lie algebra element x ⊗ t m . It will be clear from the context if x(m) is an operator or an element of sl(3).
The standard sl(3)-modules have structures of vertex operator algebra and modules and this result is stated below: Theorem 2.1 ( [FZ]; cf. [DL], [Li1] and [LL]) The standard module L(kΛ 0 ) has a natural vertex operator algebra structure. The set of the level k standard sl(3)-modules provides a complete list of irreducible L(kΛ 0 )-modules (up to equivalence). [FZ], [DL], [Li1] and [LL]). We have the following products of operators:

The vertex operator map
for any λ ∈ P , α a simple root and m ∈ Z and for any λ ∈ P , h ∈ h and n = 0.

Principal subspaces
In [FS1]- [FS2] B. Feigin and A. Stoyanovsky introduced certain substructures, called principal subspaces, of the standard modules for sl(n) with n ≥ 2. Consider L(Λ) a level k standard module for sl(3). Then Λ = k 0 Λ 0 + k 1 Λ 1 + k 2 Λ 2 , where k 0 , k 1 , k 2 ∈ N whose sum is k (cf. (2.9)). The principal subspace of L(Λ), denoted by W (Λ), is defined as follows: where U (n) is the universal enveloping algebra ofn (recall (2.3)) and v Λ is a highest weight vector of L(Λ). Let us consider the natural surjective map: Denote by I Λ the annihilator of the highest weight vector v Λ in U (n), This is a left ideal of U (n). Hence the principal subspace W (Λ) is isomorphic (as linear spaces) with the quotient space U (n)/I Λ .
Recall from Section 2 the operators x α (m) for any root α and m ∈ Z. We consider the following formal infinite sums: for any t ∈ Z. It will be convenient to truncate R t as follows: where j = 1, 2 and m is a fixed (not necessarily negative) integer. We shall often view R as elements of the algebra U (n), rather than as endomorphisms of a sl(3)-module. It will be clear from the context when expressions such as (3.6) are understood as elements of U (n) or as operators. Denote by U (n) the completion of U (n) in the sense of [LW3] or [MP1]. Let us denote by J the two-sided ideal of U (n) generated by R t for t ∈ Z. In order to obtain exact sequences and q-difference equations in the next section, it is important to have a description of the ideals I k 0 Λ 0 +k 1 Λ 1 +k 2 Λ 2 . This problem is equivalent to finding a presentation of the principal subspaces W (k 0 Λ 0 + k 1 Λ 1 + k 2 Λ 2 ).
The presentation of the principal subspaces of the standard modules for sl(2) was proved in [FS1]- [FS2]. In [FS2] the authors announced a presentation of the principal subspaces W (kΛ 0 ) of the standard representations L(kΛ 0 ) of sl(n) for n ≥ 3. We have given a precise description of the ideals I Λ corresponding to the level 1 standard modules L(Λ) of sl(n) for n ≥ 3 in [C1].
As a consequence of this statement we obtain the discrepancy between the defining ideals of the principal subspaces.
Corollary 3.1 We have (3.10) and In order to prove the above conjecture we can follow the idea of the proof of a presentation of the principal subspaces of the level 1 standard modules for sl(n) for n ≥ 3 developed in [C1]. Since this proof of a presentation of the principal subspaces of higher-level representations is very technical we omit it. We will instead prove this conjecture in [CalLM4] by using a different and new approach.

Exact sequences and q-difference equations
In this section we prove our main results. Mainly, we construct exact sequences of maps between principal subspaces of certain standard sl(3)-modules. We derive systems of qdifference equations satisfied by the graded dimensions of the principal subspaces W (iΛ 0 + (k − i)Λ j ) for 0 ≤ i ≤ k and j = 1, 2. We also obtain new formulas for the graded dimensions of W (iΛ 1 + (k − i)Λ 2 ).
Take Λ = k 0 Λ 0 +k 1 Λ 1 +k 2 Λ 2 , where k 0 , k 1 , k 2 are nonnegative integers with k 0 +k 1 +k 2 = k and take L(Λ) a standard module of level k. It is known (cf. [FZ], [DL], [LL]) that L(Λ) is graded with respect to a standard action of the Virasoro algebra operator L(0) and that and where L(Λ) s+h Λ is the weight space of L(Λ) of weight s+h Λ . This is known as grading by weight. The space L(Λ) has also gradings by charge given by the eigenvalues of the operators λ 1 and λ 2 (thought as λ 1 (0) and λ 2 (0)). The gradings by charge are compatible with the weight grading. Thus L(Λ) decomposes as where L(Λ) r 1 + λ 1 ,Λ ,r 2 + λ 2 ,Λ ;s+h Λ is the subspace of L(Λ) consisting of the vectors of charges r 1 + λ 1 , Λ , r 2 + λ 2 , Λ and of weight s + h Λ . Now we restrict these gradings to the principal subspace W (Λ) ⊂ L(Λ) and thus we have We consider the graded dimension (i.e. the generating function of the dimensions of the homogeneous subspaces) of the space W (Λ): where x 1 , x 2 and q are formal variables. To avoid the factor x λ 1 ,Λ 1 x λ 2 ,Λ 2 q h Λ we use slightly modified graded dimensions as follows: (4.6) (4.7) We shall also use the notation W (Λ) ′ r 1 ,r 2 ;k = W (Λ) r 1 + λ 1 ,Λ ,r 2 + λ 2 ,Λ ;k+h Λ . (4.8) We now recall from [C1] the automorphisms e λ : V P −→ V P for any weight λ. Consider e λ (k) = e λ ⊗ · · · ⊗ e λ k times , a linear automorphism for any λ ∈ P . Thus it follows that for any positive root α, m 1 , . . . , m r ∈ Z and v ∈ V ⊗k P (cf. (2.16)). We also have if each m j = 0 and h j ∈ h (recall (2.17)). One can also consider maps of the following type: for any λ ∈ P and 0 ≤ k 0 ≤ k, where Id is the identity map. Let k 0 and k 1 be nonnegative integers whose sum is k. The next results show that there exist maps of type (4.9) between certain standard modules, and in particular between their corresponding principal subspaces.
Lemma 4.1 Let i be an integer such that 0 ≤ i ≤ k.

We have an injective linear map between principal subspaces
(4.14) 3. If i = k then (4.14) is a linear isomorphism Moreover, we obtain the following relation between the graded dimensions of the principal subspaces W (kΛ 0 ) and W (kΛ 1 ): which is equivalent with (4.17) Proof: 1. We view L(iΛ 0 + (k − i)Λ 2 ) and L((k − i)Λ 0 + iΛ 1 ) embedded in V ⊗k P (cf. (2.13)). We have where v i 1 ,...,i k = v Λ i 1 ⊗ · · · ⊗ v Λ i k with exactly i indices equal to 0 and k − i indices equal to 2. We may and do assume that the first i indices are 0 and the other k − i indices are 2. By using our identifications (2.8) we have and [C1]). Thus it follows that where a is a nonzero constant, so that Now by (4.10), (4.11), (4.12) and (4.19) we obtain the linear map (4.20) 2. Let us restrict the map (4.20) to the principal subspace W (iΛ 0 + (k − i)Λ 2 ). Using similar arguments with U (n) instead of U ( sl (3)), we obtain a linear map which is clearly injective.
3. Let us take i = k in (4.14). By (4.18) we have (4.10)) we obtain that the linear map is surjective and thus it is a linear isomorphism. The isomorphism (4.22) does not preserve weight and charge. Let W (kΛ 0 ) r 1 ,r 2 ;s with r 1 , r 2 , s ∈ N be an homogeneous subspace of W (kΛ 0 ). The map (4.22) increases the charge corresponding to λ j by kΛ 1 , λ j for j = 1, 2. For any w ∈ W (kΛ 0 ) r 1 ,r 2 ;s the homogeneous element e λ 1 (k) (w) has weight s + r 1 + h kΛ 1 . Hence we obtain an isomorphism between homogeneous spaces e λ 1 (k) : W (kΛ 0 ) ′ r 1 ,r 2 ;s −→ W (kΛ 1 ) ′ r 1 ,r 2 ;s+r 1 , which gives the relation between the graded dimensions We have a result completely analogous to the previous lemma: Lemma 4.2 Let i be an integer such that 0 ≤ i ≤ k.
1. The restriction of e λ 2 (k) to L(iΛ 0 + (k − i)Λ 1 ) lies in L((k − i)Λ 0 + iΛ 2 ). 2. At the level of the principal subspaces we have the following injection: (4.23) 3. If i = k then (4.23) is a linear isomorphism In particular, we obtain which is equivalent with Our main goal is to obtain exact sequences of maps between principal subspaces. We have already seen in Lemmas 4.1 and 4.2 that the maps e λ 1 (k) and e λ 2 (k) are examples in this direction. In order to construct more general exact sequences and thereby to obtain qdifference equations we introduce maps of type e λ (k) with λ = λ 1 and λ = λ 2 between principal subspaces.
Intertwining operators among standard sl(3)-modules and fusion rules (the dimensions of the vector spaces of intertwining operators of a certain type) are important tools in our work. In this paper we follow [DL] for the construction of distinguished intertwining operators and for some of their relevant properties.
Let i be an integer with 1 ≤ i ≤ k. By using Chapter 13 of [DL] there exists a nonzero intertwining operator . (4.35) In fact, the dimension of the vector space of the intertwining operators of type (4.35) (i.e. fusion rule) is one (cf. [DL], [FZ], [Li3]; see also [BMW], [F] and [FW]). In this work we use only intertwining operators whose fusion rules equal one. We use the notation (cf. [FHL], [FZ] and [Li3]). As a consequence of the Jacobi identity we obtain (4.36) so that the coefficients of Y(v (k−1)Λ 0 +Λ 1 , x) commute with the action of U (n) (cf. [DL]; see also [G1], [CLM1]- [CLM2] and [C1]). Now we take the constant term of x −h 3 +h 1 +h 2 Y(v (k−1)Λ 0 +Λ 1 , x) and we denote it by with the following property: where b is a nonzero scalar. By (4.36) we immediately have Similarly, there exists a nonzero intertwining operator of type and there also exists a linear map associated to the intertwining operator Y(v (k−1)Λ 0 +Λ 2 , x) of type (4.40), satisfying similar properties to those of the map (4.37). The standard modules L(kΛ 0 ), L(kΛ 1 ) and L(kΛ 2 ) are "group-like" elements in the fusion ring at level k. Such modules are also called simple currents (cf. [Li2]).
We now prove our main theorem that gives two families of i exact sequences of maps between principal subspaces for 1 ≤ i ≤ k.
Remark 4.1 The chain property of the sequence (4.42) follows from (4.47), (4.50), (4.51) and (4.52). The chain property follows also by a different argument, as we have just seen in the proof of the previous theorem. We do not have an elementary proof that does not use the description of the ideals I iΛ 0 +(k−i)Λ 1 and I (i−1)Λ 0 +(k−i+1)Λ 1 for the exactness of the sequence (4.42). Similar assertions are of course true for the sequence (4.43).
Recall from Lemmas 4.1 and 4.2 the equations and (4.56) Given the exact sequences from Theorem 4.1 we now derive a system of q-difference equations that characterize the graded dimensions of W (iΛ 0 + (k − i)Λ j ) for 1 ≤ i ≤ k and j = 1, 2.
Theorem 4.2 For any i such that 1 ≤ i < k we have the following q-difference equations: Proof: We first prove that the following q-difference equations hold: and The q-equation (4.60) follows by using the exact sequence (4.42). Formula (4.61) is obtained similarly from the exactness of the sequence (4.43).
Let us consider the homogeneous subspace W (iΛ 1 + (k − i)Λ 2 ) ′ r 1 ,r 2 ;s (recall our notation (4.8)). From (4.30) and (4.31) we notice that the map e λ (k) increases the charge corresponding to λ 1 by i and preserves the charge corresponding to λ 2 . Also, for any w ∈ W (iΛ 1 + (k − i)Λ 2 ) ′ r 1 ,r 2 ;s the vector e λ (k) (w) has weight s + r 1 − r 2 + i. Hence we have By (4.38) and (4.39) we obtain The exacteness of the sequence (4.42) implies the equation (4.60) for any i with 1 ≤ i ≤ k. Easy computations show that (4.60) and (4.61) yield the following equations: for all i with 1 ≤ i ≤ k. Now assume that 1 ≤ i < k. We take i instead of k − i in (4.66) and equate the expression of χ ′ W (iΛ 1 +(k−i)Λ 2 ) (x 1 , x 2 ; q) from the resulting formula with the expression of χ ′ W (iΛ 1 +(k−i)Λ 2 ) (x 1 , x 2 ; q) from (4.65) and thus we obtain By multiplying all terms of this expression by x i 1 and substituing x 1 with x 1 q and x 2 with x 2 q −1 we obtain the four-term q-difference equation (4.57).
In a similar way, we obtain (4.58) by taking k − i instead of i in (4.65), equating the expressions of χ ′ W ((k−i)Λ 1 +iΛ 2 ) (x 1 , x 2 ; q) and doing the corresponding substitutions for x 1 and x 2 . Note that (4.58) can be derived from (4.57) by using an isomorphism induced from the Dynkin diagram automorphism of sl(3) that interchanges α 1 and α 2 , and thus it interchanges Λ 1 and Λ 2 .
The particular case k = i = 1 of Theorem 4.2 is listed below.
Remark 4.2 By taking k = i = 1 in (4.60) and (4.61) and by using (4.55) and (4.56) for k = 1 we recover the recursions satisfied by the graded dimension of the principal subspace W (Λ 0 ) of the level 1 vacuum representation of sl(3) obtained in [C1] (see also [C2]): and ). These recursions are equivalent with Graded dimensions of the principal subspaces W (iΛ 0 + (k − i)Λ j ) with 0 ≤ i ≤ k and j = 1, 2 were computed in [G1] by constructing quasiparticle bases of these principal subspace: where j t = 0 for 0 ≤ t ≤ i and j t = j for i < t ≤ k, j = 1, 2, and where we use the notation (q) m = (1 − q) · · · (1 − q m ) for any nonnegative integer m. Following an argument similar to the one developed in Section 7.3 in [A] one can show that (4.69) satisfies the system of equations (4.55), (4.56), (4.57), (4.58) and (4.59).
The q-difference equations (4.65) and (4.66) combined with formulas (4.69) give the graded dimensions of the principal subspaces corresponding to highest weights of the form iΛ 1 + (k − i)Λ 2 for any i with 1 ≤ i ≤ k.
It is interesting to consider the problem of computing the graded dimension of W (k 0 Λ 0 + k 1 Λ 1 + k 2 Λ 2 ), where k 0 , k 1 , k 2 are any positive integers whose sum is k, as a combination of the graded dimensions already obtained. One attempt is given by the next theorem, whose proof is completely analogous to that of Theorem 4.1.
Theorem 4.3 There are natural exact sequences However, the maps Id (k 0 ) ⊗ e λ (k−k 0 ) and Id (k 0 ) ⊗ e β (k−k 0 ) , restricted to the corresponding homogeneous spaces, do not shift the weight of a homogeneous element by a scalar, so that our method used before in Theorem 4.2 to obtain q-difference equations does not extend to this case.