Explicit fermionic-type character formulas for Feigin–Stoyanovsky type subspaces of standard $\mathfrak{sl}(\ell+1,\mathbb{C})^{\sim}$-modules in rank $\ell\geq 3$

For the affine Lie algebra $\widetilde{\mathfrak{sl}(\ell+1,\mathbb{C})}$ of type $A_\ell^{(1)}$, the Feigin–Stoyanovsky type subspace $W(\Lambda)=U(\widetilde{\mathfrak{g}}_1)v_\Lambda$ of a standard module $L(\Lambda)$ admits combinatorial quasi-particle bases for every rank and level, and exact sequences among these subspaces yield a system of recurrence relations for their multigraded characters that is valid for all $\ell$. Solving this system, Jerković (2012) obtained explicit fermionic-type character formulas for all highest weights and all levels only in rank $\ell=2$ ($\mathfrak{sl}(3)$). Level-one formulas are known for every $\ell$ (Trupčević), and bosonic-type (Weyl-group-sum) formulas are known for every $\ell$, level and highest weight (Feigin–Jimbo–Loktev–Miwa–Mukhin, with an alternative proof by Makhlin via Brion's theorem). The open problem is to derive explicit fermionic-type character formulas for $W(\Lambda)$ in ranks $\ell\geq 3$ — equivalently, to solve the general-rank recurrence system in quasi-particle form, where the substantive open content lies at levels $k\geq 2$.

Background

Let $\mathfrak{sl}(\ell+1,\mathbb{C})$ be the Lie algebra of traceless $(\ell+1)\times(\ell+1)$ matrices and let $\widetilde{\mathfrak{sl}(\ell+1,\mathbb{C})}=\mathfrak{sl}(\ell+1,\mathbb{C})\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\oplus\mathbb{C}d$ be its affine Lie algebra (type $A_\ell^{(1)}$), the central extension of the Lie algebra of polynomial loops with values in $\mathfrak{sl}(\ell+1,\mathbb{C})$; $c$ is a central element and $d$ the degree operator. Its irreducible integrable highest-weight modules $L(\Lambda)$ with dominant integral highest weight $\Lambda=k_0\Lambda_0+\cdots+k_\ell\Lambda_\ell$, $k_i\in\mathbb{Z}_{\geq 0}$, are called standard modules, and the eigenvalue $k=\Lambda(c)=k_0+\cdots+k_\ell$ of $c$ is the level of the module. Writing $x(n)=x\otimes t^n$ for the $n$-th loop mode of $x$, the Feigin–Stoyanovsky (FS) type subspace of $L(\Lambda)$ is

$$W(\Lambda)=U(\widetilde{\mathfrak{g}}_1)\cdot v_\Lambda,$$

where $v_\Lambda$ is a highest-weight vector, $\widetilde{\mathfrak{g}}_1=\mathfrak{g}_1\otimes\mathbb{C}[t,t^{-1}]$, and $\mathfrak{g}_1$ is the span of the root vectors of the positive roots $\gamma_i=\alpha_i+\cdots+\alpha_\ell$ for $i=1,\dots,\ell$ — exactly the positive roots whose pairing with the fundamental weight $\omega_\ell$ equals $1$ ($\omega_\ell$ is minuscule, meaning its pairing with every root lies in $\{-1,0,1\}$). Since $\mathfrak{g}_1$ is commutative, $\widetilde{\mathfrak{g}}_1$ is an infinite-dimensional commutative algebra, and $W(\Lambda)$ is spanned by monomial vectors $x_{\gamma_{i_1}}(-n_1)\cdots x_{\gamma_{i_s}}(-n_s)v_\Lambda$ with $n_j\geq 1$; the roots $\gamma_i$ are called colors and the monomials quasi-particles. These subspaces were introduced by Feigin and Stoyanovsky in their functional-model study of current-algebra representations; the closely related principal subspaces $U(\mathfrak{n}\otimes\mathbb{C}[t,t^{-1}])v_\Lambda$, generated by the modes of the full nilpotent subalgebra $\mathfrak{n}$ of $\mathfrak{sl}(\ell+1,\mathbb{C})$ (all positive-root vectors), were also introduced by Feigin and Stoyanovsky and studied combinatorially by Georgiev.

The formal character

$$\chi(W(\Lambda))(z_1,\dots,z_\ell;q)=\sum \dim W^{m,n_1,\dots,n_\ell}\, q^m z_1^{n_1}\cdots z_\ell^{n_\ell}$$

records the degree $m$ (sum of mode numbers) and the color content $n_i$ (number of quasi-particles of color $\gamma_i$) of the basis vectors. A character formula is called fermionic, or of quasi-particle type, when it is written as a restricted sum over occupation numbers with $q$-factorials $(q)_n=\prod_{j=1}^{n}(1-q^j)$ in the denominators — the shape dictated by quasi-particle statistics, analogous to sums over Bethe-type occupation numbers in statistical mechanics — and bosonic when it is written as a product-type or Weyl-group-sum expression, analogous to the Weyl–Kac character formula. For instance, Jerković's rank-two theorem states that for every level $k$ and every $\Lambda=k_0\Lambda_0+k_1\Lambda_1+k_2\Lambda_2$,

$$\chi(W(\Lambda))(z_1,z_2;q)=\sum_{n_1,n_2\geq 0}\ \sum\frac{q^{\sum_i N_{1,i}^2+N_{2,i}^2+N_{1,i}N_{2,i}}\,L_{k_0,k_1,k_2}(q)}{\prod_{i=1}^{k}(q)_{N_{1,i}-N_{1,i+1}}(q)_{N_{2,i}-N_{2,i-1}}}\,z_1^{n_1}z_2^{n_2},$$

where the inner sum runs over $N_{1,1}\geq\cdots\geq N_{1,k}\geq 0$ with $\sum_i N_{1,i}=n_1$ and $N_{2,k}\geq\cdots\geq N_{2,1}\geq 0$ with $\sum_i N_{2,i}=n_2$ (with $N_{1,k+1}=N_{2,0}=0$), and $L_{k_0,k_1,k_2}(q)$ is an explicit 'linear term' depending on the highest weight: a finite sum of monomials $q^{\text{linear combination of the } N_{i,j}}$ multiplied by explicit factors of the form $(1-q^{m})$. Formulas of this type are the fermionic side of the Rogers–Ramanujan-type ($q$-series) identities that arise from affine Lie algebra representation theory.

The state of knowledge divides cleanly. Fermionic-type formulas are known for $\ell=1$ at all levels (the original quasi-particle formulas of Feigin and Stoyanovsky), for $\ell=2$ at all levels and all highest weights (the principally specialized formulas for $(k,3)$-admissible configurations of Feigin–Jimbo–Miwa–Mukhin–Takeyama — principally specialized meaning that the color variables are specialized so that a single variable $q$ records the total degree — together with a bijection to rigged partitions, combinatorial data consisting of an integer sequence equipped with nonnegative labels, and Jerković's multigraded solution of the recurrence system), and at level $k=1$ for every $\ell$ (Trupčević). Combinatorial quasi-particle bases of $W(\Lambda)$ are known for all ranks and all levels (Trupčević), and exact sequences among the subspaces $W(\Lambda)$ at fixed level — obtained from vertex-operator-algebra structures and intertwining operators, which realize the affine fusion product — yield a system of recurrence relations for the characters valid for every $\ell$ (Jerković). On the bosonic side, Feigin–Jimbo–Loktev–Miwa–Mukhin proved a monomial basis and a bosonic-type character formula for the FS subspace of $L(\lambda)$ of $\widehat{\mathfrak{sl}_n}$ for every $n$, level and highest weight, with an alternative proof by Makhlin via Brion's theorem on lattice points of convex polytopes. What is missing is the fermionic-type formula in ranks $\ell\geq 3$ — a solution of the general-rank recurrence system.

Problem Statement

Derive explicit fermionic-type (quasi-particle) character formulas for the Feigin–Stoyanovsky type subspaces $W(\Lambda)=U(\widetilde{\mathfrak{g}}_1)v_\Lambda$ of the standard modules $L(\Lambda)$ of the affine Lie algebra $\widetilde{\mathfrak{sl}(\ell+1,\mathbb{C})}$ (type $A_\ell^{(1)}$) for all ranks $\ell\geq 3$, all levels $k\geq 1$ and all dominant integral highest weights $\Lambda=k_0\Lambda_0+\cdots+k_\ell\Lambda_\ell$; that is, solve the system of recurrence relations for the multigraded characters $\chi(W(\Lambda))(z_1,\dots,z_\ell;q)$ established by Jerković — equivalently, evaluate the generating function of the admissible quasi-particle monomial bases of Trupčević — in closed fermionic form, in the same sense in which Jerković's theorem solves the $\ell=2$ system: a sum over occupation numbers with explicit difference-type summation conditions, explicit quadratic and highest-weight-dependent linear $q$-exponents, and $q$-factorial denominators. The problem is stated in its full source generality; note that the level-one case is already settled for every $\ell$ (Trupčević), so the substantive open content lies at levels $k\geq 2$.

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