Fermionic sum sides for $\mathcal{W}_r$ minimal-model characters and $r$-cylindric partition generating functions at ranks $r\geq 4$

The generating functions of $r$-cylindric partitions with infinitely long rows are, up to a factor $(q;q)_\infty$, the normalized characters of the $\mathcal{W}_r$ minimal models $\mathcal{M}_r^{r,r+d}$, and their product sides are known for every rank through Borodin's hook-length formula. Their fermionic sum sides — nested $q$-series sums with manifestly non-negative coefficients, i.e. the sum sides of the higher-rank Andrews–Gordon–Bressoud (Rogers–Ramanujan-type) identities — are known only for $r=2$ (Andrews–Gordon–Bressoud) and $r=3$ (Andrews–Schilling–Warnaar; Feigin–Foda–Welsh). Foda and Welsh stated in 2016 that "the $\mathcal{W}_r$ sum sides, for $r=4,5,\cdots$, are yet to be found", and the problem remains open: all subsequent progress on cylindric sum sides concerns rank 2, rank 3 (still partly conjectural), or the infinite-level limit, while Warnaar (2023) explicitly restates the arbitrary-rank $A_{r-1}^{(1)}$ Andrews–Gordon identities as an important open problem. The problem asks for explicit fermionic multi-sum expressions for the $r$-cylindric partition generating functions / $\mathcal{W}_r$ minimal-model characters at every rank $r\ge4$, together with proofs of the resulting sum–product identities against Borodin's product formula.

Background

The Rogers–Ramanujan identities and their Andrews–Gordon and Bressoud generalizations equate a product side with a sum side that is a nested multiple sum with manifestly non-negative coefficients. Such identities have two readings. In representation theory, the sum sides are fermionic character formulae: characters of Virasoro minimal models (two-dimensional conformal field theories with finitely many irreducible representations, labelled by coprime integers $p<p'$) written as quasi-particle sums, as opposed to the alternating-sign ("bosonic") series produced by free-boson oscillator sums. In statistical mechanics, both sides arise as generating functions of one-dimensional lattice paths that describe local order parameters of exactly solved solid-on-solid lattice models (two-dimensional models whose local degrees of freedom are integer heights constrained by interaction rules around each face) computed by corner transfer matrices (a technique reducing one-point functions of two-dimensional models to one-dimensional path sums); the Andrews–Baxter–Forrester eight-vertex solid-on-solid model is the prototype, and fermionic forms underlie the thermodynamic Bethe ansatz (a finite-temperature method that derives integral equations for a model from its exact scattering data).

The $\mathcal{W}_r$ algebras are chiral (holomorphic-sector) symmetry algebras with $r-1$ generating currents of spins $2,3,\ldots,r$ — the Virasoro algebra is the case $r=2$ — and their minimal models $\mathcal{M}_r^{p,p'}$ are the higher-rank counterparts of the Virasoro minimal models. Foda and Welsh (2016) connected the characters of $\mathcal{M}_r^{r,r+d}$ to cylindric partitions: plane partitions (finite arrays of non-negative integers, weakly decreasing along rows and down columns) with $r$ rows that additionally obey the cylindric condition $\pi_{r,j}\ge\pi_{1,j+d}$, so that the array can be wrapped on a cylinder of circumference $r+d$; the relative shifts of the rows define a profile. The generating function $Z^{(r)}_\Lambda(q)=\sum_\pi q^{|\pi|}$, summed over cylindric partitions with infinitely long rows and profile encoded by a level-$d$ dominant integral weight $\Lambda=\sum_i m_i\Lambda_i$ (non-negative integers $m_i$ with $\sum_i m_i=d$) of the affine Kac–Moody algebra $\widehat{\mathfrak{sl}}_r=A^{(1)}_{r-1}$ ("level" being the weight's value on the canonical central element), satisfies

$$Z^{(r)}_\Lambda(q)=\frac{\mathrm{Pr}\,\chi^{\widehat{\mathfrak{sl}}_r}_\Lambda}{(q^r;q^r)_\infty}=\frac{\chi^{\mathcal{W}_r}_\Lambda}{(q;q)_\infty},$$

where $(a;q)_\infty=\prod_{n\ge0}(1-aq^n)$ is the $q$-Pochhammer symbol, $\mathrm{Pr}\,\chi$ is the principally specialized character (all specialization variables set equal to $q$), and $\chi^{\mathcal{W}_r}_\Lambda$ is the normalized character of the minimal model $\mathcal{M}_r^{r,r+d}$, defined when $\gcd(r,d)=1$. The product side is known for all $r$ by Borodin's formula

$$Z^{(r)}_\Lambda(q)=\frac{1}{(q^{r+d};q^{r+d})_\infty}\prod_{\substack{1\le i<j\le r+d\\ \omega_i>\omega_j}}\frac{1}{(q^{j-i};q^{r+d})_\infty}\prod_{\substack{1\le i<j\le r+d\\ \omega_i<\omega_j}}\frac{1}{(q^{r+d-j+i};q^{r+d})_\infty},$$

a hook-length-type product determined by the binary word $\omega=\omega(\Lambda)$ with $r$ zeros and $d$ ones that encodes the profile.

For $r=2$, Foda and Welsh re-derived the Andrews–Gordon and Bressoud identities by mapping 2-cylindric partitions through two-string abaci (bead-and-runner encodings of partitions) to decorated versions of Bressoud's restricted lattice paths, and transforming between paths obeying different restrictions so as to obtain sum expressions with manifestly non-negative coefficients; equating these with Borodin's product and cancelling the common factor $(q;q)_\infty^{-1}$ yields the identities. For $r=3$, fermionic sum sides are known from the $A_2$ Bailey lemma of Andrews, Schilling and Warnaar — whose Rogers–Ramanujan-type identities express principally specialized characters of level-$d$ standard $A^{(1)}_2$-modules as multiple sums — and from the construction of Feigin, Foda and Welsh out of combinations of Virasoro characters. For $r\ge4$ nothing comparable exists, and this is precisely the gap the source states.

Problem Statement

Find explicit fermionic expressions — that is, $q$-series of the form $\sum_{n_1,\dots,n_N\ge0} q^{Q(n_1,\dots,n_N)}/\prod_{i=1}^{N}(q;q)_{n_i}$, where $(q;q)_n=\prod_{j=1}^{n}(1-q^j)$ and $Q$ is a quadratic form in the summation variables $n_1,\dots,n_N$, so that every coefficient is manifestly non-negative — for the generating functions $Z^{(r)}_\Lambda(q)$ of $r$-cylindric partitions with infinitely long rows, at every rank $r\ge4$, where $\Lambda$ ranges over the dominant integral weights of $A^{(1)}_{r-1}$ of arbitrary level $d\ge1$ (equivalently, for the normalized characters $\chi^{\mathcal{W}_r}_\Lambda=(q;q)_\infty Z^{(r)}_\Lambda(q)$ of the minimal models $\mathcal{M}_r^{r,r+d}$, which are defined when $\gcd(r,d)=1$), and prove the resulting sum–product identities

$$\text{fermionic sum side} = \text{Borodin's product side}$$

for every dominant integral weight $\Lambda$ of level $d\ge1$ (when $\gcd(r,d)>1$ the identity is required in the cylindric-partition reading only, since the minimal-model character is then undefined). These are the rank-$r\ge4$ analogues of the Andrews–Gordon–Bressoud identities (the $r=2$ case) and of the Andrews–Schilling–Warnaar and Feigin–Foda–Welsh identities (the $r=3$ case); a sum side at rank $r$ is expected to involve at least $r-1$ nested summation variables carrying the $A_{r-1}$ structure, and a rewriting of the product side of Borodin's formula, or of the series defining $Z^{(r)}_\Lambda$, is not such a fermionic expression. The method is unrestricted: the source's route through $r$-string abaci and Bressoud-type transforms is one proposed template, not a requirement. A general-rank solution would in particular subsume the currently conjectural manifestly positive formulas for cylindric partitions of rank 3 and level $d$ not divisible by 3.

References

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