The ground state of the dense O(1) loop model on a semi-infinite cylinder is a vector with components ψπ\psi_\pi indexed by link patterns (noncrossing matchings) π\pi; the Razumov–Stroganov theorem now identifies each ψπ\psi_\pi with the number AπA_\pi of fully packed loop (FPL) configurations with connectivity π\pi. Fonseca and Zinn-Justin (2009) computed two infinite families of partial sums of these components and related them to the enumeration of non-intersecting lattice paths (NILPs) and totally symmetric self-complementary plane partitions (TSSCPPs), but left the constructive description incomplete. The open problem comprises three linked goals: (1) a simple closed product formula for the NILP enumerant Fp,r=ψ(()r)p+1\mathcal{F}_{p,r}=\psi_{(()^r)_{p+1}} at loop weight τ=1\tau=1, analogous to Krattenthaler's product for the companion enumerant Gp,r\mathcal{G}_{p,r}; (2) a general closed-form expression, or effective direct computation method, for the individual components ψπ=Aπ\psi_\pi=A_\pi for arbitrary link patterns π\pi, including the inverse basis-change coefficients relating them to the integral aa-basis of the qKZ solution space; (3) an explicit natural bijection between TSSCPPs and alternating sign matrices (ASMs). The Razumov–Stroganov conjecture was settled by Cantini and Sportiello (2011); the three goals above remain open.

Background

An alternating sign matrix (ASM) is a square matrix with entries in {0,1,−1}\{0,1,-1\} whose rows and columns each sum to 1 and whose nonzero entries alternate in sign along every row and column. A fully packed loop (FPL) configuration is a covering of the edges of an n×nn\times n square grid by loop segments in which every internal vertex has exactly two occupied edges, with boundary edges alternately occupied and empty (the domain-wall boundary conditions of the six-vertex model); the loops then pair up the 2n2n occupied boundary points, and the induced noncrossing perfect matching π\pi of these 2n2n points, read around the boundary, is a link pattern, with AπA_\pi denoting the number of FPL configurations realizing π\pi. Since each FPL configuration determines an ASM (and vice versa), the total ∑πAπ\sum_\pi A_\pi equals the number of n×nn\times n ASMs, famously ∏j=0n−1(3j+1)!/(n+j)!\prod_{j=0}^{n-1}(3j+1)!/(n+j)!.

The O(1) loop model (also called the completely packed loop model) is the τ=1\tau=1 case of the O(τ)O(\tau) loop model on a square lattice wrapped on a semi-infinite cylinder of perimeter 2n2n, whose states are the link patterns. Its transfer matrix is generated by the Temperley–Lieb generators eie_i (each eie_i acts on sites i,i+1i,i+1 by drawing a little arch connecting them, giving each closed loop a weight τ\tau), and the Hamiltonian Hn=−∑i=12neiH_n=-\sum_{i=1}^{2n} e_i, obtained from the transfer matrix of a single row of plaquettes, has a one-dimensional ground state with eigenvalue −2n-2n whose components ψπ\psi_\pi, indexed by link patterns and normalized so that the minimal component — that of the fully nested 'rainbow' pattern ()n()_n, for which there's provably a unique FPL configuration — equals 1, are what the model's combinatorics is about. Razumov and Stroganov conjectured in 2004 that ψπ=Aπ\psi_\pi=A_\pi exactly, in the homogeneous normalization. Cantini and Sportiello proved this in 2011 by a purely combinatorial argument based on the gyration operation (an alternating rotation of the FPL loops that cyclically shifts link patterns), after partial results: the multi-parameter sum rule of Di Francesco and Zinn-Justin (2005) and the conjecture for infinite families of link patterns (Zinn-Justin, 2006). The theorem equates the numbers but supplies neither closed formulae for them nor a bijective correspondence.

A second, algebraic face of the same objects comes from the quantum Knizhnik–Zamolodchikov (qKZ) equation at level 1, a system of difference equations in spectral parameters z1,…,z2nz_1,\dots,z_{2n} acting on the loop-model states: the homogeneous components ψπ\psi_\pi are the z1=⋯=z2n=1z_1=\cdots=z_{2n}=1 limits of polynomial solutions Ψπ(z1,…,z2n)\Psi_\pi(z_1,\dots,z_{2n}) which satisfy the wheel-vanishing condition (vanishing whenever zi=qzj=q2zkz_i=q z_j=q^2 z_k with q=e2πi/3q=e^{2\pi i/3}, a 'wheel'), forming a space of dimension equal to the Catalan number (the number of link patterns of 2n2n points); Fonseca and Zinn-Justin (2009) exploited an 'a-basis' Φa\Phi_a of integral (contour-integral) representations of this space.

A totally symmetric self-complementary plane partition (TSSCPP) is a plane partition (an array of weakly decreasing nonnegative integers, viewed as a stack of cubes) inside a 2n×2n×2n2n\times 2n\times 2n box that is invariant under every permutation of the three coordinate axes (total symmetry) and coincides with the complement of its image under flipping the box end-to-end (self-complementarity). Mills, Robbins and Rumsey conjectured, and Andrews proved in 1994 by non-bijective determinant evaluations, that TSSCPPs are equinumerous with n×nn\times n ASMs; finding an explicit natural bijection has been open ever since. TSSCPPs are in bijection with families of non-intersecting lattice paths (NILPs), which the Lindström–Gessel–Viennot (LGV) lemma enumerates by determinants.

Fonseca and Zinn-Justin (2009) studied the components ψ(π)p\psi_{(\pi)_p} (a matching π\pi surrounded by pp nested arches) and ψ(pα\psi_{(_p\alpha} (pp initial openings), proving sum rules such as ∑π of size 2rτoπψ(π)p=ψ(()r)p+1\sum_{\pi \text{ of size } 2r} \tau^{o_\pi}\psi_{(\pi)_p}=\psi_{(()^r)_{p+1}}, where ()r()^r denotes rr juxtaposed small arches and oπo_\pi counts arches opening at even sites. They identified the NILP enumerant Fp,r=ψ(()r)p+1\mathcal{F}_{p,r}=\psi_{(()^r)_{p+1}} as counting TSSCPPs with a fixed central hexagon of size 2p2p ('punctured' TSSCPPs) and gave an explicit contour-integral formula for it; for the companion enumerant Gp,r=∑αψ(pα\mathcal{G}_{p,r}=\sum_\alpha\psi_{(_p\alpha} they quoted Krattenthaler's closed product of factorials at τ=1\tau=1 (with separate even/odd rr cases). They observed that, by contrast, there does not seem to be a simple closed formula for Fp,r\mathcal{F}_{p,r} even at τ=1\tau=1. They also proved a conjecture on consecutive-point disconnection probabilities and the large-pp limit of one of Zuber's conjectures, and Zuber's numerical data for small pp exhibit non-factorized expressions.

Problem Statement

The common objective is an exact, explicit and constructive description of the O(1) loop-model ground state and of the TSSCPP/FPL/ASM correspondence — completing, constructively, the equality of numbers established by the Razumov–Stroganov theorem. Within this objective, the source paper by Fonseca and Zinn-Justin leaves three concrete goals.

(G1) Determine a simple closed product formula — of the same hypergeometric/factorial type as Krattenthaler's product for Gp,r\mathcal{G}_{p,r} — for the enumerant Fp,r=ψ(()r)p+1=A(()r)p+1\mathcal{F}_{p,r}=\psi_{(()^r)_{p+1}}=A_{(()^r)_{p+1}} at loop weight τ=1\tau=1, valid for all nonnegative integers p,rp,r.

(G2) Find a general closed-form expression — or, alternatively, an effective direct computation method that avoids exhaustive enumeration and transfer-matrix diagonalization from scratch — for the individual ground-state components ψπ=Aπ\psi_\pi=A_\pi for an arbitrary link pattern π\pi of arbitrary size; a method counts as effective here if it computes AπA_\pi for a pattern of size nn in time polynomial in nn, so that an explicit closed-form expression and a polynomial-time algorithm are treated as resolving (G2) on equal footing. In the qKZ language of the source, this includes a direct way to compute the inverse basis-change coefficients C~π,a\tilde C_{\pi,a} relating the components to the integral aa-basis Φa\Phi_a for arbitrary π\pi and aa, going beyond the triangularity, recursion, and special-family Ansätze available so far.

(G3) Construct an explicit natural bijection between the totally symmetric self-complementary plane partitions in a 2n×2n×2n2n\times 2n\times 2n box and the n×nn\times n alternating sign matrices (equivalently, FPL configurations) for every nn — the long-standing open bijection problem originating with Mills, Robbins and Rumsey and with Andrews's 1994 non-bijective proof of equinumerosity — ideally one respecting natural statistics on both sides.

These deliverables are linked but independent: (G1), (G2) and (G3) are separate open problems, and resolving any one of them is sufficient to complete the task. In particular, (G1) is the two-parameter subfamily of (G2) singled out by the source paper, so that a general solution to (G2) also supplies the special values Fp,r=A(()r)p+1\mathcal{F}_{p,r}=A_{(()^r)_{p+1}} but discharges (G1) only if those values emerge as a simple product formula of the type demanded in (G1); (G3) is the bijective face of the same enumeration, and a bijection explaining the ground-state components combinatorially is the sense in which the source results were described as 'a small step towards' it.

References

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