Interpretation and independence of the non-$T$-system relations of the $\widehat{GL}_3$ 3Q-system, and multiple orthogonal polynomials in the general three-series case

Addabbo and Bergvelt introduced a discrete integrable system (the $\widehat{GL}_3$ system, or 3Q-system) arising from Birkhoff factorizations of loop-group elements acting on the three-component semi-infinite Fock space, and showed that it contains four quadratic difference relations among tau-functions $\tau_{k,\ell}^{(\alpha,\beta)}$: two are $T$-system relations after a change of variables, while the other two are believed to be independent of them and of each other, with no known conceptual interpretation. The same construction yields type II multiple orthogonal polynomials only under the simplifying restriction that one of the defining series $C^{(\alpha-\beta)}(z)$ or $E^{(\beta)}(z)$ vanishes (the case $E^{(\beta)}(z)=0$ is the one used below). The open problem is to establish or refute the independence of the two non-$T$-system relations (of the $T$-system relations and of each other), give them a conceptual meaning (cluster-algebraic, representation-theoretic, or combinatorial), and construct the multiple-orthogonal-polynomial theory when all three series are present.

Background

A $Q$-system is a family of quadratic difference recurrences, first identified for the characters of Kirillov–Reshetikhin modules (special finite-dimensional modules of quantum affine algebras), in which products of neighboring terms are expressed in terms of squares and shifted terms; a $T$-system is the analogous recurrence for transfer matrices or characters related by Hirota bilinear equations. Di Francesco and Kedem showed that $Q$-systems are instances of cluster algebras (algebras generated by mutating distinguished variables under exchange relations), and Kuniba, Nakanishi, and Suzuki survey the general $T$-/Q-system theory and its representation-theoretic meaning. Separately, in the two-component (i.e. $GL_2$-type) semi-infinite fermionic Fock space — the Hilbert-space completion of a wedge product whose tau-functions encode determinants — the same circle of ideas produces the $A_{\infty/2}$ $Q$-system and (bi)orthogonal polynomials: the key mechanism is that the tau-functions are Hankel-type determinants in the moments, and ratios of shifted determinants are monic polynomials satisfying orthogonality relations. Addabbo and Bergvelt generalized this machine to the three-component Fock space, on which a central extension of the loop group of $GL_3$ (the group of $GL_3$-valued formal Laurent series in $z^{-1}$; the extension is denoted $\widehat{GL}_3$ in their program) acts by a fermionic representation. Starting from three arbitrary formal series $C(z)$, $D(z)$, $E(z)$ in $z^{-1}$ (with coefficient-shifted variants $C^{(\alpha-\beta)}(z)$, $D^{(\alpha)}(z)$, $E^{(\beta)}(z)$), they form the group element $g_{C,D,E}^{(\alpha,\beta)}$, lifting the lower-unipotent $3\times 3$ matrix whose subdiagonal entries are those three series; act on it with the shift (translation) operators $T_1^{-k}T_2^{-\ell}$; and take Birkhoff factorizations — the factorization of a loop-group element into a part holomorphic at the origin and a part holomorphic at infinity, in the sense of Segal and Wilson, whose matrix coefficients define the tau-functions $\tau_{k,\ell}^{(\alpha,\beta)}$. Four quadratic relations among these tau-functions follow (equations (1)–(4) of arXiv:1604.02190, restated as Theorem 3.8 of the SIGMA companion). Two of them, (2) and (4), become $T$-system relations after a change of variables; the authors state their belief that (1) and (3) are independent of these and of each other, and that they are searching for other contexts in which they arise. Multiple orthogonal polynomials are polynomials orthogonal with respect to several moment functionals simultaneously (type II: one polynomial made orthogonal to low powers of $z$ against each of several measures, with Riemann–Hilbert formulations surveyed by Van Assche, Geronimo, and Kuijlaars). Under the restriction $E^{(\beta)}(z)=0$, the tau-functions $\tau_{k,\ell}^{(\alpha,\beta)}$ degenerate to block-Hankel determinants in the $C$- and $D$-moments, and their ratios yield type II multiple orthogonal polynomials; when all three series are present, the tau-functions are infinite sums and this determinant structure is lost.

Problem Statement

Consider the $\widehat{GL}_3$ discrete system (the 3Q-system) of Addabbo–Bergvelt: the four quadratic difference relations among the tau-functions $\tau_{k,\ell}^{(\alpha,\beta)}$, for $k,\ell\ge0$ and $\alpha,\beta\in\mathbb{Z}$, produced by Birkhoff factorization of $T_2^{-\ell}T_1^{-k}g_{C,D,E}^{(\alpha,\beta)}$, where $g_{C,D,E}^{(\alpha,\beta)}$ is the loop-group element lifting the lower-unipotent $3\times 3$ matrix with subdiagonal entries $C^{(\alpha-\beta)}(z)$, $D^{(\alpha)}(z)$, $E^{(\beta)}(z)$ — arbitrary formal series in $z^{-1}$ with coefficient variables $c_i$, $d_i$, $e_i$. Negative-index tau-functions, which occur in the boundary instances $k=0$ or $\ell=0$ of the relations below, follow the source's convention $\tau_{k,\ell}^{(\alpha,\beta)}=0$ whenever $k<0$ or $\ell<0$: Theorem 3.1 of arXiv:1604.02190 expresses $\tau_{k,\ell}^{(\alpha,\beta)}$ as a sum over nonnegative integers $n_c$, $n_d$, $n_e$ with $n_c+n_d=k$ and $n_e+n_d=\ell$, so the sum is empty for negative $k$ or $\ell$, matching the $GL_2$ convention $\tau_{-1}^{(\alpha)}=0$. Explicitly (equations (1)–(4) of arXiv:1604.02190; the same system is restated as Theorem 3.8 of SIGMA 15 (2019) 023, which uses a different tau-function normalization and consequently prints the analogues of (2) and (4) with the opposite sign on the cross term):

$$\text{(1)}\quad (\tau_{k,\ell}^{(\alpha+1,\beta)})^2=\tau_{k,\ell}^{(\alpha,\beta)}\tau_{k,\ell}^{(\alpha+2,\beta)}+\tau_{k+1,\ell+1}^{(\alpha,\beta)}\tau_{k-1,\ell-1}^{(\alpha+2,\beta)}-\tau_{k+1,\ell}^{(\alpha,\beta)}\tau_{k-1,\ell}^{(\alpha+2,\beta)}$$ $$\text{(2)}\quad \tau_{k,\ell}^{(\alpha+1,\beta)}\tau_{k,\ell-1}^{(\alpha+2,\beta)}=\tau_{k-1,\ell-1}^{(\alpha+2,\beta)}\tau_{k+1,\ell}^{(\alpha+1,\beta)}+\tau_{k,\ell-1}^{(\alpha+1,\beta)}\tau_{k,\ell}^{(\alpha+2,\beta)}$$ $$\text{(3)}\quad (\tau_{k,\ell}^{(\alpha,\beta+1)})^2=\tau_{k,\ell}^{(\alpha,\beta)}\tau_{k,\ell}^{(\alpha,\beta+2)}-\tau_{k,\ell-1}^{(\alpha,\beta+2)}\tau_{k,\ell+1}^{(\alpha,\beta)}-\tau_{k+1,\ell}^{(\alpha,\beta+2)}\tau_{k-1,\ell}^{(\alpha,\beta)}$$ $$\text{(4)}\quad \tau_{k-1,\ell}^{(\alpha,\beta+1)}\tau_{k,\ell+1}^{(\alpha,\beta)}=\tau_{k-1,\ell}^{(\alpha,\beta)}\tau_{k,\ell+1}^{(\alpha,\beta+1)}+\tau_{k-1,\ell+1}^{(\alpha,\beta)}\tau_{k,\ell}^{(\alpha,\beta+1)}$$

Two of the relations, (2) and (4), are $T$-system relations after a change of variables — (2) in the $k,\ell$ variables for fixed $\beta$, and (4) in the $k,\ell$ variables for fixed $\alpha$. The problem has two linked components, which together complete the rank-three analogue of the $GL_2$ correspondence between $Q$/T-systems, cluster and character structure, and orthogonal polynomials:

(A) Independence and interpretation of relations (1) and (3). Establish or refute the authors' belief, stated for the tau-functions produced by the Birkhoff-factorization construction above from arbitrary formal series $C$, $D$, $E$, that relations (1) and (3) are independent of each other and are not implied by the $T$-system relations (2) and (4); and give these two rank-three quadratic relations a conceptual interpretation — a realization as exchange relations of a cluster algebra, an interpretation as character or fermionic matrix-element relations of a representation-theoretic object (in the way $T$-systems encode Kirillov–Reshetikhin characters), a combinatorial model, or an occurrence within another identified integrable structure.

(B) Multiple orthogonal polynomials in the general three-series case. Remove the restriction $E^{(\beta)}(z)=0$: when all three series $C$, $D$, $E$ are present, the tau-functions are infinite sums rather than block-Hankel determinants, and the existing argument that ratios of shifted tau-functions are finite-degree polynomials with definite multiple-orthogonality properties no longer applies. Construct the corresponding theory — define finite-degree polynomial objects from these tau-functions, specify the moment functionals or a Riemann–Hilbert/biorthogonality framework, and prove the analogue of the type II multiple orthogonality relations. The reference target is the source's restricted-case construction (Section 5.2 of arXiv:1604.02190): under $E^{(\beta)}(z)=0$, ratios of shifted tau-functions give monic degree-$k$ polynomials $p_{k,\ell}^{(\alpha,\beta)}(z)$ satisfying $\langle p_{k,\ell}^{(\alpha,\beta)},z^{n}\rangle_{C}=0$ for $0\le n<k-\ell$ and $\langle p_{k,\ell}^{(\alpha,\beta)},z^{n}\rangle_{D}=0$ for $0\le n<\ell$ against the two bilinear forms $\langle\cdot,\cdot\rangle_{C}$ and $\langle\cdot,\cdot\rangle_{D}$ induced by the $C$- and $D$-moments; the general three-series theory should reproduce an analogue of these finite-degree and multiple-orthogonality properties.

References

1. D. Addabbo and M. Bergvelt, Generalizations of Q-systems and Orthogonal Polynomials from Representation Theory, Contemporary Mathematics 695 (2017) 1–13, DOI 10.1090/CONM/695/13991, arXiv:1604.02190, https://arxiv.org/abs/1604.02190
2. D. Addabbo and M. Bergvelt, τ-Functions, Birkhoff Factorizations and Difference Equations, SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) 15 (2019) 023, DOI 10.3842/SIGMA.2019.023, arXiv:1605.00192, https://arxiv.org/abs/1605.00192
3. D. Addabbo and M. Bergvelt, Difference Hierarchies for nT τ-functions, International Journal of Mathematics 29 (2018) 1850090, DOI 10.1142/S0129167X18500908, arXiv:1611.10340, https://arxiv.org/abs/1611.10340
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