Commutativity of antidiagonal quasi-determinants in the reflection equation algebra for arbitrary rank (Talalaev's Hypothesis 2.6)

The reflection equation (RE) algebra of the Drinfeld–Jimbo $R$-matrix of $GL_n$ is a $q$-deformation of $U(\mathfrak{gl}_n)$ arising in the theory of integrable systems with a boundary. Talalaev (2024) proved that for matrix orders $n=2$ and $n=3$ the quantum determinant of this algebra factors into a family of pairwise commuting antidiagonal principal quasi-determinants $I_{n-k}=|L^{(k)}|_{n-k,k+1}$, each computed on the successive upper-right blocks of the generating matrix, whose ordered product $Y=\prod_{k=1}^n I_k$ is central and proportional to the quantum determinant, and he explicitly posed the same statement for arbitrary $n$ as the open Hypothesis 2.6. The construction is the RE-algebra analogue of the classical Gelfand–Retakh factorization of the RTT quantum determinant and quantizes the Deift–Li–Nanda–Tomei integrals of the full Toda system. The problem — prove or refute, for arbitrary $n$, the commutativity of the $I_k$ in the localization together with the centrality and determinant-proportionality of their product — remains open.

Background

In the $R$-matrix (RTT) approach to quantum groups one starts from a solution $R$ of the quantum Yang–Baxter equation and defines the RTT algebra as the algebra generated by the entries of a matrix $T$ subject to the quadratic relations $RT_1T_2=T_1T_2R$. When $R$ is of Hecke type, deforming the matrix transposition flip, this algebra carries a central quantum determinant — the noncommutative analogue of $\det T$. Gelfand and Retakh (1991), developing Dieudonné's noncommutative determinant, showed that for the standard Drinfeld–Jimbo $R$-matrix of $GL_n$ the quantum determinant of the RTT algebra equals an ordered product of principal quasi-determinants of $T$ (one for each successive northwest corner block), and that these factors commute with one another. A quasi-determinant is the noncommutative analogue of a Schur complement: for a matrix $A=(a_{ij})$ over an associative ring, $|A|_{pq}=a_{pq}-r_p(A^{pq})^{-1}c_q$, where $A^{pq}$ is $A$ with row $p$ and column $q$ deleted, $r_p$ is the $p$-th row without its $q$-th entry, and $c_q$ is the $q$-th column without its $p$-th entry; it is defined whenever $A^{pq}$ is invertible, i.e. after a suitable localization. Quasi-determinants satisfy noncommutative analogues of heredity, Plücker and Sylvester identities and play the role of Plücker coordinates on quantum flag varieties.

The reflection equation (RE) algebra is a sibling construction introduced by Cherednik (1984) in the study of factorized scattering on a half-line. For the Drinfeld–Jimbo $R$-matrix $R=\sum_{i<j}(q-q^{-1})\,E_{ii}\otimes E_{jj}+\sum_{i,j}q^{\delta(i,j)}\,E_{ij}\otimes E_{ji}\in\mathrm{Mat}_n^{\otimes 2}$ (with $E_{ij}$ the matrix unit and $q$ a generic deformation parameter), it is the algebra $\mathcal{A}$ generated by the entries $a_{ij}$ of a generating matrix $L=\sum_{i,j}E_{ij}\otimes a_{ij}$ subject to the reflection equation $RL_1RL_1=L_1RL_1R$ in $\mathrm{Mat}_n^{\otimes 2}\otimes\mathcal{A}$, where $L_1=\sum_{i,j}E_{ij}\otimes 1\otimes a_{ij}$. Unlike the RTT algebra, $\mathcal{A}$ is a flat deformation of the universal enveloping algebra $U(\mathfrak{gl}_n)$, has a polynomial center deforming the $GL(n)$-invariants, and is a comodule algebra over the RTT algebra — a quantum homogeneous space. The structure theory of these algebras, including their quantum determinants and central elements, was developed by Gurevich, Pyatov and Saponov (1997) and Gurevich and Saponov (2021); Krob and Leclerc (1995) established the corresponding minor identities in the RTT setting.

The physical motivation is the full Toda system. Deift, Li, Nanda and Tomei (1986) proved integrability of the Toda flows on generic adjoint orbits — with phase space realized by symmetric matrices carrying the Kirillov–Kostant Poisson bracket — by the chopping procedure: from the Lax matrix $L$ one forms the characteristic polynomials $\Delta_k(\lambda)$ of the successive upper-right blocks obtained by deleting the first $k$ columns and the last $k$ rows, and the roots (equivalently, suitable ratios of coefficients) of all the $\Delta_k$ Poisson-commute, yielding a complete involutive family. The classical limit of the reflection equation near $L=1+\varepsilon U$ is a quadratic $r$-matrix Poisson bracket on $\mathfrak{gl}_n$ of the type displayed by Sokolov and Talalaev (2025), so the RE algebra quantizes exactly this Poisson structure, and the antidiagonal quasi-determinants of the RE algebra are the natural quantum counterparts of the Deift–Li–Nanda–Tomei integrals.

Talalaev (2024) realized this program for small rank: he proved that for $n=2$ (elementary) and $n=3$ (the paper's main result) the elements $I_{n-k}=|L^{(k)}|_{n-k,k+1}$ — quasi-determinants of the successive upper-right blocks $L^{(k)}$, highlighted at the antidiagonal corner — commute in the localization of the RE algebra, and that their ordered product $Y$ is central and proportional to the quantum determinant. The $n=3$ centrality computation used central-element formulas communicated privately by P. Pyatov. The paper then states, as Hypothesis 2.6, the identical claim for arbitrary $n$, together with the expectation that the factorization reflects the Bruhat-cell geometry of the RE algebra as a quantum homogeneous space (cf. Berenstein and Retakh's noncommutative double Bruhat cells). Related but distinct settings in which quasideterminant factorizations are known include the RTT algebra (Gelfand–Retakh, all $n$, diagonal corner) and twisted Yangians — coideal subalgebras defined by a twisted reflection equation, whose Sklyanin determinants admit quasideterminant expressions and minor identities (Jing–Zhang 2024, 2025) — while the classical antidiagonal-minor theory for quadratic Poisson brackets on $\mathfrak{gl}_n$ was developed by Sokolov–Talalaev (2025). None of these covers the RE algebra at general rank.

Problem Statement

Prove or refute Hypothesis 2.6 of Talalaev (2024). Fix an integer $n\ge 2$ and let $q$ be an indeterminate (or a generic complex parameter, avoiding roots of unity). Let $R\in\mathrm{Mat}_n^{\otimes 2}$ be the Drinfeld–Jimbo $R$-matrix of $GL_n$, $R=\sum_{i<j}(q-q^{-1})\,E_{ii}\otimes E_{jj}+\sum_{i,j}q^{\delta(i,j)}\,E_{ij}\otimes E_{ji},$ (with $\delta(i,j)$ the Kronecker delta, $E_{ij}$ the standard matrix units, and $q$ the deformation parameter), and let $\mathcal{A}$ be the reflection equation algebra: the associative algebra generated by $a_{ij}$, $1\le i,j\le n$, with generating matrix $L=\sum_{i,j}E_{ij}\otimes a_{ij}$ subject to $RL_1RL_1=L_1RL_1R,\qquad L_1=\sum_{i,j}E_{ij}\otimes 1\otimes a_{ij}.$ For $k=0,\dots,n-1$ let $L^{(k)}$ be the upper-right $(n-k)\times(n-k)$ block of $L$ obtained by deleting the lower $k$ rows and the left $k$ columns (so $L^{(k)}$ occupies rows $1,\dots,n-k$ and columns $k+1,\dots,n$ of $L$), and set $I_{n-k}=|L^{(k)}|_{n-k,k+1},$ the quasi-determinant of $L^{(k)}$ highlighted at its bottom-left, i.e. antidiagonal, corner — the entry of $L$ at row $n-k$ and column $k+1$; here $|A|_{pq}=a_{pq}-r_p(A^{pq})^{-1}c_q$ is the Gelfand–Retakh quasi-determinant, with $A^{pq}$ the matrix $A$ with row $p$ and column $q$ deleted, $r_p$ the $p$-th row without its $q$-th entry, and $c_q$ the $q$-th column without its $p$-th entry. Each $I_{n-k}$ is an element of the localization of $\mathcal{A}$ in which the submatrices $A^{pq}$ appearing in these quasi-determinants are inverted.

The claims to be settled, for arbitrary $n$ (the cases $n=2$ and $n=3$ are proved in the source): (i) the $n$ elements $I_1,\dots,I_n$ pairwise commute in this localization of $\mathcal{A}$; (ii) their ordered product $Y=\prod_{k=1}^{n} I_k$ lies in the center of the localized algebra and is proportional (up to a scalar in the ground field) to the quantum determinant of the RE algebra $\mathcal{A}$ (the central determinant-type element of the matrix $L$, in the sense of the determinant theory of reflection equation algebras due to Gurevich–Pyatov–Saponov).

A complete answer either proves both claims for every $n\ge 2$ at generic $q$, or refutes the hypothesis by exhibiting, for some $n\ge 4$, a concrete instance — matrices satisfying the reflection equation, or certified elements of the RE algebra over $\mathbb{C}(q)$ — in which claim (i) or claim (ii) fails.

References

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