Tableau-sum versus Wronskian-determinant T-functions for the $U_q(\widehat{gl}(M|N))$ T-system: the general Young-diagram case of Tsuboi's Conjecture 3.5

Transfer-matrix eigenvalues (T-functions) of integrable spin chains with quantum affine superalgebra symmetry $U_q(\widehat{gl}(M|N))$ admit two a priori different closed forms in terms of Baxter Q-functions: (i) tableau sums, weighted sums over admissible Young-tableau fillings whose entries label bosonic and fermionic flavors, and (ii) compact Wronskian-type (Casoratian) determinants of fundamental Q-functions at $q$-shifted spectral parameters. Tsuboi (2010) proved that the two coincide only for single-box diagrams, rectangular diagrams, and purely bosonic flavor sets, and posed the general equivalence as Conjecture 3.5 of that paper. The problem is to prove, or refute by an explicit counterexample, the identity $\mathcal{T}_\mu^{B_m,F_n}(x)=\mathsf{F}_\mu^{B_m\times F_n}(x)$, together with its barred companion $\overline{\mathcal{T}}_\mu^{B_m,F_n}(x)=\overline{\mathsf{F}}_\mu^{B_m\times F_n}(x)$, for every non-skew Young diagram $\mu$ and every choice of bosonic and fermionic flavor sets, under the standard normalization and the graded QQ-relations. The genuinely supersymmetric, non-rectangular regime is the open part: Tsuboi's 2024 folding paper still records it verbatim as a conjecture. The identity is the missing certification that compact determinant formulas compute transfer-matrix eigenvalues for arbitrary covariant (mixed) representations of $U_q(\widehat{gl}(M|N))$.

Background

An integrable quantum spin chain is solved through a commuting family of transfer matrices $T_s^{(a)}(x)$, labeled by a Dynkin-diagram node $a$ and a non-negative integer level $s$ and depending on a spectral parameter $x$; here the parameter is multiplicative, so all shifts below mean $x\mapsto xq^k$. The eigenvalues of these transfer matrices obey the T-system, a discrete Hirota equation $(T_s^{(a)})^2=T_{s+1}^{(a)}T_{s-1}^{(a)}+T_s^{(a+1)}T_s^{(a-1)}$ with model-dependent boundary conditions. The finest building blocks are Baxter Q-functions $\mathcal{Q}_I(x)$, one for each subset $I$ of the flavor set $\{1,\dots,M+N\}$; they satisfy graded QQ-relations of quantum-Wronskian type, for example $(z_i-z_j)\mathcal{Q}_I(x)\mathcal{Q}_{I\sqcup\{i,j\}}(x)=z_i\mathcal{Q}_{I\sqcup\{i\}}(xq^{p_i})\mathcal{Q}_{I\sqcup\{j\}}(xq^{-p_i})-z_j\mathcal{Q}_{I\sqcup\{i\}}(xq^{-p_i})\mathcal{Q}_{I\sqcup\{j\}}(xq^{p_i})$ for two flavors of equal statistics, where $p_i=+1$ ($-1$) for a bosonic (fermionic) flavor and $z_i$ is a twist parameter; nested Bethe ansatz equations are the zero conditions on the $\mathcal{Q}_I$.

For supersymmetric chains based on the quantum affine superalgebra $U_q(\widehat{gl}(M|N))$ (generic $q$, not a root of unity; $M$ bosonic and $N$ fermionic flavors), the admissible labels $(a,s)$ fill a hook-shaped domain (the $[M,N]$-hook), and the relevant auxiliary spaces include tensor products of $m$ bosonic defining representations ($B_m$) and $n$ fermionic ones ($F_n$) — the mixed representations. Two closed descriptions of the same T-functions exist. First, by the supersymmetric Bazhanov-Reshetikhin (analytic Bethe ansatz) formula, a T-function is a tableau sum: a weighted sum over admissible fillings of a Young diagram $\mu$ with entries from $B_m\cup F_n$, each box contributing a fundamental Q-function at a $q$-shifted argument and each fermionic entry carrying a sign; when the Q-functions specialize to complete supersymmetric symmetric functions these sums reduce to supersymmetric Schur (hook Schur) functions, i.e. supercharacters (graded traces in which fermionic weight spaces enter with a minus sign) of covariant $gl(M|N)$ modules, and supersymmetric Jacobi-Trudi-type determinant formulas for them are known. Second, Tsuboi (2010) derived compact Wronskian-type (Casoratian) determinant expressions $\mathcal{T}_\mu^{B_m,F_n}(x)$ — determinants whose rows are fundamental Q-functions at ladders of $q$-shifted arguments — valid without any reference to the tableau structure, and used them to derive finite-order Baxter equations and Bäcklund transformations for the supersymmetric T-system.

For the purely bosonic case ($n=0$) the coincidence of the two descriptions is a known combinatorial fact — the tableau representation of Macdonald's ninth (factorial) variation of Schur functions, Theorem 3.2 of Nakagawa, Noumi, Shirakawa and Yamada. Tsuboi proved the supersymmetric coincidence for the single box $\mu=(1)$ (direct case analysis over $m,n$) and for rectangular diagrams $\mu=(s^a)$ (both sides satisfy the same functional relations with the same boundary conditions), and verified it by computer algebra in many further cases. The general statement is Conjecture 3.5 (p. 426 of the published paper): $\mathcal{T}_\mu^{B_m,F_n}(x)=\mathsf{F}_\mu^{B_m\times F_n}(x)$ under the normalization $\mathcal{Q}_\emptyset(x)=1$ and the graded QQ-relations, together with the barred (contragredient-type) companion $\overline{\mathcal{T}}_\mu^{B_m,F_n}(x)=\overline{\mathsf{F}}_\mu^{B_m\times F_n}(x)$; here $\mathsf{F}$ is the tableau-sum function regularized by a pole-killing Q-function prefactor (with a 180-degree rotation of the diagram), not the raw tableau sum. A subtle obstruction to a naive proof: finite-dimensional covariant $gl(M|N)$ modules can be atypical — the highest weight is orthogonal to an isotropic odd root, so the induced Kac module is reducible — and their supercharacters are not single Weyl-type determinants, so a character-theoretic argument that works for typical (infinite-dimensional) modules does not cover the open regime.

Problem Statement

Prove, or refute by an explicit counterexample, Conjecture 3.5 of Tsuboi (2010) in its full generality. For the quantum affine superalgebra $U_q(\widehat{gl}(M|N))$ with generic $q$ and twist parameters $z_i$, let $B_m$ be a set of $m$ bosonic flavors ($p_i=+1$) and $F_n$ a set of $n$ fermionic flavors ($p_i=-1$), and let $\mu$ be an arbitrary non-skew Young diagram. Let $\mathsf{F}_\mu^{B_m\times F_n}(x)$ denote the tableau-sum T-function — the weighted sum over admissible tableaux of shape $\mu$ with entries from $B_m\cup F_n$, each box contributing a fundamental Baxter Q-function $\mathcal{Q}_I(x)$ at a $q$-shifted argument with signs from fermionic entries — regularized by the pole-killing prefactor fixed in the source paper (equivalently, by the Q-function prefactor with the 180-degree-rotated diagram). Let $\mathcal{T}_\mu^{B_m,F_n}(x)$ denote the Wronskian-type (Casoratian) determinant of the fundamental Q-functions associated with $B_m\cup F_n$ at $q$-shifted arguments, divided by the value of the empty-diagram determinant at $x=0$ (the source paper's normalization, under which both sides reduce to the same Q-function at $\mu=\emptyset$). Then under the normalization $\mathcal{Q}_\emptyset(x)=1$ and the graded QQ-relations among the fundamental Q-functions as fixed in the source paper, establish that

$$\mathcal{T}_\mu^{B_m,F_n}(x)=\mathsf{F}_\mu^{B_m\times F_n}(x)$$

holds for every non-skew Young diagram $\mu$ and every $(m,n)$ — in particular in the open regime $m,n\ge 1$ with $\mu$ non-rectangular, where only computer-algebra verification currently exists — together with the barred companion identity $\overline{\mathcal{T}}_\mu^{B_m,F_n}(x)=\overline{\mathsf{F}}_\mu^{B_m\times F_n}(x)$ under the barred normalization and QQ-relations. Any sound route is admissible: a combinatorial proof via supersymmetric (factorial) Schur function identities, a representation-theoretic proof via supercharacters of covariant $U_q(\widehat{gl}(M|N))$ modules, or an operator-level proof via traces of Q-operators. A refutation must supply one explicit instance — a concrete $\mu$, $(M,N)$, $(m,n)$ with $m,n\ge 1$ and $\mu$ non-rectangular, twist values, and explicit Q-function data satisfying the stated QQ-relations and normalization — for which the two sides differ.

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