The functional-equation characterization of the $gl_{1|2}$ transfer-matrix spectrum

Let $T_{1}^{(K)}(\lambda)$ be the transfer matrix of the general $gl_{1|2}$-graded Yang–Baxter algebra with twisted boundary conditions. Let $t_{1}(\lambda \mid \{x_a\})$ be the polynomial defined above, excluding the trivial solution $x_1=\cdots=x_{\mathsf N}=0$, and let the associated higher polynomials satisfy the fusion relations. The inner-boundary condition is

$$(-1)^{\mathcal N}\operatorname{Ber}(\lambda)t_{\mathcal N}^{(\mathcal M+1)}(\lambda+\eta\mid\{x_a\})=t_{\mathcal N+1}^{(\mathcal M)}(\lambda\mid\{x_a\}),$$

for all $\lambda\in\mathbb C$, and the null out-boundary conditions are

$$t_{\mathcal N+m}^{(\mathcal M+n)}(\lambda\mid\{x_a\})=0,$$

for all $\lambda\in\mathbb C$ and $n,m\geq 1$. The functional-equation characterization conjecture. For $\mathcal M=1$ and $\mathcal N=2$, the polynomial $t_{1}(\lambda\mid\{x_a\})$ is an eigenvalue of $T_{1}^{(K)}(\lambda)$ if and only if the associated higher polynomials satisfy, in addition to the fusion relations, the inner-boundary condition and the null out-boundary conditions above. This conjecture is the proposed complete characterization of the transfer-matrix spectrum for the $gl_{1|2}$-graded Yang–Baxter algebra. It is proved for some special classes of twist matrices and verified for quantum chains with two and three sites, while only initial motivation is given for general representations.