Finite-dimensional irreducible $\square_q$-modules and their Drinfel'd polynomials

Let $\mathbb{F}$ denote an algebraically closed field with characteristic $0$, and let $q$ denote a nonzero scalar in $\mathbb{F}$ that is not a root of unity. Let $\mathbb{Z}_4$ denote the cyclic group of order $4$. Let $\square_q$ denote the unital associative $\mathbb{F}$-algebra defined by generators $\{x_i\}_{i\in \mathbb{Z}_4}$ and relations \begin{gather*} \frac{qx_ix_{i+1}-q^{-1}x_{i+1}x_i}{q-q^{-1}}=1, \\ x_i^3x_{i+2}-[3]_qx_i^2x_{i+2}x_i+[3]_qx_ix_{i+2}x_i^2-x_{i+2}x_i^3=0, \end{gather*} where $[3]_q=(q^3-q^{-3})/(q-q^{-1})$. There exists an automorphism $\rho$ of $\square_q$ that sends $x_i\mapsto x_{i+1}$ for $i\in \mathbb{Z}_4$. Let $V$ denote a finite-dimensional irreducible $\square_q$-module of type $1$. To $V$ we attach a polynomial called the Drinfel'd polynomial. In our main result, we explain how the following are related: (i) the Drinfel'd polynomial for the $\square_q$-module $V$; (ii) the Drinfel'd polynomial for the $\square_q$-module $V$ twisted via $\rho$. Specifically, we show that the roots of (i) are the inverses of the roots of (ii). We discuss how $\square_q$ is related to the quantum loop algebra $U_q(L(\mathfrak{sl}_2))$, its positive part $U_q^+$, the $q$-tetrahedron algebra $\boxtimes_q$, and the $q$-geometric tridiagonal pairs.