A $q$-version of the relation between the hypercube, the Krawtchouk chain and Dicke states

It is shown how the spin chain based on the dual $q$-Krawtchouk polynomials is connected to a weighted hypercube through the use of $q$-Dicke states. The representation theoretic underpinnings based on the quantum algebra $U_q(\mathfrak{s}\mathfrak{u}(2))$ are emphasized.