Representations of the $U_q(u_{4,1})$ and a $q$-polynomial that determines baryon mass sum rules

With quantum groups $U_q(s{u}_n)$ taken as classifying symmetries for hadrons of $n$ flavors, we calculate within irreducible representation $D_{12}^{+}(p-1,p-3,p-4;p,p-2)$ ($p\in \mathbf{Z}$) of ‘dynamical’ quantum group $U_q(u_{4,1})$ the masses of baryons ${\frac{1}{2}}^{+}$ that belong to ${\it 20}$-pletof$U_q(su_4)$\mathrm{.}Theobtained$q$-analogofmassrelation(MR)for$U_q(su_3)$-octet contains unexpected mass-dependent term multiplied by the factor $\frac{A_q}{B_q}$ where $A_q,$ $B_q$ are certain polynomials (resp. of 7-th and 6-th order) in the variable $q+q^{-1}\equiv [2{]}_q$. Both values $q=1$ and $q=e^{\frac{i\pi }{6}}$ turn the polynomial $A_q$ into zero. But, while $q=1$ results in well-known Gell-Mann–Okubo (GMO) baryon MR, the second root of $A_q$ reduces the $q$-MR to some novel mass sum rule which has irrational coefficients and which holds, for empirical masses, even with better accuracy than GMO mass sum rule.