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

With quantum groups $U_q(su_n)$ taken as classifying symmetries for hadrons of $n$ flavors, we calculate within irreducible representation $D^+_{12}(p-1,p-3,p-4;p,p-2)$ ($p \in {\bf Z}$) of 'dynamical' quantum group $U_q(u_{4,1})$ the masses of baryons ${1\over 2}^+$ that belong to ${\it 20}$-plet of $U_q(su_4)$. The obtained $q$-analog of mass relation (MR) for $U_q(su_3)$-octet contains unexpected mass-dependent term multiplied by the factor ${A_q\over B_q}$ 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 $q=e^{i\pi \over 6}$ 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.