[en] We discover that the exact string scattering amplitudes (SSA) of three tachyons and one arbitrary string state, or the Lauricella SSA (LSSA), in the 26D open bosonic string theory can be expressed in terms of the basis functions in the infinite dimensional representation space of the $SL(K+3,\mathbb{C})$ group. In addition, we find that the $K+2$ recurrence relations among the LSSA discovered by the present authors previously can be used to reproduce the Cartan subalgebra and simple root system of the $SL(K+3,\mathbb{C})$ group with rank $K+2$. As a result, the $SL(K+3,\mathbb{C})$ group can be used to solve all the LSSA and express them in terms of one amplitude. As an application in the hard scattering limit, the $SL(K+3,\mathbb{C})$ group can be used to directly prove Gross conjecture [1], [2], [3], which was previously corrected and proved by the method of decoupling of zero norm states [4], [5], [6], [7], [8], [9], [10].