  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,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,C) group with rank K+2. As a
result, the SL(K+3,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,C) group can be used to directly prove Gross conjecture [1-3], which was
previously corrected and proved by the method of decoupling of zero norm states
[4-10].
