  We determine explicitly all structure constants of the whole chiral BRST
cohomology ring in $D=2$ string theory including both the discrete states and
tachyon states. This is made possible by establishing several identities for
Schur polynomials with operator argument and exploring associativity.
Furthermore we find that the (chiral) symmetry algebra of the charges obtained
by using the descent equations can actually be read off from the cohomology
ring structure by simple operation involving the ghost field $b$. We also
determine the enlarged symmetry algebra which contains the charges having ghost
number $-1$ and $1$. Finally the complete symmetry transformation rules are
derived for closed string discrete states by carefully combining the left and
right sectors. It turns out that the new states introduced recently by Witten
and Zwiebach are naturally created when symmetries act on the old states.
