Principal subspaces for quantum affine algebra $U_q(A_n^{(1)})$

We consider principal subspace W({\Lambda}) of integrable highest weight module L({\Lambda}) for quantum affine algebra $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_{n+1})$. We introduce quantum analogues of the quasi-particles associated with the principal subspaces for ${\widehat{\mathfrak{s}\mathfrak{l}}}_{n+1}$ and discover certain relations among them. By using these relations we find, for certain highest weight {\Lambda}, combinatorial bases of principal subspace W({\Lambda}) in terms of monomials of quantum quasi-particles.