Generations and anti-generations of \(N=2\) string theory, as well as the Yukawa coupling among them, are described by chiral algebras. These algebras are thoroughly investigated in this paper which is a step in the study of the interplay between geometric and algebraic structures in \(N=2\) string theory. They are studied in the context of \(N=2\) superconformal field theories based on quotients of Lie algebras [\textit{D. Kastor}, \textit{E. Martinec} and \textit{Z. Qiu}, Phys. Lett. B 200, 434 (1988); \textit{Y. Kazama} and \textit{H. Suzuki}, University of Tokyo, preprint, UT-Komaba 88-8 (1988)]. Since the chiral algebras are finite- dimensional, their Poincaré series [\textit{R. P. Stanley}, Combinatorics and commutative algebra (Birkhäuser, Boston 1983; Zbl 0537.13009)] become polynomials: the Poincaré polynomials. It is shown that Poincaré polynomials are sums of quotients of polynomials of a well- defined form.