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In the the study of fractional quantum Hall states, a certain clusteringcondition involving up to four integers has been identified. We give a simpleproof that particular Jack polynomials with $\alpha = - (r-1)/(k+1)$, $(r-1)$and $(k+1)$ relatively prime, and with partition given in terms of itsfrequencies by $[n_00^{(r-1)s}k 0 ^{r-1}k 0 ^{r-1}k \cdots 0 ^{r-1} m]$ satisfythis clustering condition. Our proof makes essential use of the fact that theseJack polynomials are translationally invariant. We also consider nonsymmetricJack polynomials, symmetric and nonsymmetric generalized Hermite and Laguerrepolynomials, and Macdonald polynomials from the viewpoint of the clustering.

Jack polynomial fractional quantum Hall states and their generalizations