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We construct the ($\beta$-deformed) partition function hierarchies with$W$-representations. Based on the $W$-representations, we analyze thesuperintegrability property and derive their character expansions with respectto the Schur functions and Jack polynomials, respectively. Some well knownsuperintegrable matrix models such as the Gaussian hermitian one-matrix model(in the external field), $N\times N$ complex matrix model, $\beta$-deformedGaussian hermitian and rectangular complex matrix models are contained in theconstructed hierarchies.

Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations