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

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