Superintegrability for ($$\beta $$-deformed) partition function hierarchies with W-representations
Rui Wang, Fan Liu, Chun-Hong Zhang, Wei‐Zhong Zhao
Abstract
Abstract We construct the ( $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> -deformed) partition function hierarchies with W -representations. Based on the W -representations, we analyze the superintegrability property and derive their character expansions with respect to the Schur functions and Jack polynomials, respectively. Some well known superintegrable matrix models such as the Gaussian hermitian one-matrix model (in the external field), $$N\times N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mo>×</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math> complex matrix model, $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> -deformed Gaussian hermitian and rectangular complex matrix models are contained in the constructed hierarchies.