| Abstract: |
| Matrix models have wide applications in mathematics and physics. In the study of matrix models, the superintegrability means that the average of a properly chosen symmetric function is proportional to ratios of symmetric functions on a proper locus, i.e., $\langle$ character $\rangle$~character. $W$-representations of the matrix models realize the partition functions by acting on elementary functions with exponents of the given $W$-operators. In this talk, I will introduce our recent works on how to derive the superintegrability of several matrix models from their $W$-representations. Meanwhile, we construct the partition function hierarchies with $W$-representations, and present their character expansions with respect to the Schur (Jack) polynomials. For the negative branch of hierarchies, it gives the $\tau$-functions of the KP hierarchy. The $W$-operators in the positive branch of hierarchies can be related to the many-body systems. |
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