Proving superintegrability in $β$-deformed eigenvalue models

Abstract

In this note we provide proofs of various expressions for expectation values of symmetric polynomials in β-deformed eigenvalue models with quadratic, linear, and logarithmic potentials. The relations we derive are also referred to as superintegrability. Our work completes proofs of superintegrability statements conjectured earlier in literature.

Supersymmetric partition function hierarchies and character expansions

“…The superintegrability for matrix models has attracted much attention (see [10] and the references therein, [11][12][13][14][15][16][17][18][19]). Here the superintegrability means that for the character expansions of the matrix models, the average of a properly chosen symmetric function is proportional to ratios of symmetric functions on a proper locus, i.e., < character >∼ character.…”