New insights into superintegrability from unitary matrix models

Proving superintegrability in $\beta$-deformed eigenvalue models

“…(Note that P λ {xδ k,2 } = x |λ|/2 P λ {δ k,2 } and P λ {xδ k,1 } = x |λ| P λ {δ k,1 }.) On the other hand, again by the Cauchy identity (37), one can expand the partition function (6) as…”

“…Further deformation that involves two parameters, commonly denoted q and t, and corresponding superintegrability statement analogous to (3), however involving Macdonald polynomials, were conjectured in [3,4]. For a review of these developments and some recent results see [5,6]. The methods of the proof that we discuss in this paper do not immediately generalize to the (q, t)-deformed case, therefore we leave it for future consideration.…”

“…Recently the superintegrability has been analyzed in much detail in matrix models, where these extra conditions take form of the string equation and Virasoro constraints. One manifestation of superintegrability in this context is the statement that expectation values of various symmetric functions can be explicitly expressed also in terms of analogous symmetric functions evaluated with appropriate arguments [1][2][3][4][5][6][7][8]. As symmetric functions in this context can be identified with appropriate characters, such relations are often presented schematically as 〈char ac t er〉 ∼ char ac t er.…”

Proving superintegrability in $β$-deformed eigenvalue models

“…Recently the superintegrability has been analyzed in much detail in matrix models, where these extra conditions take form of the string equation and Virasoro constraints. One manifestation of superintegrability in this context is the statement that expectation values of various symmetric functions can be explicitly expressed also in terms of analogous symmetric functions evaluated with appropriate arguments [1][2][3][4][5][6][7][8]. As symmetric functions in this context can be identified with appropriate characters, such relations are often presented schematically as character ∼ character.…”

“…Further deformation that involves two parameters, commonly denoted q and t, and corresponding superintegrability statement analogous to (1.3), however involving Macdonald polynomials, were conjectured in [3,4]. For a review of these developments and some recent results see [5,6]. The methods of the proof that we discuss in this paper do not immediately generalize to the (q, t)-deformed case, therefore we leave it for future consideration.…”