The q-Gelfand–Tsetlin graph, Gibbs measures and q-Toeplitz matrices

The problem of the description of finite factor representations of the infinite-dimensional unitary group investigated by Voiculescu in 1976, is equivalent to the description of all totally positive Toeplitz matrices. Vershik–Kerov showed that this problem is also equivalent to the description of the simplex of central (i.e. possessing a certain Gibbs property) measures on paths in the Gelfand–Tsetlin graph. We study a quantum version of the latter problem. We introduce a notion of a q-centrality and describe the simplex of all q-central measures on paths in the Gelfand–Tsetlin graph. Conjecturally, q-central measures are related to representations of the quantized universal enveloping algebra Uϵ(gl∞). We also define a class of q-Toeplitz matrices and show that extreme q-central measures correspond to q-Toeplitz matrices with non-negative minors. Finally, our results can be viewed as a classification theorem for certain Gibbs measures on rhombus tilings of the halfplane.We use a class of q-interpolation polynomials related to Schur functions. One of the key ingredients of our proofs is the binomial formula for these polynomials proved by Okounkov.