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

Understand

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.