THE BOUNDARY OF THE GELFAND-TSETLIN GRAPH: NEW PROOF OF BORODIN-OLSHANSKI'S FORMULA, AND ITS q-ANALOGUE

In a recent paper, Borodin and Olshanski have presented a novel proof of the celebrated Edrei-Voiculescu theorem which describes the boundary of the Gelfand-Tsetlin graph as a region in an infinite-dimensional coordinate space. This graph encodes branching of irreducible characters of finite-dimensional unitary groups. Points of the boundary of the Gelfand-Tsetlin graph can be identified with finite in-decomposable (=extreme) characters of the infinite-dimensional unitary group. An equivalent description identifies the boundary with the set of doubly in totally nonnegative sequences. A principal ingredient of Borodin-Olshanski's proof is a new explicit determinantal formula for the Humber of semi-standard Young tableaux of a given skew shape (or of Gelfand-Tsetlin schemes of trapezoidal shape). We present a simpler and more direct derivation of that formula using the Cauchy-Binet summation involving the inverse Vandermonde matrix. We also obtain a q-generalization of that formula, namely, a new explicit determinantal formula for arbitrary q-specializations of skew Schur polynomials. Its particular case is related to the q-Gelfand-Tsetlin graph and q-Toeplitz matrices introduced and studied by Gorin.