# The Boundary of the Gelfand-Tsetlin Graph: New Proof of Borodin-Olshanski’s Formula, and its q-analogue

### 2012/08/15

*Moscow Mathematical Journal, 14 (2014) no. 1, 121-160* •

arXiv:1208.3443 [math.CO]

In the recent paper [arXiv:1109.1412], 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 indecomposable (= extreme)
characters of the infinite-dimensional unitary group. An equivalent description
identifies the boundary with the set of doubly infinite totally nonnegative
sequences.

A principal ingredient of Borodin-Olshanski’s proof is a new explicit
determinantal formula for the number 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 [arXiv:1011.1769].