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We study the combinatorial structure of the irreducible characters of theclassical groups ${\rm GL}_n(\mathbb{C})$, ${\rm SO}_{2n+1}(\mathbb{C})$, ${\rmSp}_{2n}(\mathbb{C})$, ${\rm SO}_{2n}(\mathbb{C})$ and the "non-classical" oddsymplectic group ${\rm Sp}_{2n+1}(\mathbb{C})$, finding new connections to theprobabilistic model of Last Passage Percolation (LPP). Perturbing theexpressions of these characters as generating functions of Gelfand-Tsetlinpatterns, we produce two families of symmetric polynomials that interpolatebetween characters of ${\rm Sp}_{2n}(\mathbb{C})$ and ${\rmSO}_{2n+1}(\mathbb{C})$ and between characters of ${\rm SO}_{2n}(\mathbb{C})$and ${\rm SO}_{2n+1}(\mathbb{C})$. We identify the first family as aone-parameter specialization of Koornwinder polynomials, for which we thusprovide a novel combinatorial structure; on the other hand, the second familyappears to be new. We next develop a method of Gelfand-Tsetlin patterndecomposition to establish identities between all these polynomials that, inthe case of irreducible characters, can be viewed as branching rules. Throughthese formulas we connect orthogonal and symplectic characters, and moregenerally the interpolating polynomials, to LPP models with various symmetries,thus going beyond the link with classical Schur polynomials originally found byBaik and Rains (Duke Math. J., 2001). Taking the scaling limit of the LPPmodels, we finally provide an explanation of why the Tracy-Widom GOE and GSEdistributions from random matrix theory admit formulations in terms of bothFredholm determinants and Fredholm Pfaffians.

Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation