On Solutions of the Fuji–Suzuki–Tsuda System

We derive Fredholm determinant and series representation of the tau function of the Fuji–Suzuki–Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlevé VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also initiate the study of algebraic braid group dynamics of semi-degenerate monodromy, and obtain as a byproduct a direct isomonodromic proof of the AGT-W relation for ${c=N-1}$.