It is known that random matrix distributions such as those that describe the largest eignevalue of the Gaussian Orthogonal and Symplectic ensembles (GOE, GSE) admit two types of representations: one in terms of a Fredholm Pfaffian and one in terms of a Fredholm determinant. The equality of the two sets of expressions has so far been established via involved computations of linear algebraic nature. We provide a structural explanation of this duality via links (old and new) between the model of last passage percolation and the irreducible characters of classical groups, in particular the general linear, symplectic and orthogonal groups, and by studying, combinatorially, how their representations decompose when restricted to certain subgroups. Based on joint work with Elia Bisi.
