Representation Theory and Combinatorics Arising from Determinants

Originating from the unified treatment of probability distributions, the -determinant is a one-parameter interpolation of the determinant and permanent of a matrix.While it generally does not have the invariance properties of the determinant, the irreducible representations of the associated general linear (Lie) groups define interesting invariants that unveil a new and rich mathematical theory with abundant unsolved problems, related to symmetric functions, representation theory, combinatorics, number theory, probability and other areas.