A method for finding irreducible representations of finite groups using gradient descent is presented. Mappings between finite groups and matrices are represented as vectors in Euclidean space and a non-negative loss function is defined, which is zero exactly at irreducible unitary representations. The connection between gradient descent and solving differential equations is presented. Each irreducible representation is expressed as a limit of the gradient flow of a smooth function with suitable initial parameters. Numerical results for cyclic and dihedral groups are studied. The method is extended to finding actions and graph isomorphisms via a smooth family of distributions over mappings of finite sets. A smooth family of distributions over the inversion table is defined, offering an alternative to the Sinkhorn algorithm.
|
