The Schrödinger bridge problem: given an initial and final probability distribution, find the most likely stochastic process connecting them. On Euclidean space, the solution is well-characterized — the optimal process is a Brownian motion with a drift that interpolates between the two distributions, and the drift minimizes the relative entropy with respect to the reference process.
On a compact connected Lie group, the problem acquires algebraic structure. The state space has both a metric (from the Riemannian structure) and a group operation (from the Lie structure), and the Schrödinger bridge must respect both. The reference process is Brownian motion on the group — diffusion generated by the Laplace-Beltrami operator — and the bridge modifies this diffusion minimally to connect the prescribed endpoints.
The paper derives the Schrödinger bridge on compact Lie groups using the group's representation theory. The heat kernel on the group decomposes into characters of irreducible representations, and this decomposition carries through to the bridge's optimal drift. The drift is expressed in terms of the group's representation-theoretic data — dimensions of representations, Casimir eigenvalues, character values — rather than in terms of coordinates or geodesics.
The representation-theoretic form is exact and explicit for any compact group, including non-Abelian groups where direct computation of geodesics and heat kernels is intractable. The algebra replaces the geometry: instead of solving a PDE on a curved manifold, you evaluate a sum over representations. The bridge's structure is algebraic because the group's structure is algebraic, and the stochastic process inherits the algebraic organization.