| Recent work of Sturmfels and Xu establishes an intriguing connection between the Hilbert functions of important varieties from phylogenetic algebraic geometry and a genus 0, sl_2(\C) case of the celebrated Verlinde formula from mathematical physics. We will discuss how this relationship can be constructed from the deformation theory of moduli stacks of principal bundles over a curve, and extend it to higher genus. We will also discuss how similar combinatorial patterns emerge in the presence of more general Lie algebras. |