Highest Weight Generating Functions for Hilbert Series

Understand

We develop a new method for representing Hilbert series based on the highest weight Dynkin labels of their underlying symmetry groups.

• The method draws on plethystic functions and character generating functions along with Weyl integration.
• We give explicit examples showing how the use of such highest weight generating functions (HWGs) permits an efficient encoding and analysis of the Hilbert series of the vacuum moduli spaces of classical and exceptional SQCD theories and also of the moduli spaces of instantons.
• We identify how the HWGs of gauge invariant operators of a selection of classical and exceptional SQCD theories result from the interaction under symmetrisation between a product group and the invariant tensors of its gauge group.