Five-brane webs and highest weight representations

We consider BPS-counting functions Z(N,M) of N parallel M5-branes probing a transverse Z(M) orbifold geometry. These brane web con fi gurations can be dualised into a class of toric non-compact Calabi-Yau threefolds which have the structure of an elliptic fi bration over (affine)A(N-1). We make this symmetry of Z(N,M) manifest in particular regions of the parameter space of the setup: we argue that for specific choices of the deformation parameters, the supercharges of the system acquire particular holonomy charges which lead to in fi nitely many cancellations among states contributing to the partition function. The resulting (simplified) Z(N,M) can be written as a sum over weights forming a single irreducible representation of the Lie algebra a(N-1) (or its affine counterpart). We show this behaviour explicitly for an extensive list of examples for specific values of (N, M) and generalise the arising pattern for generic con fi gurations. Finally, for a particular compact M5-brane setup we use this form of the partition function to make the duality N <-> M manifest.