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We consider Little String Theories (LSTs) that are engineered by $N$ parallelM5-branes probing a transverse $\mathbb{Z}_M$ geometry. By exploiting a dualdescription in terms of F-theory compactified on a toric Calabi-Yau threefold$X_{N,M}$, we establish numerous symmetries that leave the BPS partitionfunction $\mathcal{Z}_{N,M}$ invariant. They furthemore act in anon-perturbative fashion from the point of view of the low energy quiver gaugetheory associated with the LST. We present different group theoreticalorganisations of these symmetries, thereby generalising the results of[arXiv:1811.03387] to the case of generic $M \geq 1$. We also provide aMathematica package that allows to represent them in terms of matrices that actlinearly on the K\"ahler parameters of $X_{N,M}$. From the perspective of dualrealisations of the LSTs the symmetries found here act in highly nontrivialways: as an example, we consider a formulation of $\mathcal{Z}_{N,M}$ in termsof correlation functions of a vertex operator algebra, whose commutationrelations are governed by an affine quiver algebra. We show the impact of thesymmetry transformations on the latter and discuss invariance of$\mathcal{Z}_{N,M}$ from this perspective for concrete examples.

Non-perturbative Symmetries of Little Strings and Affine Quiver Algebras