arXiv:2610.35916v1 Announce Type: new Abstract: We give an alternative proof of Mongardi's result that every finite group of symplectic automorphisms of a hyperk\"ahler manifold of $K3^{[n]}$-type acts trivially on the discriminant groups of its second cohomology lattice and its coinvariant lattice. The proof combines sphere-packing bounds, Vinberg's reflection chambers and structure results on even $2$-elementary lattices.
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