Abstract
We extend the theory of bilinear forms on the Green ring of a finite group developed by Benson and Parker to the context of the Grothendieck group of a triangulated category with Auslander–Reiten triangles, taking only relations given by direct sum decompositions. We examine the non-degeneracy of the bilinear form given by dimensions of homomorphisms, and show that the form may be modified to give a Hermitian form for which the standard basis given by indecomposable objects has a dual basis given by Auslander–Reiten triangles. An application is given to the homotopy category of perfect complexes over a symmetric algebra, with a consequence analogous to a result of Erdmann and Kerner.
To Dave Benson on his 60th birthday.
The author was supported by Simons Foundation award 282425.
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Webb, P. (2018). Bilinear Forms on Grothendieck Groups of Triangulated Categories. In: Carlson, J., Iyengar, S., Pevtsova, J. (eds) Geometric and Topological Aspects of the Representation Theory of Finite Groups. PSSW 2016. Springer Proceedings in Mathematics & Statistics, vol 242. Springer, Cham. https://doi.org/10.1007/978-3-319-94033-5_19
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DOI: https://doi.org/10.1007/978-3-319-94033-5_19
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