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.
|Original language||English (US)|
|Title of host publication||Geometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016|
|Editors||Srikanth B. Iyengar, Julia Pevtsova, Jon F. Carlson|
|Publisher||Springer New York LLC|
|Number of pages||16|
|State||Published - 2018|
|Event||PIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016 - Vancouver, Canada|
Duration: Jul 27 2016 → Aug 5 2016
|Name||Springer Proceedings in Mathematics and Statistics|
|Other||PIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016|
|Period||7/27/16 → 8/5/16|
Bibliographical noteFunding Information:
The author was supported by Simons Foundation award 282425.
© 2018, Springer International Publishing AG.
- Auslander–Reiten triangle
- Green ring
- Perfect complex
- Symmetric algebra