## 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.

Original language | English (US) |
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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 |

Pages | 465-480 |

Number of pages | 16 |

ISBN (Print) | 9783319940328 |

DOIs | |

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 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 242 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

Other | PIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016 |
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Country | Canada |

City | Vancouver |

Period | 7/27/16 → 8/5/16 |

### Bibliographical note

Funding Information:The author was supported by Simons Foundation award 282425.

## Keywords

- Auslander–Reiten triangle
- Green ring
- Perfect complex
- Symmetric algebra