Hom-Orthogonal Partial Tilting Modules for Dynkin Quivers

Abstract

We count the number of isomorphism classes of hom-orthogonal partial tilting modules over path algebras of Dynkin quiver of type 𝔸 _n , 𝔻 _n , 𝔼 _n . This number is independent on the choice of an orientation of the arrows, and the number for 𝔸 _n or 𝔻 _n -type can be expressed as a special value of a hypergeometric function. As a consequence of our theorem, we obtain a minimum value of the number of basic relative invariants of corresponding regular prehomogeneous vector spaces.

Key Words:

• Dynkin quiver
• Hom-orthogonal partial tilting module
• Regular prehomogeneous vector space

2010 Mathematics Subject Classification:

• Primary 16D80
• Secondary 16G20, 11S90

ACKNOWLEDGMENT

The authors would like to thank Tokuji Araya and Osamu Iyama for their helpful comments on the maximum number of pairwise non-isomorphic indecomposable direct summands of hom-orthogonal partial tilting modules.

Notes

Communicated by D. Zacharia.