## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 43, Number 1 (2019), 71-111.

### Regular prehomogeneous vector spaces for valued Dynkin quivers

Tomohiro Kamiyoshi, Yoshiteru Kurosawa, Hiroshi Nagase, and Makoto Nagura

#### Abstract

We introduce regular prehomogeneous vector spaces associated with an arbitrary valued Dynkin graph $(\Gamma, \boldsymbol{v})$ having a fixed oriented modulation $(𝔐, \Omega)$ over the ground field $K$. Here $K$ is of characteristic zero, but it may not be algebraically closed. We will construct a fundamental theory of such prehomogeneous vector spaces.

Each generic point of a regular prehomogeneous vector space corresponds to a hom-orthogonal partial tilting $\Lambda$-module, where $\Lambda$ is the tensor $K$-algebra of $(𝔐, \Omega)$. We count the number of isomorphism classes of hom-orthogonal partial tilting $\Lambda$-modules of type $\mathbf{B}_n$, $\mathbf{C}_n$, $\mathbf{F}_4$ and $\mathbf{G}_2$. As a consequence of our theorem, we estimate lower and upper bounds for the number of basic relative invariants of regular prehomogeneous vector spaces for any valued Dynkin quiver.

#### Article information

**Source**

Tsukuba J. Math., Volume 43, Number 1 (2019), 71-111.

**Dates**

First available in Project Euclid: 25 October 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1571968822

**Digital Object Identifier**

doi:10.21099/tkbjm/1571968822

**Mathematical Reviews number (MathSciNet)**

MR4023315

**Subjects**

Primary: 16D80: Other classes of modules and ideals [See also 16G50]

Secondary: 16G20: Representations of quivers and partially ordered sets 11S90: Prehomogeneous vector spaces

**Keywords**

hom-orthogonal partial tilting module valued Dynkin quiver regular prehomogeneous vector space

#### Citation

Kamiyoshi, Tomohiro; Kurosawa, Yoshiteru; Nagase, Hiroshi; Nagura, Makoto. Regular prehomogeneous vector spaces for valued Dynkin quivers. Tsukuba J. Math. 43 (2019), no. 1, 71--111. doi:10.21099/tkbjm/1571968822. https://projecteuclid.org/euclid.tkbjm/1571968822