Tsukuba Journal of Mathematics

Regular prehomogeneous vector spaces for valued Dynkin quivers

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

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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


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