**Authors:**

(1) Wahei Hara;

(2) Yuki Hirano.

## Table of Links

- Abstract and Intro
- Exchanges and Mutations of modifying modules
- Quasi-symmetric representation and GIT quotient
- Main results
- Applications to Calabi-Yau complete intersections
- Appendix A. Matrix factorizations
- Appendix B. List of Notation
- References

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Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan

*Email address:* **wahei.hara@ipmu.jp**

Tokyo University of Agriculture and Technology, 2-24-16 Nakacho, Koganei, Tokyo 184-8588, Japan

*Email address:* **hirano@go.tuat.ac.jp**

This paper is available on arxiv under CC0 1.0 DEED license.