**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

## 3. Quasi-symmetric representation and GIT quotient

3.1. **Quasi-symmetric representations and magic windows.** This section recalls fundamental properties of derived categories of GIT quotients arising from quasi-symmetric representations, which are established in [HSa] and [SV1]. We freely use notation from Section 1.6.

and then it associates the GIT quotient stack [Xss(ℓ)/G].

**Proposition 3.10** ([HSa, Proposition 6.2]). *There is an equivalence of groupoids*

**Proposition 3.13** ([HSa, Proposition 6.5]). *There is an equivalence*

*extending the equivalence in Proposition 3.10.*

(3) This follows from (2).

The following is elementary, but we give a proof for the convenience of the reader

Proof. If W is trivial, the results are obvious. Thus, assume that W ̸= 1

The following result proves that this map is bijective.

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