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. Authors: (1) Wahei Hara; (2) Yuki Hirano. Authors: 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 Abstract and Intro Abstract and Intro Exchanges and Mutations of modifying modules Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Quasi-symmetric representation and GIT quotient Main results Main results Applications to Calabi-Yau complete intersections Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix A. Matrix factorizations Appendix B. List of Notation Appendix B. List of Notation References 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. Quasi-symmetric representations and magic windows. 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 3.10 There is an equivalence of groupoids Proposition 3.13 ([HSa, Proposition 6.5]). There is an equivalence Proposition 3.13 There is an equivalence extending the equivalence in Proposition 3.10. 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. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv

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Mutations of noncommutative crepant resolutions: Appendix A. Matrix factorizations

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