Mutata

mutata

How Do You Go About Creating Biological Weapons That Cannot Be Countered?

The following piece claims to reduce an artist's cognitive overload in graphics design software using a simplified digital compositing node graph aka Pipeline. Proceed at your own risk. Content Overview An Example of Cognitive Overload A Mad Proposal A Mad Comparison to Digital Compositing Node Graph What Changed? Nothing, Really An Insane User Interface Then, things Start to get Crazier Digital Compositing Paradigm in a Pipeline Omg! Advantages. Lol! You must be Mad too An Example of Cognitive Overload Imagine a graphic artist performing a series of photo edits: 1. Apply Disc Blur to an image.
2. Apply Sepia Tone to Disc Blur.
3. Blend the Sepia Tone with the original. The intent of the artist usually consists of a sequence of actions to be translated into operations  carried out with the graphics software. For example, the process of the photo edits above may involve creating a new document, selecting the optimal image options in the application menu, creating Layers to apply a Disc Blur, changing properties in a side panel, bringing up an object dialog box, and repeating the process to add additional layers for the Sepia Tone/Blend to achieve the desired effect. This approach of user interface and when exacerbated with an increasing number of steps results in and . NON-STREAMLINED disrupts the flow of thoughts of the artist Cognitive Overload loss of creative productivity We are OK with the above. What's WRONG? A Mad Proposal We the user interface into a linear sequence of actions that can be compounded or stacked like a digital compositing node graph but with simplification to express how to achieve the desired outcome succinctly, like below: streamline Pipeline

1. Read Image
2. Disc Blur
3. Sepia Tone
4. Blend (2,3) A Pipeline is to the user interface of graphics design software like what Markdown is to HTML. A Mad Comparison to Digital Compositing Node Graph To someone familiar with Digital Compositing, they will immediately recognize that the list of steps, the Pipeline, above is "almost" identical to a digital compositing node graph. A digital compositing node graph is shown below: Read Image
|         |
|         |
|         V
|         Disc Blur
|         |
|         |
|         V
|         Sepia Tone
|         |
|         |
V         V
   Blend What Changed? Nothing, Really A Pipeline is indeed similar to a digital compositing node graph with a subtle yet significant change. Instead of displaying a full node graph graphically, with two lines from the Read Image node and Sepia Tone node connecting to the Blend node. The Pipeline simplifies the node graph by just showing a list of steps and a spreadsheet-like function for the final Blend step. 4. Blend (2,3) This creativity of arranging a digital compositing node graph as a Pipeline enables the intent of the artist to be conveyed as a succinct list of steps, following the artist's flow of thoughts from start to end, without a graphical map. The Pipeline also guides the artist into thinking of what to do next in a streamlined manner. Without using a traditional node graph, more information can be conveyed in a smaller amount of space linearly, providing a larger contextual overview to the creative mind. An Insane User Interface A Pipeline, in software, maps directly to a list and is easy to implement. The list removes the need for a complicated graphical map that may require (scrolling) to identify the right node to work on. The properties screen of a node can also be brought up intuitively and naturally by drilling into the node. panning Pipeline

1. Read Image
2. Disc Blur  ----> Drill into the Properties screen
3. Sepia Tone
4. Blend (2,3) PROTOTYPE - Fits into a mobile phone Then, Things Start to get Crazier No loss of information The important thing to note is the above creativity does not lose any information. The simplified node graph above, the Pipeline, in a mathematical sense, can be fully represented as a node graph (Directed Acyclic Graph) in any manner or form. For example, if we have the following node graph for video processing. Read media Read media 
|          |
V          V
Premult    Color Correction 
|          |
|          V 
|          Transform 
|          |
|          V 
|          Gaussian Blur 
|          | 
V          V 
Merge 
|
V 
Viewer It can be flattened, streamlined, and represented as follows: Read media
|
V
Premult

Read media
|
V
Color Correction
|
V
Transform
|
V
Gaussian Blur

Merge
|
V
Viewer Note the break from the first series after the 'Premult' node and the second break after the 'Gaussian Blur' node. Pipeline

1. Read media         >
2. Premult            >

3. Read media         >
4. Color Correction   >
5. Transform          >
6. Blur               >

7. Merge (2,6)        >
8. Viewer             > The space between node 2 and node 3 represents the break, while the '>' is optional and used to indicate the availability of a properties screen. Digital Compositing Paradigm in a Pipeline The paradigm of node-based digital compositing where we can lay out steps in a progression and make modifications to an earlier step to change the outcome is completely kept. Each node can be easily reused by referencing it with a node number as in Step 7, and if required, used further in a later step, saving time and effort. In the same way that a digital compositing node graph is broken into a series of nodes in a graph, a Pipeline breaks it into a series of actions or steps. A Pipeline, from another angle, is thus like a single-column spreadsheet, a metaphor that many computer users are quite familiar with. Viewing the output of each step: Viewer Cycling The arrangement in a series of steps enables the output of each progression step to be viewed easily, for example, by long pressing on a node on the Pipeline. This enables an artist to step through the progression to reinforce his or her cognitive working memory of what has been performed. Connecting the steps: Auto-Chaining Each node on the Pipeline that requires an input can be assumed to be taking the output of the previous node automatically. For example, the 'Premult' node in Step 2 is assumed to automatically take the output from 'Read Media' in Step 1. Connecting the steps like a node graph: Reverse Compositing The auto-chaining or automatic preceding node referencing behavior can also be altered by enabling the artist to make changes in the properties of a node. For example, in Step 7, the artist can choose from its properties screen any of the previous nodes, such as nodes 2 and 6, for blending. Omg! Advantages. Lol! Modular Extensibility A Pipeline, being almost identical to a digital compositing node graph, enables it to achieve the same modular extensibility in software development. Software developers can think of developing graphical effects, filters, animations, and 3D operations modularly as a node. Each node is accompanied by a properties user interface that does not interfere with other parts of the software. A node and its user interface can thus be developed and tested independently while knowing ultimately that it can be easily integrated into a full node graph, aka the Pipeline, to achieve any complex graphics operation. Such a design will benefit the open-source community greatly where many different developers can build an open-source Pipeline, the Open Pipeline, together in a distributed and collaborative manner. Reduce Information Overload To start using a graphics application, artists usually need to learn the way of achieving it through documentation. Often, the documentation requires an artist to follow a series of screenshots, explaining where and how to click or tap on different parts of the screen using a non-streamlined user interface offered by the graphics application. The non-streamlined user interface makes the information hard for an artist to remember and may require the artist to constantly refer to the documentation. Over time, this introduces an additional cost that, when accumulated, can lead to . Information Overload Succinct Self-Describing Pipeline The Open Pipeline reduces Information Overload as it succinctly describes what needs to be done and allows all necessary details to be shown in context nearby. This significantly benefits documentation: tutorials, articles, or books. Open Pipeline

1. Disc Blur: radius = 8
2. Sepia Tone: intensity = 1
3. Blend (1,2) The details of the radius for Disc Blur are shown as part of the node in the Pipeline. Moreover, besides being , it is common for the above description to supporting screenshots as it is self-explanatory. succinct NOT REQUIRE You must be Mad too If you have followed this article till here, find out more about the author's banishment due to his 'NOT SO' , and where he is living in exile. His ‘BLUFF’ of implementing a node that ‘CAN BE INTEGRATED EASILY’ into the Pipeline. And him performing a MAGIC TRICK. REALLY. He compares himself with the Roman poet Ovid by quoting on ‘how shapes changed into new bodies’ in Metamorphoses: Open Pipeline Photo Curve Photo Fire Effect – OVID In nova fert animus mutatas dicere formas corpora; di, coeptis (nam vos mutastis et illas) adspirate meis primaque ab origine mundi ad mea perpetuum deducite tempora carmen. Sigh, I give up He further mumbles about animating a graphics object with a node. METAMORPHOSIS The lead image for this article was generated by HackerNoon's AI Image Generator via the prompt "cognitive overload".

Revolutionary User Interface Design: Tackling Cognitive Overload in Graphics Software

Insect Adventures by Jean-Henri Fabre and Louise Hasbrouck Zimm, is part of the HackerNoon Books Series. You can jump to any chapter in this book here . THE GEOMETRY OF THE SPIDER’S WEB CHAPTER XXV. THE GEOMETRY OF THE SPIDER’S WEB [This chapter, one of the most wonderful in Fabre’s books, is included in a simplified form in this volume, on account of its interest to such younger readers as have studied geometry.] WHEN we look at the webs of the Garden Spiders, especially those of the Silky Spider and the Banded Spider, we notice first that the spokes or radii are equally spaced; the angles formed by each consecutive pair are of the same value; and this in spite of their number, which in the webs of the Silky Spider sometimes exceeds forty. We know in what a strange way the Spider weaves her web and divides the area of the web into a large number of equal parts or sectors, a number which is almost always the same in the work of each species of Spider. The Spider darts here and there when laying her spokes as if she had no plan, and this irresponsible way of working produces a beautiful web like the rose-window in a church, a web which no designer could have drawn better with compasses. We shall also notice that, in each sector, the various chords, parts of the angular spiral, are parallel to one another and gradually draw closer together as they near the center. With the two radiating lines that frame them they form obtuse angles on one side and acute angles on the other; and these angles remain constant in the same sector, because the chords are parallel. There is more than this: these same angles, the obtuse as well as the acute, do not alter in value, from one sector to another, as far as the eye can judge. Taken as a whole, therefore, the spiral consists of a series of cross-bars intersecting the several radiating lines obliquely at angles of equal value. By this characteristic we recognize what geometricians have named the “logarithmic spiral.” It is famous in science. The logarithmic spiral describes an endless number of circuits around its pole, to which it constantly draws nearer without ever being able to reach it. We could not see such a line, the whole of it, even with our best philosophical instruments. It exists only in the imagination of scientists. But the Spider knows it, and winds her spiral in the same way, and very accurately at that. Another property of this spiral is that if one in imagination winds a flexible thread around it, then unwinds the thread, keeping it taut the while, its free end will describe a spiral similar at all points to the original. The curve will merely have changed places. Jacques Bernouilli, the professor of mathematics who discovered this magnificent theorem, had engraved on his tomb, as one of his proudest titles to fame, the spiral and its double, made by the unwinding of the thread. Written underneath it was the sentence: Eadem mutata resurgo. “I rise again like unto myself.” It was a splendid flight of fancy which showed his belief in immortality. Now is this logarithmic spiral, with its curious properties, merely an idea of the geometricians? Is it a mere dream, an abstract riddle? No, it is a reality in the service of life, a method of construction often employed by animals in their architecture. The Mollusk never makes its shell without reference to the scientific curve. The first-born of the species knew it and put it into practice; it was as perfect in the dawn of creation as it can be to-day. There are perfect examples of this spiral found in the shells of fossils. To this day, the last representative of an ancient tribe, the Nautilus of the Southern Seas, remains faithful to the old design, and still whirls its spiral logarithmically, as did its ancestors in the earliest ages of the world’s existence. Even in the stagnant waters of our grassy ditches, a tiny Shellfish, no bigger than a duckweed, rolls its shell in the same manner. The common snail-shell is constructed according to logarithmic laws. Where do these creatures pick up this science? We are told that the Mollusk is descended from the Worm. One day the Worm, rendered frisky by the sun, brandished its tail and twisted it into a corkscrew for sheer glee. There and then the plan of the future spiral shell was discovered. This is what is taught quite seriously, in these days, as the very last word in science. But the Spider will have none of this theory. For she is not related to the Worm; and yet she is familiar with the logarithmic spiral and uses it in her web, in a simpler form. The Mollusk has years in which to build her spiral, so she makes it very perfectly. The Spider has only an hour at the most to spread her net, so she makes only a skeleton of the curve; but she knows the same line dear to the Snail. What guides her? Nothing but an inborn skill, whose effects the animal is no more able to control than the flower is able to control the arrangement of its petals and stamens. The Spider practices higher geometry without knowing or caring. The thing works of itself and takes its way from an instinct imposed upon creation at the start. The stone thrown by the hand returns to earth describing a certain curve; the dead leaf torn and wafted away by a breath of wind makes its journey from the tree to the ground with a similar curve. The curve is known to science and is called the “parabola.” The geometricians speculate still more about this curve; they imagine it rolling on an indefinite straight line and ask what course the focus of the curve follows. The answer comes that the focus of the parabola describes a “catenary,” a line whose algebraic symbol is so complicated that a numeral will not express it. The nearest it can get is this terrible sum: The geometricians do not attempt to refer to it by this number; they give it a letter, e. Is this line imaginary? Not at all; you may see the catenary frequently. It is the shape taken by a flexible cord when held at each end and relaxed; it is the line that governs the shape of a sail filled out by the wind. All this answers to the number e. What a quantity of abstruse science for a bit of string! Let us not be surprised. A pellet of shot swinging at the end of a thread, a drop of dew trickling down a straw, a splash of water rippling under the kisses of the air, a mere trifle, after all, becomes tremendously complicated when we wish to examine it with the eye of calculation. We need the club of Hercules to crush a fly. Our methods of mathematical investigation are certainly ingenious; we cannot too much admire the mighty brains that have invented them; but how slow and laborious they seem when compared with the smallest actual things! Shall we never be able to inquire into reality in a simpler fashion? Shall we be intelligent enough some day to do without all these heavy formulæ? Why not? Here we have the magic number e reappearing, written on a Spider’s thread. On a misty morning the sticky threads are laden with tiny drops, and, bending under the burden, have become so many catenaries, so many chains of limpid gems, graceful chaplets arranged in exquisite order and following the curve of a swing. If the sun pierce the mist, the whole lights up with rainbow-colored fires and becomes a dazzling cluster of diamonds. The number e is in its glory. Geometry, that is to say, the science of harmony in space, rules over everything. We find it in the arrangement of the scales of a fir-cone, as in the arrangement of a Spider’s sticky snare; we find it in the spiral of a snail-shell, in the chaplet of a Spider’s thread, as in the orbit of a planet; it is everywhere, as perfect in the world of atoms as in the world of immensities. And this universal geometry tells us of a Universal Geometrician, whose divine compass has measured all things. I prefer that, as an explanation of the logarithmic curve of the Nautilus and the Garden Spiders, to the Worm screwing up the tip of its tail. It may not perhaps be in agreement with some latter-day teaching, but it takes a loftier flight. About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Jean-Henri Fabre and Louise Hasbrouck Zimm (2014). Insect Adventures. Urbana, Illinois: Project Gutenberg. Retrieved October https://www.gutenberg.org/cache/epub/45812/pg45812-images.html This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org , located at https://www.gutenberg.org/policy/license.html .

THE GEOMETRY OF THE SPIDER’S WEB

THE GEOMETRY OF THE EPEIRA’S WEB

SECONDARY SEXUAL CHARACTERS OF MAN

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 5. Applications to Calabi-Yau complete intersections Therefore, (5.A) and (5.B) gives an equivalence Proposition 5.1. The restriction of (5.F to a magic window and the functor (5.G) are equivalences. Since the bottom functor is an equivalence by Theorem A.5, so is (5.G). for derived factorization categories is an equivalence by Theorem A.5. The following shows that the equivalences of magic windows generating the group action (5.D) correspond to mutation functors between noncommutative matrix factorizations. Proof. We only show that the left square commutes, since the commutativity of the right one follows from a similar argument. Consider the following diagram commutes, where the vertical equivalences are the compositions of (5.C) and (5.H). Lemma 5.5. There is an isomorphism where the first isomorphism follows from Lemma A.6. This finishes the proof. The following is a generalization of [KO, Theorem 8.5], which we prove by a similar argument as in loc. cit. Lemma 5.6. The following diagram commutes. Thus it is enough to show that there is a natural isomorphism By Lemma 5.6, there is an isomorphism Proof of Corollary 5.3. For simplicity, write Therefore, the assertion follows from Theorem 5.2. This paper is available on arxiv under CC0 1.0 DEED license. [1] Although [HSh] only discusses complexes, there are similar functors and semi-orthogonal decompositions for matrix factorizations by [BFK2], and so a similar argument as in [HSh] works in our setting. 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 5. Applications to Calabi-Yau complete intersections Therefore, (5.A) and (5.B) gives an equivalence Proposition 5.1. The restriction of (5.F Proposition 5.1. The restriction of (5.F to a magic window and the functor (5.G) are equivalences. are equivalences. Since the bottom functor is an equivalence by Theorem A.5, so is (5.G). for derived factorization categories is an equivalence by Theorem A.5. The following shows that the equivalences of magic windows generating the group action (5.D) correspond to mutation functors between noncommutative matrix factorizations. Proof . We only show that the left square commutes, since the commutativity of the right one follows from a similar argument. Consider the following diagram Proof commutes, where the vertical equivalences are the compositions of (5.C) and (5.H). commutes, where the vertical equivalences are the compositions of and Lemma 5.5. There is an isomorphism Lemma 5.5. There is an isomorphism where the first isomorphism follows from Lemma A.6. This finishes the proof. The following is a generalization of [KO, Theorem 8.5], which we prove by a similar argument as in loc. cit. Lemma 5.6. The following diagram commutes. Lemma 5.6. The following diagram commutes. Thus it is enough to show that there is a natural isomorphism By Lemma 5.6, there is an isomorphism Proof of Corollary 5.3. For simplicity, write Proof of Corollary 5.3. Therefore, the assertion follows from Theorem 5.2. 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 [1] Although [HSh] only discusses complexes, there are similar functors and semi-orthogonal decompositions for matrix factorizations by [BFK2], and so a similar argument as in [HSh] works in our setting.

Mutations of noncommutative crepant resolutions:  Applications to Calabi-Yau complete intersections

নন-কমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: ক্যালাবি-ইয়াউ সম্পূর্ণ ইন্টারসেকশনে আবেদন

Mutationen nichtkommutativer Crepant-Auflösungen: Anwendungen auf vollständige Calabi-Yau-Schnittpunkte

Mutaciones de resoluciones crepantes no conmutativas: aplicaciones a intersecciones completas Calabi-Yau

Mutations de résolutions crépantes non commutatives : applications aux intersections complètes de Calabi-Yau

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: कैलाबी-याउ पूर्ण प्रतिच्छेदन पर अनुप्रयोग

非可換クレパント分解の突然変異：カラビ・ヤウ完全交差への応用

비가환적 크레판트 해결의 돌연변이: Calabi-Yau 완전 교차점에 적용

Mutações de resoluções crepantes não comutativas: Aplicações a interseções completas de Calabi-Yau

Мутации некоммутативных крепантных резольвент: приложения к полным пересечениям Калаби-Яу

Değişmeyen crepant çözümlerin mutasyonları: Calabi-Yau tam kavşaklarına uygulamalar

Đột biến của độ phân giải crepant không giao hoán: Ứng dụng cho giao lộ hoàn chỉnh Calabi-Yau

非交换 crepant 解析的突变：应用于 Calabi-Yau 完全交集

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 Appendix B. List of Notation 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 Appendix B. List of Notation 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

Mutations of noncommutative crepant resolutions: Appendix B. List of Notation

ননকমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: পরিশিষ্ট বি. নোটেশনের তালিকা

Mutationen nichtkommutativer Crepant-Resolutionen: Anhang B. Liste der Notation

Mutaciones de resoluciones crepantes no conmutativas: Apéndice B. Lista de notación

Mutations de résolutions crépantes non commutatives : Annexe B. Liste des notations

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: परिशिष्ट बी. संकेतन की सूची

非可換クレパント分解の変形: 付録 B. 表記法の一覧

비가환적 crepant 해상도의 돌연변이: 부록 B. 표기법 목록

Mutações de resoluções crepantes não comutativas: Apêndice B. Lista de notação

Мутации некоммутативных крепантных резольвент: Приложение Б. Список обозначений

Değişmeyen crepant çözünürlüklerin mutasyonları: Ek B. Gösterim Listesi

Đột biến của độ phân giải crepant không giao hoán: Phụ lục B. Danh sách ký hiệu

非交换 crepant 解析的突变：附录 B。符号列表

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 4. Main results 4.1. Wall crossing and tilting equivalence. This section shows that wall-crossings of magic windows correspond to equivalences that are induced by tilting modules. Proof. By Teleman’s quantization theorem [Tel], for all k ∈ Z, the natural restriction map induces an isomorphism of equivalences is commutative. Proof. (1) The adjunction gives an isomorphism Therefore we only need to prove that the right hand sides of (4.E) and (4.F) are isomorphic functors. But this follows from a natural isomorphism Lemma 4.8. Notation is same as above. (2) This also follows from Lemma 3.19 and the fact that µδ,δ′ is a bijection. (3) This is a consequence of (2). For each F ∈ F(δ,δ′) Theorem 4.9. Notation is same as above. 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 4. Main results 4.1. Wall crossing and tilting equivalence. This section shows that wall-crossings of magic windows correspond to equivalences that are induced by tilting modules. Wall crossing and tilting equivalence. Proof. By Teleman’s quantization theorem [Tel], for all k ∈ Z, the natural restriction map induces an isomorphism of equivalences is commutative. of equivalences is commutative. Proof . (1) The adjunction gives an isomorphism Proof Therefore we only need to prove that the right hand sides of (4.E) and (4.F) are isomorphic functors. But this follows from a natural isomorphism Lemma 4.8. Notation is same as above. Lemma 4.8. Notation is same as above. (2) This also follows from Lemma 3.19 and the fact that µδ,δ′ is a bijection. (3) This is a consequence of (2). For each F ∈ F(δ,δ′) Theorem 4.9. Notation is same as above. Theorem 4.9. Notation is same as above. 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

Mutations of noncommutative crepant resolutions: Main results

নন-কমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: প্রধান ফলাফল

Mutationen nichtkommutativer Crepant-Auflösungen: Hauptergebnisse

Mutaciones de resoluciones crepantes no conmutativas: principales resultados

Mutations de résolutions crépantes non commutatives : principaux résultats

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: मुख्य परिणाम

非可換クレパント分解の変異: 主な結果

비가환적 크레판트 해상도의 돌연변이: 주요 결과

Mutações de resoluções crepantes não comutativas: Principais resultados

Мутации некоммутативных крепантных резолюций: основные результаты

Değişmeyen crepant çözünürlüklerin mutasyonları: Ana sonuçlar

Đột biến của độ phân giải crepant không giao hoán: Kết quả chính

非交换 crepant 解析的突变：主要结果

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 References [BFK1] M. Ballard, D. Favero and L. Katzarkov, A category of kernels for equivariant factorizations and its implications for Hodge theory. Publ. Math. Inst. Hautes Etudes Sci. ´ 120, 1–111 (2014). 36, 38 [BFK2] M. Ballard, D. Favero and L. Katzarkov, Variation of geometric invariant theory quotients and derived categories, J. Reine Angew. Math. 746, 235–303 (2019). 32, 35 [BDFIK] M. Ballard, D. Deliu, D. Favero, M. U. Isik, and L. Katzarkov, Resolutions in factorization categories. Adv. Math. 295, 195–249 (2016). 34, 36 [BLS] D. Bergh, V. A. Lunts, O. M. Schn¨urer1, Geometricity for derived categories of algebraic stacks, Selecta Math. (N.S.) 22 (2016), no. 4, 2535–2568. 9 [Bri] T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), no. 3, 613–632. 1 [BH] W. Bruns and J. Herzog, Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics, 39. 7, 12 [Che] J.-C. Chen, Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities, J. Differential Geom. 61 (2002), no. 2, 227–261. 1 [Har1] W. Hara, Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops. Adv. Math.318(2017), 355–410. 2, 4, 31 [Har2] W. Hara, On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists. Matematiche (Catania) 77(2022), no.2, 329–371. 2, 4, 10, 30, 31 [Hal] D. Halpern-Leistner, The derived category of a GIT quotient. J. Amer. Math. Soc. 28 (2015), no. 3, 871–912. 23 [HSa] D. Halpern-Leistner and S. V. Sam, Combinatorial constructions of derived equivalences. J. Amer. Math. Soc. 33, no. 3, 871–912 (2020). 1, 2, 3, 12, 13, 14, 16, 21, 31, 32, 33 [HSh] D. Halpern-Leistner and I. Shipman, Autoequivalences of derived categories via geometric invariant theory. Adv. Math. 303, 1264–1299 (2016). 34, 35 [HN] A. Higashitani and Y. Nakajima, Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions. Selecta Math. (N.S.) 25(2019), no.5, Paper No. 78, 25 pp. 2, 4 [Hir1] Y. Hirano, Equivalences of derived factorization categories of gauged Landau-Ginzburg models. Adv. Math. 306, 200–278 (2017). 36 [Hir2] Y. Hirano, Derived Kn¨orrer periodicity and Orlov’s theorem for gauged Landau-Ginzburg models. Compos. Math. 153, no. 5, 973–1007 (2017). 32 [Hir3] Y. Hirano, Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix Factorizations . SIGMA Symmetry Integrability Geom. Methods Appl. 17, Paper No. 055, 43 pp (2021). 5, 36, 37, 38 [HW1] Y. Hirano and M. Wemyss, Faithful actions from hyperplane arrangements, Geom. Topol. 22 (2018), no. 6, 3395–3433. 1 [HW2] Y. Hirano and M. Wemyss, Stability conditions for 3-fold flops, arXiv:1907.09742. 1, 3 [HR] J. Hall, D. Rydh, Perfect complexes on algebraic stacks, Compos. Math. 153 (2017), no. 11, 2318–2367. 9 [Isi] M. U. Isik, Equivalence of the derived category of a variety with a singularity category, Int. Math. Res. Not. IMRN (2013), no. 12, 2787–2808. 32 [IR] O. Iyama and I. Reiten, Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras. Am. J. Math. 130 (4), 1087–1149 (2008). 5 [IW1] O. Iyama and M. Wemyss, Maximal modifications and Auslander-Reiten duality for non-isolated singularities. Invent. Math. 197 (2014), no. 3, 521–586. 1, 2, 6, 7, 8, 11 [IW2] O. Iyama and M. Wemyss, Tits cones intersections and applications, preprint. 3 [Kaw] Y. Kawamata, Flops connect minimal models, Publ. RIMS, 44, 419–423 (2008). 1 [KO] N. Koseki and G. Ouchi, Perverse schobers and Orlov equivalences, Eur. J. Math. 9, no. 2, Paper No. 32, 38 pp (2023). 36 [Nak] Y. Nakajima, Mutations of splitting maximal modifying modules: the case of reflexive polygons. Int. Math. Res. Not. IMRN(2019), no.2, 470–550. 2 [OT] C. Okonek and A. Teleman, Graded tilting for gauged Landau-Ginzubrg models and geometric applications. arXiv:1907.10099. 5, 37 [Pos] L. Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. Mem. Amer. Math. Soc. 212 (2011), no. 966. 36 [Shi] I. Shipman, A geometric approach to Orlov’s theorem, Compos. Math. 148, no. 5, 1365-1389 (2012). 32 [SV1] S. ˇ Spenko and M. Van den Bergh, ˇ Non-commutative resolutions of quotient singularities for reductive groups. Invent. Math. 210, no. 1, 3–67 (2017). 1, 12, 21, 22, 25 [SV2] S. ˇ Spenko and M. Van den Bergh, ˇ Non-commutative crepant resolutions for some toric singularities I. Int. Math. Res. Not. IMRN(2020), no.21, 8120–8138. 10 [SV3] S. ˇ Spenko and M. Van den Bergh, ˇ Non-commutative crepant resolutions for some toric singularities. II. J. Noncommut. Geom. 14 (2020), no. 1, 73–103. 9 [SV4] S. ˇ Spenko and M. Van den Bergh, ˇ Tilting bundles on hypertoric varieties. Int. Math. Res. Not. IMRN(2021), no.2, 1034–1042. 31 [SV5] S. ˇ Spenko and M. Van den Bergh, J.-P. Bell, ˇ On the noncommutative Bondal-Orlov conjecture for some toric varieties. Math. Z. 300(2022), no.1, 1055–1068. 2 [Sta] The Stacks Project Authors, Stacks Project. https://stacks.math.columbia.edu 12 [Tel] C. Teleman, The quantization conjecture revisited. Ann. of Math. (2) 152 (2000), no. 1, 1–43. 23, 24 [Van1] M. Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455. 1 [Van2] M. Van den Bergh, Non-commutative crepant resolutions. The Legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2004) 1, 2 [Van3] M. Van den Bergh, Non-commutative crepant resolutions, an overview. arXiv:2207.09703. 1, 6, 26 [Wem] M. Wemyss, Flops and Clusters in the Homological Minimal Model Program, Invent. Math. 211 (2018), no. 2, 435–521. 1, 2, 30, 31 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. 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 References [BFK1] M. Ballard, D. Favero and L. Katzarkov, A category of kernels for equivariant factorizations and its implications for Hodge theory. Publ. Math. Inst. Hautes Etudes Sci. ´ 120, 1–111 (2014). 36, 38 [BFK2] M. Ballard, D. Favero and L. Katzarkov, Variation of geometric invariant theory quotients and derived categories, J. Reine Angew. Math. 746, 235–303 (2019). 32, 35 [BDFIK] M. Ballard, D. Deliu, D. Favero, M. U. Isik, and L. Katzarkov, Resolutions in factorization categories. Adv. Math. 295, 195–249 (2016). 34, 36 [BLS] D. Bergh, V. A. Lunts, O. M. Schn¨urer1, Geometricity for derived categories of algebraic stacks, Selecta Math. (N.S.) 22 (2016), no. 4, 2535–2568. 9 [Bri] T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), no. 3, 613–632. 1 [BH] W. Bruns and J. Herzog, Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics, 39. 7, 12 [Che] J.-C. Chen, Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities, J. Differential Geom. 61 (2002), no. 2, 227–261. 1 [Har1] W. Hara, Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops. Adv. Math.318(2017), 355–410. 2, 4, 31 [Har2] W. Hara, On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists. Matematiche (Catania) 77(2022), no.2, 329–371. 2, 4, 10, 30, 31 [Hal] D. Halpern-Leistner, The derived category of a GIT quotient. J. Amer. Math. Soc. 28 (2015), no. 3, 871–912. 23 [HSa] D. Halpern-Leistner and S. V. Sam, Combinatorial constructions of derived equivalences. J. Amer. Math. Soc. 33, no. 3, 871–912 (2020). 1, 2, 3, 12, 13, 14, 16, 21, 31, 32, 33 [HSh] D. Halpern-Leistner and I. Shipman, Autoequivalences of derived categories via geometric invariant theory. Adv. Math. 303, 1264–1299 (2016). 34, 35 [HN] A. Higashitani and Y. Nakajima, Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions. Selecta Math. (N.S.) 25(2019), no.5, Paper No. 78, 25 pp. 2, 4 [Hir1] Y. Hirano, Equivalences of derived factorization categories of gauged Landau-Ginzburg models. Adv. Math. 306, 200–278 (2017). 36 [Hir2] Y. Hirano, Derived Kn¨orrer periodicity and Orlov’s theorem for gauged Landau-Ginzburg models. Compos. Math. 153, no. 5, 973–1007 (2017). 32 [Hir3] Y. Hirano, Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix Factorizations . SIGMA Symmetry Integrability Geom. Methods Appl. 17, Paper No. 055, 43 pp (2021). 5, 36, 37, 38 [HW1] Y. Hirano and M. Wemyss, Faithful actions from hyperplane arrangements, Geom. Topol. 22 (2018), no. 6, 3395–3433. 1 [HW2] Y. Hirano and M. Wemyss, Stability conditions for 3-fold flops, arXiv:1907.09742. 1, 3 [HR] J. Hall, D. Rydh, Perfect complexes on algebraic stacks, Compos. Math. 153 (2017), no. 11, 2318–2367. 9 [Isi] M. U. Isik, Equivalence of the derived category of a variety with a singularity category, Int. Math. Res. Not. IMRN (2013), no. 12, 2787–2808. 32 [IR] O. Iyama and I. Reiten, Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras. Am. J. Math. 130 (4), 1087–1149 (2008). 5 [IW1] O. Iyama and M. Wemyss, Maximal modifications and Auslander-Reiten duality for non-isolated singularities. Invent. Math. 197 (2014), no. 3, 521–586. 1, 2, 6, 7, 8, 11 [IW2] O. Iyama and M. Wemyss, Tits cones intersections and applications, preprint. 3 [Kaw] Y. Kawamata, Flops connect minimal models, Publ. RIMS, 44, 419–423 (2008). 1 [KO] N. Koseki and G. Ouchi, Perverse schobers and Orlov equivalences, Eur. J. Math. 9, no. 2, Paper No. 32, 38 pp (2023). 36 [Nak] Y. Nakajima, Mutations of splitting maximal modifying modules: the case of reflexive polygons. Int. Math. Res. Not. IMRN(2019), no.2, 470–550. 2 [OT] C. Okonek and A. Teleman, Graded tilting for gauged Landau-Ginzubrg models and geometric applications. arXiv:1907.10099. 5, 37 [Pos] L. Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. Mem. Amer. Math. Soc. 212 (2011), no. 966. 36 [Shi] I. Shipman, A geometric approach to Orlov’s theorem, Compos. Math. 148, no. 5, 1365-1389 (2012). 32 [SV1] S. ˇ Spenko and M. Van den Bergh, ˇ Non-commutative resolutions of quotient singularities for reductive groups. Invent. Math. 210, no. 1, 3–67 (2017). 1, 12, 21, 22, 25 [SV2] S. ˇ Spenko and M. Van den Bergh, ˇ Non-commutative crepant resolutions for some toric singularities I. Int. Math. Res. Not. IMRN(2020), no.21, 8120–8138. 10 [SV3] S. ˇ Spenko and M. Van den Bergh, ˇ Non-commutative crepant resolutions for some toric singularities. II. J. Noncommut. Geom. 14 (2020), no. 1, 73–103. 9 [SV4] S. ˇ Spenko and M. Van den Bergh, ˇ Tilting bundles on hypertoric varieties. Int. Math. Res. Not. IMRN(2021), no.2, 1034–1042. 31 [SV5] S. ˇ Spenko and M. Van den Bergh, J.-P. Bell, ˇ On the noncommutative Bondal-Orlov conjecture for some toric varieties. Math. Z. 300(2022), no.1, 1055–1068. 2 [Sta] The Stacks Project Authors, Stacks Project. https://stacks.math.columbia.edu 12 [Tel] C. Teleman, The quantization conjecture revisited. Ann. of Math. (2) 152 (2000), no. 1, 1–43. 23, 24 [Van1] M. Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455. 1 [Van2] M. Van den Bergh, Non-commutative crepant resolutions. The Legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2004) 1, 2 [Van3] M. Van den Bergh, Non-commutative crepant resolutions, an overview. arXiv:2207.09703. 1, 6, 26 [Wem] M. Wemyss, Flops and Clusters in the Homological Minimal Model Program, Invent. Math. 211 (2018), no. 2, 435–521. 1, 2, 30, 31 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 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 Email address: hirano@go.tuat.ac.jp 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

Mutations of noncommutative crepant resolutions: References

নন-কমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: রেফারেন্স

Mutationen nichtkommutativer Crepant-Auflösungen: Referenzen

Mutaciones de resoluciones crepantes no conmutativas: referencias

Mutations de résolutions crépantes non commutatives : références

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: संदर्भ

非可換クレパント解決の突然変異: 参考文献

비가환적 크레판트 해상도의 돌연변이: 참고자료

Mutações de resoluções crepantes não comutativas: Referências

Мутации некоммутативных крепантных резолюций: Литература.

Değişmeyen crepant çözünürlüklerin mutasyonları: Referanslar

Đột biến của độ phân giải crepant không giao hoán: Tài liệu tham khảo

非交换 crepant 解析的突变：参考文献

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 1. Introduction 1.1. Backgrounds. A crepant resolution is one of the best modifications of singularities. This can be regarded as a higher dimensional analog of minimal resolutions of surface singularities, and in the terminology from minimal model theory, a crepant resolution can be paraphrased as a smooth minimal model of the singularity. As a noncommutative analog of the notion of crepant resolutions, Van den Bergh introduced noncommutative crepant resolutions (= NCCRs) [Van2, Van3]. In both commutative and noncommutative cases, the existence of such a resolution is not always true. There are two large classes of singularities for which the study of NCCRs (and crepant resolutions) is well established. One is the class of quotient singularities arising from quasi-symmetric representations of reductive groups, which was first studied in [SV1], and the other is the class of (3-fold) compound du Val singularities, studied in [Van1, Wem]. To investigate the latter class, Iyama and Wemyss [IW1] introduced an operation called mutation, which produce a new NCCR from the original one. By Kawamata [Kaw], it is known that all minimal models (and hence all crepant resolutions) are connected by iterated flops, and mutations can be regarded as a noncommutative counterpart of flops. Indeed, it is proved in [Wem] that the derived equivalences associated to 3-fold flops, which are established in [Bri, Che], correspond to derived equivalences associated to mutations of NCCRs. This interpretation and the technique of mutations of NCCRs provides the main ingredients for the study of Bridgeland stability conditions for 3-fold flops [HW1, HW2]. The main purpose of this paper is to import such a technology established by [IW1] to deepen the study of NCCRs for quotient singularities arising from quasi-symmetric representations, by studying the combinatorics associated to the representation, and by accessing to the ideas from [HSa, SV1]. 1.2. Exchanges and mutations of modifying modules. The present section, Section 1.3, and Section 1.4 explain the setting of this paper, and recall some terminologies, notations, and known results that are needed to state our results. The precise statements of main results are given in Section 1.5. Let R be a normal equidimensional Gorenstein ring. A finitely generated reflexive Rmodule M is said to be modifying if the endomorphism ring EndR(M) is Cohen-Macaulay as an R-module. A noncommutative crepant resolution (=NCCR) of R is the endomorphism ring Λ = EndR(M) of some modifying R-module M such that the global dimension of Λ is finite. If EndR(M) is an NCCR, we say that M gives an NCCR. The following is one of the central problems about NCCRs. Conjecture 1.1 ([Van2]). Let R be an equidimensional normal Gorenstein ring. Then all crepant resolutions and all NCCRs of R are derived equivalent. In relation to this derived equivalence problem, Iyama and Wemys It is natural to ask if two given NCCRs are connected by (iterated) mutations or not. It is known that, for many types of singularities, their natural NCCRs are actually connected by mutations [Har1, Har2, HN, Nak, SV5, Wem]. One of the main aims of this paper is to present a similar result for NCCRs associated to the quotient of quasi-symmetric representations, which are recalled in the next section. will be constructed, and using this yields The following is our main result. Acknowledgements. W. H. would like to thank Prof. Michael Wemyss for discussions and comments. W. H. was supported by EPSRC grant EP/R034826/1 and by ERC Consolidator Grant 101001227 (MMiMMa). Y. H. was supported by JSPS KAKENHI 19K14502. 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 1. Introduction 1.1. Backgrounds. A crepant resolution is one of the best modifications of singularities. This can be regarded as a higher dimensional analog of minimal resolutions of surface singularities, and in the terminology from minimal model theory, a crepant resolution can be paraphrased as a smooth minimal model of the singularity. 1.1. Backgrounds. As a noncommutative analog of the notion of crepant resolutions, Van den Bergh introduced noncommutative crepant resolutions (= NCCRs) [Van2, Van3]. In both commutative and noncommutative cases, the existence of such a resolution is not always true. There are two large classes of singularities for which the study of NCCRs (and crepant resolutions) is well established. One is the class of quotient singularities arising from quasi-symmetric representations of reductive groups, which was first studied in [SV1], and the other is the class of (3-fold) compound du Val singularities, studied in [Van1, Wem]. To investigate the latter class, Iyama and Wemyss [IW1] introduced an operation called mutation, which produce a new NCCR from the original one. By Kawamata [Kaw], it is known that all minimal models (and hence all crepant resolutions) are connected by iterated flops, and mutations can be regarded as a noncommutative counterpart of flops. Indeed, it is proved in [Wem] that the derived equivalences associated to 3-fold flops, which are established in [Bri, Che], correspond to derived equivalences associated to mutations of NCCRs. This interpretation and the technique of mutations of NCCRs provides the main ingredients for the study of Bridgeland stability conditions for 3-fold flops [HW1, HW2]. The main purpose of this paper is to import such a technology established by [IW1] to deepen the study of NCCRs for quotient singularities arising from quasi-symmetric representations, by studying the combinatorics associated to the representation, and by accessing to the ideas from [HSa, SV1]. 1.2. Exchanges and mutations of modifying modules. The present section, Section 1.3, and Section 1.4 explain the setting of this paper, and recall some terminologies, notations, and known results that are needed to state our results. The precise statements of main results are given in Section 1.5. Exchanges and mutations of modifying modules. Let R be a normal equidimensional Gorenstein ring. A finitely generated reflexive Rmodule M is said to be modifying if the endomorphism ring EndR(M) is Cohen-Macaulay as an R-module. A noncommutative crepant resolution (=NCCR) of R is the endomorphism ring Λ = EndR(M) of some modifying R-module M such that the global dimension of Λ is finite. If EndR(M) is an NCCR, we say that M gives an NCCR. The following is one of the central problems about NCCRs. Conjecture 1.1 ([Van2]). Let R be an equidimensional normal Gorenstein ring. Then all crepant resolutions and all NCCRs of R are derived equivalent. In relation to this derived equivalence problem, Iyama and Wemys Conjecture 1.1 It is natural to ask if two given NCCRs are connected by (iterated) mutations or not. It is known that, for many types of singularities, their natural NCCRs are actually connected by mutations [Har1, Har2, HN, Nak, SV5, Wem]. One of the main aims of this paper is to present a similar result for NCCRs associated to the quotient of quasi-symmetric representations, which are recalled in the next section. will be constructed, and using this yields The following is our main result. Acknowledgements . W. H. would like to thank Prof. Michael Wemyss for discussions and comments. W. H. was supported by EPSRC grant EP/R034826/1 and by ERC Consolidator Grant 101001227 (MMiMMa). Y. H. was supported by JSPS KAKENHI 19K14502. Acknowledgements 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

Mutations of noncommutative crepant resolutions: Abstract and Intro

নন-কমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: অ্যাবস্ট্রাক্ট এবং ইন্ট্রো

Mutationen nichtkommutativer Crepant-Auflösungen: Zusammenfassung und Einführung

Mutaciones de resoluciones crepantes no conmutativas: resumen e introducción

Mutations de résolutions crépantes non commutatives : résumé et introduction

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: सार और परिचय

非可換クレパント解像度の突然変異: 概要と序論

비가환적 크레판트 해결의 돌연변이: 초록 및 소개

Mutações de resoluções crepantes não comutativas: Resumo e introdução

Мутации некоммутативных крепантных резолюций: Аннотация и введение

Değişmeyen crepant çözünürlüklerin mutasyonları: Özet ve Giriş

Đột biến của độ phân giải crepant không giao hoán: Tóm tắt và Giới thiệu

非交换性 crepant 解析的突变：摘要和简介

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