Multiplicative Invariant Fields of Dimension ≤6

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials Pub Date : 2023-03-01 DOI:10.1090/memo/1403

A. Hoshi, M. Kang, A. Yamasaki

{"title":"Multiplicative Invariant Fields of Dimension ≤6","authors":"A. Hoshi, M. Kang, A. Yamasaki","doi":"10.1090/memo/1403","DOIUrl":null,"url":null,"abstract":"<p>The finite subgroups of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L 4 left-parenthesis double-struck upper Z right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mn>4</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">GL_4(\\mathbb {Z})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are classified up to conjugation in Brown, Büllow, Neubüser, Wondratscheck, and Zassenhaus (1978); in particular, there exist <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"710\">\n <mml:semantics>\n <mml:mn>710</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">710</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> non-conjugate finite groups in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L 4 left-parenthesis double-struck upper Z right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mn>4</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">GL_4(\\mathbb {Z})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Each finite group <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L 4 left-parenthesis double-struck upper Z right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mn>4</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">GL_4(\\mathbb {Z})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> acts naturally on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z Superscript circled-plus 4\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>⊕</mml:mo>\n <mml:mn>4</mml:mn>\n </mml:mrow>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Z}^{\\oplus 4}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>; thus we get a faithful <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-lattice <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\n <mml:semantics>\n <mml:mi>M</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal r normal a normal n normal k Subscript double-struck upper Z Baseline upper M equals 4\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">r</mml:mi>\n <mml:mi mathvariant=\"normal\">a</mml:mi>\n <mml:mi mathvariant=\"normal\">n</mml:mi>\n <mml:mi mathvariant=\"normal\">k</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mi>M</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mn>4</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {rank}_\\mathbb {Z} M=4</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. In this way, there are exactly <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"710\">\n <mml:semantics>\n <mml:mn>710</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">710</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> such lattices. Given a <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-lattice <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\n <mml:semantics>\n <mml:mi>M</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal r normal a normal n normal k Subscript double-struck upper Z Baseline upper M equals 4\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">r</mml:mi>\n <mml:mi mathvariant=\"normal\">a</mml:mi>\n <mml:mi mathvariant=\"normal\">n</mml:mi>\n <mml:mi mathvariant=\"normal\">k</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mi>M</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mn>4</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {rank}_\\mathbb {Z} M=4</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, the group <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> acts on the rational function field <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper C left-parenthesis upper M right-parenthesis colon-equal double-struck upper C left-parenthesis x 1 comma x 2 comma x 3 comma x 4 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>M</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>≔</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>x</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>x</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>x</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>x</mml:mi>\n <mml:mn>4</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {C}(M)≔\\mathbb {C}(x_1,x_2,x_3,x_4)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> by multiplicative actions, i.e. purely monomial automorphisms over <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper C\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {C}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We are concerned with the rationality problem of the fixed field <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper C left-parenthesis upper M right-parenthesis Superscript upper G\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>M</mml:mi>\n <mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mi>G</mml:mi>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {C}(M)^G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. A tool of our investigation is the unramified Brauer group of the field <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper C left-parenthesis upper M right-parenthesis Superscript upper G\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n <mml:mo str","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/memo/1403","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}

引用次数: 0

Abstract

The finite subgroups of G L 4 ( Z ) GL_4(\mathbb {Z}) are classified up to conjugation in Brown, Büllow, Neubüser, Wondratscheck, and Zassenhaus (1978); in particular, there exist 710 710 non-conjugate finite groups in G L 4 ( Z ) GL_4(\mathbb {Z}) . Each finite group G G of G L 4 ( Z ) GL_4(\mathbb {Z}) acts naturally on Z ⊕ 4 \mathbb {Z}^{\oplus 4} ; thus we get a faithful G G -lattice M M with r a n k Z M = 4 \mathrm {rank}_\mathbb {Z} M=4 . In this way, there are exactly 710 710 such lattices. Given a G G -lattice M M with r a n k Z M = 4 \mathrm {rank}_\mathbb {Z} M=4 , the group G G acts on the rational function field C ( M ) ≔ C ( x 1 , x 2 , x 3 , x 4 ) \mathbb {C}(M)≔\mathbb {C}(x_1,x_2,x_3,x_4) by multiplicative actions, i.e. purely monomial automorphisms over C \mathbb {C} . We are concerned with the rationality problem of the fixed field C ( M ) G \mathbb {C}(M)^G . A tool of our investigation is the unramified Brauer group of the field C

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维数≤6的乘不变域

在Brown，Büllow，Neubüser，Wondratscheck和Zassenhaus（1978）中，对G L 4（Z）GL_4（\mathbb｛Z｝）的有限子群进行了共轭分类；特别地，在G L4（Z）GL_4（\mathbb｛Z｝）中存在710 710个非共轭有限群。G L4（Z）GL_4（\mathbb｛Z｝）的每个有限群G G自然作用于ZŞ4\mathbb{Z｝^｛\oplus 4｝；因此我们得到了一个忠实的G G-格M M，其中r a n k Z M=4{rank}_\mathb{Z}M=4。通过这种方式，正好有710 710个这样的晶格。给定一个具有r a n k Z M=4\ mathrm的G G-格M M{rank}_\mathbb｛Z｝M=4，群G G通过乘法作用作用于有理函数域C（M）≔C（x1，x2，x3，x4）\mathbb｛C｝（M）\ mathbb{C}（x_1，x_2，x_3，x_4），即C\mathbb｛C｝上的纯单体自同构。我们讨论了固定域C（M）G\mathbb｛C｝（M）^G的合理性问题。我们研究的一个工具是域C

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