Pub Date : 2023-12-11DOI: 10.1017/s1446788723000216
NGUYEN N. HUNG, ALEXANDER MORETÓ, LUCIA MOROTTI
We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $mathsf {S}_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group G, with nonlinear irreducible characters of G as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by three. Lastly, the result for $mathsf {S}_n$ is applied to prove the nonequivalence of the metrics on permutations induced from faithful irreducible characters of the group.
我们研究有限群中不可还原字符之间的零共享行为。对于对称群 $mathsf {S}_n$，除了一个例外，任何两个不可还字符都至少有一个公共零点。为了进一步探讨这一现象，我们引入了有限群 G 的公共零图，以 G 的非线性不可还原字符为顶点，并以边连接在某些公共群元素上消失的字符。我们证明，对于可解群和简单群，该图的连通分量数以三为界。最后，我们应用 $mathsf {S}_n$ 的结果来证明由群的忠实不可还原字符诱导的排列上的度量的非等价性。
{"title":"COMMON ZEROS OF IRREDUCIBLE CHARACTERS","authors":"NGUYEN N. HUNG, ALEXANDER MORETÓ, LUCIA MOROTTI","doi":"10.1017/s1446788723000216","DOIUrl":"https://doi.org/10.1017/s1446788723000216","url":null,"abstract":"<p>We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231208131534917-0583:S1446788723000216:S1446788723000216_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {S}_n$</span></span></img></span></span>, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce <span>the common-zero graph</span> of a finite group <span>G</span>, with nonlinear irreducible characters of <span>G</span> as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by three. Lastly, the result for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231208131534917-0583:S1446788723000216:S1446788723000216_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {S}_n$</span></span></img></span></span> is applied to prove the nonequivalence of the metrics on permutations induced from faithful irreducible characters of the group.</p>","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"30 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138569372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-05DOI: 10.1017/s1446788723000186
MAKOTO KAWASHIMA
In this article, we prove a generalized Rodrigues formula for a wide class of holonomic Laurent series, which yields a new linear independence criterion concerning their values at algebraic points. This generalization yields a new construction of Padé approximations including those for Gauss hypergeometric functions. In particular, we obtain a linear independence criterion over a number field concerning values of Gauss hypergeometric functions, allowing the parameters of Gauss hypergeometric functions to vary.
{"title":"RODRIGUES FORMULA AND LINEAR INDEPENDENCE FOR VALUES OF HYPERGEOMETRIC FUNCTIONS WITH VARYING PARAMETERS","authors":"MAKOTO KAWASHIMA","doi":"10.1017/s1446788723000186","DOIUrl":"https://doi.org/10.1017/s1446788723000186","url":null,"abstract":"In this article, we prove a generalized Rodrigues formula for a wide class of holonomic Laurent series, which yields a new linear independence criterion concerning their values at algebraic points. This generalization yields a new construction of Padé approximations including those for Gauss hypergeometric functions. In particular, we obtain a linear independence criterion over a number field concerning values of Gauss hypergeometric functions, allowing <jats:italic>the parameters of Gauss hypergeometric functions to vary.</jats:italic>","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"28 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138531150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-09DOI: 10.1017/s1446788722000349
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"JAZ volume 115 issue 3 Cover and Front matter","authors":"","doi":"10.1017/s1446788722000349","DOIUrl":"https://doi.org/10.1017/s1446788722000349","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":" 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135242710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-09DOI: 10.1017/s1446788722000350
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"JAZ volume 115 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s1446788722000350","DOIUrl":"https://doi.org/10.1017/s1446788722000350","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":" 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135242708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-09DOI: 10.1017/s1446788722000362
{"title":"INDEX","authors":"","doi":"10.1017/s1446788722000362","DOIUrl":"https://doi.org/10.1017/s1446788722000362","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"1 1","pages":"431 - 431"},"PeriodicalIF":0.7,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139282345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-06DOI: 10.1017/s1446788723000149
SHOUDONG MAN, GUOQING ZHANG
Abstract Let $G=(V, E)$ be a locally finite graph with the vertex set V and the edge set E , where both V and E are infinite sets. By dividing the graph G into a sequence of finite subgraphs, the existence of a sequence of local solutions to several equations involving the p -Laplacian and the poly-Laplacian systems is confirmed on each subgraph, and the global existence for each equation on graph G is derived by the convergence of these local solutions. Such results extend the recent work of Grigor’yan, Lin and Yang [ J. Differential Equations , 261 (2016), 4924–4943; Rev. Mat. Complut. , 35 (2022), 791–813]. The method in this paper also provides an idea for investigating similar problems on infinite graphs.
{"title":"SOME GLOBAL EXISTENCE RESULTS ON LOCALLY FINITE GRAPHS","authors":"SHOUDONG MAN, GUOQING ZHANG","doi":"10.1017/s1446788723000149","DOIUrl":"https://doi.org/10.1017/s1446788723000149","url":null,"abstract":"Abstract Let $G=(V, E)$ be a locally finite graph with the vertex set V and the edge set E , where both V and E are infinite sets. By dividing the graph G into a sequence of finite subgraphs, the existence of a sequence of local solutions to several equations involving the p -Laplacian and the poly-Laplacian systems is confirmed on each subgraph, and the global existence for each equation on graph G is derived by the convergence of these local solutions. Such results extend the recent work of Grigor’yan, Lin and Yang [ J. Differential Equations , 261 (2016), 4924–4943; Rev. Mat. Complut. , 35 (2022), 791–813]. The method in this paper also provides an idea for investigating similar problems on infinite graphs.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135636457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-27DOI: 10.1017/s1446788723000137
EDMUND HENG, KIE SENG NGE
Abstract We show that a certain category of bimodules over a finite-dimensional quiver algebra known as a type B zigzag algebra is a quotient category of the category of type B Soergel bimodules. This leads to an alternate proof of Rouquier’s conjecture on the faithfulness of the 2-braid groups for type B .
{"title":"FAITHFULNESS OF THE 2-BRAID GROUP VIA ZIGZAG ALGEBRA IN TYPE B","authors":"EDMUND HENG, KIE SENG NGE","doi":"10.1017/s1446788723000137","DOIUrl":"https://doi.org/10.1017/s1446788723000137","url":null,"abstract":"Abstract We show that a certain category of bimodules over a finite-dimensional quiver algebra known as a type B zigzag algebra is a quotient category of the category of type B Soergel bimodules. This leads to an alternate proof of Rouquier’s conjecture on the faithfulness of the 2-braid groups for type B .","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"62 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136235258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-18DOI: 10.1017/s1446788723000101
Thomas Gobet
Abstract Several finite complex reflection groups have a braid group that is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order k for some $kgeq 2$ , and meridians are mapped to reflections. We study all possible quotients of torus knot groups obtained by requiring meridians to have finite order. Using the theory of J -groups of Achar and Aubert [‘On rank 2 complex reflection groups’, Comm. Algebra 36 (6) (2008), 2092–2132], we show that these groups behave like (in general, infinite) complex reflection groups of rank two. The large family of ‘toric reflection groups’ that we obtain includes, among others, all finite complex reflection groups of rank two with a single conjugacy class of reflecting hyperplanes, as well as Coxeter’s truncations of the $3$ -strand braid group. We classify these toric reflection groups and explain why the corresponding torus knot group can be naturally considered as its braid group. In particular, this yields a new infinite family of reflection-like groups admitting braid groups that are Garside groups. Moreover, we show that a toric reflection group has cyclic center by showing that the quotient by the center is isomorphic to the alternating subgroup of a Coxeter group of rank three. To this end we use the fact that the center of the alternating subgroup of an irreducible, infinite Coxeter group of rank at least three is trivial. Several ingredients of the proofs are purely Coxeter-theoretic, and might be of independent interest.
几个有限复反射群都有一个与环面结群同构的辫群。反射组是从环面结组中获得的，通过声明子午线对某些$kgeq 2$具有k阶，并且子午线被映射到反射。研究了通过要求子午线有有限阶而得到的环面结群的所有可能商。利用Achar和Aubert的J群理论[' On rank 2 complex reflection groups '， Comm.代数36(6)(2008)，2092-2132]，我们证明了这些群的行为类似于(一般来说，无限的)秩2的复反射群。我们得到的“环反射群”大族，除其他外，包括所有具有单一反射超平面共轭类的二阶有限复反射群，以及$3$ -strand辫群的Coxeter截断。我们对这些环面反射群进行了分类，并解释了为什么相应的环面结群可以自然地被认为是它的编织群。特别地，这产生了一个新的无限类反射群，承认辫群是Garside群。此外，通过证明中心的商同构于3阶Coxeter群的交替子群，证明了一个环反射群具有循环中心。为此，我们利用了这样一个事实，即秩至少为3的不可约无限Coxeter群的交替子群的中心是平凡的。证明的一些成分是纯粹的辅助理论，可能是独立的兴趣。
{"title":"TORIC REFLECTION GROUPS","authors":"Thomas Gobet","doi":"10.1017/s1446788723000101","DOIUrl":"https://doi.org/10.1017/s1446788723000101","url":null,"abstract":"Abstract Several finite complex reflection groups have a braid group that is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order k for some $kgeq 2$ , and meridians are mapped to reflections. We study all possible quotients of torus knot groups obtained by requiring meridians to have finite order. Using the theory of J -groups of Achar and Aubert [‘On rank 2 complex reflection groups’, Comm. Algebra 36 (6) (2008), 2092–2132], we show that these groups behave like (in general, infinite) complex reflection groups of rank two. The large family of ‘toric reflection groups’ that we obtain includes, among others, all finite complex reflection groups of rank two with a single conjugacy class of reflecting hyperplanes, as well as Coxeter’s truncations of the $3$ -strand braid group. We classify these toric reflection groups and explain why the corresponding torus knot group can be naturally considered as its braid group. In particular, this yields a new infinite family of reflection-like groups admitting braid groups that are Garside groups. Moreover, we show that a toric reflection group has cyclic center by showing that the quotient by the center is isomorphic to the alternating subgroup of a Coxeter group of rank three. To this end we use the fact that the center of the alternating subgroup of an irreducible, infinite Coxeter group of rank at least three is trivial. Several ingredients of the proofs are purely Coxeter-theoretic, and might be of independent interest.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135823697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-18DOI: 10.1017/s1446788723000125
VISHVESH KUMAR, MICHAEL RUZHANSKY
Abstract The main purpose of this paper is to prove Hörmander’s $L^p$ – $L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing the Paley inequality and Hausdorff–Young–Paley inequality for commutative hypergroups. We show the $L^p$ – $L^q$ boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli–Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the $L^p$ – $L^q$ norms of the heat kernel for generalised radial Laplacian.
摘要本文的主要目的是证明交换超群上傅里叶乘子Hörmander的$L^p$ - $L^q$有界性。我们通过建立可交换超群的Paley不等式和Hausdorff-Young-Paley不等式来实现这一目标。通过检验ch - trim -切超群的结果，我们证明了广义径向拉普拉斯算子的谱乘子的有界性。因此，我们得到了广义径向拉普拉斯热核的L^p$ - L^q$范数的嵌入定理和时间渐近性。
{"title":"– MULTIPLIERS ON COMMUTATIVE HYPERGROUPS","authors":"VISHVESH KUMAR, MICHAEL RUZHANSKY","doi":"10.1017/s1446788723000125","DOIUrl":"https://doi.org/10.1017/s1446788723000125","url":null,"abstract":"Abstract The main purpose of this paper is to prove Hörmander’s $L^p$ – $L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing the Paley inequality and Hausdorff–Young–Paley inequality for commutative hypergroups. We show the $L^p$ – $L^q$ boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli–Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the $L^p$ – $L^q$ norms of the heat kernel for generalised radial Laplacian.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"154 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135885044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-19DOI: 10.1017/s1446788723000113
MICHAELA CULLY-HUGILL, ADRIAN W. DUDEK
Abstract This paper gives an explicit version of Selberg’s mean-value estimate for the prime number theorem in intervals, assuming the Riemann hypothesis [25]. Two applications are given to short-interval results for primes and for Goldbach numbers. Under the Riemann hypothesis, we show there exists a prime in $(y,y+32,277log ^2 y]$ for at least half the $yin [x,2x]$ for all $xgeq 2$ , and at least one Goldbach number in $(x,x+9696 log ^2 x]$ for all $xgeq 2$ .
{"title":"AN EXPLICIT MEAN-VALUE ESTIMATE FOR THE PRIME NUMBER THEOREM IN INTERVALS","authors":"MICHAELA CULLY-HUGILL, ADRIAN W. DUDEK","doi":"10.1017/s1446788723000113","DOIUrl":"https://doi.org/10.1017/s1446788723000113","url":null,"abstract":"Abstract This paper gives an explicit version of Selberg’s mean-value estimate for the prime number theorem in intervals, assuming the Riemann hypothesis [25]. Two applications are given to short-interval results for primes and for Goldbach numbers. Under the Riemann hypothesis, we show there exists a prime in $(y,y+32,277log ^2 y]$ for at least half the $yin [x,2x]$ for all $xgeq 2$ , and at least one Goldbach number in $(x,x+9696 log ^2 x]$ for all $xgeq 2$ .","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135015074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$mathsf {S}_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group G, with nonlinear irreducible characters of G as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by three. Lastly, the result for 
$mathsf {S}_n$ is applied to prove the nonequivalence of the metrics on permutations induced from faithful irreducible characters of the group.
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号