{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
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