{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T00:03:04Z","timestamp":1768262584431,"version":"3.49.0"},"reference-count":45,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2013]]},"abstract":"<jats:p>Following our paper [Linear Algebra Appl. 433(2010), 699\u2013717], we present a framework and computational tools for the Coxeter spectral classification of finite posets<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>J<\/mml:mi><mml:mo>\u2261<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>J<\/mml:mi><mml:mo>,<\/mml:mo><mml:mo>\u2aaf<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by Drozd [Funct. Anal. Appl. 8(1974), 219\u2013225]. We are mainly interested in a Coxeter spectral classification of posets<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:math>such that the symmetric Gram matrix<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:msub><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>:<\/mml:mo><mml:mo>=<\/mml:mo><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mo>\/<\/mml:mo><mml:mrow><mml:mn>2<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mo stretchy=\"false\">[<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><mml:mrow><mml:mtext>t<\/mml:mtext><mml:mtext>r<\/mml:mtext><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">]<\/mml:mo><mml:mo>\u2208<\/mml:mo><mml:msub><mml:mrow><mml:mi>\ud835\udd44<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u211a<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>is positive semidefinite, where<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>\u2208<\/mml:mo><mml:msub><mml:mrow><mml:mi>\ud835\udd44<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u2124<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>is the incidence matrix of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:math>. Following the idea of Drozd mentioned earlier, we associate to<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:math>its Coxeter matrix<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:msub><mml:mrow><mml:mtext>Cox<\/mml:mtext><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>:<\/mml:mo><mml:mo>=<\/mml:mo><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>\u00b7<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mtext>t<\/mml:mtext><mml:mtext>r<\/mml:mtext><\/mml:mrow><\/mml:msubsup><\/mml:math>, its Coxeter spectrum<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">s<\/mml:mi><mml:mi mathvariant=\"bold\">p<\/mml:mi><mml:mi mathvariant=\"bold\">e<\/mml:mi><mml:mi mathvariant=\"bold\">c<\/mml:mi><mml:mi mathvariant=\"bold\">c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>, a Coxeter polynomial<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:msub><mml:mrow><mml:mtext>cox<\/mml:mtext><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2208<\/mml:mo><mml:mi>\u2124<\/mml:mi><mml:mo stretchy=\"false\">[<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo stretchy=\"false\">]<\/mml:mo><\/mml:math>, and a Coxeter number<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:msub><mml:mrow><mml:mi>\u2009\u2009<\/mml:mi><mml:mi mathvariant=\"bold-italic\">c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>. In case<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mrow><mml:msub><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>is positive semi-definite, we also associate to<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:math>a reduced Coxeter number<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mrow><mml:msub><mml:mrow><mml:mi>\u2009\u2009<\/mml:mi><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">c<\/mml:mi><\/mml:mrow><mml:mo>\u030c<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>, and the defect homomorphism<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:msub><mml:mrow><mml:mo>\u2202<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>:<\/mml:mo><mml:msup><mml:mrow><mml:mi>\u2124<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>\u2192<\/mml:mo><mml:mi>\u2124<\/mml:mi><\/mml:math>. In this case, the Coxeter spectrum<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">s<\/mml:mi><mml:mi mathvariant=\"bold\">p<\/mml:mi><mml:mi mathvariant=\"bold\">e<\/mml:mi><mml:mi mathvariant=\"bold\">c<\/mml:mi><mml:mi mathvariant=\"bold\">c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>is a subset of the unit circle and consists of roots of unity. In case<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mrow><mml:msub><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>is positive semi-definite of corank one, we relate the Coxeter spectral properties of the posets<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:math>with the Coxeter spectral properties of a simply laced Euclidean diagram<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M18\"><mml:mi>D<\/mml:mi><mml:mi>J<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mo stretchy=\"false\">{<\/mml:mo><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>\ud835\udd3b<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>\ud835\udd3c<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mrow><mml:mn>6<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>\ud835\udd3c<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mrow><mml:mn>7<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>\ud835\udd3c<\/mml:mi><\/mml:mrow><mml:mo>\u0303<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mrow><mml:mn>8<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:math>associated with<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M19\"><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:math>. Our aim of the Coxeter spectral analysis of such posets<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M20\"><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:math>is to answer the question when the Coxeter type<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M21\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">C<\/mml:mi><mml:mi mathvariant=\"bold\">t<\/mml:mi><mml:mi mathvariant=\"bold\">y<\/mml:mi><mml:mi mathvariant=\"bold\">p<\/mml:mi><mml:mi mathvariant=\"bold\">e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>:<\/mml:mo><mml:mo>=<\/mml:mo><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">s<\/mml:mi><mml:mi mathvariant=\"bold\">p<\/mml:mi><mml:mi mathvariant=\"bold\">e<\/mml:mi><mml:mi mathvariant=\"bold\">c<\/mml:mi><mml:mi mathvariant=\"bold\">c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u2009\u2009<\/mml:mi><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold\">c<\/mml:mi><\/mml:mrow><mml:mo>\u030c<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M22\"><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:math>determines its incidence matrix<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M23\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>(and, hence, the poset<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M24\"><mml:mrow><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:math>) uniquely, up to a<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M25\"><mml:mrow><mml:mi>\u2124<\/mml:mi><\/mml:mrow><\/mml:math>-congruency. In connection with this question, we also discuss the problem studied by Horn and Sergeichuk [Linear Algebra Appl. 389(2004), 347\u2013353], if for any<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M26\"><mml:mrow><mml:mi>\u2124<\/mml:mi><\/mml:mrow><\/mml:math>-invertible matrix<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M27\"><mml:mi>A<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:msub><mml:mrow><mml:mi>\ud835\udd44<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u2124<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, there is<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M28\"><mml:mi>B<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:msub><mml:mrow><mml:mi>\ud835\udd44<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u2124<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>such that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M29\"><mml:msup><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mtext>t<\/mml:mtext><mml:mtext>r<\/mml:mtext><\/mml:mrow><\/mml:msup><mml:mo>=<\/mml:mo><mml:msup><mml:mrow><mml:mi>B<\/mml:mi><\/mml:mrow><mml:mrow><mml:mtext>t<\/mml:mtext><mml:mtext>r<\/mml:mtext><\/mml:mrow><\/mml:msup><mml:mo>\u00b7<\/mml:mo><mml:mi>A<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mi>B<\/mml:mi><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M30\"><mml:msup><mml:mrow><mml:mi>B<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>=<\/mml:mo><mml:mi>E<\/mml:mi><\/mml:math>is the identity matrix.<\/jats:p>","DOI":"10.1155\/2013\/743734","type":"journal-article","created":{"date-parts":[[2013,3,21]],"date-time":"2013-03-21T21:07:12Z","timestamp":1363900032000},"page":"1-22","source":"Crossref","is-referenced-by-count":16,"title":["A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots"],"prefix":"10.1155","volume":"2013","author":[{"given":"Daniel","family":"Simson","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Ulica Chopina 12\/18, 87-100 Toru\u0144, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4139-9276","authenticated-orcid":true,"given":"Katarzyna","family":"Zaj\u0105c","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Ulica Chopina 12\/18, 87-100 Toru\u0144, Poland"}]}],"member":"311","reference":[{"key":"33","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2010.03.041"},{"key":"36","volume":"123","year":"2013","journal-title":"Fundamenta Informatica"},{"key":"37","volume":"125","year":"2013","journal-title":"Fundamenta Informaticae"},{"key":"38","volume":"27","year":"2013","journal-title":"SIAM Journal on Discrete Mathematics"},{"key":"44","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(82)90033-6"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(02)00391-9"},{"issue":"5","key":"5a","first-page":"38","volume":"3","year":"1991","journal-title":"Algebra i Analiz"},{"key":"5b","first-page":"973","volume":"3","year":"1992","journal-title":"St. Petersburg Mathematical Journal"},{"issue":"2","key":"6","first-page":"31","volume":"12","year":"2011","journal-title":"Algebra and Discrete Mathematics"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2008.06.038"},{"issue":"2","key":"8","first-page":"20","volume":"2","year":"2005","journal-title":"Algebra and Discrete Mathematics"},{"key":"9","first-page":"17","volume":"2","year":"2006","journal-title":"Algebra and Discrete Mathematics"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1007\/s11253-009-0147-7"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1007\/s11253-009-0245-6"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2009.02.033"},{"key":"14","series-title":"Encyclopaedia of Mathematical Sciences","volume":"73","year":"1992"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2011.10.045"},{"key":"16","doi-asserted-by":"crossref","first-page":"83","DOI":"10.4064\/cm127-1-6","volume":"127","year":"2012","journal-title":"Colloquium Mathematicum"},{"issue":"4","key":"20","doi-asserted-by":"crossref","first-page":"437","DOI":"10.3233\/FI-2011-572","volume":"111","year":"2011","journal-title":"Fundamenta Informaticae"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.4064\/cm87-1-3"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.3233\/FI-2012-731"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2007.08.002"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.4064\/cm89-1-10"},{"key":"26","first-page":"3","volume-title":"Integral weakly positive forms","year":"1978"},{"key":"29","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(00)00150-6"},{"key":"30","volume":"4","year":"1992"},{"key":"31","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(93)90138-J"},{"key":"32","doi-asserted-by":"publisher","DOI":"10.4064\/cm115-2-9"},{"key":"42","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(03)00299-1"},{"issue":"1","key":"43","volume":"14","year":"2007","journal-title":"Electronic Journal of Combinatorics"},{"key":"45","doi-asserted-by":"publisher","DOI":"10.1080\/00927878908823853"},{"issue":"3","key":"13","first-page":"219","volume":"8","year":"1974","journal-title":"Functional Analysis and its Applications"},{"key":"27"},{"issue":"4","key":"35","doi-asserted-by":"crossref","first-page":"425","DOI":"10.3233\/FI-2011-520","volume":"109","year":"2011","journal-title":"Fundamenta Informaticae"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2004.03.010"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511614309"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.4064\/cm120-2-8"},{"key":"39","series-title":"Tubes and Concealed Algebras of Euclidean Type, London Mathematical Society Student Texts 71","year":"2007"},{"issue":"4","key":"40","first-page":"389","volume":"83","year":"2008","journal-title":"Fundamenta Informaticae"},{"key":"34","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2010.02.029"},{"issue":"4","key":"3","first-page":"339","volume":"17","year":"1999","journal-title":"Expositiones Mathematicae"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2005.03.036"},{"key":"17"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2010.06.052"},{"key":"41"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2013\/743734.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2013\/743734.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2013\/743734.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,7,24]],"date-time":"2020-07-24T05:24:42Z","timestamp":1595568282000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2013\/743734\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"references-count":45,"alternative-id":["743734","743734"],"URL":"https:\/\/doi.org\/10.1155\/2013\/743734","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013]]}}}