najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2023-14-4-405-412najo-114Research ArticleMATHEMATICSМАТЕМАТИКАIrreducible characters of the icosahedral groupНеприводимые характеры группы икосаэдраhttps://orcid.org/0000-0002-8489-7665КанемицуС.KanemitsuS.

S. Kanemitsu – Sanmenxia SUDA New Energy Research Institute, No. 1, Taiyang Road, Sanmenxia Economic Development Zone

Sanmenxia, Henan, 472000, P. R. China

omnikanemitsu@yahoo.comhttps://orcid.org/0000-0003-1739-9639МехтаДжейMehtaJ.

Jay Mehta – Department of Mathematics, Sardar Patel University

 Vallabh Vidyanagar, Gujarat 388 120

jay_mehta@spuvvn.eduhttps://orcid.org/0000-0003-0433-8917СанЮ.SunY.

Y. Sun – Graduate School of Engrg., Kyushu Inst. Tech.

1-1Sensuicho Tobata, Kitakyushu 804-8555

sun@ele.kyutech.ac.jpSanmenxia SUDA New Energy Research InstituteChinaDepartment of Mathematics, Sardar Patel UniversityIndiaGraduate School of Engrg., Kyushu Inst. Tech.Japan202303062025144405412Copyright © Kanemitsu S., Mehta J., Sun Y., 20252025Канемицу С., Мехта Д., Сан Ю.Kanemitsu S., Mehta J., Sun Y.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/114

To study point groups, their irreducible characters are essential. The table of irreducible characters of the icosahedral group A5 is usually obtained by using its duality to the dodecahedral group. It seems that there is no literature which gives a routine computational way to complete it. In the works of Harter and Allen, a computational method is given and the character table up to the tetrahedral group A4 using the group algebra table and linear algebra. In this paper, we employ their method with the aid of computer programming to complete the table. The method is applicable to any other more complicated groups. 

Для изучения точечных групп существенны их неприводимые характеры. Таблицу неприводимых характеров икосаэдрической группы А5 обычно получают с помощью ее двойственности к додекаэдрической группе. Кажется, что не существует литературы, которая бы предлагала рутинный вычислительный способ его выполнения. В работах Хартера и Аллена дан вычислительный метод и таблица характеров с точностью до тетраэдрической группы А4 с использованием таблицы групповой алгебры и линейной алгебры. В этой статье мы используем их метод с помощью компьютерного программирования для заполнения таблицы. Метод применим и к любым другим, более сложным группам.

группа икосаэдранеприводимое представлениепростые характерырегулярное представлениесобственные значенияicosahedral groupirreducible representationsimple charactersregular representationeigenvaluesReferencesAndova V., Kardos F., Skrekovski R. Mathematical aspects of fullerenes. Ars Mathematica Contemporanea, 2016, 11, P. 353–379.Andova V., Kardos F., Skrekovski R. Mathematical aspects of fullerenes. Ars Mathematica Contemporanea, 2016, 11, P. 353–379.Dresselhaus M.S. Dresselhaus D. and Saito Sh. Group-theoretical concepts for C60 and other fullerenes., Material Sci. Engrg., 1993, B19, P. 122– 128.Dresselhaus M.S. Dresselhaus D. and Saito Sh. Group-theoretical concepts for C60 and other fullerenes., Material Sci. Engrg., 1993, B19, P. 122– 128.Nikolaev A.V., Plakhutin B.N. C60 fullerene as a pseudoatom of the icosahedral symmetry. Russian Chemical Reviews, 2010, 79(9), P. 729–755.Nikolaev A.V., Plakhutin B.N. C60 fullerene as a pseudoatom of the icosahedral symmetry. Russian Chemical Reviews, 2010, 79(9), P. 729–755.Pleshakov I.V., Proskurina O.V., Gerasimov V.I., Kuz’min Yu.I. Optical anisotropy in fullerene-containing polymer composites induced by magnetic field. Nanosystems: Phys. Chem. Math., 2022, 13(5), P. 503–508.Pleshakov I.V., Proskurina O.V., Gerasimov V.I., Kuz’min Yu.I. Optical anisotropy in fullerene-containing polymer composites induced by magnetic field. Nanosystems: Phys. Chem. Math., 2022, 13(5), P. 503–508.Akizuki Y. Abstract Algebra, Kyoritsu-shuppan, Tokyo, 1956.Akizuki Y. Abstract Algebra, Kyoritsu-shuppan, Tokyo, 1956.Conway J.H. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Cambridge UP, Cambridge, 1985.Conway J.H. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Cambridge UP, Cambridge, 1985.Lomont J.S. Applications of finite groups, Academic Press, New York, 1959.Lomont J.S. Applications of finite groups, Academic Press, New York, 1959.Harter W.G. Applications of algebraic representation theory. PhD Thesis, Univ. of Calif., Irvine, 1967.Harter W.G. Applications of algebraic representation theory. PhD Thesis, Univ. of Calif., Irvine, 1967.Harter W.G. Algebraic theory of ray representations of finite groups. J. Math. Phys., 1968, 10, P. 739–752.Harter W.G. Algebraic theory of ray representations of finite groups. J. Math. Phys., 1968, 10, P. 739–752.Allen G. An efficient method for computation of character tables of finite groups. NASA Techn. Rept. (TN D-4763), Nat. Aerodyn. & space adm., Washington, 1968.Allen G. An efficient method for computation of character tables of finite groups. NASA Techn. Rept. (TN D-4763), Nat. Aerodyn. & space adm., Washington, 1968.Kanemitsu S., Kuzumaki T. and Liu J.Y. Intermediate abstract algebra, Preprint 2024.Kanemitsu S., Kuzumaki T. and Liu J.Y. Intermediate abstract algebra, Preprint 2024.

The authors declare that there are no conflicts of interest present.