The first theorem of Andrey Roiter

The first theorem of Andrey Roiter

The article contains memories about A.V. Roiter.

Збережено в:
Бібліографічні деталі
Дата:2012
Автор: Lichtman, A.I.
Формат: Стаття
Мова:English
Опубліковано: Інститут прикладної математики і механіки НАН України 2012
Назва видання:Algebra and Discrete Mathematics
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/152241
Теги: Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:The first theorem of Andrey Roiter / A.I. Lichtman // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 236–238. — Бібліогр.: 3 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id irk-123456789-152241
record_format dspace
spelling irk-123456789-1522412019-06-10T01:26:21Z The first theorem of Andrey Roiter Lichtman, A.I. The article contains memories about A.V. Roiter. 2012 Article The first theorem of Andrey Roiter / A.I. Lichtman // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 236–238. — Бібліогр.: 3 назв. — англ. 1726-3255 2010 MSC:20C10. http://dspace.nbuv.gov.ua/handle/123456789/152241 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The article contains memories about A.V. Roiter.
format Article
author Lichtman, A.I.
spellingShingle Lichtman, A.I.
The first theorem of Andrey Roiter
Algebra and Discrete Mathematics
author_facet Lichtman, A.I.
author_sort Lichtman, A.I.
title The first theorem of Andrey Roiter
title_short The first theorem of Andrey Roiter
title_full The first theorem of Andrey Roiter
title_fullStr The first theorem of Andrey Roiter
title_full_unstemmed The first theorem of Andrey Roiter
title_sort first theorem of andrey roiter
publisher Інститут прикладної математики і механіки НАН України
publishDate 2012
url http://dspace.nbuv.gov.ua/handle/123456789/152241
citation_txt The first theorem of Andrey Roiter / A.I. Lichtman // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 236–238. — Бібліогр.: 3 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT lichtmanai thefirsttheoremofandreyroiter
AT lichtmanai firsttheoremofandreyroiter
first_indexed 2025-07-13T02:37:57Z
last_indexed 2025-07-13T02:37:57Z
_version_ 1837497614944174080
fulltext Jo ur na l A lg eb ra D is cr et e M at h.Algebra and Discrete Mathematics SURVEY ARTICLE Volume 14 (2012). Number 2. pp. 236 – 238 c© Journal “Algebra and Discrete Mathematics” The first theorem of Andrey Roiter A. I. Lichtman Communicated by V. V. Kirichenko Abstract. The article contains memories about A.V. Roiter. I first met Andrey in the Fall of 1959, in Uzhgorod, at a conference organized by I.R. Shafarevich and D.K. Faddeev. Most of the talks at the conference were on algebraic geometry, homological algebra and algebraic topology, but there were also two survey talks on representation theory–by D.K. Faddeev on unimodular representations, and by S.D. Berman on modular representations of finite groups. In his talk Faddeev quoted a number of important theorems on uni- modular representations-the classical Jordan-Zassenhaus theorem, the theorems of Maranda on p-adic representations, and described in detail the recent problem which arose in connection with the results of Diederichsen on unimodular representations of cyclic groups. In fact, Diederichsen proved in 1938 (see [1]) that if G is a cyclic group of prime order p then the number of indecomposable representation is finite and it is equal to 2h + 1 where h is the number of the ideal classes of the pth cyclotomic field Z(ǫ); another proof of this result was given by Reiner in [2]. The situation for cyclic groups of higher order was considerably more complicated, by this time the only fact that was known was that the paper of Diederichsen contained an infinite series of indecomposable representations for the cyclic group of order 4. 2010 MSC: 20C10. Key words and phrases: matrices, unimodular representations. Jo ur na l A lg eb ra D is cr et e M at h. A. I. Lichtman 237 Faddeev announced that his student, A.V. Roiter, proved that the cyclic group of order 4 had only 9 indecomposable representations, so if this was true then Diederichsen’s result was incorrect. Roiter’s paper did not appear yet, at this time it was very difficult to get a preprint because the Soviet government believed that copy machines were a threat for the existence of the “dictatorship of the proletariat”, so we could not read the proof of Roiter’s theorem; this proof could not be presented in one hour talk because it was based on quite tedious computations. Faddeev wrote on the blackboard the nine matrices which gave all the indecomposable representations of the cyclic group of order 4:      1 1 0 1 0 0 −1 0 0 1 0 0 0 0 0 −1      ,      1 2 0 0 0 0 −1 1 0 1 0 0 0 0 0 −1      ,      1 1 0 0 0 0 −1 1 0 1 0 0 0 0 0 −1      ,    0 −1 1 1 0 0 0 0 −1    ,    1 1 0 0 0 −1 0 1 0    , ( 0 −1 1 0 ) , ( 1 1 0 −1 ) , ( −1 ) , ( 1 ) . Everyone, including Faddeev, was quite surprised by this result. Fad- deev said that he understood that seven of Roiter’s representations were obtained in the ideals of the integral group ring, but there were two mysterious representations (“tainstvennie predstavleniya”). Andrey Roiter’s paper appeared a year later, in 1960 in [3]. His proof was correct and he pointed out that Diederichsen’s paper contained a mistake. Andrey did not mention in his paper that while working on his theorem he did not know about Diederichsen’s paper, it was Faddeev who said that Andrey learned about the existence of this paper after he had already proven his theorem. We all highly value every mathematical fact which expands our knowl- edge or changes our view. However when I think about Roiter’s first theorem I value not only its content and importance but I remember also that he proved it when he was an undegraduate student who just began his research activity and that at the time no one in the Soviet Union worked on these types of problems. This was really a remarkable achievement! Jo ur na l A lg eb ra D is cr et e M at h. 238 The first theorem of Andrey Roiter References [1] F.E. Diederichsen, Über die Ausreduction ganzzahliger Gruppendarstellungen bei arithmetischer Äquivalenz, Hamb. Abh. 14 (1938), 357-412. [2] I. Reiner, Integral representations of cyclic groups pf prime order, Proc. Amer. Math. Soc. 8 (1957), 142-146. [3] A.V. Roiter, On representations of cyclic group of order 4 by integral matrices, Vestn. Leningr. Univ. 19 (1960), 65-74. Contact information A. I. Lichtman Emeritus Professor, University of Wisconsin E-Mail: lichtman@uwp.edu Received by the editors: 27.10.2012 and in final form 31.10.2012.

Схожі ресурси