{"id":602,"date":"2015-01-03T13:52:53","date_gmt":"2015-01-03T18:52:53","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/sin\/?page_id=602"},"modified":"2026-03-19T08:10:54","modified_gmt":"2026-03-19T12:10:54","slug":"representation-theory-of-finite-groups-mas739609cb-2","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/sin\/courses\/representation-theory-of-finite-groups-mas739609cb-2\/","title":{"rendered":"Representation Theory of Finite Groups (MAS7396\/09CB)"},"content":{"rendered":"\r\n
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Representation Theory of Finite Groups (MAS7396\/09CB)<\/h1>\n\n\n\n

Instructor<\/h3>\n\n\n\n

Prof. Peter Sin <\/a><\/p>\n\n\n\n

Time and Location<\/h3>\n\n\n\n

MWF Per. 7, Little 237.<\/p>\n\n\n\n

Office Hours:<\/h3>\n\n\n\n

(Tentative) MTW Per. 6,\u00a0 Little 432, and by appointment.<\/p>\n\n\n\n\n\n

Description and Goals<\/h3>\n\n\n\n

Group representation theory is the study of group
\nactions on vector spaces where each group element acts as a linear transformation.
\nWe will concentrate the case of finite groups, but we will consider
\nvector spaces both of characteristic 0 (“ordinary” representation theory)
\nand of characteristic p (“modular” representation theory). The aim will be
\nto alternate between developing the theory and examining examples.\n<\/p>\n\n\n\n\n\n

Part I. Generalities.<\/h3>\n\n\n\n

We shall begin with general background on modules that apply in all cases and work out
\nsome small examples. Topics include:
\nGroup algebras; modules and representations; direct sums and indecomposable modules;
\nHom spaces,tensor products; canonical isomorphisms; induction and restriction,
\npermutation modules; the trace map, tensor identity, Nakayama relations (Frobius reciprocity); Mackey formula.\n<\/p>\n\n\n\n\n\n

Part II. Character Theory.<\/h3>\n\n\n\n

In the ordinary theory all modules are semisimple and the theory can be further developed
\nin the form of character theory. Highlights will include Burnside’s p^aq^b Theorem and
\nFrobenius’s Theorem, and Brauer’s characterization of characters. These
\nare spectacular applications of character theory to answer questions about groups.<\/p>\n\n\n\n\n\n

Part III. Modular representation theory.<\/h3>\n\n\n\n

The modular theory is more subtle and homological algebra comes into play.
\nWe shall consider projective modules, extensions, and filtrations.
\nThe ordinary and modular theories are tied together by the study of modules
\nover a local principal ideal domain whose quotient field is of characteristic zero
\nand whose residue field is of characteristic p. This leads to Brauer’s block theory, whereby ordinary and modular representations are grouped into “blocks”.
\nBlock theory leads to deeper arithmetical results on characters through
\nthe study of modular representations, and also has many fascinating
\nproblems of its own. We hope to end by discussing a famous
\nconjecture of Alperin which is now one of the main focal points in this subject. <\/p>\n\n\n\n\n\n\n\n\n\n

UF Grades Policy<\/h3>\n\n\n\n

The UF regulations on grades are here: https:\/\/catalog.ufl.edu\/ugrad\/current\/regulations\/info\/grades.aspx <\/a><\/p>\n\n\n\n\n\n

The UF policy on minus grades is here:\u00a0 https:\/\/catalog.ufl.edu\/UGRD\/academic-regulations\/grades-grading-policies\/#:~:text=No%20grades%20will%20be%20calculated,credit%20is%20awarded%20only%20once.&text=Colleges%20may%20not%20accept%20grade,credit%20in%20the%20same%20field..<\/a><\/p>\n\n\n\n\n\n\n\n\n\n

Attendance and Late Policy<\/h3>\n\n\n\n

Attendance is recommended.<\/p>\n\n\n\n\n\n

The UF policy on attendance is here: https:\/\/catalog.ufl.edu\/ugrad\/current\/regulations\/info\/attendance.aspx<\/a><\/p>\n\n\n\n

Special accommodations<\/h3>\n\n\n\n

Students requesting classroom accommodation must first register with the Dean of Student Office. The Dean of Students Office will provide documentation to the student who must then provide this documentation to the instructor when requesting accommodation.<\/p>\n\n\n\n

Honor Code<\/h3>\n\n\n\n

UF students are bound by The Honor Pledge which states, \u201cWe, the members of the University of Florida community, pledge to hold ourselves and our peers to the highest standards of honor and integrity by abiding by the Honor Code.
\nOn all work submitted for credit by students at the University of Florida, the following pledge is either
\nrequired or implied: \u201cOn my honor, I have neither given nor received unauthorized aid in doing this
\nassignment.\u201d<\/p>\n\n\n\n\n\n

The Honor Code (http:\/\/www.dso.ufl.edu\/sccr\/process\/student-conduct-honor-code\/<\/a>)
\nspecifies a number of behaviors that are in violation of this code and the possible sanctions.
\nFurthermore, you are obliged to report any condition that facilitates academic misconduct to appropriate
\npersonnel. If you have any questions or concerns, please consult with the instructor of this class.<\/p>\n\n\n\n

Course Evaluations<\/h3>\n\n\n\n

Students are expected to provide feedback on the quality of instruction in this course based on 10 criteria. These evaluations are conducted online at https:\/\/evaluations.ufl.edu<\/a>. Evaluations are typically open during the last two or three weeks of the semester, but students will be given specific times when they are open. Summary results of these assessments are available to students at https:\/\/evaluations.ufl.edu\/results\/.<\/a>
\n<\/a><\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t<\/div>\r\n<\/section>\r\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":142,"featured_media":0,"parent":8,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"featured_post":"","footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-602","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/sin\/wp-json\/wp\/v2\/pages\/602","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/people.clas.ufl.edu\/sin\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/people.clas.ufl.edu\/sin\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/sin\/wp-json\/wp\/v2\/users\/142"}],"replies":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/sin\/wp-json\/wp\/v2\/comments?post=602"}],"version-history":[{"count":2,"href":"https:\/\/people.clas.ufl.edu\/sin\/wp-json\/wp\/v2\/pages\/602\/revisions"}],"predecessor-version":[{"id":1469,"href":"https:\/\/people.clas.ufl.edu\/sin\/wp-json\/wp\/v2\/pages\/602\/revisions\/1469"}],"up":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/sin\/wp-json\/wp\/v2\/pages\/8"}],"wp:attachment":[{"href":"https:\/\/people.clas.ufl.edu\/sin\/wp-json\/wp\/v2\/media?parent=602"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}