{"id":867,"date":"2022-09-05T16:24:03","date_gmt":"2022-09-05T13:24:03","guid":{"rendered":"http:\/\/me.hse.ru\/esmirnov\/?page_id=867"},"modified":"2022-12-22T16:36:07","modified_gmt":"2022-12-22T13:36:07","slug":"reptheory","status":"publish","type":"page","link":"http:\/\/me.hse.ru\/esmirnov\/reptheory\/","title":{"rendered":"Representation Theory, Fall 2022"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">\u5317\u4eac\u5927\u5b66<\/h2>\n\n\n\n<p><strong>Instructor:<\/strong> Evgeny Smirnov, <a href=\"mailto:esmirnov@hse.ru\">esmirnov@hse.ru<\/a>, <a href=\"https:\/\/www.hse.ru\/en\/staff\/esmirnov\">https:\/\/www.hse.ru\/en\/staff\/esmirnov<\/a><\/p>\n\n\n\n<p><strong>Course assistant:<\/strong> Wang Yihan (\u738b\u4e00\u6db5), <a href=\"mailto:2001110016@pku.edu.cn\">2001110016@pku.edu.cn<\/a><\/p>\n\n\n\n<p><strong>Time:<\/strong> Mon 15:10-17:00, Thu 18:40-20:30 CST (GMT+8).<\/p>\n\n\n\n<p><strong>Lectures<\/strong><\/p>\n\n\n\n<ul>\n<li><strong>Lecture 1, 05\/09\/22.<\/strong> What is representation theory? Main definitions. Irreducibility and indecomposability, Maschke&#8217;s theorem. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides01.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 2, 08\/09\/22.<\/strong> Morphisms of representations. Schur&#8217;s lemma, uniqueness of decomposition into irreducibles. Representations of abelian groups and S<sub>3<\/sub>. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides02.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 3, 12\/09\/22.<\/strong> Multilinear maps. Tensor product of vector spaces, universality. Symmetric powers. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides03.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 4, 15\/09\/22.<\/strong> Exterior powers. Characters of representations, their properties. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides04.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 5, 19\/09\/22.<\/strong> Orthonormality of characters. Characters form a basis in class functions. <a href=\"https:\/\/www.mccme.ru\/~smirnoff\/teaching\/2223\/rt_slides05.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 6, 22\/09\/22. <\/strong>Representations of S<sub>4<\/sub>. Character tables, tensor products. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides06.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 7, 26\/09\/22.<\/strong> Representations of A<sub>4<\/sub>, S<sub>5<\/sub>, and A<sub>5<\/sub>. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides07.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 8, 29\/09\/22.<\/strong> Representations of A<sub>5<\/sub> and symmetries of dodecahedron. Irreducibility of exterior powers of the tautological representation of S<sub>n<\/sub>. The group algebra. <strong><a href=\"https:\/\/www.mccme.ru\/~smirnoff\/teaching\/2223\/rt_slides08.pdf\">Blackboard<\/a> | <a href=\"https:\/\/www.mccme.ru\/~smirnoff\/teaching\/2223\/rt_pictures08.pdf\">Pictures<\/a><\/strong><\/li>\n\n\n\n<li><strong>Lecture 9, 06\/10\/22. <\/strong>Induced representations. Frobenius reciprocity. <strong><a href=\"https:\/\/www.mccme.ru\/~smirnoff\/teaching\/2223\/rt_slides09.pdf\">Blackboard<\/a><\/strong><\/li>\n\n\n\n<li><strong>Lecture 10, 13\/10\/22.<\/strong> Representations of S<sub>d<\/sub>: construction of Young symmetrizers. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides10.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 11, 17\/10\/22.<\/strong> Representations of S<sub>d<\/sub>. The Frobenius character formula. <strong><a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides11.pdf\">Blackboard<\/a><\/strong><\/li>\n\n\n\n<li><strong>Lecture 12, 20\/10\/22.<\/strong> Euler&#8217;s pentagonal theorem. Gaussian binomial coefficients. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides12.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 13, 24\/10\/22. <\/strong>Quiver representations. Indecomposable representations. Examples. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides13.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 14, 27\/10\/22.<\/strong> Gabriel&#8217;s theorem. Dynkin diagrams and positive definite Cartan matrices. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides14.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 15, 31\/10\/22.<\/strong> Roots. Simple and positive roots. Reflection functors. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides15.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 16, 03\/11\/22.<\/strong> Coxeter elements. Proof of Gabriel&#8217;s theorem. Path algebra of a quiver. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides16.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 17, 07\/11\/22.<\/strong> Representations of associative algebras. The density theorem. Representations of direct sums of matrix algebras. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides17.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 18, 14\/11\/22.<\/strong> Radical. Semisimple algebras. Jordan-H\u00f6lder theorem. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides18.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 19, 17\/11\/22. <\/strong>Krull-Schmidt theorem. Extensions, Ext<sup>1<\/sup>(W,V). <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides19.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 20, 21\/11\/22.<\/strong> Algebraic integers. Frobenius divisibility theorem. Burnside&#8217;s theorem. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides20.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 21, 24\/11\/22.<\/strong> Lie groups, tangent spaces. The exponential map. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides21.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 22, 28\/11\/22.<\/strong> Lie algebras. Adjoint representation. Epimorphism SU<sub>2<\/sub>-&gt;SO<sub>3<\/sub>. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides22.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n\n\n\n<li><strong>Lecture 23, 01\/12\/22. <\/strong>Compact Lie groups, complete reducibility. Weyl&#8217;s unitary trick. Representations of sl(2).<strong> <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides23.pdf\">Blackboard<\/a><\/strong><\/li>\n\n\n\n<li><strong>Lecture 24, 05\/12\/22. <\/strong>Universal enveloping algebra. Double centralizer theorem. Schur-Weyl duality. <a href=\"https:\/\/users.mccme.ru\/smirnoff\/teaching\/2223\/rt_slides24.pdf\"><strong>Blackboard<\/strong><\/a><\/li>\n<\/ul>\n\n\n\n<p><strong>Videos<\/strong> are accessible here: <a rel=\"noreferrer noopener\" href=\"https:\/\/disk.pku.edu.cn\/link\/EFB541F8F4E3C0D5723B713D7F4BECCE\" target=\"_blank\">https:\/\/disk.pku.edu.cn:443\/link\/EFB541F8F4E3C0D5723B713D7F4BECCE<\/a><\/p>\n\n\n\n<p><strong>Textbooks<\/strong><\/p>\n\n\n\n<ol>\n<li>William Fulton, Joe Harris. Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. Springer, 1991. <em>This will be our main textbook<\/em><\/li>\n\n\n\n<li>Jean-Pierre Serre. Linear Representations of Finite Groups. Graduate Texts in Mathematics. Vol. 42. Springer, 1996.<\/li>\n\n\n\n<li>Pavel Etingof et al. Introduction to Representation Theory. AMS, 2011. Available at the author&#8217;s website: <a href=\"https:\/\/math.mit.edu\/~etingof\/repb.pdf\">https:\/\/math.mit.edu\/~etingof\/repb.pdf<\/a><\/li>\n\n\n\n<li>J.N. Bernstein, I.M. Gelfand, V.A.Ponomarev. Coxeter functors and Gabriel&#8217;s theorem. <a href=\"https:\/\/www.math.uni-bielefeld.de\/~sek\/kau\/bgp.pdf\">https:\/\/www.math.uni-bielefeld.de\/~sek\/kau\/bgp.pdf<\/a><\/li>\n\n\n\n<li>Ernest B. Vinberg. A Course in Algebra. Graduate Texts in Mathematics, Vol. 56. AMS, 2003.<\/li>\n<\/ol>\n\n\n\n<p><strong>Notes and other links<\/strong><\/p>\n\n\n\n<ol>\n<li>J.J.Jiang&#8217;s <a href=\"https:\/\/mp.weixin.qq.com\/s\/ajwuV-ciW4fhU_lEjPA-Hw\">notes on semisimplicity<\/a> (in Chinese)<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u5317\u4eac\u5927\u5b66 Instructor: Evgeny Smirnov, esmirnov@hse.ru, https:\/\/www.hse.ru\/en\/staff\/esmirnov Course assistant: Wang Yihan (\u738b\u4e00\u6db5), 2001110016@pku.edu.cn Time: Mon 15:10-17:00, Thu 18:40-20:30 CST (GMT+8). Lectures Videos are accessible here: https:\/\/disk.pku.edu.cn:443\/link\/EFB541F8F4E3C0D5723B713D7F4BECCE Textbooks Notes and other links<\/p>\n","protected":false},"author":18,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"templates\/page-no-sidebar.php","meta":{"footnotes":""},"_links":{"self":[{"href":"http:\/\/me.hse.ru\/esmirnov\/wp-json\/wp\/v2\/pages\/867"}],"collection":[{"href":"http:\/\/me.hse.ru\/esmirnov\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/me.hse.ru\/esmirnov\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/me.hse.ru\/esmirnov\/wp-json\/wp\/v2\/users\/18"}],"replies":[{"embeddable":true,"href":"http:\/\/me.hse.ru\/esmirnov\/wp-json\/wp\/v2\/comments?post=867"}],"version-history":[{"count":31,"href":"http:\/\/me.hse.ru\/esmirnov\/wp-json\/wp\/v2\/pages\/867\/revisions"}],"predecessor-version":[{"id":985,"href":"http:\/\/me.hse.ru\/esmirnov\/wp-json\/wp\/v2\/pages\/867\/revisions\/985"}],"wp:attachment":[{"href":"http:\/\/me.hse.ru\/esmirnov\/wp-json\/wp\/v2\/media?parent=867"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}