-1+1829 
Introduction to Group Representations

Group representations are where group theory meets linear algebra, and important applications arise in various math subjects (number theory, analysis, algebraic geometry), physics, and chemistry.

We consider the basic representation theory of finite groups. Goals include a look at Fourier series/analysis using groups and elementary character theory.

Lectures
  1. Week 1 - Representation Theory Basics

    Representations are defined, as are subrepresentations and irreducibility. Links include Linear Algebra Review, video RT1, Problem Set 1, and optional background videos.

  2. Week 2 - Unitary Representations

    Unitary representations are those that preserve an inner product. We show that any representation can be made unitary and, in turn, is fully reducible. Links include Linear Algebra Review 2, Solution Set 1, Problems Set 2 (with solutions), video RT2, and optional background videos.

  3. Week 3 - Equivalence and Examples

    We introduce the notion of equivalence of representations and present some low-dimensional examples, including a look at one-dimensional representations in general. Links include Solution Set 2, video RT3, Problem Set 3, and an option background video.

  4. Week 4 - Constructions from Linear Algebra

    We present some methods for constructing new representations from old ones. Techniques include direct sums, dual spaces, and tensors. Links include Solution Set 3, Linear Algebra Review 3, Problem Set 4, and videos RT4.1, RT4.1.1.

  5. Week 5 - Schur's Lemma

    Schur's lemma is a powerful tool for working with irreducible representations. Consequences include uniqueness results for equivalences and invariant forms. Links include Solution Set 4, Problem Set 5, and videos RT4.2. and RT5.

  6. Week 6 - Representations on Function Spaces

    For the rest of the course, we explore the connections between representations and functions. Here we explore basic properties, noting Fourier series as the prototype for groups (or vice versa if new). Links include Solution Set 5, Problem Set 6, and video RT6.

  7. Week 7 - Finite Abelian Groups: Characters

    We consider the vector space of functions on finite abelian G as a representation space for G. An orthonormal basis is given by the characters of G. Links include Solution Set 6, Problem Set 7, videos RT7 and GT18.1. (Dihedral Group review), an updated glossary, and a list of main results for analysis on groups.

  8. Week 8 - Finite Abelian Groups: Fourier Analysis

    Using the orthonormal basis of characters fof L^2 (G), we apply Fourier's Trick and Parseval's Identity. We also define convolution, and we apply convolution to obtain projection operators onto irreducible types in representations. Links include videos RT7.2 and RT7.3, Solution Set 7, and Problem Set 8.

  9. Week 9 - Finite Groups: Matrix Coefficients and L^2 (G)

    We repeat the story for finite abelian groups for general finite groups. We give orthonormal bases for L^2 (G) and Class(G), and, in turn, obtain important numeric information about G. The main result is the Schur Orthogonality Relations. Links include RT8.1 and RT8.2, Solution Set 8, and Problem Set 9.

  10. Week 10 - Finite Groups 2: Projection to Irreducibles

    Now that we can identify multiplicities of irreducibles in a given representation using characters, we give a formula for the orthogonal projections to the spaces for each irreducible type. Links include video RT8.3, Solution Set 9, and Problem Set 10.

  11. Week 11 - Basic Tensor Analysis

    Now that we have a basic theory of irreducibles using characters, we turn to the question of decomposing tensor product representations. In particular, we focus on 2-tensors, and the special cases of alternating and symmetric 2-tensors. we give character formulas for each case. Links include video RT9, Solution Set 10, and Problem Set 11.

  12. Week 12 - Application: Normal Modes (last class)

    We finish with an application of tensor analysis - finding normal modes of oscillating systems. In particular, we consider mass-spring systems in dimensions one and two. Links include RT9.1 and Solution Set 11.

Prerequisites

One semester of group theory - group actions and conjugacy classes are needed. No need for Sylow Theory that I can see.

One semester of linear algebra covering up to inner products, orthonormal bases, and orthogonal projections. A second course will help greatly, but further required topics will be addressed in the class.

I'll try to pitch the course towards advanced undergraduates, which means a heavy focus on teaching through examples.

Syllabus

The course will be run out of the subreddit /r/grouptheory2012 and material will be hosted at mathdoctorbob.org. Class will start the week of June 10 and should run from 9-12 weeks.

Tentative list of topics:

  1. Basic Definitions and Concepts

  2. Unitary Representations

  3. Equivalence and Examples

  4. Constructions from Linear Algebra

  5. Schur's Lemma

  6. Fourier series and the circle group

  7. Analysis on finite abelian groups

  8. Analysis on finite groups

  9. Symmetric and alternating tensors

The format will be the same as for the Group Theory class: each week links to materials are provided, and comments and questions can be sent directly to me or the subreddit. Of course, since students can add in at any time, questions are welcome on any covered sections.

Additional information

As with the group theory class, I don't have a book in mind. Fulton and Harris' Representation Theory is great (lots of exercises with hints, and Lie theory in the second half) if you have the background. Most of what I want to do is in the first 30 pages, but the analysis results are often pushed off to the exercises.

Terras' Fourier Analysis on Finite Groups and Applications is a good source for the specific direction of the course and provides a great deal of background and real-world applications.

Teacher qualifications

SchurThing - Over a decade teaching experience. Ph.D. from Stony Brook on Representations of Semi-simple Lie Groups. Post-docs include IAS and MIT.

See the U.Reddit class on Group Theory for previous work.

Latest Update
Group Representations - Last Lecture (2012-08-21 19:03:29)

"Hi All,

I just put up the last lecture and problem set for this course. Next week will be the last solution set.

If you are just signing up, no hurry. The course is designed to be taken at your own pace, and I'll keep an eye on the subreddit for questions once the course is archived.

Any comments or questions are appreciated.

Best, Bob (SchurThing)"

Roster
1. black_white
2. Aztek_Pr0phet
3. chngr
4. nbouscal
5. wuncidunci
6. Qbit42
7. ebbee
8. adamimos
9. edwardt
10. skaray
11. Venchinzo
12. dhttn
13. pkd858
14. Sidereus_Nuncius
15. DeinMutti
16. rengineerman
17. wizzyfizzy
18. Ananda.Chanda
19. procrastinator_94
20. AutoPterydactyl
21. gregm
22. jeromeblackridge
23. zentropia
24. videogamer321
25. asawingmotion
26. ImNotEvenARedditor
27. whacko_jacko
28. cwrw2005
29. hydr0genic
30. travisestes
31. crankprof
32. ding
33. crazdfanatic
34. fatalisticfatalist
35. wsliang
36. abcdenisse
37. rehnn
38. hungryhungryhuman
39. kimolas
40. sahanrohana
41. drgonzo28
42. chocolate_
43. PekPekWarrior
44. frechet
45. adimit
46. chixuclub
47. lbarnett
48. MikeyJ231
49. schwiiz
50. miolar
51. sujayakar314
52. plumbium
53. Wonder
54. lanin
55. mathmatt
56. darkainur
57. zeriod
58. jwcross
59. dhawansoumya
60. RRavier1
61. tkaczek
62. SiukeyP
63. everhostile
64. liverb
65. OlCrustacean
66. clemonsx90
67. nutmegmagi
68. Pfannkuchen
69. warwicka
70. goldayce
71. patternsonascreen
72. Deimor
73. danoph
74. syang
75. shadonra
76. tertl3
77. BigBubba417
78. Monobrow02
79. andromedae
80. mjarret
81. raganer
82. sklarzm
83. blumsky1985
84. TortoiseDream
85. krennylavitz
86. domdeuce
87. maymoo
88. fpinzn
89. Adlq
90. mayf2786
91. legopelle
92. jaylab12345
93. masterchip27
94. BorderedHessian
95. dopplerdog
96. TaceFatue
97. helasraizam
98. SPxChairman
99. stemcele
100. fermisea
101. TheRealDJ
102. Zedrein
103. englishmustard
104. msoltysiak
105. redditstudentfrmcanada
106. AtlasAnimated
107. RandomAwesomeName
108. rmayer3
109. jjoonathan
110. dandor
111. Atif
112. yanivfr
113. Illemagnus
114. kakashi_
115. humon2
116. violaep
117. sk3ptic
118. letmefinish
119. varen
120. najera_
121. spqr11
122. N0tAUsername
123. tumbleweed
124. cupofchupchups
125. cheesegreyter
126. Watcentral
127. wollff
128. whywork7
129. GilTheARM
130. anvil
131. lu6cifer
132. skadamat
133. tengil
134. metalliska
135. rdm5181
136. tbear9011
137. Catalanist
138. teh4x
139. dieek
140. danielsmw
141. RuNZ
142. grungleshnorts
143. st4rx0r
144. andyval
145. nielse63
146. marchdown
147. floatingdecimal
148. sccrstud92
149. dunkinsticks
150. kipperk
151. ihusker_42
152. yend
153. yend
154. murfs
155. rajeeves
156. Mamemoo
157. pianoplayer98
158. happytaquito
159. lecafard
160. ryelacey
161. shai251
162. Grammernatze
163. Dalen07
164. kurin
165. RorySBarnes
166. rpgcubed
167. qmynd
168. mishkaechoes
169. caujin
170. sirachman
171. MaliciousMalus
172. aboojoo
173. protossible
174. farcus
175. Abagosu
176. garnade31
177. r4gnar0k
178. Bezout
179. varghese1990
180. homerunnerd
181. quaz4r
182. trompwnerer
183. JAKarpis
184. gfixler
185. Fiserfully
186. MoreThanLuck
187. 253F584C
188. svacha
189. DIRTYBRIEFS
190. western_schmoobris
191. math
192. duobei
193. TheSnowKing
194. zandorf
195. Dr_Jan-Itor
196. mooglefrooglian
197. ravik
198. donaldruok
199. jey
200. SurDin
201. iampseo
202. Superfish
203. wydell
204. SubmitQuery
205. cybelechild
206. 4thguy
207. hgat
208. galaris
209. mahalo1984
210. moomin42
211. zissou
212. RyVal1
213. phillycheece2
214. sensei_von_bonzai
215. antitree
216. jsvana
217. prankishasa
218. Driver1676
219. xaustinx
220. kyzic
221. Beldin
222. ang
223. physicsvanawesome
224. belisariusthe7th
225. theslimde
226. orangejuice982
227. bluepickles
228. nitellubv
229. PalmPaperCut
230. Darth_Learner
231. ykandel
232. Clammy
233. PennyLane91
234. KingHavana
235. commanderblingbling
236. AntonA
237. nopculture
238. SunSatION
239. Alphafain
240. Hbou
241. romistrub
242. TheMagicPancake
243. iKarampa
244. Shazaaming
245. whitehatcat
246. liutanyu
247. thompsdy
248. arktemplar
249. probablytom
250. bob
251. pg1770
252. baloghr
253. elberethf
254. djm
255. Jehuty
256. jetblackswan
257. retric
258. jamesmcm
259. singdawg
260. maverickrhetoric
261. quinn_winters
262. esanya
263. nirajanrk
264. cusickw
265. profryan
266. jamjam
267. AdamMcLean
268. dddekim
269. whinycat
270. fastzhong
271. TheVoluntarist
272. revolutionx897
273. 55five55
274. bitewhite
275. sishgahh
276. grlldcheese
277. fugglor
278. Skyflier0
279. sennmen
280. harioldridge
281. blitser
282. Mystique
283. pennacchio
284. Janus
285. toporider
286. sockshoelemma
287. clembo99
288. iwanttolearnmaths
289. stcollin
290. theblueprint
291. wandabee
292. ThomasM
293. rozzer
294. mikeypuff
295. palutz
296. SnapSnag
297. xkjc
298. sumguysr
299. wsankey
300. tobbe
301. chumofchance
302. -________________-
303. vmm5596
304. Nicole
305. ejk314
306. peterson
307. fatosholgacher
308. ABawane
309. cool8137
310. thomston
311. yammd
312. Ophashias
313. AnnonMerritt
314. devilasks
315. fattybake
316. FerrousMan
317. eipi10
318. leia.n.w.
319. dstilwell
320. skittlles
321. jumi99
322. allxk
323. yahachutebya
324. mkay510
325. cpryby
326. arvindsridhar
327. chiwakii
328. TheMaxul
329. ikeamoah
330. proteanbeing
331. steeve149
332. tenoman
333. lpyoshikawa
334. Zoccihedron
335. aoper
336. kannickel
337. Sagen
338. last_name_on_reddit
339. kevstew
340. ScicoSuitS
341. llcooldre
342. Mulien
343. spencerutt
344. francis91
345. dwhan89
346. Futurephysicist
347. pardeepsachdeva
348. cowraiser
349. zbadgett
350. Gelassenheit
351. danielsouza1985
352. xander7b
353. AmundsenJunior
354. unclebodin
355. ThatSnail
356. shutup
357. ashwin
358. rcochrane
359. Mixedmath
360. hsolmaz
361. gkigongo
362. LuminiferousEther
363. froben
364. pmiszcz
365. Trkrtrk
366. Somatostatin
367. johnsonye
368. Gilgethan
369. qzpmwxin
370. thegauntlet
371. totocus
372. chinsi
373. StarDazzle18
374. ziarkaen
375. blfang
376. saputello
377. arthur.ravet
378. geldave
379. burp3141
380. moosebot
381. mathematician_
382. GGMUDarc
383. ctwiz
384. sloppy421
385. somehokie
386. andiderp
387. Krebbet
388. cowzzwoc
389. josemariaruiz
390. shalombi
391. srslySHENANIGANS
392. walterlewout
393. _rhythmx
394. linkizcool
395. bolgovr
396. salvorhardin75
397. hareycarey
398. farazhaider
399. Nightriser
400. msashley018
401. Smyds
402. learning2013
403. zero-lag
404. divindaiana
405. kbravo
406. sakattack
407. xyendor
408. greenhairedfae
409. gpetrou
410. crazwomanyo
411. sessiongkz
412. d3banjan
413. meanturing
414. pequod
415. alekart
416. pandaman2123
417. lessac
418. camelite
419. statsn00b
420. hiker
421. ivanistheone
422. foxor
423. jgoot
424. yellowflash
425. pmelendezu
426. tosterbind
427. zwatcher
428. hyperionsshrike
429. asportking
430. ncms1990
431. Nobu
432. alexvas
433. sir_eggerton
434. Orisue
435. cauchy
436. ohaipeople
437. hjmb
438. Zaborg90
439. mike7006
440. azzaleib
441. mmahesh
442. mirorakonto
443. revskill
444. citenonsite
445. iazid
446. davetherave823
447. phub4r
448. rkrishnasanka
449. j4cc3b
450. Nicole.k.d
451. Hamza20697
452. mathiaskberg
453. ashokamaurya
454. moebiusbender
455. simonr93
456. 74300291
457. dthommy
458. ArsonistNightfire
459. phujck
460. aaaangeline
461. w8cycle
462. Tarinaky
463. valvimore
464. jjshinobi
465. emmabruns
466. crystalyw
467. neevor
468. Tim
469. hath995
470. emartin
471. profzoom
472. ChesterHiggenbothum
473. rexpan
474. morrowsend
475. rinkajgoyal
476. paob
477. silvarus
478. dr.quackster
479. aoife21
480. sridhar
481. rinkaj
482. pootpoot
483. xujieren
484. FlyingSheeps
485. Joshuac
486. rruncie
487. wlkler
488. dimosbattu
489. RATwashingM
490. szabba
491. Batchmuffin
492. shmookashmuk
493. mathemagicianalgebro
494. dpeach
495. algorhythmmm
496. cuto9
497. unfixedac0rn
498. aamirazeem
499. Rowran
500. earthy
501. Trim_Toy
502. tanniche
503. Livingandlearning
504. apmechev
505. raptorshoelace
506. gaolathe
507. douggery
508. not_a_liberal_guy
509. philipp_mueller
510. likes_things
511. Avich
512. flyingraccoon
513. likevenice
514. Browl
515. Bucket_of_water
516. packersfan25
517. wildsebastian
518. maurajbg
519. roundedge
520. ryanmcgrim
521. SleeperFactory
522. TwilieIsBestPony
523. Jstew
524. Rdslnc
525. nochoa3
526. ncastell77
527. EngineerKid101
528. nathanpl
529. ryustreetfighter
530. viper1092
531. isaaclee96
532. pariefracture
533. sin4life
534. osculator
535. softmango
536. ramcha
537. Air_Scythe
538. soaker
539. LQTMS
540. Yourfavdinosaur
541. Axlrize
542. Haroon_kalam
543. bartonsmart
544. pyroholic
545. ilovedolphins123
546. pando
547. thomasbomb
548. tsaurav08
549. geometrydude
550. murphymj
551. blakedev
552. alexandergunnarson
553. ljtiuo
554. backwardinduction
555. chilukrn
556. andrewglaucus
557. November
558. user1996
559. raikrahul
560. dylanmathews
561. TheExiled
562. imperialist315
563. Gutenberg
564. bixiedust102
565. max00paul
566. anandsgh
567. cumar
568. vinay1997
569. aubwilson
570. noiamstefan
571. rmgrave10der
572. pi37
573. russoa
574. lookingforsome
575. chaworth
576. liebstraum
577. gaurav1729
578. Emilio94
579. georgeman007
580. thegreenduck
581. chandreshwar
582. jgould
583. JacktheFlying
584. hectorthecollector
585. NiceIRL
586. quantumite
587. Nischal
588. WiserMelon
589. SteveB526
590. lx003n
591. rajendra4343
592. tgallow5
593. admiralgeneral
594. tekno
595. rere
596. Birdmav
597. ilr1
598. rafaelGeo
599. axfwk17
600. LordDaedalus
601. Julian
602. timrabb211
603. blackcats31
604. freedumb123
605. ficar
606. Dragonduck
607. myreprise
608. tjbecker
609. tillyt
610. eponymic23
611. packard910
612. CharlesJCoopper
613. PavanePrincess
614. ktsenthil
615. ValiantEntity
616. Rizhiy
617. mejjem
618. abc123def
619. Yenniesaurus
620. victoremnm
621. leonbasin
622. Banach-Tarski
623. aaidan
624. Shobhana
625. Shobhana
626. gn0m0n
627. MisterE
628. ifr33m4n
629. jwhitlark
630. Vnuke
631. sabbysaur
632. rampant_juju
633. NeeBo
634. brentshermana
635. atmospheric
636. joehigh
637. anaran
638. tpkalman
639. timlovelock
640. bpalkmim
641. foxboy
642. ishouldread
643. aatxe
644. Bloodrazor
645. timjstewart
646. L2BSmart
647. foo
648. frwdrik
649. aDisciple
650. nunofilipemargarido
651. Bennymoonbase
652. pinguhash
653. mtilles69
654. konfou
655. Troglodite
656. jarebear
657. Richo1337
658. maka_maria
659. RMnumbers
660. noamtwx
661. wwalid2050
662. hmeier
663. drumchaps
664. wjzhu
665. cryptguy
666. aeknel
667. bbmacarthur
668. zeroms
669. crichm03
670. RicoSpartan
671. reemaj
672. atap
673. dschwartz266
674. mpmpie
675. JimSunkes
676. HexMerck
677. benshats
678. Threscher
679. martyg
680. Tessinator
681. gerardlouw
682. swapanj162
683. Snejana
684. justin_ecklund
685. Suzanne2578
686. sintrastes
687. antone
688. shitzfez
689. natasakalp
690. kingosraigi
691. Ichikoa
692. Theguybehindthescreen
693. haraf_ijhal
694. lemoskid
695. lazypakman
696. fauxhemian5
697. namesare4tombstones
698. vincentmakreel
699. TrollyMike
700. DumbleDaz
701. Reathur
702. nono
703. eKro
704. Usagi
705. jmrkumar
706. JayMendez
707. tpdemixxx
708. asingh21
709. burtgervis
710. jtormey
711. analy7er
712. fezspinkguitar
713. EntropicalForest
714. BTizzle23
715. Higs
716. 20sat92
717. Cori83
718. DylanSA
719. kbanks4130
720. catomno
721. hooray4alex
722. hderms
723. bendaizer
724. vct
725. ruadhri299
726. jimmmyboy
727. uberstalker
728. hannibal
729. outterspaceproduct
730. riogunawan007
731. WhackAMolE
732. takumf
733. xtransient
734. JulianDelphiki2
735. ndroock1
736. maacks
737. petemonkeyguts
738. dopyuu
739. Cody72
740. Ollie64
741. Kulikant
742. aBRRICK
743. Nanodive
744. thotar
745. Giedrius
746. msoup
747. Joecorgi
748. eprairie
749. kitty_zombie1
750. thunderkrak
751. Mystery
752. kayama
753. WarGy_Luremus
754. rajrahul
755. wutabum17
756. samiiina
757. Tri2Find
758. duran
759. burubaxair
760. EvilEgg
761. Yankee
762. pjtoha
763. Andres
764. kiadams2393
765. pkmnkiller41
766. Scream
767. seanknight
768. pmcgee
769. giftgirl2305
770. opaedo
771. evdoyle
772. thedanielchacko
773. belgerath
774. scantron_amazing
775. scottyboy1119
776. __hsams__
777. rggmcn
778. hardy1234
779. heimu177
780. Mete0r1t3
781. soueuls
782. openbriefcase
783. shaynee
784. sevenstarsofstef
785. VonFoerster
786. dpdocs21
787. keenlearner
788. CampCarthage
789. GreybeardJosh
790. meisel
791. inmesia
792. blakjak02
793. chanchalsahoo
794. HaroldJIncandenza
795. ItsMe
796. nemke
797. JoshuaWeinberger
798. Gargantuon
799. vrewilson
800. RalphWhite
801. imshinebright_
802. dereddit
803. schmiter
804. fiercedeity91
805. jacoph
806. asm
807. Halkilt
808. dokalanyi
809. ddorksey
810. meehigh
811. some1son
812. nitsud
813. haneensa
814. MikeyyZ
815. mozammil26
816. JohnSmithson
817. AdebayoSamuel
818. Alicia
819. Nikhilnaughtystudent
820. AllYourBaseRBlong2Us
821. harsh2803
822. Markoff_Real
823. umaraxif
824. mostafaosama123
825. Omy251
826. ahmd
827. somethinpoetic21
828. megaleeker
829. noobto
830. shihaolin096
831. Gundas3073
832. loosersparadise
833. greenredidunno
834. JuNozil
835. Rattlehead
836. fusar007
837. dintdot
838. charly
839. charly3D
840. arisagf
841. neudimol
842. nukeduke
843. vasjaforutube
844. bsisman
845. rainakacb
846. jhanak
847. Gilligan-Uni
848. emilyf11
849. Haymaker33
850. allnitegyme
851. swaggerrrr
852. theflyingbrant
853. wizon
854. jlramirez43
855. onitek
856. alifathi5005
857. JerryKitich
858. geletu
859. scholarlyBrute
860. elcamlost
861. dtlawhon
862. Schmucko
863. tungto
864. elspark
865. nhok_conan
866. prateekchandrajha
867. brandon_ball_z
868. chibipammy
869. Willis_l96
870. bdobba
871. neoezekiel
872. jakeofcurtis
873. okyan0
874. gus
875. mirrorman867
876. s0fist
877. burakterzi1988
878. mvresh
879. Hedrew
880. miharroes
881. Yassine
882. djofficial
883. gradsunca
884. p3t3r.tnie
885. ChuTung
886. hydrogendos
887. ikenwreed
888. hemingwaywouldntbeimpressed
889. msamdgirl01
890. Athar
891. mcwillzyx
892. qcwinters
893. binary_student
894. dit7ya
895. pfrcks
896. vijaypradhan
897. asilentsound
898. kmmadan7
899. vdtudzharska
900. ramiregui
901. imbarbarryn
902. ipk
903. mmonica
904. kaydea
905. cinnabon
906. ceater259
907. ksimp126
908. everlin
909. rohit1212
910. kersenn
911. daisyrose
912. euphorialive
913. eladieu
914. tbakermo
915. Khy2134
916. elifseneng
917. lankie
918. lankie
919. CobaltSthenia
920. Vrinda
921. manidou
922. kodumuru_niranjan
923. julinoreddit
924. yashkandukuri
925. spookyfoster
926. monwaneri
927. DIPANSHU
928. levonsumit
929. g618034
930. dfgfhsr
931. marklevinson123
932. hsams
933. Emma0311
934. melb93
935. grandpadeadpool
936. DMedinov
937. videshpa
938. TBLIXXXY
939. Dibya
940. Hannah01292
941. aryfletchy
942. Edd
943. BrittaniPotter06
944. michaelmalatesta
945. Adthelad3455
946. devansheethakkar
947. frengky
948. Mkeita23
949. mikey1410
950. khier
951. Everstorm
952. zxia31