-1+1780 
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. addemf
394. _rhythmx
395. linkizcool
396. bolgovr
397. salvorhardin75
398. hareycarey
399. farazhaider
400. Nightriser
401. msashley018
402. Smyds
403. learning2013
404. zero-lag
405. divindaiana
406. kbravo
407. sakattack
408. xyendor
409. greenhairedfae
410. gpetrou
411. crazwomanyo
412. sessiongkz
413. d3banjan
414. meanturing
415. pequod
416. alekart
417. pandaman2123
418. lessac
419. camelite
420. statsn00b
421. hiker
422. ivanistheone
423. foxor
424. jgoot
425. yellowflash
426. pmelendezu
427. tosterbind
428. zwatcher
429. hyperionsshrike
430. asportking
431. ncms1990
432. Nobu
433. alexvas
434. sir_eggerton
435. Orisue
436. cauchy
437. ohaipeople
438. hjmb
439. Zaborg90
440. mike7006
441. azzaleib
442. mmahesh
443. mirorakonto
444. revskill
445. citenonsite
446. iazid
447. davetherave823
448. phub4r
449. rkrishnasanka
450. j4cc3b
451. Nicole.k.d
452. Hamza20697
453. mathiaskberg
454. ashokamaurya
455. moebiusbender
456. simonr93
457. 74300291
458. dthommy
459. ArsonistNightfire
460. phujck
461. aaaangeline
462. w8cycle
463. Tarinaky
464. valvimore
465. jjshinobi
466. emmabruns
467. crystalyw
468. neevor
469. Tim
470. hath995
471. emartin
472. profzoom
473. ChesterHiggenbothum
474. rexpan
475. morrowsend
476. rinkajgoyal
477. paob
478. silvarus
479. dr.quackster
480. aoife21
481. sridhar
482. rinkaj
483. pootpoot
484. xujieren
485. FlyingSheeps
486. Joshuac
487. rruncie
488. wlkler
489. dimosbattu
490. RATwashingM
491. szabba
492. Batchmuffin
493. shmookashmuk
494. mathemagicianalgebro
495. dpeach
496. algorhythmmm
497. cuto9
498. unfixedac0rn
499. aamirazeem
500. Rowran
501. earthy
502. Trim_Toy
503. tanniche
504. Livingandlearning
505. apmechev
506. raptorshoelace
507. gaolathe
508. douggery
509. not_a_liberal_guy
510. philipp_mueller
511. likes_things
512. Avich
513. flyingraccoon
514. likevenice
515. Browl
516. Bucket_of_water
517. packersfan25
518. wildsebastian
519. maurajbg
520. roundedge
521. ryanmcgrim
522. SleeperFactory
523. TwilieIsBestPony
524. Jstew
525. Rdslnc
526. nochoa3
527. ncastell77
528. EngineerKid101
529. nathanpl
530. ryustreetfighter
531. viper1092
532. isaaclee96
533. pariefracture
534. sin4life
535. osculator
536. softmango
537. ramcha
538. Air_Scythe
539. soaker
540. LQTMS
541. Yourfavdinosaur
542. Axlrize
543. Haroon_kalam
544. bartonsmart
545. pyroholic
546. ilovedolphins123
547. pando
548. thomasbomb
549. tsaurav08
550. geometrydude
551. murphymj
552. blakedev
553. alexandergunnarson
554. ljtiuo
555. backwardinduction
556. chilukrn
557. andrewglaucus
558. November
559. user1996
560. raikrahul
561. dylanmathews
562. TheExiled
563. imperialist315
564. Gutenberg
565. bixiedust102
566. max00paul
567. anandsgh
568. cumar
569. vinay1997
570. aubwilson
571. noiamstefan
572. rmgrave10der
573. pi37
574. russoa
575. lookingforsome
576. chaworth
577. liebstraum
578. gaurav1729
579. Emilio94
580. georgeman007
581. thegreenduck
582. chandreshwar
583. jgould
584. JacktheFlying
585. hectorthecollector
586. NiceIRL
587. quantumite
588. Nischal
589. WiserMelon
590. SteveB526
591. lx003n
592. rajendra4343
593. tgallow5
594. admiralgeneral
595. tekno
596. rere
597. Birdmav
598. ilr1
599. rafaelGeo
600. axfwk17
601. LordDaedalus
602. Julian
603. timrabb211
604. blackcats31
605. freedumb123
606. ficar
607. Dragonduck
608. myreprise
609. tjbecker
610. tillyt
611. eponymic23
612. packard910
613. CharlesJCoopper
614. PavanePrincess
615. ktsenthil
616. ValiantEntity
617. Rizhiy
618. mejjem
619. abc123def
620. Yenniesaurus
621. victoremnm
622. leonbasin
623. Banach-Tarski
624. aaidan
625. Shobhana
626. Shobhana
627. gn0m0n
628. MisterE
629. ifr33m4n
630. jwhitlark
631. Vnuke
632. sabbysaur
633. rampant_juju
634. NeeBo
635. brentshermana
636. atmospheric
637. joehigh
638. anaran
639. tpkalman
640. timlovelock
641. bpalkmim
642. foxboy
643. ishouldread
644. aatxe
645. Bloodrazor
646. timjstewart
647. L2BSmart
648. foo
649. frwdrik
650. aDisciple
651. nunofilipemargarido
652. Bennymoonbase
653. pinguhash
654. mtilles69
655. konfou
656. Troglodite
657. jarebear
658. Richo1337
659. maka_maria
660. RMnumbers
661. noamtwx
662. wwalid2050
663. hmeier
664. drumchaps
665. wjzhu
666. cryptguy
667. aeknel
668. bbmacarthur
669. zeroms
670. crichm03
671. RicoSpartan
672. reemaj
673. atap
674. dschwartz266
675. mpmpie
676. JimSunkes
677. HexMerck
678. benshats
679. Threscher
680. martyg
681. Tessinator
682. gerardlouw
683. swapanj162
684. Snejana
685. justin_ecklund
686. Suzanne2578
687. sintrastes
688. antone
689. shitzfez
690. natasakalp
691. kingosraigi
692. Ichikoa
693. Theguybehindthescreen
694. haraf_ijhal
695. lemoskid
696. lazypakman
697. fauxhemian5
698. namesare4tombstones
699. vincentmakreel
700. TrollyMike
701. DumbleDaz
702. Reathur
703. nono
704. eKro
705. Usagi
706. jmrkumar
707. JayMendez
708. tpdemixxx
709. asingh21
710. burtgervis
711. jtormey
712. analy7er
713. fezspinkguitar
714. EntropicalForest
715. BTizzle23
716. Higs
717. 20sat92
718. Cori83
719. DylanSA
720. kbanks4130
721. catomno
722. hooray4alex
723. hderms
724. bendaizer
725. vct
726. ruadhri299
727. jimmmyboy
728. uberstalker
729. hannibal
730. outterspaceproduct
731. riogunawan007
732. WhackAMolE
733. takumf
734. xtransient
735. JulianDelphiki2
736. ndroock1
737. maacks
738. petemonkeyguts
739. dopyuu
740. Cody72
741. Ollie64
742. Kulikant
743. aBRRICK
744. Nanodive
745. thotar
746. Giedrius
747. msoup
748. Joecorgi
749. eprairie
750. kitty_zombie1
751. thunderkrak
752. Mystery
753. kayama
754. WarGy_Luremus
755. rajrahul
756. wutabum17
757. samiiina
758. Tri2Find
759. duran
760. burubaxair
761. EvilEgg
762. Yankee
763. pjtoha
764. Andres
765. kiadams2393
766. pkmnkiller41
767. Scream
768. seanknight
769. pmcgee
770. giftgirl2305
771. opaedo
772. evdoyle
773. thedanielchacko
774. belgerath
775. scantron_amazing
776. scottyboy1119
777. __hsams__
778. rggmcn
779. hardy1234
780. heimu177
781. Mete0r1t3
782. soueuls
783. openbriefcase
784. shaynee
785. sevenstarsofstef
786. VonFoerster
787. dpdocs21
788. keenlearner
789. CampCarthage
790. GreybeardJosh
791. meisel
792. inmesia
793. blakjak02
794. chanchalsahoo
795. HaroldJIncandenza
796. ItsMe
797. nemke
798. JoshuaWeinberger
799. Gargantuon
800. vrewilson
801. RalphWhite
802. imshinebright_
803. dereddit
804. schmiter
805. fiercedeity91
806. jacoph
807. asm
808. Halkilt
809. dokalanyi
810. ddorksey
811. meehigh
812. some1son
813. nitsud
814. haneensa
815. MikeyyZ
816. mozammil26
817. JohnSmithson
818. AdebayoSamuel
819. Alicia
820. Nikhilnaughtystudent
821. AllYourBaseRBlong2Us
822. harsh2803
823. Markoff_Real
824. umaraxif
825. mostafaosama123
826. Omy251
827. ahmd
828. somethinpoetic21
829. megaleeker
830. noobto
831. shihaolin096
832. Gundas3073
833. loosersparadise
834. greenredidunno
835. JuNozil
836. Rattlehead
837. fusar007
838. dintdot
839. charly
840. charly3D
841. arisagf
842. neudimol
843. nukeduke
844. vasjaforutube
845. bsisman
846. rainakacb
847. jhanak
848. Gilligan-Uni
849. emilyf11
850. Haymaker33
851. allnitegyme
852. swaggerrrr
853. theflyingbrant
854. wizon
855. jlramirez43
856. onitek
857. alifathi5005
858. JerryKitich
859. geletu
860. scholarlyBrute
861. elcamlost
862. dtlawhon
863. Schmucko
864. tungto
865. elspark
866. nhok_conan
867. prateekchandrajha
868. brandon_ball_z
869. chibipammy
870. Willis_l96
871. bdobba
872. neoezekiel
873. jakeofcurtis
874. okyan0
875. gus
876. mirrorman867
877. s0fist
878. burakterzi1988
879. mvresh
880. Hedrew
881. miharroes
882. Yassine
883. djofficial
884. gradsunca
885. p3t3r.tnie
886. ChuTung
887. hydrogendos
888. ikenwreed
889. hemingwaywouldntbeimpressed
890. msamdgirl01
891. Athar
892. mcwillzyx
893. qcwinters