-1+1633 
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. karjala
115. kakashi_
116. humon2
117. violaep
118. sk3ptic
119. letmefinish
120. varen
121. najera_
122. spqr11
123. N0tAUsername
124. tumbleweed
125. cupofchupchups
126. cheesegreyter
127. Watcentral
128. wollff
129. whywork7
130. GilTheARM
131. anvil
132. lu6cifer
133. skadamat
134. tengil
135. metalliska
136. rdm5181
137. tbear9011
138. Catalanist
139. teh4x
140. dieek
141. danielsmw
142. RuNZ
143. grungleshnorts
144. st4rx0r
145. andyval
146. nielse63
147. marchdown
148. floatingdecimal
149. sccrstud92
150. dunkinsticks
151. kipperk
152. ihusker_42
153. yend
154. yend
155. murfs
156. rajeeves
157. Mamemoo
158. pianoplayer98
159. happytaquito
160. lecafard
161. ryelacey
162. shai251
163. Grammernatze
164. Dalen07
165. kurin
166. RorySBarnes
167. rpgcubed
168. qmynd
169. mishkaechoes
170. caujin
171. sirachman
172. MaliciousMalus
173. aboojoo
174. protossible
175. farcus
176. Abagosu
177. garnade31
178. r4gnar0k
179. Bezout
180. varghese1990
181. homerunnerd
182. quaz4r
183. trompwnerer
184. JAKarpis
185. gfixler
186. Fiserfully
187. MoreThanLuck
188. 253F584C
189. svacha
190. DIRTYBRIEFS
191. western_schmoobris
192. math
193. duobei
194. TheSnowKing
195. zandorf
196. Dr_Jan-Itor
197. mooglefrooglian
198. ravik
199. donaldruok
200. jey
201. SurDin
202. iampseo
203. Superfish
204. wydell
205. SubmitQuery
206. cybelechild
207. 4thguy
208. hgat
209. galaris
210. mahalo1984
211. moomin42
212. zissou
213. RyVal1
214. phillycheece2
215. sensei_von_bonzai
216. antitree
217. jsvana
218. prankishasa
219. Driver1676
220. xaustinx
221. kyzic
222. Beldin
223. ang
224. physicsvanawesome
225. belisariusthe7th
226. theslimde
227. orangejuice982
228. bluepickles
229. nitellubv
230. PalmPaperCut
231. Darth_Learner
232. ykandel
233. Clammy
234. PennyLane91
235. KingHavana
236. commanderblingbling
237. AntonA
238. nopculture
239. SunSatION
240. Alphafain
241. Hbou
242. romistrub
243. Ollie64
244. TheMagicPancake
245. iKarampa
246. Shazaaming
247. whitehatcat
248. liutanyu
249. thompsdy
250. arktemplar
251. probablytom
252. bob
253. pg1770
254. baloghr
255. elberethf
256. djm
257. Jehuty
258. jetblackswan
259. retric
260. jamesmcm
261. singdawg
262. maverickrhetoric
263. quinn_winters
264. esanya
265. nirajanrk
266. cusickw
267. profryan
268. jamjam
269. AdamMcLean
270. dddekim
271. whinycat
272. fastzhong
273. TheVoluntarist
274. revolutionx897
275. 55five55
276. bitewhite
277. sishgahh
278. grlldcheese
279. fugglor
280. Skyflier0
281. sennmen
282. harioldridge
283. blitser
284. Mystique
285. pennacchio
286. Janus
287. toporider
288. sockshoelemma
289. clembo99
290. iwanttolearnmaths
291. stcollin
292. theblueprint
293. wandabee
294. ThomasM
295. rozzer
296. mikeypuff
297. palutz
298. SnapSnag
299. xkjc
300. sumguysr
301. wsankey
302. tobbe
303. chumofchance
304. -________________-
305. vmm5596
306. Nicole
307. ejk314
308. peterson
309. fatosholgacher
310. ABawane
311. cool8137
312. thomston
313. yammd
314. Ophashias
315. AnnonMerritt
316. devilasks
317. fattybake
318. FerrousMan
319. eipi10
320. leia.n.w.
321. dstilwell
322. skittlles
323. jumi99
324. allxk
325. yahachutebya
326. mkay510
327. cpryby
328. arvindsridhar
329. chiwakii
330. TheMaxul
331. ikeamoah
332. proteanbeing
333. steeve149
334. tenoman
335. lpyoshikawa
336. Zoccihedron
337. aoper
338. kannickel
339. Sagen
340. last_name_on_reddit
341. kevstew
342. ScicoSuitS
343. llcooldre
344. Mulien
345. spencerutt
346. francis91
347. dwhan89
348. Futurephysicist
349. pardeepsachdeva
350. cowraiser
351. zbadgett
352. Gelassenheit
353. danielsouza1985
354. xander7b
355. AmundsenJunior
356. unclebodin
357. ThatSnail
358. shutup
359. ashwin
360. rcochrane
361. Mixedmath
362. hsolmaz
363. gkigongo
364. LuminiferousEther
365. froben
366. pmiszcz
367. Trkrtrk
368. Somatostatin
369. johnsonye
370. Gilgethan
371. qzpmwxin
372. thegauntlet
373. totocus
374. chinsi
375. StarDazzle18
376. ziarkaen
377. blfang
378. saputello
379. arthur.ravet
380. geldave
381. burp3141
382. moosebot
383. mathematician_
384. GGMUDarc
385. ctwiz
386. sloppy421
387. somehokie
388. andiderp
389. Krebbet
390. cowzzwoc
391. josemariaruiz
392. shalombi
393. srslySHENANIGANS
394. walterlewout
395. addemf
396. _rhythmx
397. linkizcool
398. bolgovr
399. salvorhardin75
400. hareycarey
401. farazhaider
402. Nightriser
403. msashley018
404. Smyds
405. learning2013
406. zero-lag
407. divindaiana
408. kbravo
409. sakattack
410. xyendor
411. greenhairedfae
412. gpetrou
413. crazwomanyo
414. sessiongkz
415. d3banjan
416. meanturing
417. pequod
418. alekart
419. pandaman2123
420. lessac
421. camelite
422. statsn00b
423. hiker
424. ivanistheone
425. foxor
426. jgoot
427. yellowflash
428. pmelendezu
429. tosterbind
430. zwatcher
431. hyperionsshrike
432. asportking
433. ncms1990
434. Nobu
435. alexvas
436. sir_eggerton
437. Orisue
438. cauchy
439. ohaipeople
440. hjmb
441. Zaborg90
442. mike7006
443. azzaleib
444. mmahesh
445. mirorakonto
446. revskill
447. citenonsite
448. iazid
449. davetherave823
450. phub4r
451. rkrishnasanka
452. j4cc3b
453. Nicole.k.d
454. Hamza20697
455. mathiaskberg
456. ashokamaurya
457. moebiusbender
458. simonr93
459. 74300291
460. dthommy
461. ArsonistNightfire
462. phujck
463. aaaangeline
464. w8cycle
465. Tarinaky
466. valvimore
467. jjshinobi
468. emmabruns
469. crystalyw
470. neevor
471. Tim
472. hath995
473. emartin
474. profzoom
475. ChesterHiggenbothum
476. rexpan
477. morrowsend
478. rinkajgoyal
479. paob
480. silvarus
481. dr.quackster
482. aoife21
483. sridhar
484. rinkaj
485. pootpoot
486. xujieren
487. FlyingSheeps
488. Joshuac
489. rruncie
490. wlkler
491. dimosbattu
492. RATwashingM
493. szabba
494. Batchmuffin
495. shmookashmuk
496. mathemagicianalgebro
497. dpeach
498. algorhythmmm
499. cuto9
500. unfixedac0rn
501. aamirazeem
502. Rowran
503. earthy
504. Trim_Toy
505. tanniche
506. Livingandlearning
507. apmechev
508. raptorshoelace
509. gaolathe
510. douggery
511. not_a_liberal_guy
512. philipp_mueller
513. likes_things
514. Avich
515. flyingraccoon
516. likevenice
517. Browl
518. Bucket_of_water
519. packersfan25
520. wildsebastian
521. maurajbg
522. roundedge
523. ryanmcgrim
524. SleeperFactory
525. TwilieIsBestPony
526. Jstew
527. Rdslnc
528. nochoa3
529. ncastell77
530. EngineerKid101
531. nathanpl
532. ryustreetfighter
533. viper1092
534. isaaclee96
535. pariefracture
536. sin4life
537. osculator
538. softmango
539. ramcha
540. Air_Scythe
541. soaker
542. LQTMS
543. Yourfavdinosaur
544. Axlrize
545. Haroon_kalam
546. bartonsmart
547. pyroholic
548. ilovedolphins123
549. pando
550. thomasbomb
551. tsaurav08
552. geometrydude
553. murphymj
554. blakedev
555. alexandergunnarson
556. ljtiuo
557. backwardinduction
558. chilukrn
559. andrewglaucus
560. November
561. user1996
562. raikrahul
563. dylanmathews
564. TheExiled
565. imperialist315
566. Gutenberg
567. bixiedust102
568. max00paul
569. anandsgh
570. cumar
571. vinay1997
572. aubwilson
573. noiamstefan
574. rmgrave10der
575. pi37
576. russoa
577. lookingforsome
578. chaworth
579. liebstraum
580. gaurav1729
581. Emilio94
582. georgeman007
583. thegreenduck
584. chandreshwar
585. jgould
586. JacktheFlying
587. hectorthecollector
588. NiceIRL
589. quantumite
590. Nischal
591. WiserMelon
592. SteveB526
593. lx003n
594. rajendra4343
595. tgallow5
596. admiralgeneral
597. tekno
598. rere
599. Birdmav
600. ilr1
601. rafaelGeo
602. axfwk17
603. LordDaedalus
604. Julian
605. timrabb211
606. blackcats31
607. freedumb123
608. ficar
609. Dragonduck
610. myreprise
611. tjbecker
612. tillyt
613. eponymic23
614. packard910
615. CharlesJCoopper
616. PavanePrincess
617. ktsenthil
618. ValiantEntity
619. Rizhiy
620. mejjem
621. abc123def
622. Yenniesaurus
623. victoremnm
624. leonbasin
625. Banach-Tarski
626. aaidan
627. Shobhana
628. Shobhana
629. gn0m0n
630. MisterE
631. ifr33m4n
632. hamitgo
633. jwhitlark
634. Vnuke
635. sabbysaur
636. rampant_juju
637. NeeBo
638. brentshermana
639. atmospheric
640. joehigh
641. anaran
642. tpkalman
643. timlovelock
644. bpalkmim
645. foxboy
646. ishouldread
647. aatxe
648. Bloodrazor
649. timjstewart
650. L2BSmart
651. foo
652. frwdrik
653. aDisciple
654. nunofilipemargarido
655. Bennymoonbase
656. pinguhash
657. mtilles69
658. konfou
659. Troglodite
660. jarebear
661. Richo1337
662. maka_maria
663. RMnumbers
664. noamtwx
665. wwalid2050
666. hmeier
667. drumchaps
668. wjzhu
669. cryptguy
670. aeknel
671. bbmacarthur
672. zeroms
673. crichm03
674. RicoSpartan
675. reemaj
676. atap
677. dschwartz266
678. mpmpie
679. JimSunkes
680. HexMerck
681. benshats
682. Threscher
683. martyg
684. Tessinator
685. gerardlouw
686. swapanj162
687. Snejana
688. justin_ecklund
689. Suzanne2578
690. sintrastes
691. antone
692. shitzfez
693. natasakalp
694. kingosraigi
695. Ichikoa
696. Theguybehindthescreen
697. haraf_ijhal
698. lemoskid
699. lazypakman
700. fauxhemian5
701. namesare4tombstones
702. vincentmakreel
703. TrollyMike
704. DumbleDaz
705. Reathur
706. nono
707. eKro
708. Usagi
709. jmrkumar
710. JayMendez
711. tpdemixxx
712. asingh21
713. burtgervis
714. jtormey
715. analy7er
716. fezspinkguitar
717. EntropicalForest
718. BTizzle23
719. Higs
720. 20sat92
721. Cori83
722. DylanSA