representations of finite groups - mustafa hajij › wp...of-finite-groups.pdf · two...
TRANSCRIPT
![Page 1: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/1.jpg)
Representations of Finite Groups
Vigre Seminar Mustafa Hajij
![Page 2: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/2.jpg)
Definition
![Page 3: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/3.jpg)
Definition
![Page 4: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/4.jpg)
Some representations of cyclic groups
![Page 5: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/5.jpg)
Some representations of cyclic groups
![Page 6: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/6.jpg)
Some representations of cyclic groups
![Page 7: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/7.jpg)
Some representations of S3
![Page 8: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/8.jpg)
Irreducible representations
![Page 9: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/9.jpg)
Irreducible representations
![Page 10: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/10.jpg)
Irreducible representations
![Page 11: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/11.jpg)
Irreducible representations of a cyclic group
![Page 12: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/12.jpg)
Irreducible representations of a cyclic group
![Page 13: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/13.jpg)
Irreducible representations of a cyclic group
![Page 14: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/14.jpg)
Irreducible representations of a cyclic group
![Page 15: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/15.jpg)
Irreducible representations of S3
![Page 16: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/16.jpg)
Irreducible representations of S3
![Page 17: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/17.jpg)
Irreducible representations of S3
![Page 18: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/18.jpg)
Irreducible representations of S3
![Page 19: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/19.jpg)
Irreducible representations of S3
![Page 20: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/20.jpg)
Direct Sum of Representations
![Page 21: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/21.jpg)
Direct Sum of Representations
![Page 22: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/22.jpg)
Maschke’s Theorem
![Page 23: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/23.jpg)
Maschke’s Theorem
![Page 24: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/24.jpg)
Tensor product of representations
![Page 25: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/25.jpg)
Tensor product of representations
![Page 26: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/26.jpg)
Tensor product of representations
![Page 27: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/27.jpg)
Group Characters
![Page 28: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/28.jpg)
Group Characters
![Page 29: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/29.jpg)
Group Characters
![Page 30: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/30.jpg)
Group Characters
![Page 31: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/31.jpg)
Elementary properties of group characters
![Page 32: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/32.jpg)
Elementary properties of group characters
![Page 33: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/33.jpg)
Elementary properties of group characters
![Page 34: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/34.jpg)
Elementary properties of group characters
![Page 35: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/35.jpg)
Inner product of characters
![Page 36: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/36.jpg)
Inner product of characters
![Page 37: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/37.jpg)
Inner product of characters
![Page 38: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/38.jpg)
Inner product of characters
![Page 39: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/39.jpg)
Inner product of characters
![Page 40: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/40.jpg)
Inner product of characters
![Page 41: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/41.jpg)
Inner product of characters
![Page 42: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/42.jpg)
Inner product of characters
![Page 43: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/43.jpg)
Inner product of characters
![Page 44: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/44.jpg)
Character Table
![Page 45: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/45.jpg)
Character Table
![Page 46: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/46.jpg)
Character Table
![Page 47: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/47.jpg)
Character Table
![Page 48: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/48.jpg)
Character Table
![Page 49: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/49.jpg)
Character Table
![Page 50: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/50.jpg)
The Character Table for S3
![Page 51: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/51.jpg)
The Character Table for S3
![Page 52: Representations of Finite Groups - Mustafa Hajij › wp...of-finite-groups.pdf · Two representations : G —+ GL(V) and 7T2 : G —+ GL(W) are said to be equivalent if there eazsts](https://reader036.vdocuments.us/reader036/viewer/2022070805/5f039c637e708231d409e798/html5/thumbnails/52.jpg)
Thank You