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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thank You

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