Archive for the ‘The Princeton Companion to Mathematics’ Category

IV.9 Representation Theory by Ian Grojnowski, Noncompact Groups, Groups in Characteristic p and Lie Algebras: Pages 428 – 431.

Thursday, March 10th, 2011

Representation Theory by Ian Grojnowski.

Interlude: The Philosophical lessons of “The Character Table Is Square”.

Coda: The Langlands Program.

Readings: “Local Representation Theory” and “Automorphic Forms, Representations and L-functions”.

IV.9 Representation Theory by Ian Grojnowski, Noncompact Groups, Groups in Characteristic p and Lie Algebras: Pages 426 – 427.

Tuesday, March 1st, 2011

Representation Theory by Ian Grojnowski.

Noncompact Groups, Groups in Characteristic p and Lie Algebras.

IV.9 Representation Theory by Ian Grojnowski, Fourier Analysis: Pages 425 – 426.

Sunday, February 27th, 2011

IV.9 Representation Theory by Ian Grojnowski, Fourier Analysis: Pages 425 – 426.

IV.9 Representation Theory by Ian Grojnowski, Why Vector Spaces?: Page 424 – 425.

Friday, February 25th, 2011

Representation Theory by Ian Grojnowski.

Why Vector Spaces?

The number of partitions of n is about (1/4n√(3))epÖ(2n/3).

IV.9 Representation Theory by Ian Grojnowski, Why Vector Spaces?: Page 423.

Monday, January 24th, 2011

Representation Theory by Ian Grojnowski.

Why Vector Spaces?

Theorem: Let G be a finite group.  then the characters of the irreducible representations form an orthonormal basis of K-sub-G, the vector space of all complex valued functions on G that are constant functions on each conjugacy class.  This means the character table is square.

IV.9 Representation Theory by Ian Grojnowski, Why Vector Spaces?: Page 422.

Friday, January 21st, 2011

Representation Theory by Ian Grojnowski.

Why Vector Spaces?

The analogy to Fourier transform decomposition.

IV.9 Representation Theory by Ian Grojnowski, Why Vector Spaces?: Page 421.

Thursday, January 20th, 2011

Representation Theory by Ian Grojnowski.

Why Vector Spaces?

Definition: A representation of a group G on a vector space V is a homomorphism from G to GL(V).  GL(V) is the set of linear maps from V to V.

If G is the dihedral group D8 acting on the set X of ordered pairs of vertices of the square, of which there are 16, then there are three orbits of G on X.

IV.8 Moduli Spaces by David. D. Ben-Zvi, Invariants from Moduli Spaces, Modular Forms and Further Reading, IV.9 Representation Theory by Ian Grojnowski, Introduction and Why Vector Spaces?: Pages 418 – 420.

Monday, January 17th, 2011

IV.8 Moduli Spaces by David. D. Ben-Zvi.

Invariants from moduli spaces.

Modular forms.

IV.9 Representation Theory by Ian Grojnowski.

Introduction.

Why vector spaces?

IV.8 Moduli Spaces by David. D. Ben-Zvi, Moduli Spaces and Jacobians, Further Directions and Deformations and Degenerations: Page 418.

Friday, January 14th, 2011

IV.8 Moduli Spaces by David. D. Ben-Zvi.

Moduli spaces and Jacobians.

Further directions.

Deformations and degenerations.

IV.8 Moduli Spaces by David. D. Ben-Zvi, Digression “Abstract Nonsense” and Moduli Spaces and Representations: Page 417.

Thursday, January 13th, 2011

Moduli Spaces by David. D. Ben-Zvi.

Digression “Abstract Nonsense”.

Moduli spaces and representations.