My OperaPrerequisites
Friday, November 18, 2005 11:02:19 PM
The set of lecture notes about Character Theory from David Towers is very beautiful written. These notes deserve a book published by Springer or even better (is that possible ?! ). If I could make a wish today it would be about following lectures from David Towers about Representation Theory some day not too far away. Who knows? Anything is possible. Back to reality.
Continuing on characters. In words...
A function from a Group to the field of complex numbers which is constant on the conjugacy classes of G is called a class function. G -> C.
The set of class functions together with the addition of class functions and multiplication with a scalar is a vector space.
So we can speak of the vector space of characters.
The characters of the irreducible CG modules of a representation form a basis of the character vector space.
Any character can be expressed as a linear combination of the irreducible characters.
I am now in Lecture 2 of the Book. "Characters". I understand the main theorem of this lecture. ( That is I understand the proof. I don't think I am able to reproduce the proof yet. But that's not what I am trying to achieve.) The characters of the irreducible representations form an orthonormal basis for the space of class functions on G. ( More about the book. It has a lot of exercises. I don't like math books which hide essential theorems behind exercises... "for the reader to discover", or other crap-reasons why the author was too lazy to solve the problems himself. Some books have lots of exercises but have no ( hints to ) answers. These are the worst books imo. Representation Theory has some answers, some hints, just enough. Excellent. ) The main difference between the book and the lecture notes ( just considering the part about Character Theory of course ) is that the book assumes a LOT more knowledge from Linear Algebra than the notes. The authors are aware of this since they included at least six appendices. Appendix B is about multilinear algebra and describes the topics: Tensor Product, Exterior and Symmetric Powers, Duals and Contractions. An opportunity to get into these topics more deeply. Appendix B has a few definitions, not more than that. The book by Halmos, Finite-dimensional vector spaces is an excellent abstract treatment of Linear Algebra. The modern Linear Algebra books by for example Strang omit essential topics like Symmetric Powers and such.
Will continue on 'calculating Character Tables'. Have to do some speed-sessions on multilinear algebra as well.
Continuing on characters. In words...
A function from a Group to the field of complex numbers which is constant on the conjugacy classes of G is called a class function. G -> C.
The set of class functions together with the addition of class functions and multiplication with a scalar is a vector space.
So we can speak of the vector space of characters.
The characters of the irreducible CG modules of a representation form a basis of the character vector space.
Any character can be expressed as a linear combination of the irreducible characters.
I am now in Lecture 2 of the Book. "Characters". I understand the main theorem of this lecture. ( That is I understand the proof. I don't think I am able to reproduce the proof yet. But that's not what I am trying to achieve.) The characters of the irreducible representations form an orthonormal basis for the space of class functions on G. ( More about the book. It has a lot of exercises. I don't like math books which hide essential theorems behind exercises... "for the reader to discover", or other crap-reasons why the author was too lazy to solve the problems himself. Some books have lots of exercises but have no ( hints to ) answers. These are the worst books imo. Representation Theory has some answers, some hints, just enough. Excellent. ) The main difference between the book and the lecture notes ( just considering the part about Character Theory of course ) is that the book assumes a LOT more knowledge from Linear Algebra than the notes. The authors are aware of this since they included at least six appendices. Appendix B is about multilinear algebra and describes the topics: Tensor Product, Exterior and Symmetric Powers, Duals and Contractions. An opportunity to get into these topics more deeply. Appendix B has a few definitions, not more than that. The book by Halmos, Finite-dimensional vector spaces is an excellent abstract treatment of Linear Algebra. The modern Linear Algebra books by for example Strang omit essential topics like Symmetric Powers and such.
Will continue on 'calculating Character Tables'. Have to do some speed-sessions on multilinear algebra as well.
