Program
Introduction to classical Invariant Theory | |
Student No.： | 50 |
Time： | Mon/Wed 15:20-17:00, 2014-04-21~ 2014-05-14 (except for April 28 and 30) |
Instructor： | Claudio Procesi [University of Roma - La Sapienza] |
Place： | Conference Room3 , Floor 2, Jin Chun Yuan West Building |
Starting Date： | 2014-4-21 |
Ending Date： | 2014-5-14 |
Description:
Prerequisite:
A good knowledge of linear algebra and some basic abstract algebra
Reference:
C. ~Procesi, Lie Groups. An approach through invariants and representations,
pp.{xxiv+596}, Springer, Universitext, 2007.
Invariant theory was born in the 19^th century together with the foundations of geometry. It provides formulas and structural information for invariant or equivariant properties of algebraic and geometric objects under some group of symmetries which usually is associated to the given geometry.
At the beginning invariant theory is very computational and formulas oriented, here the main names are Cayley, Sylvester, Clebsch, Gordan, Capelli, Deruyts. The initial geometric theory is due to Hilbert and it goes along with the foundations of commutative algebra. In classical invariant theory, classical groups (in the sense of H. Weyl) are analyzed and the theory of invariants has a strict relationship with representation theory. Here the names are Frobenius, Schur, Young, Brauer, Weyl.
In the course aspects of these two theories are presented.
topics:
1. Groups and invariants, Hilbert's 14^th problem, Hilbert Theory
2. Invariants of classical groups, tensor representations
3. The Schur--Weyl duality
4. Symmetric functions Schur functions characters
5. Young symmetrizers
6. Fist and second fundamental theorem of invariant theory for classical groups
7. The role of standard tableaux