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(श्रीपाद म. गर्गॆ) Shripad M. Garge [Home page] |
Title: Invariant theory of classical groups Abstract: Let a group $G$ act linearly on a finite dimensional complex vector space $V$. It is natural to ask for the $G$-invariant complex valued polynomial functions on $V$. These functions form a ring in a natural way. The two fundamental theorems in invariant theory of $G$ give generators for this ring as a complex algebra and the relations amongst these generators. As an example, the symmetric group $S_n$ acts by permuting a basis of $\mathbb{C}^n$ and the corresponding invariants are generated as a polynomial ring by the elementary symmetric functions. So the generators are the elementary symmetric functions and there are no relations amongst them. Our aim in this lecture series is to understand the invariants for the natural action of $GL(V)$ on $V$ and on related vector spaces. If there is time, we will also study the analogous theorems for the orthogonal and symplectic groups. We will be using the online notes by Kraft and Procesi: `Classical invariant theory: A primer' which can be downloaded here. Venue: Room 216, Department of Mathematics. Lecture schedule:
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