Description

Invariant Theory has gone through many ups and down throughout its two centuries history. Computations was the main focus until Hilbert, who was the first to prove that the invariants of the classical groups are finitely generated. These works feature such results as the Nullstellensatz and the Basis Theorem, and were a crucial step in the development of modern Commutative Algebra and Algebraic Geometry. After leaving the spotlight for a few decades, Invariant Theory came back in force in the middle of the twentieth century and greatly benefited from the advances made by Commutative Algebra and Algebraic Geometry. Apart from being a source of interesting and difficult questions on its own, Invariant Theory is an invaluable source of examples for testing all sorts of theories.

In this project you will learn the basic algorithms for computing rings of invariants. We will focus on linear actions of finite groups on finite dimensional vector spaces. Along the way, you will become familiar with many important concepts of commutative algebra, and in particular the theory of Gröbner bases, the key computational tool in many algorithms. This will be done through experimentation by hand, but also with the help of a computer and the software packages MAGMA and Macaulay2.

Resources

[1] Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. Third edition. Undergraduate Texts in Mathematics. Springer, New York, 2007.

[2] Harm Derksen and Gregor Kemper, Computational invariant theory. Invariant Theory and Algebraic Transformation Groups, I. Encyclopaedia of Mathematical Sciences, 130. Springer-Verlag, Berlin, 2002.

[3] Mara Neusel, Invariant theory. Student Mathematical Library, 36. American Mathematical Society, Providence, RI, 2007.

[4] Bernd Sturmfels, Algorithms in invariant theory. Second edition. Texts and Monographs in Symbolic Computation. SpringerWienNewYork, Vienna, 2008.