In the fall of 1997 I gave a series of lectures at Queen’s University on algorithms in invariant theory of finite groups. This article is an expanded version of the material presented there. The main topic is the calculation of the invariant ring of a finite group acting on a polynomial ring by linear transformations of the indeterminates. By “calculation” I mean finding a finite system of gene...

The discovery of symmetry in nature is one of the most fundamental and universal intellectual achievements. The mathematical language for analyzing symmetries is the theory of groups and their invariants: objects or phenomena of interest and their properties are described mathematically in terms of solutions of systems of equations, involving numerical functions that depend on chosen coordinate...

If V is a faithful module for a nite group G over a eld of characteristic p > 0, then the ring of invariants need not be Cohen-Macaulay if p divides the order of G. In this article the cohomology of G is used to study the question of Cohen-Macaulayness of the invariant ring. Let R = S(V) be the polynomial ring on which G acts. Then the main result can be stated as follow: If H r (G; R) 6 = 0 fo...

Working over an algebraically closed base field k of characteristic 2, the ring of invariants R is studied, where G is the orthogonal group O(n) or the special orthogonal group SO(n), acting naturally on the coordinate ring R of the m-fold direct sum k⊕ · · · ⊕ k of the standard vector representation. It is proved for O(2), O(3) = SO(3), SO(4), and O(4), that there exists an m-linear invariant ...

Working over an algebraically closed base field k of characteristic 2, the ring of invariants R is studied, where G is the orthogonal group O(n) or the special orthogonal group SO(n), acting naturally on the coordinate ring R of the m-fold direct sum k⊕ · · · ⊕ k of the standard vector representation. It is proved for O(2), O(3) = SO(3), SO(4), and O(4), that there exists an m-linear invariant ...

The moduli space of the ordinary non-singular quartic curves over fields of characteristic 2 is isomorphic to a certain open subset of an affine variety, whose coordinate ring in turn is given as the invariant algebra of a certain module of the finite group GL3(F2). We derive a complete description of this invariant algebra by combining theoretical analysis with application of specially tailore...

Let K be the field of fractions of a Henselian discrete valuation ring OK . Let XK/K be a smooth proper geometrically connected scheme admitting a regular model X/OK . We show that the index δ(XK/K) of XK/K can be explicitly computed using data pertaining only to the special fiber Xk/k of the model X. We give two proofs of this theorem, using two moving lemmas. One moving lemma pertains to hori...

We propose an algorithm for computing invariant rings of algebraic groups which act linearly on affine space, provided that degree bounds for the generators are known. The groups need not be finite nor reductive, in particular, the algorithm does not use a Reynolds operator. If an invariant ring is not finitely generated the algorithm can be used to compute invariants up to a given degree.

Nagahara and Kishimoto [i] studied free ring extensions B(x) of degree n n for some integer n over a ring B with i, where x b, cx xp(c) for all c and n-I some b in B (p automorphism of B), and {i, x x is a basis. Parimala and Sridharan [2], and the author investigated a class of free ring extensions called generalized quaternlon algebras in which b -I and p is of order 2. The purpose of the pre...