Definitions

# Invariant theory

Invariant theory is a branch of abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group.

Invariant theory of finite groups has intimate connections with Galois theory. One of the first major results was the main theorem on the symmetric functions that described the invariants of the symmetric group Sn acting on the polynomial ring R[x1, …, xn] by permutations of the variables. More generally, Chevalley–Shephard–Todd theorem characterizes finite groups whose algebra of invariants is a polynomial ring. Modern research in invariant theory of finite groups emphasizes "effective" results, such as explicit bounds on the degrees of the generators. The case of positive characteristic, ideologically close to modular representation theory, is an area of active study, with links to algebraic topology.

Invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, the theories of quadratic forms and determinants. Another subject with strong mutual influence was projective geometry, where invariant theory was expected to play a major role in organizing the material. One of the highlights of this relationship is the symbolic method. Representation theory of semisimple Lie groups has its roots in invariant theory.

David Hilbert's work on the question of the finite generation of the algebra of invariants (1890) resulted in the creation of a new mathematical discipline, abstract algebra. A later paper of Hilbert (1893) dealt with the same questions in more constructive and geometric ways, but remained virtually unknown until David Mumford brought these ideas back to life in the 1960s, in a considerably more general and modern form, in his geometric invariant theory. In large measure due to the influence of Mumford, the subject of invariant theory is presently seen to encompass the theory of actions of linear algebraic groups on affine and projective varieties. A distinct strand of invariant theory, going back to the classical constructive and combinatorial methods of the nineteenth century, has been developed by Gian-Carlo Rota and his school. A prominent example of this circle of ideas is given by the theory of standard monomials.

## The nineteenth century origins

Classically, the term "invariant theory" refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. This was a major field of study in the latter part of the nineteenth century, when it appeared that progress in this particular field (out of any number of possible mathematical formulations of invariance with respect to symmetry) was the key algorithmic discipline. Despite some heroic efforts that promise was not fulfilled but many spin-off advances are connected to this research. Current theories relating to the symmetric group and symmetric functions, commutative algebra, moduli spaces and the representations of Lie groups are rooted in this area.

In greater detail, given a finite-dimensional vector space V of dimension n we can consider the symmetric algebra S(Sr(V)) of the polynomials of degree r over V, and the action on it of GL(V). It is actually more accurate to consider the projective representation of GL(V), or representations of SL(V), if we are going to speak of invariants: that's because a scalar multiple of the identity will act on a tensor of rank r in S(V) through the r-th power 'weight' of the scalar. The point is then to define the subalgebra of invariants I(Sr(V)) for the (projective) action. We are, in classical language, looking at invariants of n-ary r-ics, where n is the dimension of V. (This is not the same as finding invariants of SL(V) on S(V); this is an uninteresting problem as the only such invariants are constants.)

It is customary to say that the work of David Hilbert, proving abstractly that I(V) was finitely presented, put an end to classical invariant theory. That is far from being true: the classical epoch in the subject may have continued to the final publications of Alfred Young, more than 50 years later. Explicit calculations for particular purposes have been known in modern times (for example Shioda, with the binary octavics).

## Geometric invariant theory

The modern formulation of geometric invariant theory is due to David Mumford, and emphasizes the construction of a quotient by the group action that should capture invariant information through its coordinate ring. It is a subtle theory, in that success is obtained by excluding some 'bad' orbits and identifying others with 'good' orbits. In a separate development the symbolic method of invariant theory, an apparently heuristic combinatorial notation, has been rehabilitated.

## References

• Grace, J. H.; and Young, Alfred (1903). The algebra of invariants. Cambridge: Cambridge University Press. A classic monograph.
• Grosshans, Frank D. (1997). Algebraic homogeneous spaces and invariant theory. New York: Springer. ISBN 3-540-63628-5.
• Neusel, Mara D.; and Smith, Larry (2002). Invariant Theory of Finite Groups. Providence, RI: American Mathematical Society. ISBN 0-8218-2916-5. A recent resource for learning about modular invariants of finite groups.
• Olver, Peter J. (1999). Classical invariant theory. Cambridge: Cambridge University Press. ISBN 0-521-55821-2. An undergraduate level introduction to the classical theory of invariants of binary forms, including the Omega process starting at page 87.
• Springer, T. A. (1977). Invariant Theory. New York: Springer. ISBN 0-387-08242-5. An older but still useful survey.
• Sturmfels, Bernd (1993). Algorithms in Invariant Theory. New York: Springer. ISBN 0-387-82445-6. A beautiful introduction to the theory of invariants of finite groups and techniques for computing them using Gröbner bases.

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