Invariant theory
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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...
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, the 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.
