Classical Invariant Theory

Abstract

For a linear algebraic group G and a regular representation (ρ,V) of G, the basic problem of invariant theory is to describe the G-invariant elements (⊗^k V)^G of the k-fold tensor product for all k. If G is a reductive, then a solution to this problem for (ρ ^*, V ^*) leads to a determination of the polynomial invariants P(V)^G. When G ⊂ GL(W) is a classical group and V = W ^k ⊕ (W ^*)^l (k copies of W and l copies of W ^*), explicit and elegant solutions to the basic problem of invariant theory, known as the first fundamental theorem (FFT) of invariant theory for G, were found by Schur, Weyl, Brauer, and others. The fundamental case is G = GL(V) acting on V . Following Schur and Weyl, we turn the problem of finding tensor invariants into the problem of finding the operators commuting with the action of GL(V) on ⊗^k V, which we solved in Chapter 4. This gives an FFT for GL(V) in terms of complete contractions of vectors and covectors. When G is the orthogonal or symplectic group we first find all polynomial invariants of at most dim V vectors. We then use this special case to transform the general problem of tensor invariants for an arbitrary number of vectors into an invariant problem for GL(V) of the type we have already solved.

The FFT furnishes generators for the commutant of the action of a classi cal group G on the exterior algebra ∧V of the defining representation; the general duality theorem from Chapter 4 gives the G-isotypic decomposition of ∧V. This furnishes irreducible representations for each fundamental weight of the special linear group and the symplectic group. For G = SO(V) it gives representations for all G-integral fundamental weights (the spin representations for the half-integral fundamental weights of so(V) will be constructed in Chapter 6 using the Clifford algebra). Irreducible representations with arbitrary dominant integral highest weights are obtained as iterated Cartan products of the fundamental representations. In Chapters 9 and 10 we shall return to this construction and obtain a precise description of the tensor subspaces on which the irreducible representations are realized.

Combining the FFT with the general duality theorem we obtain Howe duality for the classical groups: When V is a multiple of the basic representation of a classical group G and $\mathbb{D}$(V) is the algebra of polynomial-coefficient differential operators on V , then the commuting algebra $\mathbb{D}$(V)^G is generated by a Lie algebra of differential operators isomorphic to ǵ, where ǵ is another classical Lie algebra (the Howe dual to g = Lie(G)).The general duality theorem from Chapter 4 then sets up a correspondence between the irreducible regular (finite-dimensional) representations of G occurring in P(V) and certain irreducible representations of ǵ (generally infinite-dimensional). As a special case, we obtain the classical theory of spherical harmonics. The chapter concludes with several more applications of the FFT.