Symmetry, Representations, and Invariants pp 225-300 | Cite as

# 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**.

## Keywords

Irreducible Representation Classical Group Hilbert Series Regular Representation Symplectic Group## Preview

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