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Remarks on classical invariant theory. (English) Zbl 0674.15021
As it is well known, the theorem on tensor invariants for $$GL_ n$$ is directly available by using the double commutant theorem. Recently it was shown by M. Atijah, R. Bott and V. Patodi [Inv. Math. 19, 279-330 (1973; Zbl 0257.58008)] how the result on tensor invariants for orthogonal groups $$0_ n$$ could be reduced by a fairly quick argument to the result for $$GL_ n$$, and a similar argument goes through for the symplectic groups $$SP_{2n}$$. The main purpose of the present discussion is to point out that this unified viewpoint is in fact capable of considerable extension. The starting point is the following theorem which is the extension of the first fundamental theorem of classical invariant theory to some new context:
Theorem. Let G be a classical group, and let U and W be G-modules formed by taking direct sums of the basic module for G (and, if $$G=GL_ n$$, of the contragradient module). Consider the resulting action on the associative algebra (with 1) S(U)$$\otimes \Lambda (W)$$. Then the algebra of G-invariants is generated by the invariants of degree 2 (with respect to natural grading).
Here are the specifics of the extended viewpoint of the author:
(a) The argument used to pass from tensor invariants to polynomial invariants in many variables applies equally well to the decomposition of mixed algebras of partially symmetric, partially antisymmetric tensors.
(b) By means of a process of “doubling the variables”, the result in (a) yields a description of the endormorphisms (of a certain reasonable type) commuting with a classical action on such an algebra.
(c) The result of (b) may be viewed as a “duality theorem” for commuting subalgebras of a certain naturally occuring graded Lie algebra.
(d) Particular cases of the above general scheme yield many classical computations in multilinear algebra, often with additional structure and insight. As the examples it is given the following:
(i) the full isotypic decomposition of the polynomials in k variables for a classical action (“theory of spherical harmonics”),
(ii) the exactness of the polynomial de Rham and Dolbeault complexes, and the homotopy between the identity and projection onto constants,
(iii) the algebra behind the structure of the cohomology of Kähler manifolds (sometimes called Hodge theory) as a special case of the analogue of (i) for exterior algebras,
(iv) the cohomology of the unitary group,
(v) the invariants of the adjoint action of the classical groups,
(vi) elucidation of the Capelli identity.
Reviewer: V.L.Popov

MSC:
 15A72 Vector and tensor algebra, theory of invariants 20G05 Representation theory for linear algebraic groups 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 22E70 Applications of Lie groups to the sciences; explicit representations 20C35 Applications of group representations to physics and other areas of science 20G45 Applications of linear algebraic groups to the sciences 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations
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