Renormalization theory in quantum gravity, among other applications, continues to stimulate many attempts to calculate asymptotic expansions of heat kernels and other Green functions of differential operators. Computer algebra systems now make it possible to carry these calculations to high orders, where the number of terms is very large. To be understandable and usable, the result of the calculation must be put into a standard form; because of the subtleties of tensor symmetry, to specify a basis set of independent terms is a non-trivial problem. This problem can be solved by applying some representation theory of the symmetric, general linear and orthogonal groups. In this work the authors treat the case of scalars or tensors formed from the Riemann tensor (of a torsionless, metric-compatible connection) by covariant differentiation, multiplication and contraction. (The same methods may be applied readily to other tensors.) The authors have determined the number of independent homogeneous scalar monomials of each order and degree up to order twelve in derivatives of the metric, and exhibited a basis for these invariants up through order eight. For tensors of higher rank, they present bases through order six; in that case some effort is required to match the familiar classical tensor expressions (usually supporting reducible representations) against the lists of irreducible representations provided by the more abstract group theory. Finally, the analysis yields (more easily for scalars than for tensors) an understanding of linear dependences in low dimensions among otherwise distinct tensors.

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We derive group branching laws for formal characters of subgroups Hπ of GL(n) leaving invariant an arbitrary tensor T π of Young symmetry type π where π is an integer partition. The branchings GL(n) ↓ GL(n − 1) , GL(n) ↓ O(n) and GL(2n) ↓ Sp(2n) fixing a vector vi , a symmetric tensor gij = gji and an antisymmetric tensor fij = −fji , respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function sπ ≡ {π} by the basic M series of complete symmetric functions and the L = M −1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras, and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains π -generalized Newell-Littlewood formulae, and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for H 1 3 , H21 , and H3 , showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine, and in some instances non reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2-cocycles. The algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for GL(n) formal characters. Modification rules are beyond the scope of the present paper, but are briefly discussed. applications of Kronecker products is provided by [13]. A necessary concomitant of these techniques is the automation provided by a symbolic computer package such as SCHUR c [36]. Finally, some of these techniques have been found to generalize to the representation theory of non-compact groups [19,17,18] In a recent paper [10] the role of symmetric functions in relation to group representation theory has been re-considered from the viewpoint of the underlying Hopf algebraic structure. This structure is in fact well known in the combinatorial literature [34,33,31]. In [10] the formalism of branching rules was aligned with certain endomorphisms on the algebra of symmetric functions, called branching operators, derived from 1-cochains, for which the multiplicative cohomology of Sweedler [32] provides a natural analytical setting and classification. Standard branchings from generic symmetric functions to symmetric functions of orthogonal or symplectic type (the classic Newell-Littlewood theorems for the group reduction from GL(n) ↓ O(n) or GL(n) ↓ Sp(n) ) were found to be derived from certain 2-cocycles (for which associativity is guaranteed).This result then prompts the more general question of classifying arbitrary, non-cohomologous, 2cocycles, and the nature of any associated character theory and of the algebraic or group structures which might be entrained therewith. While an ab initio approa...

Modification rules, expressible in terms of the removal of continuous boundary hooks, are derived which relate nonstandard irreducible representations (IR's) of the unitary, orthogonal, and symplectic groups in n dimensions to standard IR's. Tensorial methods are used to derive procedures for reducing the outer products of IR's of U(n), O(n), and Sp(n), and for reducing general IR's of U(n) specified by composite Young tableaux with respect to the subgroups O(n) and Sp(n). In these derivations the conjugacy relationship between the orthogonal and the symplectic groups is fully exploited. The results taken in conjunction with the modification rules are valid for all n.

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