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T is a symmetric tensor (related to the energy-momentum tensor) if the dual source Tˆ vanishes, since R ∈ 42 (M) in that case. 15). 13) Tensor Gauge Fields in Arbitrary Representations of GL(D, R) 49 Let us now stress some peculiar features of D = 4 dimensional spacetime. From our previous experience with electromagnetism and our definition of Hodge duality, we naturally expect this dimension to be privileged. In fact, the analogy between linearized gravity and electromagnetism is closer in four dimensions because less independent equations are involved : ∗R has the same symmetries as the Riemann tensor, thus the dual gauge field h˜ is a symmetric tensor in 22 (M).

P (x) in the entries of a column of Y and another entry of Y which is on the right of the column, vanishes. Using the previous definitions of multiforms, Hodge dual and trace operators, this set of conditions gives Proposition 2 (Schur module). Let α be a multiform in lj ≤ li < D , l1 ,... ,lS (M). [S] If ∀ i, j ∈ {1, . . , S} : i ≤ j , then one obtains the equivalence Trij { ∗i α } = 0 ∀ i, j : 1 ≤ i < j ≤ S ⇐⇒ α∈ (l1 ,... ,lS ) (M) . (S) Indeed, condition (i) is satisfied since α is a multiform.

A general explanation of this fact will be given at the end of the next section. The dual linearized Riemann tensor is invariant under the transformation (D−4,1) (D−3,0) ∼ D−3 (M) . 28), is represented by 48 X. Bekaert, N. Boulanger ∂ .. .. ∂ .. .. 38) respectively read (up to coefficient redefinitions) (i−1) (i) (i) S = dL S +d A , (i) S∈ (i−1) (i) A = −dL A , (D−3−i,1) (M) , (2) (i) A∈ (i = 2, . . 10) D−2−i (M) . These reducibilities are a direct consequence of Corollary 2. 2. Comparison with electrodynamics.