## Note on reversion, rotation and exponentiation in dimensions five and six.(English)Zbl 1335.15034

Authors’ abstract: The explicit matrix realizations of the reversion anti-automorphism and the spin group depend on the set of matrices chosen to represent a basis of 1-vectors for a given Clifford algebra. On the other hand, there are iterative procedures to obtain bases of 1-vectors for higher-dimensional Clifford algebras, starting from those for lower-dimensional ones. For a basis of 1-vectors for $$\mathrm{Cl} (0, 5)$$, obtained by applying such procedures to the Pauli basis of 1-vectors for $$\mathrm{Cl}(3,0)$$, we find that the matrix form of reversion involves neither of the two standard representations of the symplectic bilinear form. However, by making use of the relation between $$4 \times 4$$ real matrices and the tensor product of the quaternions with themselves, the matrix form of reversion for this basis of 1-vectors is identified. The corresponding version of the Lie algebra of the spin group has useful matrix properties which are explored. Next, the form of reversion for a basis of 1-vectors for $$\mathrm{Cl}(0,6)$$ obtained iteratively from $$\mathrm{Cl}(0,0)$$ is obtained. This is then applied to the task of computing exponentials of $$5 \times 5$$ and $$6 \times 6$$ real skew-symmetric matrices in closed form by reducing this to the simpler task of computing exponentials of certain $$4 \times 4$$ matrices. For the latter purpose, closed form expressions for the minimal polynomials of these $$4 \times 4$$ matrices are obtained, without having to compute their eigenstructure. Finally, a novel representation of $$\mathrm{Sp}(4)$$ is provided which may be of independent interest. Among the byproducts of this work are natural interpretations for some members of an orthogonal basis for $$M(4, R)$$ provided by the isomorphism with the quaternion tensor product, and a first principles approach to the spin groups in dimensions five and six.

### MSC:

 15A66 Clifford algebras, spinors 22E70 Applications of Lie groups to the sciences; explicit representations 15A16 Matrix exponential and similar functions of matrices 15A69 Multilinear algebra, tensor calculus 11E88 Quadratic spaces; Clifford algebras 15B33 Matrices over special rings (quaternions, finite fields, etc.) 15B57 Hermitian, skew-Hermitian, and related matrices
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