The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, we present an introduction to the main ideas of Clifford algebra, with the main goal to develop a package for Clifford algebra calculations for the computer algebra program Mathematica∗. The Cliffor...

CLIFFORD performs various computations in Graßmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in C (B) the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUM based on Chevalley’s recursive formula, and cmulRS based on a non-recursive Rot...

CLIFFORD performs various computations in Graßmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in Cl(B) the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for Clifford product are implemented: cmulNUM based on Chevalley’s recursive formula, and cmulRS based on non-recursive Rota-Stei...

Kravchuk polynomials arise as orthogonal polynomials with respect to the binomial distribution and have numerous applications in harmonic analysis, statistics, coding theory, and quantum probability. The relationship between Kravchuk polynomials and Clifford algebras is multifaceted. In this paper, Kravchuk polynomials are discovered as traces of conjugation operators in Clifford algebras, and ...

Knill introduced a generalization of stabilizer codes, in this note called Clifford codes. It remained unclear whether or not Clifford codes can be superior to stabilizer codes. We show that Clifford codes are stabilizer codes provided that the abstract error group has an abelian index group. In particular, if the errors are modelled by tensor products of Pauli matrices, then the associated Cli...

In this note, we describe the Gel’fand-Tsetlin procedure for the construction of an orthogonal basis in spaces of Hermitean monogenic polynomials of a fixed bidegree. The algorithm is based on the Cauchy-Kowalewski extension theorem and the Fischer decomposition in Hermitean Clifford analysis.

We define and study the ternary analogues of Clifford algebras. It is proved that the ternary Clifford algebra with N generators is isomorphic to the subalgebra of the elements of grade zero of the ternary Clifford algebra with N+1 generators. In the case N = 3 the ternary commutator of cubic matrices induced by the ternary commutator of the elements of grade zero is derived. We apply the terna...