Norms and generating functions in Clifford algebras (Q2425790)

The objective of this paper is to discuss some facts concerning norms, formal power series and generating functions in Clifford algebras, and some applications of the formalism in graph theory. Let \(\mathcal{C}\ell_{p,q}\) be the Clifford algebra of the vector space \(\mathbb{R}^{p,q}\), with \(p+q=n\). Given \(u \in \mathcal{C}\ell_{p,q}\), its power \(u^m\) is the Clifford-algebraic coefficient of \(t^m\) in the series expansion on \((1-tu)^{-1}\). Conditions on \(t \in \mathbb{R}\) for the existence of \((1-tu)^{-1}\) are given, and an explicit formulation of the generating function is obtained. A ``Clifford-Frobenius'' norm of an \(m\times m\) matrix \(A\) with entries in \(\mathcal{C}\ell_{p,q}\) is given, and norm inequalities and conditions for the existence of \((I-tA)^{-1}\) are determined. As an application, the author discusses a method for counting a graph's \(k\)-cycles using the anti-commutative basis vectors of a Clifford algebra of arbitrary signature.