# Combinatorial Clifford Algebra

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Combinatorial Aspects of Clifford Algebra, by Lars Svensson and Ambjörn Naeve (2002), presented at the International Workshop on Applications of Geometric Algebra, Cambridge, 5-6 Sept. 2002.

Svensson and Naeve (2002): Combinatorial aspects of Clifford Algebra

Abstract

In this paper we focus on some combinatorial aspects of Clifford algebra and show how this algebra allows combinatorial theorems – like e.g., Sperner’s lemma – to be “built into the algebraic background,” and become part of the structure of the algebra itself. We also give an example of how cumbersome combinatorial proofs can be “mechanized” and carried out in a purely computational manner.

Introduction

In his monumental and groundbreaking Ausdehnungslehre from 1844, Hermann Grassmann set out to build an “algebra for everything” – an algebra which he illustrated by various geometric examples. Being both far ahead of his time and on the outside of the academic mathematical community, Grassmann’s ideas received little attention during his own lifetime. However, during the last years of Grassmann’s life (late 1870s), his ideas were taken up by William Kingdon Clifford, who developed the algebra that today bears his name.

In more recent times, mathematicians and physicists – notably Marcel Riesz, Gian-Carlo Rota, and David Hestenes – have rediscovered and continued this development. Hestenes has focused on the geometric aspects of Clifford algebra – introducing the synonymous term geometric algebra – and shown how it provides a powerful geometric language that serves as a bridge between mathematics and physics.

In this paper we aim to connect with Grassmann’s original ideas and follow Rota by focusing on the purely combinatorial aspects of Clifford algebra.

Some notation and background

Since we are only interested in combinatorial and algebraic aspects of Clifford algebra, we will allow our scalars to lie in an arbitrary commutative ring $\, \mathcal {R} \,$ with unit element. We will also take a slightly different point of view regarding the Clifford algebra and its interpretations.

Let $\, X \,$ be a finite set which is totally ordered, i.e., $\, X = \{x_1, \ldots, x_n\} \,$ where $\, x_1 < x_2 < \ldots < x_n$. We will identify the $\, k$-base-blades $x_{n_1} x_{n_2} \ldots x_{n_k},$ where $\, n_1 < n_2 < \ldots < n_k \leq n,$ with the $\, k$-subsets $\, \{x_{n_1}, \ldots, x_{n_k}\} \subseteq X \,$ and we will denote the pseudoscalar $\, x_1 x_2 \ldots x_n \,$ by $\, X \,$ (by an unproblematic change of context). Moreover, the ring unit $\, 1 \,$ is identified with the empty set $\, \emptyset$.

We will view the Clifford algebra $\, C_l(X) \,$ as the free $\mathcal {R}$-module
generated by the power-set $\, \wp(X) \,$ of all subsets of $\, X \,$, i.e.,

$C_l(X) = {\oplus \atop {x \in \wp(X) } } \mathcal {R}.$

Note that if $\, X \rightarrow Y \,$ is a bijection, then $\, C_l(X) \,$ is isomorphic to $\, C_l(Y)$.

We will always assume that $\, x^2 = 1, \forall x \in X$. The set of $\, k$-vectors is denoted by ${C_l}^k(X).$ We observe that every bilinear map $\, C_l(X) \times C_l(X) \longrightarrow C_l(X) \,$ is uniquely determined by its values on $\, \wp(X) \times \wp(X)$.

NOTATION: If $\, P \,$ is a proposition, we will use $\, (P) \,$ to denote $\, 1 \,$ or $\, 0 \,$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ depending on whether $\, P \,$ is true or false.

Let $\, A,B \in \wp(X)$. The following notation will be used:

Geometric product: $\, A \, B = \epsilon \, A \, \Delta \, B, \,$ where $\, \epsilon = \pm 1 \,$ and $\, \Delta \,$ is symmetric difference.

Outer product: $\, A \, \wedge \, B \, = \, (A \cap B \, = \, \emptyset) \, A \, B.$

Inner product: $\, A \, \cdot \, B \, = \, ((A \, \subseteq \, B) \ \text {or} \ (A \, \supseteq \, B)) \, A \, B.$

Left inner product: $\; A \; \llcorner \; B \, = \, (A \, \subseteq \, B) \, A \, B.$

Scalar product: $\, A \, \ast \, B \, = \, (A \, = \, B) \, A \, B.$

Reverse: $\; A^\dagger \, = \, (-1)^\epsilon A,$ where $\, \epsilon \, = \, { {\mid A \mid } \choose { 2 } }.$

Complement: $\; \tilde {A} \, = \, A \, X^{-1}.$

All of these definitions are extended to $\, C_l(X) \,$ by linearity.

THE CONTINUATION OF THIS STORY can be found in our paper
Combinatorial Aspects of Clifford Algebra.

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