Ronald Shaw

Finite geometries and Clifford algebras

The Clifford algebra in dimension d=2m+1−1, m≥2, is treated using the finite m‐dimensional projective geometry PG(m,2) over the field of order 2. The incidence properties of the geometry help in the problem of finding a complete commuting set of operators with which to label the 2(d−1)/2 spinor states of an irreducible representation. Full details are given in the case m=3, d=15, thus generalizing previous work for the m=2, d=7 case, and various conjectures are made concerning the cases m>3.

Vector cross products in n dimensions

All the usual properties of the vector cross product a1 x a2on E 3 (three‐dimensional Euclidean space) are derived from an algebraic definition of a1 x a2. The obvious generalization of this definition yields a vector cross product a1 x ... x a n‐1 on En, and we show that all the usual properties of a1 x a2 receive natural generalizations. Even the familiar results in E 3 involving the repeated vector product (a x b) x c, and solution of the equation a x v = b, generalize in an exceedingly straightforward fashion. Finally we enquire whether or not other n‐dimensional generalizations of the three‐dimensional vector cross product are possible, pointing out exceptional features of the cases n = 7 and n = 8.

Irreducible multiplier corepresentations and generalized inducing

Wigners classification of irreducible corepresentations into three types is generalised to irreducible multiplier corepresentations. Representations of Types I, II, and III have commutants isomorphic toR,H, andC, respectively. The more general problem of relating irreducible multiplier corepresentations of a group to those of an invariant subgroup is considered, and some algebraic aspects of “generalized inducing” are described. The Wigner classification is then re-obtained as a very simple instance of the general theory.

A characterization of the primals in PG ( m , 2)

The primais of degree r in PG(m, 2) are, for each r ≤ m, shown to be characterized by the property that they have odd intersection with every r-flat.

Irreducible multiplier corepresentations of the extended Poincare group

The irreducible multiplier corepresentations of the extended Poincaré groupP are, for positive and zero mass, determined by generalized inducing from a generalized little group. This approach is compared with the previous one of Wigner. Form>0, and any spinj, a particular realization is noted which is manifestly covariant on all four components ofP. The choice of covering group forP is discussed, and reasons are given for preferring a group for whichS andT generate the quaternion group of order 8.

The polynomial degrees of Grassmann and Segre varieties over GF(2)

A recent proof that the Grassmannian G1,n,2 of lines of PG(n,2) has polynomial degree n2-1 is outlined, and is shown to yield a theorem about certain kinds of subgraphs of any (simple) graph @C=(V,E) such that |E|<|V|. Somewhat similarly, the polynomial degree of the Segre variety Sm,n,2,m=

Ternary composition algebras and Hurwitz’s theorem

Axioms are proposed for a certain ‘‘alternative’’ kind of ternary composition algebra, termed a 3Cn algebra. The axioms are shown to be (for n>2) in a simple correspondence with the axioms for a ternary vector cross product algebra. The axioms imply that n=1, 2, 4, or 8 (from which the usual Hurwitz theorem is deduced). The existence of 3C8 algebras is demonstrated by an explicit construction in four‐dimensional Hilbert space, without appeal to the properties of the algebra of octonions.

Subsets of PG(n,2) and Maximal Partial Spreads in PG(4,2)

AbstractPut θn = # {points in PG(n,2)} and φn = #{lines in PG(n,2)}. Let ψ be anypoint-subset of PG(n,2). It is shown thatthe sum of L = #{internal lines of ψ} and L′= #{external lines of ψ} is the same for all ψ having the same cardinality:[6pt] Theorem A If k is defined by k = |ψ| − θn − 1, then n

Ternary Composition Algebras. I. Structure Theorems: Definite and Neutral Signatures

L + L = phi _{n - 1} + k(k - 1)/2.