On the Clifford collineation, transform and similarity groups. (III) Generators and involutions

Abstract

In this paper the Clifford groups PCT(pm), p > 2, PCG(pm) and CS'(pm), and the factor groups ½CS' (pm), which were defined in Paper I of this series (Bolt, Room and Wall [1]), are considered as transformations of projective [pm–1] over the complex field, C. We note that the geometrical results are the same if any of the corresponding groups CT, CG or GG and CS, respectively, are considered instead.

Designs, groups and lattices

“…Bolt, Room and Wall ( [9], [10], [8]) and later Broué-Enguehard [7] described Aut(BW n ). When n = 8, it is a subgroup of index 2 in the Clifford group C k which we describe now.…”

Untitled

“…They distinguished two geometrically similar lattices L m ⊆ L ′ m in R 2 m . The automorphism group † G m = Aut(L m ) was investigated in a series of papers by Bolt, Room and Wall [8], [9], [10], [50]. G m is a subgroup of index 2 in a certain group C m of structure 2 1+2m + .O + (2m, 2).…”

“…In fact it is an immediate consequence of Runge's results that for m ≥ 3 C m has a unique harmonic invariant of degree 8 and no such invariant of degree 10 (see Corollary 4.13). The space of homogeneous invariants of degree 8 is spanned by the fourth power of the quadratic form and the complete weight enumerator of the code H 8 ⊗ F 2 F 2 m , where H 8 is the [8,4,4] Hamming code. An explicit formula for this complete weight enumerator is given in Theorem 4.14.…”

“…Theorem 6.5 shows that, apart from the center, X m is a maximal finite subgroup of U(2 m , C), and Corollary 6.6 is the analogue of Corollary 5.7. Bolt et al [8], [9], [10], [50] and Sidelnikov [45], [46], [47] also consider the group…”

Groups Associated with Certain Loci In [5]

“…(Thus L is related to the system of 27 lines on a cubic surface whose group has order 51,840 -T-2.) Properties of L and of the various Clifford groups (as well as their related projective groups) have been developed by Beverley Bolt, Room, and Wall (3), by Beverley Bolt (4), and by the author (6)(7)(8)(9)(10).…”