Re: Help understanding an argument of Phillip Hall

magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin) writes:

>I'm hoping someone can help me figure out an argument of Phillip Hall's,
>which appears in his paper "The Classification of prime-power groups",
>J. fur Reine Angew. 182 (1940), pp. 130-141.
....
> If [...] we regard K as a linear space L_4 with integers mod p as
> coefficients, we may then picture the commutator subgroup (or central)
> of a stem group of the maximal family as the carrier space L_6 of the
> Grassmannian which represents the L_2's of L_4. The subspaces L_5 of
> L_6 which give rise to the quotient groups of order p^5 which are in
> question, will then correspond to linear complexes of L_2's in L_4.
> And there are two types of these only, the special and the general.
>
>The points which give me some trouble:
>
> (1) I understand that the commutator subgroup is L_6 (as noted above).
> And "the Grassmannian which represents the L_2's of L_4" is
> surely Gr(2,4), the 2-dimensional subspaces of a 4-dimensional
> vector space. What, however, is "the carrier space [...] of the
> Grassmannian"?

Almost surely (given the clue of Hall's notation L_2 and L_4) the
6-dimensional linear space in which the Grassmannian is naturally
embedded by Plucker coordinates (that is, a 2-subspace with basis
(v_1,...,v_4), (w_1,...,w_4) has coordinates the 6 determinants
v_i w_j - v_j w_i, 1\le i < j \le 6), or, in coordinate-free
language, the second exterior power of L_4.

> (2) Again, I understand that the groups we are interested in correspond
> to 5-dimensional subspaces of this 6-dimensional space. I'm not
> sure what "correspond to linear complexes of L_2's in L_4" means,
> though.

To quote
http://www.matapp.unimib.it/~marina/ric/2.3-gen/subsection3_3_1.html
(which I found by a Google search on the conjunction of the strings
obtained by unquoting ""linear complex"" and "symplectic", a search
I knew to do because I (1) knew that "linear complex" is a term of
art from old-style projective geometry, and (2) remembered reading
an explanation of why the "symplectic group" is so called), with
symbols I can't discern elided away:

A linear complex ... is by definition the system of all lines i...
whose pl\"ckerian coordinates satisfy a given homogeneous linear
equation

(that is, in this case, a vector subspace of the second exterior
power of L_4), and

A linear complex consisting of all the lines through a given
line (the axis ... ) is called special

(the others being, of course, general).

> (3) And I do not know what the "special and general" linear complexes
> are, naturally, nor why they are the only ones.

Answered above: a mere matter of definition. (What's more to the
point, but quite easy to see, is why any two "special" complexes
are equivalent in a relevant sense.)
>
>Any and all help on this will be greatly appreciated. Thanks in advance

Lee Rudolph
.