[GAP Forum] Calculating the maximum algebra of linear transformations fixing a non-trivial subspace of a vector space

[William DeMeo]

Greetings Dursun,

This is a nice question, and I look forward to reading answers from
other members of the forum, who are more knowledgeable about invariant
subspaces than I am.  I don't have an answer for your general
question, but I have a couple of comments which may or may not be
useful to you.

First, would you be willing to restrict attention to the group of
invertible linear transformations (i.e. automorphisms of V) under
which W is invariant?  Perhaps GAP is more applicable to this problem.

Second, are you thinking about infinite dimensional vector spaces?  It
is likely that GAP could be used to answer this question in finite
dimensions, but perhaps you already have answers in this case.

Other thoughts:

Abstractly speaking, given a collection of subspaces S \subseteq
Sub[V], you could think about the algebra,

Alg(S) = \{T \in Aut[V] | T(W) \subseteq W for all W in S\}.

When S is just {W}, this is like the algebra you asked about (assuming
you're willing to restrict to invertible maps).  If my memory serves
me, I believe you can show that Alg(S) is a group... (it is certainly
a group if you take all T for which T(W)=W).

In any case, in small dimensional examples, over finite fields, you
could use GAP to search among the subgroups of Aut[V] for the group of
maps which leave W invariant.

An easy special case: If V is a 2 dimensional vector space over a
3-element field, then Aut[V] = GL(2,3), and the subgroup structure of
this group is easy to see with GAP and/or the uacalc (www.uacalc.org).

Generally speaking, I believe you can establish a dual lattice
isomorphism (Galois correspondence) between the lattice of subgroups
of Aut[V], and the lattice of (closed) subspaces of V, and in small
examples like the one mentioned above, you could probably identify the
precise correspondence between the subspaces of V and the subgroups of
Aut[V] which leave them invariant.  (If you're interested in this
abstract lattice theoretic approach, I recommend Bill Lampe's Galois
theory notes [1].)

-William

[1] http://www.math.hawaii.edu/~williamdemeo/612_Galois.pdf


On Sat, Nov 19, 2011 at 12:49 PM, Bulutoglu, Dursun A Civ USAF AETC
AFIT/ENC <Dursun.Bulutoglu at afit.edu> wrote:
> Dear Gap forum,
> Given a vector space V and a non-trivial subspace W
> I was wondering whether it is possible to calculate the maximum
> algebra of linear transformations under which W is invariant.
>
> Any theoretical or computational insight will be greatly appreciated.
>
> Dursun.
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