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## A linear algebra puzzle

I won’t get around to substantive blog posts for at least a few more days, so here is a puzzle.

We usually thinks of groups that occur in nature as permutations of a set which preserve some structure on that set. For example, the general linear group $\text{GL}_n(F)$ preserves the structure of being an $F$-vector space of dimension $n$.

What structure does the special linear group $\text{SL}_n(F)$ preserve?

(I have an answer, but I’m curious if it can be stated in a more elementary way.)

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### 11 Responses

1. Some kind of volume and orientation.

• That’s the right idea. Can you be more precise? (In particular, can you phrase your answer in terms of natural data one might attach to an object in the category of $F$-vector spaces, e.g. a morphism?)

2. The canonical answer is that SL(n) preserves a (signed) volume form. More precisely, GL(n) has a canonical action on n-dimensional space, and hence on the nth exterior power, a line. The subgroup that fixes this line pointwise is SL(n).

This answer is problemmatic in a few ways. When F has positive characteristic and n is sufficiently large, I think that some care should be taken in defining nth exterior powers. Certainly care must be taken in characteristic 2. Moreover, it’s possible for there to be “too many” points, at least for my taste. Consider, for example, when F is a finite field of size p, and n=p-1. Then all non-zero scalars have nth power equal to 1, which is not what you would expect from the case over R or C. I think that these concerns can be avoided, but I think in the smooth category, so I don’t know the details.

3. Is det(A) = 1 what you are getting at?

• This is the standard answer, but it is not a structure on an $F$-vector space.

4. SL preserves trivialization of top exterior power (i.e. morphism $F\to\det V$).

5. I only just saw this thread, and it seems to me Theo got it right, except I don’t see what is problematic. A “volume form” on an n-dimensional space $V$ is a nonzero multilinear map $f: V^n \to k$ ($k$ the ground field) such that $f(v_1, \ldots, v_n) = 0$ whenever $v_i = v_j$ for any distinct $i, j$. For any $A \in GL(n)$ we have $f \circ A^n = \det(A)f$, and this can be proven without divisions by integers greater than 1. So $A$ “preserves the volume” if and only if $A \in SL(n)$. This works over fields of arbitrary characteristic and, with a slight reformulation, even over general commutative rings (where $V$ is a free module of rank n).

• I think Theo meant that the naive definition of “alternating multilinear map” fails in characteristic $2$ (the one where you require $f(v_1, v_2, ... v_n) = - f(v_2, v_1, ... v_n)$ and so forth). I’m not sure what his second concern means.

• Sure, I figured that.

But you never said what your own answer is!

• An identification of the top exterior power with $k$. But this definition (as opposed to yours) seems mildly troubling to me in characteristic $2$.

• That was my answer too. I believe the right way to define the n-th exterior power (valid for all characteristics) is to take the n-th tensor power and divide by the subspace of tensors $v_1 \otimes \ldots \otimes v_n$ spanned by those where $v_i = v_j$ for distinct $i, j$. (Lang’s Algebra, chapter 19.)