NetNews Usenet Archive 1992 #30/NN_1992_30.iso/spool/sci/math/17213

Internet Message Format | 1992-12-21 | 2.3 KB

Path: sparky!uunet!cs.utexas.edu!swrinde!gatech!concert!duke!news.duke.edu!math.duke.edu!dkrain
From: [e-mail] (David Kraines)
Newsgroups: sci.math
Subject: Re: 1992 Putnam problems and unofficial solutions
Summary: Explicit solution
Message-ID: <[e-mail]>
Date: 19 Dec 92 18:21:13 GMT
References: <[e-mail]> <a_rubin.723836042@dn66> <[e-mail]>
Sender: [e-mail]
Lines: 41
Nntp-Posting-Host: cauchy.math.duke.edu

In article <[e-mail]>, [e-mail] (Noam Elkies) writes:
> In article <a_rubin.723836042@dn66>
> [e-mail] (Arthur Rubin) writes:
> ">Problem B6
> "
> ">Let M be a set of real n by n matrices such that
> ">(i) I \in M, where I is the n by n identity matrix;
> ">(ii) if A \in M and B \in M, then either AB \in M or -AB \in M, but not both;
> ">(iii) if A \in M and B \in M, then either AB = BA or AB = -BA;
> ">(iv) if A \in M and A \noteq I, there is at least one B \in M such that
> "
> "But can it actually contain n^2 matrices?
> "
> Yes it can: let n be a power of 2; let G be the extraspecial group of
> order 2n^2 which has a *real* (even rational) n-dimensional representation
>
> --Noam D. Elkies ([e-mail])
>   Dept. of Mathematics, Harvard University

One can easily construct an explicit set of matrices M for n = 2^k as
follows:

For n=2, let I=[1 0;0 1], A=[1 0;0 -1], B=[0 1;1 0], and C=[0 -1;1 0]
Observe that  AB=-BA=-C, AC=-CA=-B, BC=-CB=-A, and the square of each
is +-I.  Thus these 2^2 matrices satify the conditions.

Inductively build up matrices of order 2^k by taking block matrices
of the above type with entries like the above.
For example for n = 4, consider matrices such as A(B)=[B 0;0 -B].
Only the notational problems prevent me from giving the general terms.
It is trivial to show that this gives a system closed under the
operations.  For example  A(B)*C(A) =-B(C).

As noted in previous solutions, one can show that if A in M then
A^2 = +-I.  In particular all matrices in M are non singular. For
n odd, AB=-BA implies det(A)det(B)=-det(B)det(A) so A or B must
be singular.  Thus no nontrivial system exists for n odd.  A slightly
more subtle argument shows that the max n^2 can be realized only
for 2=2^k.

David Kraines, Duke University, [e-mail]