I've been looking for representations of groups, that is sets of matrices over GF(2) or GF(n) that are isomorphic to Sn, the symmetric group of order n.

One well known group is the standard representation, which works in any Field. The standard representation is the collection of matrices that have a single 1 in each row and each column and zeros elsewhere. The collection of all such nxn matrices forms a group of size n! which is isomorphic to Sn.

In a sense this set of matrices is created from the following rows.

1000...
0100...
0010...
0001...
...

Each row contains n-1 zeros and a single 1. Every matrix in the standard representation consists of n rows, each one of which comes from this set. The elements of the standard representation are also called permutation matrices. A permutation matrix is itself a permutation of the n rows given above.

I have discovered that there is another representation that works for any field. To the n rows given above, add an n+1st row consisting of all -1 s.

so now we have

-1 -1 -1 -1  ...
 1   0  0  0  ...
 0   1  0  0  ...
 0   0  1  0  ...
 0   0  0  1  ...
...

All permutations of these rows give us n+1 nxn matrices. (We have to leave off the bottom row) These matrices are all non-singular, and form a group which is isomorphic to Sn+1. This also works in GF(2) even though -1=1.

(I don't know if I'm the first to discover this representation or not. I've never seen any reference to it, but it seems odd that something this cool would have escaped everyone's notice but mine.)

Let's call the standard representation Rn and the new representation Qn. Let Tn be the transpose of Qn. Tn is also a faithful representation of Sn+1, sometimes Qn and Tn are conjugate to one another, sometimes they are not.

I'd like to make the following conjectures.

1. Any faithful representation of Sn+1 in nxn matrices over any field is conjugate to Qn or Tn.
2. If k>n+1, there is no faithful representation of Sk in nxn matrices over any field.
     (why? I think such a representation would have to permute n+2 rows. An nxn matrix can represent a
      permutation of n+1 rows, because the omitted row is uniquely determined. For n+2, there is no
      way to uniquely determine the order of the two missing rows.)

3. Any faithful representation of Sn is either
    a. Rn
    b. Qn-1
    c. Tn-1
    d. A direct sum of Rk Qi and/or Tj for k<n, i,j<n-1
    e. conjugate to a,b,c or d.

Farewell, geocaching. It was fun while it lasted. My reasons for leaving are too complicated to go into here, but let's just say, that life without geocaching has been much more peaceful than life with geocaching. Too bad.

This may just be arrogance, but I believe that much of my work is revolutionary in nature. I love inventing new algorithms, exploring new perspectives, inventing new ways of doing things. I have come to believe that the scientific community as a whole discourages revolutionary thinking. Well, not just the scientific community, the world at large hates revolutionary ideas.

Well, I've recently  been reading an excellent biography of Albert Einstein, probably the greatest revolutionary the world has ever known. Around 1905-06 he was tired of working for the Swiss patent office and was trying to find a job in academia. He sent out applications everwhere, and included in these applications several of his published papers. His applications were universally turned down. In this packet of papers were three of special note.

1. A paper that established the branch of physics known as "statistical dynamics" and used a purely mathematical approach to predict the phenomenon known as "Brownian Motion." This paper is credited as being the first to prove the existence of the atom. At the time many physicists did not believe in atoms.

2. A paper that explained the photoelectric effect by showing that light consisted of discrete particles known as photons. This paper is credited with establishing quantum theory, because it was the first to show that quantization had a noticable effect in the real world. Until that time quantization was considered to be merely a mathematical device with no physical reality. This paper won Einstein his only Nobel prize.

3. A paper on time and space establishing a theory that eventually came to be known as "The theory of special relativity." A theory which not only revolutionized physics but the whole world.

I feel your pain Albert!

If a guy with this kind of credentials can be universally turned down, what hope do we lesser revolutionarys have?

How things change! This summer I taught a course in VLSI Design Automation. I can see now that some of the work I published in that area was done when I was incredibly depressed. That's not surprising. I published one paper in 2000 that I considered to be "a mere curiosity." However, upon rereading it while presenting it in class I realized it was the starting point for an astounding amount of new research. Why didn't I see that before?

Well, at the time I was battling to try and stay at the University of South Florida. Despite the fact that it was "clearly time to leave" I didn't want to displace my children from the only home they ever knew. Consequently I was battling on every front I could think of to fix the problems and so I could remain with USF. Needless to say, these efforts were pathetically ineffective. I couldn't see it at the time, but I was spiraling into an abyss of depression in which everything I did seemed trivial and pointless.

Reading this paper for the class jarred me back to reality. I decided to do a realistic review of my past work to determine other loose ends that might be dangling. There are quite a few, actually. I think a person could put together a pretty substantial research career tying up these loose ends. Even tying up a few of them could put someone on the map. Metamorphic programming, for example. One could devote an entire career to that concept alone.

I remember leaving South Florida with the feeling that I was finished as a researcher. That I would never have another original idea as long as I lived. The truth is that there is a vast landscape in front of me with virtually unlimited possibilities. What was I thinking?