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You might have come across the book of involutions in your travels. A colleague of mine asked whether there is a natural global reason (versus ad-hoc trickery) for considering involutions in mathematics. The above book provides many situations that suggest such a global perspective. Having witnessed the extraordinary power of certain involutions in operator algebras (e.g. in Tomita-Takesaki theory), I'd be interested in hearing about such a global perspective in summary from an expert. I'm aware that this question as I've asked it risks being trite...perhaps warranting the answer "it's the simplest nontrivial symmetry" but the existence of the above book might suggest otherwise:

Question: What are some "global" reasons for considering involutions in mathematics?

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It may be the case that we are way too mathematically primitive to even start making useful use of automorphisms of order three! –  Mariano Suárez-Alvarez Apr 4 '13 at 21:21
    
!@Mariano: From where I stand, that is very reasonable! –  Jon Bannon Apr 4 '13 at 21:34
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I attended an interesting talk given by Tony O'Farrell where he made the case that products of involutions, or more generally, reversible elements in groups, occurred very naturally across mathematics. Unfortunately I don't have references to hand but he might have more on his webpage –  Yemon Choi Apr 4 '13 at 22:41
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Conversely, objects that have absolutely no symmetry are extremely common. Perhaps the reason why $\mathbb Z_2$ is so striking is that it's just the first non-trivial example of the involution idea, so we're particularly tuned to seeing and making-use of such symmetry. –  Ryan Budney Apr 4 '13 at 22:43
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This should probably be CW. –  Todd Trimble Apr 4 '13 at 23:28

2 Answers 2

I don't know the true philosophical reason, but $Z_2$ symmetry is really omnipresent in Matematics and in the nature. For example, most animals, including practically all vertebrate animals (like ourselves) have aproximately $Z_2$ symmetric bodies, and no larger group. This suggests that $Z_2$ was the favorite group of the Creator, at least in that period of his activity when we was creating advanced animals:-)

If you prefer Evolution, this $Z_2$ symmetry must somehow be explained by the survival of the fittest. I don't know exactly how, but this suggests that this is a very important group. Notice that plants, mushrooms, and simplest animals usually do not have it.

As a result of this (2-fold symmetry of animal bodies) we tend to like this kind of symmetry. Look at all our technology: cars, ships, airplanes, etc. They all have 2-fold symmetry, at least from outside (like our bodies, they also have this symmetry only outside). Once my friend, an airspace engineer, told me that there was a project of an airplane which did not have this outside 2-fold symmetry. The project was rejected for the only reason that "no one will want to fly in such an airplane". I am serious: http://en.wikipedia.org/wiki/Oblique_wing

In mathematics, from my personal perspective, it is $z\mapsto\overline{z}$ first of all. (Once I even proposed to my co-author to call one of our papers "Some applications of representation theory of $Z_2$"); the paper was full of different representations of this group, We were working on real algebraic geometry.)

This very same symmetry $z\mapsto\overline{z}$ is also hidden in Hermitian symmetry, $C^*$ algebras, all sorts of "duality" everywhere, etc. Which suggests that the Creator of the Universe always had a strong bias in favor of this particular group.

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Many plants (think of flowers and fruit) and simple animals (think of starfish, jellyfish and anemones) exhibit 3-, 5- or 6-fold symmetry, and often higher. Flowers have been "making useful use of automorphisms of order three" for ages. Jellyfish, possibly the most successful animal to ever exist on the planet, have such striking and high order radial symmetry that they are the biologist's go-to example for demonstrating this phenomenon in nature. –  Zack Wolske Apr 5 '13 at 0:43
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Maybe jellyfishy C*-algebras have stars of other orders? –  Mariano Suárez-Alvarez Apr 5 '13 at 5:33
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I don't like this answer. If we understand well how to use hammers, we are going to notice a lot of nails, but that doesn't mean nails are somehow truly ubiquitous or favoured by the gods, it just means that we recognise them when we see them. Bonus points to anyone who makes good use of Jellyfish Algebras, btw. –  Ketil Tveiten Apr 5 '13 at 9:02
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In this connection, I might add that the word itself, involution, has a botanical origin: see mathoverflow.net/questions/127332/… –  Carlo Beenakker Apr 12 '13 at 10:02
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Alexandre: I suppose the thing I object to is the word "omnipresent". Observing lots of examples of $\mathbb{Z}/2$-symmetry does not mean that it is somehow a fundamental thing, it only means that $\mathbb{Z}/2$-symmetry is a thing we are good at recognising. Most objects (or living things) in nature don't have any symmetry at all, and in a similar way, most objects in mathematics have no symmetry, it's just that we tend to work with those objects that are nice enough that we can do something with them, and having some kind of low-order symmetry is an easy way to be nice. –  Ketil Tveiten Apr 12 '13 at 16:07

It all boils down to the fact that $\mathbb{R}$ has two ends!

In all (most) mathematical processes there is some notion of direction: counting, moving (along a curve), mapping from one space into another, reading a formula from left to right... the geometry of $\mathbb{R}$ is present always in one way or other, and the flip of the negative and positive ends usually induces some sort of involution.

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An immense number of instances where involutions show up and play a significant role have absolutely nothing to do with the ends of ℝ. I would be awed to see a concrete connection between the Feit-Thompson theorem that simple finite groups have non-trivial involutions (and its central role in the classification of simple groups) and the ends of ℝ, say. –  Mariano Suárez-Alvarez Apr 5 '13 at 5:31

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