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I am trying to understand Gauge theory, but have a ways to go. I studied basic abstract algebra / group theory many years ago, so I know what groups are and yet I don't really have a good grasp of symmetries. I get the obvious geometric symmetry concepts, but apparently I have to confront with myself that I still don't have a deeper intuition for "symmetry" in general. I don't get the more general symmetries in mathematics, like:

  • A symmetric matrix is a square matrix that is equal to its transpose. (Is it just that they are equal that it's symmetric? I don't get it)
  • The symmetric group $S_n$ (on a finite set of n symbols) is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself.
  • A symmetric polynomial.
  • My goal is to understand (you don't have to explain this one, it's probably involved): In quantum mechanics, bosons have representatives that are symmetric under permutation operators, and fermions have antisymmetric representatives.
  • Set theory symmetric relation. Is this just like saying that if a graph has 2 nodes and there is an edge from left to right and right to left, that that is a form of symmetry?
  • Isometries of a space. An isometry is a distance-preserving map between metric spaces. (This seems like a case where everything can change except for distance, and yet for some reason they still call it symmetric, why?)
  • What is another complex yet practical example that is outside of geometric intuition but can be easily understood with intuition?

They say:

Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations.

To me I can't quite tell what this means. Does it mean that everything remains unchanged, or that only some things remain unchanged? For example, a geometric reflection means the position is changed, and the orientation is changed, but the scale and overall structure is unchanged. So is it that at least one property remains unchanged after some operation is applied? If so, what is an extreme example of a "symmetry" where everything changes except for just one small thing, so it is still considered symmetric? I guess an extreme example might be some sort of Lie or Gauge symmetry....

I feel if I can get a better intuition for how to apply the concept of symmetries to things outside of the geometric realm, then I will be much closer to being able to pierce some of the basics of quantum mechanics.

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  • $\begingroup$ arxiv.org/pdf/1410.6753.pdf $\endgroup$
    – Lance
    Commented Feb 11, 2023 at 12:38
  • $\begingroup$ A good introduction to gauge theory that keeps a good balance between abstract mathematical rigor and geometric intuition is in my opinion the script by Charles G. Torre of Utah State University. $\endgroup$
    – Kurt G.
    Commented Feb 11, 2023 at 17:06
  • $\begingroup$ The blog post terrytao.wordpress.com/2008/09/27/what-is-a-gauge by Tao could also be useful. $\endgroup$
    – Alp Uzman
    Commented Feb 11, 2023 at 19:24

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Your examples contain the answer to your question. Some things change, some don't, and those need not be geometric things.

A matrix is symmetric when nothing changes when you reflect it over the main diagonal.

The polynomial $p(x,y) = x^2 + xy + y^2$ is symmetric because nothing changes when you interchange the variables.

In Euclid's geometry nothing changes when you rearrange the points in the plane using a rigid motion.

In mathematics whenever you have a structure on a set you want to know the ways you can mix up the elements without affecting that structure. You call all those mixings the symmetries of the structure. They form a group since if you do two in a row you get another. Doing nothing is always a symmetry. You can always undo a symmetry. Those are the group axioms - and this is why the group axioms were invented. So it's no surprise that the noun "symmetry" and the corresponding adjective "symmetrical" have different meanings in different contexts.

If there is no structure at allon a set $S$ then nothing changes when you permute the elements using a bijection from $S$ to itself, so every bijections is a symmetry. For a set with $n$ elements we call that group of symmetries "the" symmetric group $S_n$.

Cayley's theorem says that every abstract group is in fact the group of symmetries of some mathematical structure.

You might want to read the wikipedia page on the Erlangen Program.

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  • $\begingroup$ Unfortunately this answer doesn't offer much different than standard definitions. I am looking for more concrete/hardcoded/real-world examples perhaps, or something with more intuition. $\endgroup$
    – Lance
    Commented Feb 11, 2023 at 21:07

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