In this case, you could just slide the wallpaper back down 1 foot, but if I had rotated it 60 clockwise, then you could also just rotate it 60 counterclockwise. This is a particularly nice feature of symmetry transformations: any symmetry can be undone by another symmetry. More formally, every symmetry has an inverse transformation , which is also a symmetry. Definition: The inverse of a symmetry transformation is just the symmetry transformation which undoes it by bringing the object back to its initial configuration.

Every symmetry has an inverse! Another nice feature of symmetries is that you can always combine them. If I give you two different symmetry transformations, you can always obtain a third by doing one and then doing the other, one after another.

For example, one can combine a 60 rotation with itself to obtain a rotation. This is usually referred to as composition of symmetries, but I will sometimes also use the word multiplication synonymously for reasons that will become clear in a moment. Finally, a trivial but important observation is that just leaving the wallpaper alone is a symmetry transformation as well. An aside OK, maybe I was being a bit coy in suggesting the exercise above. There are actually infinitely many symmetries of this wallpaper. For example there are infinitely many points about which I can rotate the wallpaper and leave it unchanged.

But hope is not lost; it turns out that every symmetry of the wallpaper can be obtained by taking combinations of some finite collection of symmetries. So the modified exercise might be: what finite collection of symmetries generates all the rest?

- The Mathematics of Symmetry | Stanford Pre-Collegiate Summer Institutes.
- Community Policing, Fifth Edition: A Contemporary Perspective , Fifth Edition;
- **The story of moonshine, part I: symmetry, number theory, and the monster – Brandon Rayhaun.
- 1. Introduction.

In the case of a wallpaper, we are free to slide, rotate, and reflect it along different axes, and because the pattern repeats, the wallpaper afterwards will be in the exact same configuration as before we transformed it. Groups: the algebra of symmetries This previous discussion suggests a more abstract, algebraic definition of symmetries. Definition: A group is a collection of objects called elements , , along with a way of multiplying them, , satisfying the following properties:. The idea here is that each element of a group corresponds to some kind of symmetry transformation. In this way, we are able to discuss the algebra of symmetries without lugging around a bunch of ideas having to do with geometry.

Algebra is easy! This is what math is. We found some kind of interesting phenomenon in the real world symmetry and we captured its essence in an abstract definition groups.

## Quanta Magazine

We can summarize this in a group table. What kind of symmetry does this correspond to? Well, human symmetry for one! We can imagine as abstractly representing a reflection which swaps our right and left half, and as not doing anything. As a sanity check, we know that if I perform two reflections on you, one after another, you will return to your normal unreflected self. In mathematical terms, this is the statement you learned in 5th grade,. Said in a group theoretic spirit, reflecting twice is the same thing as doing nothing to you!

This group, called , shows up all over the place in physics — for example some theories are symmetric under the reversal of time flip time twice and you end up with time flowing in the direction it started. There is maybe another group with two elements that is familiar. Does this group give us anything new?

The key to understanding this is that and are just symbols — in fact, if we take this new group table defined with addition modulo 2, erase the and write down a in its place, and erase all the instances of and and write and in their place respectively, then we recover the exact same group table as before! In other words, the way and add together is the same way and multiply. Definition: Two groups and are said to be isomorphic if there is a one-to-one correspondence of symbols in with symbols in such that replacing the symbols which appear in with their partners in recovers precisely the multiplication table of.

This is another common feature of mathematics. When we make definitions, there are often hidden redundancies, multiple objects that on the surface may look different, but are actually exactly the same up to relabeling. Summary: We captured the essence of symmetry in an algebraic definition, the group, and saw that groups are completely specified by their multiplication tables.

We also defined the notion of isomorphism, which is a more useful notion of equality between groups because it disregards artificial differences between groups, like what symbols we choose to write them down with. Of course this question seems very open-ended and difficult to answer as stated, but maybe there is a simpler question whose answer gets us most of the way there.

And of course, chemists have given us such a classification — the periodic table of elements! It turns out that even by considering these simpler groups, the answer to this question occupied the efforts of mathematicians for the greater part of the last century, and the classification now is spread out over thousands of pages in the literature.

Theorem: Every finite simple group either belongs to one of 3 families each family consisting of infinitely many finite simple groups , or is one of the 26 outlier groups, called the sporadic groups. Of these 26 sporadic groups, one stands out. The Monster group is the largest finite simple sporadic group, weighing in with a whopping elements! That is to say, the structure of most of the other sporadic groups can be, in some sense, found within the Monster. One group in particular that can be found inside the belly of the beast and will be relevant for our story later is the Thompson group.

Feel free to discard this silly mnemonic. Representations: how groups act in different dimensions Many of you have probably taken a linear algebra class in college or in high school. Because almost everything is approximately linear for example, zoom in really close on any curve and it will basically look like a line and linear algebra is an extremely well-developed and well-understood theory.

On the other hand, group theory is in general very difficult. To this end, mathematicians have found it incredibly fruitful to reduce problems of group theory to problems of linear algebra. So how exactly do we study groups in terms of linear algebra? Remember that the object that is most central to linear algebra is the matrix:.

One way to think about a representation of a group is as a box which assigns a matrix to every group element :. One shoves group elements into one side of this box and the box spits back a matrix which is meant to represent that group element.

### 1. Introduction

To give a name to this, we say that representations satisfy the homomorphism property. We described above how. So for example, the homomorphism property demands that. If you know how to multiply matrices, then you can easily verify that this is true:. Noticing that the right hand sides of both lines are the same is all it takes to prove that. What do all the other representations of look like? Remember that in asking the analogous question for molecules, chemists studied the atom. Here, we have the exact same thing.

There is a notion of an irreducible representation ; the irreducible representations of a group constitute those representations out of which all the others can be constructed, just as atoms constitute the building blocks out of which all other molecules can be constructed. Now, the chemists invented the periodic table of elements to further assist in their study of matter. Can we come up with an analogous table for representations of a particular group? The answer is yes , and it will be extremely useful for us when we encounter moonshine later on.

It turns out that there is an incredible amount of information encoded in the trace of a matrix, which is defined as the sum of the elements along the diagonal. So for example,. There is enough information baked into the trace that we will use it to characterize representations entirely. Each row will correspond to an irreducible representation , and each column will correspond to a group element. Each entry in the table will be the trace of the matrix assigned to that group element under that representation. In terms of the box analogy, the columns correspond to group elements, and the rows to different boxes.

Each entry in the table is then the trace of the matrix that you get out when you shove that group element into that box. Simple enough, no? The first irreducible representation or first box I denote with an , the second irreducible representation is denoted with an , etc. The dimension of the representation is equivalently the trace of the matrix assigned to the identity element, 1A, of the group. So the dimension of the second irreducible representation is Remember this number! Summary: To learn more about groups, we decided to cast their study into the framework of linear algebra.

We defined a representation of a group simply as an association of a matrix to every element of the group in a special way that preserves the group structure. In asking what all the representations of a group are, we learned about its irreducible representations, which are the representations out of which all others can be built.

We summarized this information in a representation table. Every entry of this table is the trace of some matrix — if we are in the column corresponding to group element and representation then the matrix is the one that maps to under the representation. We will see these important numbers come up in the study of moonshine. I always found it strange that the most elementary concepts adding, counting, etc. Yet the statement of the problem is so simple that one could explain it to a clever middle school student! As a matter of fact, there is at least one case in which the symmetric group is a group of symmetries, although "symmetry" must be taken here in an ampler sense.

In general, groups of symmetries of regular polyhedra can be realized as subgroups of the symmetric group. Any kind of group of transformations can be seen as symmetries. Then they are the rotations about the origin that preserve the perpendicular rays given by the basis. Algebraically they are permutation matrices. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Ask Question.

Asked 8 years, 3 months ago. Active 1 year, 8 months ago. Viewed 8k times. Here are my questions : What's the relation between these two concepts: "symmetric group" and "symmetry group"? What kind of " symmetry " is the symmetric group about?

Where is the name "symmetric group" from? Cloudscape 2, 9 9 silver badges 22 22 bronze badges. Jack Jack Every point is connected to every other point by equal-length edges. Though we have not seen a complete explanation of how the symmetric group got its name, at least some of the ingredients of the story are beginning to appear. Jul 3 '11 at Robert Israel Robert Israel k 23 23 gold badges silver badges bronze badges. Vladimir Sotirov Vladimir Sotirov 8, 1 1 gold badge 20 20 silver badges 51 51 bronze badges.

## Symmetry, Algebra and the Monster

Hans Lundmark Hans Lundmark Then f is even if the following equation holds for all x and -x in the domain of f :. Geometrically speaking, the graph face of an even function is symmetric with respect to the y -axis, meaning that its graph remains unchanged after reflection about the y -axis. Examples of even functions are x , x 2 , x 4 , cos x , and cosh x. Again, let f x be a real -valued function of a real variable.

Then f is odd if the following equation holds for all x and -x in the domain of f :. Geometrically, the graph of an odd function has rotational symmetry with respect to the origin , meaning that its graph remains unchanged after rotation of degrees about the origin.

### Beating the Odds for Lucky Mutations

Examples of odd functions are x , x 3 , sin x , sinh x , and erf x. This also holds true when A is infinite, but only if the integral converges. In linear algebra , a symmetric matrix is a square matrix that is equal to its transpose.

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Formally, matrix A is symmetric if. By the definition of matrix equality, which requires that the entries in all corresponding positions be equal, equal matrices must have the same dimensions matrices of different sizes or shapes cannot be equal.

Consequently, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. Every square diagonal matrix is symmetric, since all off-diagonal entries are zero. Similarly, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose.

Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them. 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 is a polynomial P X 1 , X 2 , …, X n in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting.

From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials. A theorem states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials, which implies that every symmetric polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial.

In mathematics , a symmetric tensor is tensor that is invariant under a permutation of its vector arguments:. Alternatively, an r th order symmetric tensor represented in coordinates as a quantity with r indices satisfies. The space of symmetric tensors of rank r on a finite-dimensional vector space is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V. Over fields of characteristic zero , the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V.

A related concept is that of the antisymmetric tensor or alternating form. Symmetric tensors occur widely in engineering , physics and mathematics. Given a polynomial, it may be that some of the roots are connected by various algebraic equations.