Symmetries and symmetric functions
Preliminaries and motivations
Definition. A function is even, if . A function is odd, if .
Proposition 1. Every function can be represented as a sum of even and odd functions.
Proof. Let , . Then .
Problem 1. Prove that even functions form a ring: any linear combination or product of even functions is again even. What’s wrong with odd functions?
Problem 2. Prove that any real polynomial can be written as , where are two real polynomials.
Problem 3. Is the same assertion true if we replace “polynomial” by “continuous function”? Show that any continuous function can be represented as with continuous .
Theorem. Any analytic or -smooth function defined in a neighborhood of , can be represented under the form in a (smaller, if necessary) open neighborhood of the origin, with the functions of the same type.
The difficult part of the proof of this theorem for non-analytic functions is hidden in the following Hadamard’s lemma.
Lemma. An analytic or smooth function which vanishes at the origin, , is divisible by : .
For analytic functions both the Lemma and the Theorem are very easy.
Definition. A function of two variables is symmetric, if for any values of .
Problem 4. Describe all symmetric polynomials of degree .
Answer. . [Forgotten: , and in general any other polynomial of the form . Later we will learn that any symmetric polynomial of any degree can always be presented as a polynomial in .]
Problem 5. Give the definition of an antisymmetric function of two variables in such a way that every function of two variables becomes a sum of a symmetric and antisymmetric functions.
Problem 6. Formulate and prove an analog of the statement from Problem 1 for symmetric functions.
Theorem. Any antisymmetric polynomial in two variables can be represented as , where is a symmetric polynomial in two variables.
Proof. Consider the change of variables which can easily be inverted: . Any polynomial in can be written as a polynomial in and vice versa. If is an antisymmetric polynomial, then the polynomial satisfies the property for any . This means that each polynomial obtained by “freezing” the variable , is odd in and hence is divisible by , so that . One can easily see that the quotient is again polynomial in . Thus, returning to the initial variables, we conclude that . Obviously, must be symmetric (apply the definition!).
What is common between the two examples?
Consider any set (e.g., the real line, or the plane ) and a group which consists of two transformations of into itself. One element is necessarily the identical transformation denoted by , the other (denoted by ) must be an involution: .
Real-valued functions on can be acted upon by the group : if is such a function, then the action of on is trivial (identical), and sends into by the formula . We put an asterisk to distinguish between the action of on points from the action of on functions.
Problem 7. Prove that all functions on form a linear space (denoted by ). This space is finite-dimensional, if has finitely many elements. What is the dimension? what plays the role of the zero vector in this space?
Problem 8. Prove that the map is linear.
In these terms, functions symmetric by are defined by the condition , antisymmetric — by the condition . Is there anything left out?
Problem 9. Find all numbers such that the condition defines a nontrivial function .
Theorem. Prove that any function on is a sum of a symmetric (by ) and an antisymmetric functions.
Proof. For any function consider its symmetrization by the group : . Then for any element we have , since the action of on the sum simply permutes the terms in this sum. We call the symmetric part of .
The difference is antisymmetric for : indeed, . Application of the linear operator exchanges the terms in the difference and hence changes its sign.
Problem 10. Describe symmetric and antisymmetric polynomials for the central symmetry . Check that the above theorem remains valid in this case as well.
What is special about the group ?
The definition of -symmetric function can be given in the general case when is a group acting on the set . A function is -symmetric, if for any .
Example. The constant functions taking the same value at all points of , are invariant by any group acting on . The constant functions form one-dimensional subspace in .
The notion of antisymmetric function, however, has to be modified, since the they may behave differently by action of different non-identical elements of . Still, one can suggest an analog of the decomposition of functions into symmetric (even) and odd parts.
Definition. Let be a finite group acting on , and stands for the number of elements in the group. The symmetrization (or averaging over the group) of a function is the function defined by the formula
Considered as the operator , symmetrization is a linear operator distilling from its “symmetric part”.
Definition. A function is called -even, if . A function is called -odd, if its symmetrization as above is zero, .
Lemma. For any finite group action, symmetric functions are even and vice versa.
Proof. If is symmetric, then in the above sum for all , hence . Conversely, if is equal to its average, then for any element the action by linearity is equal to . But this is the same sum as before, just with permuted terms. (Warning: this is a statement which requires thinking out! You need to use the axioms of the group to check it!).
Proposition (obvious). For any finite group action, any function admits decomposition into symmetric and odd parts. , with symmetric and odd.
The operation taking into its even and odd part is very much like projection on complementary subspaces: .
Example. Let be the set of faces of a cube (standard dice):
A function on is a marking: on each face a real number can be written (not necessarily from 1 to 6). This space is 6-dimensional (isomorphic to ), so any element can be written as a string of 6 numbers. However, this description obscures the symmetry of the dice.
The dice can be rotated around many axes, reflected in a mirror or in the center.
The dice can be rotated around many axes, reflected in a mirror or in the center. Respectively, we can consider the entire group or its proper subgroups. We have quite a few to choose from:
- The subgroup generated by the central symmetry, isomorphic to ;
- Three subgroups , generated by reflections in the three coordinate planes. Each of these subgroups is isomorphic to ;
- Three different subgroups generated by rotations around the three coordinate axis; each such subgroup is cyclic and isomorphic to ;
- Four different subgroups generated by rotations around the diagonals of the cube; each such subgroup is isomorphic to .
The only functions that are symmetric by the entire cube group, are constants (the same number is written on all six faces).
- For the first subgroup , a function is even, if its values on the opposite faces are the same, and odd, if they differ by sign. The spaces of even and off functions are both 3-dimensional.
- For each of the mirror symmetries the odd functions form a 1-dim space (values of opposite sign on two opposite faces, zeros everywhere else), while even functions constitute 5-dim susbspace.
Problem. Describe symmetric functions for each of the remaining subgroups, and find the dimension of the corresponding even and odd subspaces.
Why symmetric and antisymmetric functions may be of any interest?
Because sometimes we have to study symmetric operations on such functions, and the symmetry simplifies this task.
Problem 11. On each face of the cube a positive number is written in such a way that their sum is equal to 613. Every day each of these numbers is (simultaneously) replaced by the average of its 4 neighbors. Compute the result which will appear after one month of such manipulations with the accuracy of 1/1000.
Looks stupid, isn’t it? How can one compute the result without knowing the initial numbers? But wait.
Consider the operator , which transforms a function on the cube (“numbers on the faces”) into the average as described.Clearly, is a linear operator. If we write its -matrix, it will consist of zeros and the quarters at certain places. The problem is to compute the iterate (thirty times) and apply it to the unknown initial “vector” . If were a diagonal operator, this would be an easy task: is also diagonal with the eigenvalues being th powers of eigenvalues of . Thus we need to find all functions such that for all possible values of .
This operator “respects the symmetry”, that is, for any symmetry of the cube the identity holds, . This formula means that is interchangeable with any cube rotation. This commutativity implies many things. For instance, each eigenspace of must be invariant by all operators :
Yet our goal is the opposite: we have to find the eigenspaces of , knowing its symmetry group.
Proposition. If commutes with all elements of a group , then both -even and -odd functions form two complementary subspaces in , both invariant by and disjoint with each other.
Proof. As before, one has to write the definition of even (resp., odd) function via averaging and apply to both parts of the equality.
Now the strategy of constructing eigenspaces (subspaces on which acts as a multiplication by a scalar) is rather simple: one has to consider various subgroups and see how restrictions of on their even/odd subspaces look like.
- Application of to -odd functions is zero. Indeed, for any face its four neighbors form two pairs of opposite faces. If is -odd, then the sum of its values over these four faces is zero. Thus we have discovered a 3-dimensional subspace on which has eigenvalue .
- Consider the (invariant) subspace of -even functions and its intersection with -even/odd functions for a fixed rotation axis passing through two opposite vertices of the cube. Denote by the values on the three faces adjacent to this vertex:
the rotation permutes these faces cyclically with period 3 (and also permutes cyclically the other three invisible faces on which the same values appear, since we are inside -even subspace). A function is -even, if , and is -odd, if . In the first case all 6 numbers on the faces are equal, the corresponding doubly even functions are constants.
- Thus we have the three-dimensional subspace of -even functions represented as a sum of 1-dimensional -even subspace of constant functions and 2-dimensional -odd subspace. Clearly, the constant functions are preserved by , i.e., the corresponding eigenvalue is equal to 1. To describe the action of on the odd functions, note that its application replaces the number on one of these faces by . But since , we have , that is, is replaced by . In other words, has 1-dimensional eigenspace with the eigenvalue 1 and 2-dimensional subspace with the eigenvalue .
- The total list of eigenvalues of repeated with their multiplicities, called its spectrum, is . Without much computations we diagonalized the matrix of .
Solution of the Problem 11. Any initial set of numbers written on the faces of the cube, i.e., the initial function , can be represented as a sum of three terms, , belonging to the corresponding eigenspaces. The operator preserves , kills and multiplies by . After 30 iterations we obtain . Since , we conclude that the result is practically indistinguishable from , that is, after one month the numbers on all faces will be almost the same. To find this common number, we note that the sum of all6 numbers remains the same after application by . Thus all of the numbers will be very close to .
To complete the proof, one should be certain that the norm of the vector is not very large. This follows from the fact that the norm of the initial vector (the sum of the absolute values of all numbers) is no greater than 613, and the fact that the norm of the projection does not exceed the norm of the vector. We leave the details as an exercise.
Clearly, one can recycle the same symmetry arguments to solve an analogous problem for other Plato bodies, like tetrahedron and octahedron: in each case the operator assigns to a given face the average of values written on all adjacent 3 faces. You may think that the problem with the tetrahedron is simpler because it has only 4 faces, not 8 like the octahedron. Yet the symmetry group of the octahedron is the same as that of the cube! Indeed, centers of faces of the cube are vertices of the smaller octahedron and reciprocally. Thus quite a bit of information can be recycled if working smart.
One may question the choice of the operator as perhaps artificial. It is not! This operator is a discrete approximation of the Laplace operator which describes propagation of heat along surfaces. The eigenvalues of the Laplace operator are also the principal characteristic of sounding membranes: the collection of the corresponding numbers counted with their multiplicities characterizes main frequences and overtones of drums, loudspeakers etc. The correspondence between acoustic properties and geometry of the membranes is one of the most fascinating questions in mathematics.
For a long time it was believed that the complete spectrum (of infinitely many eigenvalues) completely characterizes the geometry. In a provocative language the question was formulated as follows, “Can one hear the shape of a drum?” by Marc Katz in 1966. However, the conjecture was disproved by John Milnor, first for 16-dimensional “drums” in a paper of record short length (half a page). Later much simpler examples of 2-dimensional membranes were constructed.
As an excellent project, one may suggest computing the spectra of the “discrete Laplacians” for all Plato bodies and compare them. Analogous problems can be posed for various graphs.