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Focus on Mathematics
- Atsushi Katsuda, Professor: Hearing the Shape of a Drum-Inverse Geometric Problem-
- Takuya Konno, Associate Professor: Understanding Symmetry
- Yutaka Ishii, Associate Professor: Mathematics in Support of the Internet
- Shin-Ichiro Ei, Visiting Professor: Patterns and Mathematics
- Hiroyuki Ochiai, Professor: Homework of Two Thousand Years Ago
- Hayato Chiba, Associate Professor: Mathematics of Synchronization Phenomena
- Mitsugu Mera, Mazda Motor Corporation: Collaboration between Number Theory and Computers
Figure 1 Symmetric figures
Many symmetric figures can be seen all around us. Some common examples of symmetry are shown in Figure 1. The human figure at the upper left is line symmetric, or in other words, left-right symmetric. The star-shaped figure to its right, meanwhile, is likewiseline symmetric, which is preserved for every one-fifth rotation of the figure. Finally, in the lowerdrawing, the symbol•••••• indicates that the sailboat figures shown continue forever to the left and right, so this drawing possesses symmetry in the sense that shifting laterally one sailboat at a time makes no difference. The above types of symmetry are often used in the natural sciences to describe homogeneous states. For example, it can be said that the molecules making up a gas are distributed uniformly within a sufficiently large space. Now, if we assume a one-dimensional space for the sake of simplicity, we can think of a sufficiently large space as being a number line. In this case, function f(x) that expresses the distribution of gas molecules in terms of position x (e.g., the number of molecules between positions x and x + c) should be symmetric with respect to lateral movement, as expressed by the following equation:
: any real number
This would indicate that P(x) is a constant function, that is, that the gas molecules are distributed uniformly.
In this way, symmetry can be expressed as follows: “Given function f(x) for some set x,x ∈ X (a number line in the above example),f(x) does not change for some type of movement (lateral movement in the above example). Conversely, we can denote the set of all movements that appear in the description of f(x) symmetry as H, which is what we turn our attention to here. Of course, each element t of H generates a one-to-one correspondence from X onto itself. We can therefore refer to t as a transformation of X. Apart from consisting of such transformations, H satisfies the following three conditions.
(i) | Identity transformation I(x) = x, x ∈ X belongs to H | |
(ii) | If transformations s, t are part of H ,f(s(t(x))) = f(t(x)) = f(x), so their combination st(x) := s(t(x)) is also part of H. | |
(iii) | If transformation t is part of H, inverse transformation t^{-1} that associates x with t(x) is also part of H. |
We call the set of transformations satisfying these three conditions a group. Symmetry can therefore be explained as something that is invariant to actionsof this group.
Figure 2 Congruent transformations of equilateral triangle
In the above example of gas molecules, fixing origin 0 in space and moving base x of H by 0 so that x + 0 = x enables a one-to-one correspondence to be established between H and X. This, however, is not true in general. For example, we consider group H consisting of congruent transformations of an equilateral triangle as shown in Figure 2. Specifically, this group consists of six transformations: turns s_{1}, s_{2}, s_{3} about the symmetry axes passing through vertices 1, 2, 3, respectively, 120-degree rotation t moving vertices 1, 2, 3 to vertices 2, 3, 1, respectively, and inverse rotation t^{-1}, plus the identity transformation I. However, there are only three triangle vertices. (Some molecules such as ammonia (NH_{3}) have such symmetry.) The reason for this is as follows. Taking vertex 1, for example, as the origin, then other than identity transformation I , there is still transformation s_{1} that does not move vertex 1 at all. In addition, transformation s_{3}, for example, moves vertex 1 to vertex 2, but the combination of s_{3} and s_{1} t = s_{3}s_{1} also moves vertex 1 to vertex 2. Denoting the group of congruent transformations that keep vertex 1 fixed as K = {I , s_{1}}, the set of transformations that move vertex 1 to vertex 2 and vertex 1 to vertex 3 are s_{3}K = {s_{3} , s_{3}s_{1}} and s_{2}K = {s_{2}, s_{1}}, respectively, or two transformations each. In this example, the number of vertices turns out to be the number of H bases (6) divided by the number of K bases (2). In this situation, we can consider H/K as a group of subsets hK = {hk | k ∈ K} instead of focusing on each base h of H. Then, by associating h(1) with hK, we can establish a one-to-one correspondence between H/K and the set of vertices X = {1, 2, 3}.
Functions having symmetry according to H in H/K space are, of course, only constant functions, but the expression X = H/K is useful in perceiving all the functions in that space. We can express functions f in H/K space in sequence as (f(1), f(2), f(3)), so a base of H, say s_{2}, would move (f(1), f(2), f(3)) to (f(s_{2}(1)), f(s_{2}(2)), f(s_{2}(3)))=(f(3), f(2), f(1)). This can be written as follows.
For example, to express the fact that f(s_{2}(1)) = f(3), (0, 0, 1) is written on the first row on the right side of this equation to indicate that f(1) and f(2) each appears 0 times and f(3) 1 time.The above arrangement of numerals in 3 rows and 3 columns is called the s_{2} relational table denotedas R(s_{2}). The diagonal component of this table, that is, the sum of all numerals on the line running from the upper left to the bottom right of the table, is called the trace of R(s_{2}) denoted as trR(s_{2}). As can be seen, trR(s_{2}) =1 here, but this indicates the number of vertices fixed by s_{2}, which in this example, is vertex 2. This relationship holds for any g ∈ H. (Please try verifying this.)
This also holds true for transformation group H and its subgroup K having a finite number of bases, and it serves as a “toy model” of the theorem known as the Lefschetz trace formula for a wide range of figures having symmetry. This expression thoroughly connects information covering all of X (R(g) trace) as to whether individual x ∈ X are fixed by g thereby expressing well the effectiveness of that symmetry.
For readers who are somewhat familiar with groups, we can write the number of fixed points in the language of H, K. Given that a certain x = hK is fixed by g ∈ H so that ghK = hK, we can restate this as follows from group conditions:
The number of fixed points therefore becomes:
Here, the summation on the right side of the equation adds subsets called conjugacy classes in K. If we now substitute H_{k} := {h ∈ H |, hkh^{-1}} = k} for k ∈ K, each term on the right can be written as follows:
Here, δ_{x,y} denotes the Kronecker delta that signifies 1 when x = y and 0 otherwise. Using this, Eq..
This, as well, serves as a toy model of the theorem known as the Selberg trace formula.
The Lefschetz trace formula and Selberg trace formula are both extensions of Eq., but each is used in completely different fields: algebraic topology for the former and representation theory or noncommutative harmonic analysis for the latter. In modern number theory, however, the Ihara-Langlands method is extremely powerful compared with the above formulas in a “Shimura variety” space. Consequently, when setting out to proveFermat’s theorem, the Shimura-Taniyama conjecture, or their various extensions, this method has come to be used as basic input. In actuality, the method is technically difficult to understand, but it is not difficult to imagine how Langlands focused on the common origins of the above two formulas to come up with the idea behind this method.