# Generalization of even / odd functions

The following four examples all have a similar structure:

1. Every function $$f:\Bbb R \to \Bbb R$$ has a unique decomposition $$f = f_e + f_o$$ where $$f_e$$ is an even function ($$f_e(-x) = f_e(x)$$) and $$f_o$$ is an odd function ($$f_o(-x) = -f_o(x)$$).

2. Every function $$g:\Bbb R\times \Bbb R\to \Bbb R$$ has a unique decomposition $$g=g_s + g_a$$ where $$g_s$$ is a symmetric function (that is, $$g_s(x,y) = g_s(y,x)$$) and $$g_a$$ is an asymmetric function (that is, $$g_a(x,y) = -g_a(y,x)$$).

3. Every real-valued matrix $$M$$ has a unique decomposition $$M = M_s + M_a$$ where $$M_s$$ is symmetric (that is, $$M_s^T = M_s$$) and $$M_a$$ is antisymmetric ($$M_a^T = -M_a$$),

4. Every complex number $$z$$ has a unique decomposition $$z = z_e+z_o$$ where $$\bar {z_e} = z_e$$ and $$\bar {z_o} = -z_o$$. Here $$z_e$$ and $$z_o$$ are simply the real and imaginary parts of $$z$$, and “real” and “pure imaginary” play the roles of “even” and “odd”.

In each case we have some space $$S$$ (real-valued functions of one or two variables, matrices, complex numbers) and an involution $$I:S\to S$$:

1. $$f(x)\leftrightarrows f(-x)$$
2. $$f(x,y)\leftrightarrows f(y,x)$$
3. $$M\leftrightarrows M^T$$
4. $$z\leftrightarrows \bar z$$

Then from $$I$$ we identify two special subclasses of $$S$$: the “even” elements, which are just the fixed points of $$I$$, and the “odd” elements, which are “negated” by $$I$$. Then every element of the space has a unique representation as a sum of an “even” and an “odd” element.

To really make sense of this we have to pin down “negated”, and I think to make it work we need something like division by two. If $$S$$ is a real vector space, as in the four examples, both of these are straightforward, and if $$I$$ is any linear map $$S\to S$$ with $$I^2=1$$, we can decompose $$x=x_e+x_o$$ where \begin{align} x_e &= \frac12\left(x + I(x)\right) \\ x_o &= \frac12\left(x - I(x)\right) \\ \end{align}

and clearly $$x_e$$ is even with respect to $$I$$ and $$x_o$$ is odd.

But it seems to me that it ought to be possible to make this work in a more general context, perhaps in several ways. For example, even in spaces where the scaling by $$\frac12$$ makes no sense, one can still solve $$x_e + x_o = x + x$$.

My questions are:

1. Is there a general name for this type of construction? Where can I find out more?
2. Is there some way to make sense of it in a less well-structured context than that of a vector space? Say, a group? Or maybe even something as general as a monoid?
3. Does this turn out to be useful or interesting in some context other than a real or complex vector space?

I tried formulating it in category-theoretic language. $$I$$ is an arrow with $$I\circ I = id_S$$, and then the “even” subobject of $$S$$ is just the equalizer of $$I$$ and $$id_S$$. But I got bogged down trying to decide what the “odd” subobject was.

• I don't really know much about this sort of thing, but if you can divide by 2, it seems "symmetric" (e.g. the "symmetric part" of something) and "skew-symmetric" are common in general. For random examples, see arxiv.org/abs/1108.4648 and jstor.org/stable/2038043 – Mark S. Apr 7 at 20:04
• That Goodaire paper looks like just the kind of thing I was hoping for. Thanks! – MJD Apr 7 at 21:21
• Note to self: Consider also that any real number $r$ has a unique decomposition $r = n+f$ where $n$ is an integer and $f\in[0,1)$. Can this be understood as a non-vector-space instance of this phenomenon, which might point the way toward a suitable generalization? Further examples: elements of $D_{2n}$ decompose uniquely into sums of rotations and reflections. Elements of $Z_{2n}$ decompose uniquely into sums $e + b$ where $e\in\{2k\mid k\in Z_{2n}\}$ and $b\in\{0, 1\}$. – MJD May 27 at 19:14

The search term you want is representation theory, specifically in this case of the cyclic group $$C_2$$ of order $$2$$. A representation of $$C_2$$ on a vector space $$V$$ over a field of characteristic not equal to $$2$$ equips $$V$$ with an involution $$I : V \to V$$, and this involution canonically decomposes $$V$$ as a direct sum of two isotypic components, namely the even component

$$V_0 = \{ v \in V : Iv = v \}$$

and the odd component

$$V_1 = \{ v \in V : Iv = -v \}.$$

The direct sum decomposition of a given general vector $$v \in V$$ is then, as you say,

$$v = \frac{v + Iv}{2} + \frac{v - Iv}{2}.$$

You can think of the even and odd components as the eigenspaces of $$I$$ also.

Many generalizations are possible. The simplest useful generalization replaces $$C_2$$ with a finite group $$G$$ (the search term here is "representation theory of finite groups"); in this case the isotypic components correspond to the irreducible representations of $$G$$. These are sensitive to the choice of ground field, particularly to 1) its characteristic and 2) which roots of unity exist over it. For example, if $$G = C_3$$ is the cyclic group of order $$3$$, then over a field of characteristic not equal to $$3$$ which also contains all third roots of unity $$1, \omega, \omega^2$$, there are three isotypic components

$$V_0 = \{ v \in V : Iv = v \}$$ $$V_1 = \{ v \in V : Iv = \omega v \}$$ $$V_2 = \{ v \in V : Iv = \omega^2 v \}$$

and the direct sum decomposition, which is considerably less obvious, is

$$v = \frac{v + Iv + I^2 v}{3} + \frac{v + \omega^2 I v + \omega I^2 v}{3} + \frac{v + \omega I v + \omega^2 I^2 v}{3}.$$

Generalizing to a cyclic group $$C_n$$ of order $$n$$ leads to the discrete Fourier transform, also sometimes known as the "roots of unity filter."