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.

Solution 1:

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."