nLab alternating multifunction

Idea

In linear algebra, alternating multilinear functions are well known, and are in many cases (over the real numbers, for example) are equivalent to antisymmetric functions?. In cases where they differ (such as in characteristic $2$), it is often the alternating functions that behave better.

Actually, being alternating is not, in itself, really about linearity, and we can abstract away to a nonlinear concept of alternating function. (That said, there is one bit of very mild linear structure that is needed: a basepoint in the target set.)

The property of being alternating is called alternation (rather than alternatingess), although in principle one could also use alternating as a noun.

Definitions

Let $X$ be a set, and let $(Y,0)$ be a pointed set (so $Y$ is a set and $0$ is one of its elements). Let $n$ be a natural number (or indeed any cardinal number). Recall that a multifunction of arity $n$ to $Y$ from $X$ is the same thing as a function to $Y$ from the $n$-fold cartesian power $X^n$.

An alternating multifunction (or simply alternating function) of arity $n$ from $X$ to $(Y,0)$ is a multifunction of arity $n$ from $X$ to $Y$ such that, whenever two of the function's arguments are equal, the value of the function is $0$. In arity $0$ or $1$, every multifunction is trivially alternating; in arity $2$, we can write this as the equational law? $f(a,a) = 0$; in arity $3$, we have the equational laws $f(a,a,b) = 0$, $f(a,b,a) = 0$, and $f(a,b,b) = 0$; etc.

Properties

There are many nice properties of alternating multilinear functions. So suppose that $X$ and $Y$ are modules over a base rig? $K$ and that $f$ is a multilinear function from $X$ to $Y$; use the usual zero element of the module $Y$ as the basepoint $0$. In the case where $Y$ is $K$ itself, we speak of an alternating form (a phrase which is usually taken to include multilinearity).

We will sometimes want to assume that scalar multiplication by $2$ is cancellable in $Y$ (which for example is always the case when $2$ is invertible in $K$, in particular when $K$ is a field of characteristic other than $2$), but only when stated. Since alternation requires looking at two arguments of $f$, we will often, when this leads to no loss of generality, assume that these are the first two arguments, writing $\vec{z}$ to represent all of the other arguments.

Proposition

An alternating multilinear function is antisymmetric?.

By multilinearity,

$$f(x+y,x+y,\vec{z}) = f(x,x,\vec{z}) + f(x,y,\vec{z}) + f(y,x,\vec{z}) + f(y,y,\vec{z}) .$$

Applying alternation, most of these terms vanish:

$$0 = 0 + f(x,y,\vec{z}) + f(y,x,\vec{z}) + 0 .$$

Therefore,

$$f(x,y,\vec{z}) + f(y,x,\vec{z}) = 0 ,$$

which is antisymmetry.

Proposition

If multiplication by $2$ is cancellable in $Y$, then an antisymmetric? function to $Y$ is alternating.

Proof

By antisymmetry,

$$f(x,x,\vec{z}) + f(x,x,\vec{z}) = 0 ,$$

or equivalently

$$2 f(x,x,\vec{z}) = 2 \cdot 0 .$$

Cancelling $2$,

$$f(x,x,\vec{z}) = 0 ,$$

which is alternation.

Remark

It is false in both directions to state in general that alternating functions and antisymmetric functions are the same, but for different reasons. An alternating function must be antisymmetric if it is multilinear, regardless of the behaviour of $2$, but not when it is nonlinear; an antisymmetric function must be alternating if multiplication by $2$ is cancellable in the target, regardless of linearity, but not when $2$ is noncancellable.

The simplest strongest-possible counterexamples are

$$(x,y \mapsto |y-x|)\colon \mathbb{R}^2 \to \mathbb{R} ,$$

which is alternating but not antisymmetric, and

$$(x,y \mapsto x y)\colon \mathbb{F}_2^2 \to \mathbb{F}_2 ,$$

which is antisymmetric but not alternating.

Of course, alternating and antisymmetric functions are the same in the context of multilinear functions to a module in which $2$ is cancellable, in particular for multilinear functions between vector fields over the real numbers.

Remark

The alternating groups are really about antisymmetric functions rather than alternating functions as such. (Whereas a symmetric function is preserved by the application of any element of the symmetric group, an antisymmetric function is preserved by and only by the elements of the alternating group.) Nevertheless, this precise distinction between ‘alternating’ and ‘antisymmetric’ is well established in the theory of vector spaces over a field of characteristic $2$ (in which multiplication by $2$ is as uncancellable as possible).

Constructive aspects

In constructive mathematics, we usually assume that the arity $n$ of $f$ has decidable equality, which is true if $n$ is a natural number (which is most common) or even a (possibly infinite) extended natural number. However, as long as the arity is equipped with an inequality, then we can state the definition: whenever equal arguments have inequal indices, the value of the multifunction $f$ there is zero. If $X$ and $Y$ are also equipped with inequalities, then $f$ is strongly alternating if, whenever its value is inequal to $0$ in $Y$, then arguments with inequal indices must be inequal in $X$. (In arity $2$, for example, if $f(a,b) \ne 0$, then $a \ne b$.) If the inequality on $Y$ is tight (so that its negation is equality in $Y$), then every strongly alternating function is alternating, but the reverse requires excluded middle in general.