How to define an antisymmetric symbol?

Question

I want to work with linear expressions involving the formal symbol $w[a_1,...,a_n]$, and I would like Mathematica to know that expressions such as

w[a,b,d] + w[a,d,b] = 0

i.e. that the symbol w is antisymmetric with respect to the swap of any of its entries. What is the best way to implement this?

Answer 1

Here's a variation of Roman's answer that doesn't require the application of ReplaceAll to simplify things:

a_w /; !OrderedQ @ Unevaluated @ a := Signature[Unevaluated[a]]Sort[Unevaluated[a]]

Your example:

w[a, b, d] + w[a, d, b]

0

Another possibility is to use the symbolic tensor capabilities of Mathematica. For example:

TensorReduce[
    w + TensorTranspose[w, {1, 3, 2}],
    Assumptions -> w ∈ Arrays[{n, n, n}, Complexes, Antisymmetric[{1,2,3}]]
]

0

Answer 2

You can do it by simplifying every mention of w to its sorted form:

w[a, b, d] + w[a, d, b] /. W_w :> Signature[W]*Sort[W]

0

{w[a, b, d], w[a, d, b], w[b, a, d], w[b, d, a], w[d, a, b], w[d, b, a]} /.
  W_w :> Signature[W]*Sort[W]

{w[a, b, d], -w[a, b, d], -w[a, b, d], w[a, b, d], w[a, b, d], -w[a, b, d]}