I have a totally antisymmetric function of n arguments, f[a,b,c,d] = -f[b,a,c,d], which I would like to be multi-linear, e.g.
f[a+x,b+y,c,d] = f[a,b,c,d] + f[x,b,c,d] + f[a,y,c,d] + f[x,y,c,d]
How can I achieve this? It's easy enough to do for one argument, e.g. I could use
f[a_Plus,b__]:= f[#,b]&/@a,
however I need this to work for all n arguments. Any ideas?
Clear[f];
f[a1___, a2_Plus, a3___] := f[a1, #, a3] & /@ a2
f[a + x, b + y, c, d]
(* f[a, b, c, d] + f[a, y, c, d] + f[x, b, c, d] + f[x, y, c, d] *)
Best way is to transfer the sorting to Simplify - ing commands
f /: Simplify[f[a_, b__]] := Signature[{a, b}]*f @@ Sort[{a, b}]
f/:Expand[ f[a___,b_Plus,c___]]:=Distribute[f[a,b,c],Plus]
I don't trust automatic definitions like
f[a_, b__] := Signature[{a, b}]*f @@ Sort[{a, b}] /; ! OrderedQ[{a, b}]
f[a___, b__Plus, c___] := f[a, #, c] & /@ b
f[a___, b_?NumericQ *c__, d___] := b f[a, c, d]
Too often such identity definitions have produced infinite recursions without control for me.
