  Recently, the diagrammatic description of soft-gluon exponentiation in
scattering amplitudes has been generalized to the multiparton case. It was
shown that the exponent of Wilson-line correlators is a sum of webs, where each
web is formed through mixing between the kinematic factors and colour factors
of a closed set of diagrams which are mutually related by permuting the gluon
attachments to the Wilson lines. In this paper we use replica trick methods, as
well as results from enumerative combinatorics, to prove that web mixing
matrices are always: (a) idempotent, thus acting as projection operators; and
(b) have zero sum rows: the elements in each row in these matrices sum up to
zero, thus removing components that are symmetric under permutation of gluon
attachments. Furthermore, in webs containing both planar and non-planar
diagrams we show that the zero sum property holds separately for these two
sets. The properties we establish here are completely general and form an
important step in elucidating the structure of exponentiation in non-Abelian
gauge theories.
