What the paper is about: The authors give a simple, explicit rule for the signature — the counts of positive and negative directions — of a canonical complex inner product that appears in a classical algebraic construction known as the Katz–Long–Moody (KLM) or multiplicative middle convolution. This inner product, called a Hermitian form, arises when one builds new representations of braid and related groups from given input data. The main result says the signature can be read off from just three spectral pieces of information: the angles of the eigenvalues (the “eigenangles”) of the input matrices, the eigenangles of their ordered product, and the chosen convolution parameter on the unit circle.

What the researchers did: Earlier work produced a recursive algorithm to compute this signature. Here the authors replace that algorithm by a closed formula that works uniformly, even in degenerate or resonant situations that had required separate treatment before. The proof is elementary: it uses linear algebra identities, block pivot (Schur complement) calculations, and an inertia formula for sums of Cayley transforms of unitary matrices. The formula explicitly tracks any kernel (directions where the form vanishes) present before passing to the usual quotient, and it describes how the signature changes when the convolution parameter crosses resonance walls.

Why this matters: The signature of the canonical form controls whether the output representation can be made unitary — that is, whether there is a positive-definite inner product it preserves. The paper characterizes exactly when the induced form on the quotient is definite, solving the “definiteness problem” posed in a companion paper. Definiteness implies the representation is unitarizable; the converse is true when the quotient representation is irreducible. Because the formula depends only on eigenangles and the parameter, it is stable across families: for fixed parameter, any unitary inputs with the same local conjugacy classes have the same signature.