The criteria for the uniqueness of a weight homomorphism of a baric algebra
Dali Zangurashvili
TL;DR
The paper addresses when a finite-dimensional baric algebra over $K$ has a unique weight homomorphism. It establishes two central criteria: first, that uniqueness corresponds to the Etherington system $x_i x_j = \sum_{k=1}^n \gamma_{ijk} x_k$ possessing a unique non-trivial solution; second, that uniqueness is equivalent to every transition matrix between semi-natural bases being row stochastic. It further shows Holgate's sufficient condition is not necessary and provides explicit examples, including a non-commutative $3$-D algebra with two weight homomorphisms, as well as an approach that ties basis transitions to a row-stochastic group structure $RS_n(K)$. A unifying framework is developed where the set of weight homomorphisms $W$ is linked to the Etherington solution sets via basis transitions, yielding a coset description that clarifies how changes of basis influence weight multiplicity in baric algebras.
Abstract
The criteria for a baric algebra $A$ (over a field $K$) to have a unique weight homomorphism are found. One of them requires a certain system of equations to have a unique non-trivial solution in the field $K$. Applying this criterion, we provide an example showing that Holgate's well-known sufficient condition for the uniqueness of a weight homomorphism is not necessary, and give also a new example of a baric algebra with two weight homomorphisms. Another criterion found in this paper asserts that a baric algebra has a unique weight homomorphism if and only if the transition matrix from any semi-natural basis $B_1$ to any semi-natural basis $B_2$ is stochastic.