Let \(\mathbf{t}\) be the treatment assignment vector, and let \(B_{\mathbf{t}}\) be the associated block factor, i.e., \(B_{\mathbf{t}}(i, j)=1\) if \(t_{i}=t_{j}\) and zero otherwise. For \(g \in \mathbb{R}\), consider the transformations

\[
\Sigma \stackrel{g}{\longmapsto} \Sigma+g^{2} B_{\mathbf{t}}
\]

for \(\Sigma\) in the space of positive definite matrices. Discuss whether these transforms determine a group action or group homomorphism (preserving identity and composition). If not, is it a semi-group homomorphism in a suitable sense? Maybe after changing \(g^{2}\) to \(e^{g}\) or \(|g|\) to maintain positivity?