  Recently in [JHEP 09 (2021) 053], Wang et al. discussed the polynomial ring
formed by flavor invariants in the leptonic sector with massive Majorana
neutrinos. They have explicitly constructed the finite generating sets of the
polynomial rings for both two-generation scenario and three-generation
scenario. However, Wang et al.'s claim of the finiteness of the generating sets
of the polynomial rings and their calculation by the approach of Hilbert series
with generalized Molien-Weyl formula are both based on their assertion that the
unitary group ${\rm U}(n,\mathbb{C})$ is reductive, which is unfortunately
incorrect. The property of being reductive is only applicable to linear
algebraic groups. And it is well-known that the unitary group ${\rm
U}(n,\mathbb{C})$ is not even a linear algebraic group. In this paper, we point
out the above issue and provide a solution to fill in the accompanying logical
gaps in [JHEP 09 (2021) 053]. Some important results from the theory of linear
algebraic group, the invariant theory of square matrices and group theory are
needed in the analysis. We also clarify some somewhat misleading or vague
statements in [JHEP 09 (2021) 053] about the scope of flavor invariants. Note
that, although built from incorrect assertion, Wang et al.'s calculation
results in [JHEP 09 (2021) 053] are nonetheless correct, which is ultimately
because the ring of invariants of ${\rm U}(n,\mathbb{C})$ is isomorphic to that
of ${\rm GL}(n,\mathbb{C})$ which is itself reductive.
