Ooms spectra of Frobenius maximal parabolics: strict unimodality and Euclidean log-concavity

Source abstract

We study the Ooms multiplicities of Frobenius maximal parabolics $W(a,b)$ of type~$A$. For $a=mb+r$ with $1\le r t$, $\gcd(b,t)=1$, and $m\ge1$. These results cover every Frobenius case with smaller block at most $8$, and also smaller block $10$. For each fixed residue class, we then prove that log-concavity of the whole family is determined by finitely many initial spectra. The finite cutoff is obtained symbolically. Exact integer verification of the resulting finite lists proves strict internal log-concavity whenever the smaller block is at most $128$, with no bound on the larger block. The unrestricted log-concavity conjecture is not proved here.