Frobenius 极大抛物子代数的 Ooms 谱：严格单峰性与欧几里得对数凹性
Ooms spectra of Frobenius maximal parabolics: strict unimodality and Euclidean log-concavity
- Lehigh University（利哈伊大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究 Frobenius 极大抛物子代数的 Ooms 谱，证明严格单峰性并借助欧几里得算法和有限截断验证对数凹性，覆盖较小块至多 128 的情形。
AI中文摘要：
我们研究类型 $A$ 的 Frobenius 极大抛物子代数 $W(a,b)$ 的 Ooms 重数。对于 $a=mb+r$ 且 $1\le r<b$，欧几里得算法将其潜在直方图与更小代数 $W(r,b-r)$ 的潜在直方图联系起来。我们证明了对每个正互素对严格单峰性：重数在特征值 $0$ 和 $1$ 处严格增加到相等的中心值，然后严格递减。这加强了 Giaquinto、Irving、Lauve 和 Mastnak 最近的极大抛物单峰性定理。我们还证明了无界族 $W(mb\pm t,b)$ 的对数凹性，其中 $t\in\{1,2,3\}$，$b>t$，$\gcd(b,t)=1$，且 $m\ge1$。这些结果覆盖了较小块至多为 $8$ 以及较小块为 $10$ 的所有 Frobenius 情形。对于每个固定的剩余类，我们随后证明整个族的对数凹性由有限多个初始谱决定。有限截断是符号化获得的。对所得有限列表的精确整数验证证明了当较小块至多为 $128$ 时的严格内部对数凹性，对较大块无限制。无限制的对数凹性猜想在此未获证明。
英文摘要：
We study the Ooms multiplicities of Frobenius maximal parabolics $W(a,b)$ of type~$A$. For $a=mb+r$ with $1\le r<b$, the Euclidean algorithm relates their potential histograms to those of the smaller algebra $W(r,b-r)$. We prove strict unimodality for every positive coprime pair: the multiplicities increase strictly to the equal central values at eigenvalues $0$ and $1$, then decrease strictly. This strengthens the recent maximal-parabolic unimodality theorem of Giaquinto, Irving, Lauve, and Mastnak. We also prove log-concavity for the unbounded families $W(mb\pm t,b)$, where $t\in\{1,2,3\}$, $b>t$, $\gcd(b,t)=1$, 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.
