发表机构
Christian-Albrechts-Universität zu Kiel; Beijing Institute of Mathematical Sciences and Applications (BIMSA)（基尔大学; 北京国际数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究计算了SL₄(ℤ)与GL₄(ℤ)的欧拉示性数，推导了其拟多项式形式、上同调的消失结果与下界，结合Horozov的工作得到对称幂相关的恒等式与猜想。
AI 中文摘要
我们计算了SL₄(ℤ)与GL₄(ℤ)在任意不可约有理最高权表示系数下的同调欧拉示性数。应用Wall公式，我们结合挠元中心化子的orbifold欧拉示性数与利用Jacobi-Trudi恒等式计算的迹，推导得到显式公式与有理生成函数。由此，这些欧拉示性数是最高权参数的拟多项式函数，总次数至多为2。我们进一步推导了次数意义下的消失结果与上同调群维数的奇偶敏感下界。将SL₄(ℤ)系数系统向GL₄(ℤ)的两种延拓给出了更紧的界，这些界在显式无穷族中线性或二次增长。对于对称幂，将我们的公式与Horozov的行列式扭曲求和项计算相结合，得到了未扭曲求和项的精确恒等式与下界，以及其上同调的次数意义下的猜想描述。
英文摘要
We compute the homological Euler characteristics of $\mathrm{SL}_4(\mathbb{Z})$ and $\mathrm{GL}_4(\mathbb{Z})$ with coefficients in arbitrary irreducible rational highest-weight representations. Applying Wall's formula, we combine the orbifold Euler characteristics of centralizers of torsion elements with traces computed using the Jacobi-Trudi identity to derive explicit formulas and rational generating functions. Consequently, these Euler characteristics are quasi-polynomial functions of the highest-weight parameters, of total degree at most two. We further derive degreewise vanishing results and parity-sensitive lower bounds for the dimensions of cohomology groups. The two extensions of an $\mathrm{SL}_4(\mathbb{Z})$-coefficient system to $\mathrm{GL}_4(\mathbb{Z})$ yield sharper bounds, which grow linearly or quadratically in explicit infinite families. For symmetric powers, combining our formulas with Horozov's calculation of the determinant-twisted summand yields exact identities and lower bounds for the untwisted summand, together with a conjectural degreewise description of its cohomology.
Comments59 pages
