Euler Characteristics of SL₄(ℤ) and GL₄(ℤ), and their cohomological consequences

@misc{bajpai2026euler,
  title = {{Euler Characteristics of $\mathrmSL_4(\mathbbZ)$ and $\mathrmGL_4(\mathbbZ)$, and their cohomological consequences}},
  author = {Jitendra Bajpai and Taiwang Deng},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29796},
  eprint = {2608.29796},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29796},
}

TY  - UNPB
TI  - Euler Characteristics of SL_4(ℤ) and GL_4(ℤ), and their cohomological consequences
AU  - Jitendra Bajpai
AU  - Taiwang Deng
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.29796
UR  - https://arxiv.org/abs/2608.29796
AB  - We compute the homological Euler characteristics of SL_4(ℤ) and GL_4(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 SL_4(ℤ)-coefficient system to GL_4(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.
ER  -