Kernel and splitting of the joint Cartier-character map

Determine the kernel of the joint character map \(\chi_{\mathrm{all}}:U_n(k)\to\bigoplus_{s\in S_{n,p}}(k^n,+)\), decide whether it equals \([U_n(k),U_n(k)]\,U_n(k)^{[p]}\), and determine whether the induced surjection from the mod-\(p\) abelianization \(Q_n(k)\) is an isomorphism; additionally, determine whether \(\chi_{\mathrm{all}}\) admits a group-theoretic section.

Background

The paper constructs homomorphisms $\chi_{r,j}:U_n(k)\to(k,+)$ and combines them into a surjective joint map $\chi_{\mathrm{all}}$ onto a countable direct sum. This proves that the mod-$p$ abelianization has countably infinite dimension, but surjectivity alone does not identify the kernel.

The authors also construct group-theoretic sections after restricting to one fixed coordinate. A section for the full joint map is unresolved because the explicit test elements associated with different coordinates need not commute. Determining the kernel and whether the joint map splits would clarify the precise structure of the mod-$p$ abelianization.