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PreprintarXiv licencearXiv

On modular vector invariant fields

  • Ying Han
  • Runxuan Zhang
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Abstract

Let 𝔽q be a finite field of any characteristic and GL(n,𝔽q) the general linear group over 𝔽q. Suppose W denotes the standard representation of GL(n,𝔽q) and GL(n,𝔽q) acts diagonally on the direct sum of W and its dual space W*. Let G be any subgroup of GL(n,𝔽q). Suppose the invariant field 𝔽q(W)G=𝔽q(f1,f2,…,fk), where f1,f2,…,fk∈𝔽q[W]G are homogeneous invariant polynomials. We prove that there exist homogeneous polynomials ℓ1,ℓ2,…,ℓn in the invariant ring 𝔽q[W⊕ W*]G such that the invariant field 𝔽q(W⊕ W*)G is generated by {f1,f2,…,fk,ℓ1,ℓ2,…,ℓn} over 𝔽q.

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Cite

BibTeXRIS
BibTeX
@misc{han2026modular,
  title = {{On modular vector invariant fields}},
  author = {Ying Han and Runxuan Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15442},
  eprint = {2610.15442},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15442},
}
RIS
TY  - UNPB
TI  - On modular vector invariant fields
AU  - Ying Han
AU  - Runxuan Zhang
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15442
UR  - https://arxiv.org/abs/2610.15442
AB  - Let 𝔽_q be a finite field of any characteristic and GL(n,𝔽_q) the general linear group over 𝔽_q. Suppose W denotes the standard representation of GL(n,𝔽_q) and GL(n,𝔽_q) acts diagonally on the direct sum of W and its dual space W^*. Let G be any subgroup of GL(n,𝔽_q). Suppose the invariant field 𝔽_q(W)^G=𝔽_q(f_1,f_2,…,f_k), where f_1,f_2,…,f_k∈𝔽_q[W]^G are homogeneous invariant polynomials. We prove that there exist homogeneous polynomials ℓ_1,ℓ_2,…,ℓ_n in the invariant ring 𝔽_q[W⊕ W^*]^G such that the invariant field 𝔽_q(W⊕ W^*)^G is generated by {f_1,f_2,…,f_k,ℓ_1,ℓ_2,…,ℓ_n} over 𝔽_q.
ER  -