Constant rank subspaces of alternating bilinear forms from Galois Theory

2023/10/05 by Ashish Kumar Gupta, Gupta, Ashish, Sugata Mandal +1
Mathematics · #12F05 #12F10 #15A63 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2310.03340

openalex publication_date 2023/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L/K be a cyclic extension of degree n = 2m. It is known that the space Alt_K(L) of alternating K-bilinear forms (skew-forms) on L decomposes into a direct sum of K-subspaces A^σi indexed by the elements of Gal(L/K) = ⟨ σ⟩. It is also known that the components A^σi can have nice constant-rank properties. We enhance and enrich these constant-rank results and show that the component A^σ often decomposes directly into a sum of constant rank subspaces, that is, subspaces all of whose non-zero skew-forms have a fixed rank r. In particular, this is always true when -1 \not ∈ L². As a result we deduce a decomposition of Alt_K(L) into subspaces of constant rank in several interesting situations. We also establish that a subspace of dimension (n)/(2) all of whose nonzero skew-forms are non-degenerate can always be found in A^σi where σⁱ has order divisible by 2.

Related

• Galois extensions and subspaces of bilinear forms with special rank properties
• Symmetric bilinear Forms and Galois Theory
• Connections between rank and dimension for subspaces of bilinear forms
• Rank-related dimension bounds for subspaces of bilinear forms over finite fields
• Rank-related dimension bounds for subspaces of symmetric bilinear forms