2024/05/12 by Tatsuya Horiguchi, Horiguchi, Tatsuya · 1 citation
Mathematics · Physics and Astronomy · #14M15 #17B22 #32S22 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2405.07247
openalex publication_date 2024/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a complex semisimple linear algebraic group. Fix a subset Θ of simple roots. Given a lower ideal I in positive roots, one can define the regular nilpotent Hessenberg variety Hess(N,I) in the full flag variety G/B. For a Θ-ideal I (which is a special lower ideal), we can define the regular nilpotent partial Hessenberg variety HessΘ(N,I) in the partial flag variety G/P. In this manuscript we first provide a summand formula and a product formula for the Poincaré polynomial of regular nilpotent partial Hessenberg varieties. It is a well-known result from Bernstein-Gelfand-Gelfand that the cohomology ring of the partial flag variety G/P is isomorphic to the invariants in the cohomology ring of the full flag variety G/B under an action of the parabolic Weyl group WΘ generated by Θ. We generalize this result to regular nilpotent partial Hessenberg varieties. More concretely, we give an isomorphism between the cohomology ring of a regular nilpotent partial Hessenberg variety HessΘ(N,I) and the WΘ-invariant subring of the cohomology ring of the regular nilpotent Hessenberg variety Hess(N,I). Furthermore, we provide a description of the cohomology ring for a regular nilpotent partial Hessenberg variety HessΘ(N,I) in terms of the WΘ-invariants in the logarithmic derivation module of the ideal arrangement AI, which is a generalization of the result by Abe-Masuda-Murai-Sato with the author.