{"paper":{"title":"Determinants of incidence and Hessian matrices arising from the vector space lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Alexandra Seceleanu, Junzo Watanabe, Saeed Nasseh","submitted_at":"2014-08-09T17:04:17Z","abstract_excerpt":"Let $\\mathcal{V}=\\bigsqcup_{i=0}^n\\mathcal{V}_i$ be the lattice of subspaces of the $n$-dimensional vector space over the finite field $\\mathbb{F}_q$ and let $\\mathcal{A}$ be the graded Gorenstein algebra defined over $\\mathbb{Q}$ which has $\\mathcal{V}$ as a $\\mathbb{Q}$ basis. Let $F$ be the Macaulay dual generator for $\\mathcal{A}$. We compute explicitly the Hessian determinant $|\\frac{\\partial ^2F}{\\partial X_i \\partial X_j}|$ evaluated at the point $X_1 = X_2 = \\cdots = X_N=1$ and relate it to the determinant of the incidence matrix between $\\mathcal{V}_1$ and $\\mathcal{V}_{n-1}$. Our exp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.2136","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}