{"paper":{"title":"The defect of generalized Fourier matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.OA"],"primary_cat":"math.CO","authors_text":"Teodor Banica","submitted_at":"2012-10-09T10:50:49Z","abstract_excerpt":"The $N\\times N$ complex Hadamard matrices form a real algebraic manifold $C_N$. We have $C_N=M_N(\\mathbb T)\\cap\\sqrt{N}U_N$, and following Tadej and \\.Zyczkowski we investigate here the computation of the enveloping tangent space $\\widetilde{T}_HC_N=T_HM_N(\\mathbb T)\\cap T_H\\sqrt{N}U_N$, and notably of its dimension $d(H)=\\dim(\\widetilde{T}_HC_N)$, called undephased defect of $H$. Our main result is an explicit formula for the defect of the Fourier matrix $F_G$ associated to an arbitrary finite abelian group $G=\\mathbb Z_{N_1}\\times...\\times\\mathbb Z_{N_r}$. We also comment on the general ques"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2556","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"}