{ "id": "1712.06737", "version": "v1", "published": "2017-12-19T01:14:50.000Z", "updated": "2017-12-19T01:14:50.000Z", "title": "Conormal Varieties on the Cominuscule Grassmannian", "authors": [ "Rahul Singh", "Venkatraman Lakshmibai" ], "categories": [ "math.AG" ], "abstract": "Let $G$ be a simply connected, almost simple group over an algebraically closed field $\\mathbf k$, and $P$ a maximal parabolic subgroup corresponding to omitting a cominuscule root. We construct a compactification $\\phi:T^*G/P\\rightarrow X(u)$, where $X(u)$ is a Schubert variety corresponding to the loop group $LG$. Let $N^*X(w)\\subset T^*G/P$ be the conormal variety of some Schubert variety $X(w)$ in $G/P$; hence we obtain that the closure of $\\phi(N^*X(w))$ in $X(u)$ is a $B$-stable compactification of $N^*X(w)$. We further show that this compactification is a Schubert subvariety of $X(u)$ if and only if $X(w_0w)\\subset G/P$ is smooth, where $w_0$ is the longest element in the Weyl group of $G$. This result is applied to compute the conormal fibre at the $0$ matrix in any determinantal variety.", "revisions": [ { "version": "v1", "updated": "2017-12-19T01:14:50.000Z" } ], "analyses": { "subjects": [ "14M15" ], "keywords": [ "conormal variety", "cominuscule grassmannian", "compactification", "determinantal variety", "maximal parabolic subgroup corresponding" ], "note": { "typesetting": "TeX", "pages": 0, "language": "en", "license": "arXiv", "status": "editable" } } }