{"id":39,"date":"2016-09-10T00:33:52","date_gmt":"2016-09-10T00:33:52","guid":{"rendered":"https:\/\/petermc.net\/blog\/?p=39"},"modified":"2016-09-10T00:40:35","modified_gmt":"2016-09-10T00:40:35","slug":"some-easy-singular-schubert-varieties","status":"publish","type":"post","link":"https:\/\/petermc.net\/blog\/2016\/09\/10\/some-easy-singular-schubert-varieties\/","title":{"rendered":"Some easy singular Schubert varieties"},"content":{"rendered":"

I write \"w\" for both an element of the Weyl group and the corresponding point in \"G\/B\". Let \"X_w\" be a Schubert variety. Then \"T_eX_w\subset is an inclusion of \"\mathfrak{b}\" representations. The latter is generated by the \"-\theta\"-weight space, where \"\theta\" is the highest root.<\/p>\n

Suppose \"w\geq, where \"s_\theta\" is the reflection corresponding to \"\theta\". Then the \"\mathbb{P}^1\" connecting \"e\" and \"s_\theta\" lies in \"X_w\" (think of the corresponding SL2<\/sub>), so the \"-\theta\"-weight space lies in \"T_eX_w\". Since this generates \"T_e(G\/B)\", we have \"T_eX_w=T_e(G\/B)\".<\/p>\n

Therefore if \"w_0\neq, then \"X_w\" is singular. This includes some of the first examples of singular Schubert varieties, for example the B2<\/sub> singular Schubert variety and one of the A3<\/sub> ones.<\/p>\n","protected":false},"excerpt":{"rendered":"

I write for both an element of the Weyl group and the corresponding point in . Let be a Schubert variety. Then is an inclusion of representations. The latter is generated by the -weight space, where is the highest root. … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6],"tags":[],"class_list":["post-39","post","type-post","status-publish","format-standard","hentry","category-maths"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7V6a7-D","_links":{"self":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/39","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/comments?post=39"}],"version-history":[{"count":6,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/39\/revisions"}],"predecessor-version":[{"id":45,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/39\/revisions\/45"}],"wp:attachment":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/media?parent=39"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/categories?post=39"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/tags?post=39"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}