{"id":246,"date":"2021-06-16T14:07:52","date_gmt":"2021-06-16T14:07:52","guid":{"rendered":"https:\/\/petermc.net\/blog\/?p=246"},"modified":"2021-06-16T14:07:52","modified_gmt":"2021-06-16T14:07:52","slug":"jucys-murphy-elements-and-induction","status":"publish","type":"post","link":"https:\/\/petermc.net\/blog\/2021\/06\/16\/jucys-murphy-elements-and-induction\/","title":{"rendered":"Jucys-Murphy elements and induction"},"content":{"rendered":"

This post concerns the representation theory of the symmetric group over the complex numbers. Recall that the irreducible representations of the symmetric group \"S_n\" are indexed by partitions of \"n\". Let \"S^\lambda\" be the irreducible representation indexed by \"\lambda\". I want to say some words about the theorem that the decomposition of the induced module \"\operatorname{Ind}_{S_{n}}^{S_{n+1}}S^\lambda\" is given by the decomposition into eigenspaces under the action of the Jucys-Murphy element.<\/p>\n

First, the relevant Jucys-Murphy element is<\/p>\n

  <\/span>   <\/span>\"\[X:=(1,n+1)+(2,n+1)+\cdots+(n,n+1)\in \mathbb{C}[S_{n+1}].\]\"<\/p>\n

The way it acts on \"\operatorname{Ind}_{S_{n}}^{S_{n+1}}S^\lambda=\mathbb{C}[S_{n+1}]\otimes_{\mathbb{C}[S_n]}S^\la\" is not as an element of \"\mathbb{C}[S_{n+1}]\" but by \"X\cdot (a\otimes v)=aX\otimes v.\" This is well-defined since \"X\" commutes with \"\mathbb{C}[S_n]\".<\/p>\n

What this action defines is a natural transformation from the functor \"\operatorname{Ind}_{S_{n}}^{S_{n+1}}\" to itself. The induction functor is (bi)-adjoint to the restriction functor and this natural transformation is even simpler to construct on the adjoint side. Recall that if \"\mathcal{F}\" and \"\mathcal{G}\" are adjoint functors, then there is an isomorphism<\/p>\n

  <\/span>   <\/span>\"\[\operatorname{End}\mathcal{F}\cong\operatorname{End}\mathcal{G}.\]\"<\/p>\n

Here \"\operatorname{End}\mathcal{F}\" refers to the natural transformations from \"\mathcal{F}\" to itself, and the map in this isomorphism is given by pre- and post-composition by the unit and counit of the adjunction.<\/p>\n

And the way that \"X\" yields a natural transformation from \"\operatorname{Res}_{S_n}^{S_{n+1}}\" to itself is very simple, it’s just by its usual action as an element of \"\mathbb{C}[S_{n+1}]\". If you transport this natural transformation to a natural transformation of the induction functor via the method I just mentioned, then you get the formula mentioned above.<\/p>\n

Now given a pair of adjoint functors \"\mathcal{F}\" and \"\mathcal{G}\", a natural transformation \"X\" from \"\mathcal{F}\" to \"\mathcal{F}\" (and hence from \"\mathcal{G}\" to \"\mathcal{G}\") and a complex number \"a\", we can define a functor \"\mathcal{F}_a\" by<\/p>\n

  <\/span>   <\/span>\"\[\mathcal{F}_a(V)=\{w\in \mathcal{F}(V)\mid Xw=aw\}.\]\"<\/p>\n

and similarly for \"\mathcal{G}\" (this requires some linearity assumptions, but they’re satisfied here. Also you could take generalised eigenspaces if you wanted to, but in our application there is no difference).<\/p>\n

When you do this, the functors \"\mathcal{F}_a\" and \"\mathcal{G}_a\" are adjoint:<\/p>\n

Proof: Both \"\operatorname{Hom}(\mathcal{F}_aV,W)\" and \"\operatorname{Hom}(V,\mathcal{G}_aW)\" are the \"a\"-eigenspace of the action of \"X\" on \"\operatorname{Hom}(\mathcal{F}V,W)\cong\operatorname{Hom}(V,\mathcal{G}W)\".<\/p>\n

Now apply this to our situation. We also use the following standard fact about the action of the Jucys-Murphy element (as developed e.g. in the Vershik-Okounkov approach):<\/p>\n

Consider the decomposition <\/p>\n

  <\/span>   <\/span>\"\[\operatorname{Res}_{S_n}^{S_{n+1}}S^\mu=\bigoplus_{\lambda\to\mu}S^\lambda.\]\"<\/p>\n

Then the Jucys-Murphy element \"X\" acts by the scalar \"c(\alpha)\" on \"S^\lambda\", where \"c(\alpha)\" is the content of the box \"\alpha\" added to \"\lambda\" to get \"\mu\".<\/p>\n

Now translating this statement via the above yoga onto the adjoint side, we get<\/p>\n

In the decomposition <\/p>\n

  <\/span>   <\/span>\"\[\operatorname{Ind}_{S_{n}}^{S_{n+1}}S^\lambda=\bigoplus_{\lambda\to\mu}S^\mu,\]\"<\/p>\n

the Jucys-Murphy element \"X\" acts by the scalar \"c(\alpha)\" on \"S^\mu\", where \"c(\alpha)\" is the content of the box \"\alpha\" added to \"\lambda\" to get \"\mu\".<\/p>\n","protected":false},"excerpt":{"rendered":"

This post concerns the representation theory of the symmetric group over the complex numbers. Recall that the irreducible representations of the symmetric group are indexed by partitions of . Let be the irreducible representation indexed by . I want to … 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":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6],"tags":[],"class_list":["post-246","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\/p7V6a7-3Y","_links":{"self":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/246","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=246"}],"version-history":[{"count":8,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/246\/revisions"}],"predecessor-version":[{"id":254,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/246\/revisions\/254"}],"wp:attachment":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/media?parent=246"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/categories?post=246"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/tags?post=246"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}