2
$\begingroup$

I heard someone said( maybe Okonov) that specht module is the intersection of two induced modules, but I do not know why.The details of my question is as follows.

Let $\lambda\vdash n$ be a partiton, the specht module $S^\lambda$ is the irreducible module of $S_n$ corresponding to the partition $\lambda$.

I guess (of course maybe I'm wrong) that $S^\lambda$ can be obtained as follows: define $$R=S_{\{1,\ldots,\lambda_1\}}\times S_{\{\lambda_1+1,\ldots,\lambda_1+\lambda_2\}}\times\cdots$$ be the Young subgroup with respect to $\lambda$, and $$C=S_{\{1,\ldots,\lambda_1'\}}\times S_{\{\lambda_1'+1,\ldots,\lambda_1'+\lambda_2'\}}\times\cdots$$ be the Young subgroup with respect to the conjugate partition $\lambda'$(transpose of $\lambda$).

Assertion: $S^\lambda$ is the intersection of $\text{Ind}_{1_R}^{S_n}$ and $\text{Ind}_{1_c\otimes\epsilon}^{S_n}$. where $\epsilon$ is the 1-dimensional sign representation of $C$.

I know the tabloid - polytabloid approach of constructing specht modules, but I wonder whether this way is equivalent to the construction depicted above.Can anyone explain me the connections? thanks.

$\endgroup$
  • 2
    $\begingroup$ If by "intersection" you mean "a (maximal) module that embeds into both modules" then yes, the Specht module has the claimed property. If I'm not mistaken, the book by James and Kerber contains a proof. $\endgroup$ – Johannes Hahn Jan 16 '15 at 12:19
  • $\begingroup$ I wrote up some notes on this for a class a few years ago. math.lsa.umich.edu/~speyer/594/Specht.pdf (stated in the form that the space of $S_n$-equivariant hom's between the inductions is one dimensional.) $\endgroup$ – David E Speyer Jan 16 '15 at 14:15
  • $\begingroup$ On the other hand, you could mean to embed the inductions into the regular rep and literally intersect them. (So $\mathrm{Ind}_{1_R}^{S_n}$ is the space of functions in $\mathbb{C} S_n$ which transform by $1_R$ for right multiplication by $R$, and likewise for the other induction.) Then this statement is not true; that intersection is usually $0$. (For example, when $\lambda = (2,1)$ if I recall correctly.) $\endgroup$ – David E Speyer Jan 16 '15 at 14:17
  • $\begingroup$ @DavidSpeyer:I find your webpage for course 665 very helpful:math.lsa.umich.edu/~speyer/665.html,but some notes are incomplete. Will they be updated further? $\endgroup$ – zemora Jan 17 '15 at 4:11
  • $\begingroup$ @zemora Sorry, they will not. On the other hand, I plan to rerun the course in Spring 2016 (which, confusingly, is called Winter Term here at Michigan). I'll have the students create notes for that course again, editing the existing notes or writing new ones as appropriate. My tentative plan is to cut web diagrams and Specht modules and spend more time on jdt and the harder parts of crystals this time. $\endgroup$ – David E Speyer Jan 19 '15 at 15:07

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.