Background

Seminormal Basis of Specht modules of $\mathfrak{S}_n$

Let $\lambda$ be a partition of $n$. A $\lambda$-tableau is a bijection $\mathfrak{t}:\lambda \to \{1,2,...,n\}$. We say a tableau, $\mathfrak{t}$, is standard if the entries are increasing along the rows and columns. We let $\mathcal{T}_{\lambda}$ denote the set of standard $\lambda$-tableaux.

Given some tableau $\mathfrak{t}$ and two integers $1\leq i < j\leq n$, we define the axial distance, $a(i,j)$, as follows: if $i$ occurs in row $i_0$ and column $i_1$ and $j$ occurs in row $j_0$ and column $j_1$, then $a(i,j)=(i_0-i_1) -(j_0-j_1)$.

If $\mathfrak{t}$ is a $\lambda$-tableau and $w \in \Sigma_n$ let $w\mathfrak{t}$ be the tableau obtained from $\mathfrak{t}$ by replacing each entry in $\mathfrak{t}$ by its image under $w$. If $\mathfrak{t}$ is a standard $\lambda$-tableau, we set $\mathfrak{t}_{i \leftrightarrow i+1}$ equal to $w \mathfrak{t}$ if this is still a standard $\lambda$-tableau, and 0 otherwise.

For a given partition $\lambda$ of $n$, the Specht module ${\mathbf{S}(\lambda)}$ has a basis given by the set of standard $\lambda$-tableaux. With respect to this basis the generators act as follows \begin{align*} {\rho_{\lambda}}(s_{i,i+1})\mathfrak{t} = \frac{1}{a(i,i+1)} \mathfrak{t} + \left(1 + \frac{1}{a(i,i+1)}\right) \mathfrak{t}_{i \leftrightarrow i+1} \end{align*}

This basis is very compatible with induction and restriction rules (see Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Arun Ram).

The Littlewood--Richardson rule

The LR rule describes the coefficients in the restriction $$\mathbf{S}(\nu)\downarrow_{\mathfrak{S}_{r_1}\times \mathfrak{S}_{r_2}} \cong \oplus c^{\nu}_{\lambda,\mu} \mathbf{S}(\lambda) \boxtimes \mathbf{S}(\mu)$$

There are many formulations of this rule. For example, the Jeu de Taquin version maps standard skew-tableaux of shape $\nu/\lambda$ to those of shape $\mu$. The LR coef, $c^{\nu}_{\lambda, \mu}$ is the cardinality of the fiber $f^{-1}(\mathfrak{t})$ for any $\mu$-tableau $\mathfrak{t}$.

So we have a map from $\nu$-tableaux to $\lambda \times \mu$-tableaux. The fibers give the LR coefficients. However, this map is not a homomorphism of Specht modules.

Question:

Is there a reference for an explicit construction of such a homomorphism? I.e. a formulation of the LR rule which is compatible with the seminormal bases of Specht modules.

These references solve the analogous problem for the general linear groups:

MR2166314 (2006h:20062)  Howe, Roger E. ;  Tan, Eng-Chye ;  Willenbring, Jeb F.
A basis for the GLn tensor product algebra.
Adv. Math.  196  (2005),  no. 2, 531--564.

MR2888167  Howe, Roger ;  Lee, Soo Teck .
Why should the Littlewood-Richardson rule be true?
Bull. Amer. Math. Soc. (N.S.)  49  (2012),  no. 2, 187--236.

MR0955587 (89j:20046)  Tokuyama, Takeshi .
Determinantal method and the Littlewood-Richardson rule.
J. Algebra  117  (1988),  no. 1, 1--18.

with some work, using Schur-Weyl duality, you should be able to solve your problem.

  • 1
    @Bruce: Without making any guess about how easy or hard it is to use Schur-Weyl duality here, I'd just add that access to the three papers varies. The first was posted (in at least some version) on the arXiv at front.math.ucdavis.edu/0407.5468, while the AMS Bulletin articles are all freely available online through the AMS webpage. The third paper is older, with access depending entirely on local libraries. – Jim Humphreys Jan 3 '13 at 23:40
  • The third paper is downloadable from sciencedirect.com/science/article/pii/0021869388902372 . – darij grinberg Jul 14 '13 at 12:04

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.

Not the answer you're looking for? Browse other questions tagged or ask your own question.