# Littlewood Richardson rule and seminormal basis of Specht modules

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.

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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.

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@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