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Let $s_{\lambda}$ be a Schur function. The set of all such functions are known to be a linear basis of the algebra of symmetric functions.

The Littlewood-Richardson coefficients $c_{\nu\mu}^{\lambda}$ are the constants appearing in the following product:

$$ s_{\nu}s_{\mu} = \displaystyle \sum_{\lambda} c_{\nu\mu}^{\lambda} s_{\lambda} $$

In this article:

http://www.kurims.kyoto-u.ac.jp/EMIS/journals/JACO/Volume4_3/v4251k50n37273n3.fulltext.pdf

Carré and Leclerc shows that $c_{\nu\mu}^{\lambda}$ is the number of domino tableaux of a certain shape $\alpha$ and content $\lambda$ such that its reading word is Yamanouchi. Here, $\alpha$ is the partition such that its 2-quotient is $(\nu,\mu)$ and its 2-core is empty. (All the definitions are in the article).

More generally, let $\lambda$ be a partition, $(\lambda^{(1)},\lambda^{(2)}, ..., \lambda^{(n)})$ be its $n$-quotient and $\lambda_{(n)}$ be its $n$-core. Stanton and White showed that when $\lambda_{(n)}$ is empty, the generating function for $n$-ribbon tableaux (a generalization of domino tableaux, which are the case $n=2$) of shape $\lambda$ correspond to the following product:

$$ s_{\lambda^{(1)}}s_{\lambda^{(2)}}...s_{\lambda^{(n)}} = \displaystyle \sum_{T \text{ a }n-\text{ribbon tableaux of shape }\lambda} x^T = \displaystyle \sum_{\gamma} c_{\lambda^{(1)}\lambda^{(2}...\lambda^{(n)}}^{\gamma} s_{\gamma} $$

Is it true that the coefficients $c_{\lambda^{(1)}\lambda^{(2)}...\lambda^{(n)}}^{\gamma}$ correspond to $n$-ribbon tableaux such that their reading word is Yamanouchi? Carré and Leclerc say that it is true for $n=2$, so this would be my guess, and a few (small) examples seem to confirm this. And if yes, where can I find a proof?

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  • $\begingroup$ For a description of the quotient (with figures and examples): symmetricfunctions.com/… $\endgroup$ Jan 25, 2022 at 7:03
  • $\begingroup$ So, one interpret the domino tableaux as tuples of tableaux (of quotient shape). These, in turn, can be interpreted as being parts of a big skew shape. There are models for the Littlewood-Richardson coefficients as counting the number of SYT's of skew shapes with some condition. The Yamanouchi condition should translate over to such a condition on the skew tableaux... $\endgroup$ Jan 25, 2022 at 7:06

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