# A particular proof of the Littlewood Richardson rule

Given $\lambda \subseteq \nu$ we define a tableau of shape $\nu\setminus \lambda$ and weight $\mu$ to be a map ${\sf T}: [\nu\setminus\lambda] \rightarrow \{1,\ldots, r\}$ such that $\mu_c=|\{ x \in [\nu\setminus\lambda] : {\sf T}(x)=c \}|$ for $c\geq 1$.

We say that a tableau of shape $\nu\setminus\lambda$ and weight $\mu$ is semistandard if the entries along the rows are weakly increasing and along the columns are strictly increasing. We let ${\rm SStd}(\nu\setminus\lambda,\mu)$ denote the set of all semistandard tableaux of shape $\nu\setminus\lambda$ and weight $\mu$.

Given a tableau weight $\mu$, we define the reverse reading word to be the sequence of integers
obtained by reading the entries of the tableau from top to bottom and right to left along successive rows. We let ${\rm Latt}(\nu\setminus\lambda,\mu)$ denote the set of semistadnard tableaux whose reverse reading word is a lattice permutation. That is, for each positive integer $j$, starting from the first entry of the reverse reading word to any other place in the word, there are at least as many entries equal to $j$ as there are equal to $(j+1)$.

We have that $$|{\rm SStd}(\nu\setminus\lambda,\mu)|= {\rm dim}_{\mathbb{C}}{\rm Hom}_{\mathfrak{S}_r}(M(\mu), S(\nu\setminus\lambda)),$$ where $M(\mu)=1{\uparrow}_{\mathfrak{S}_\mu}^{\mathfrak{S}_r}$ is the Young permutation module and $S(\nu\setminus\lambda)$ is the Skew Specht module.

We have that the number of homomorphisms which factor through the Specht module is given by $$|{\rm Latt}(\nu\setminus\lambda,\mu)|= {\rm dim}_{\mathbb{C}}{\rm Hom}_{\mathfrak{S}_r}(S(\mu), S(\nu\setminus\lambda)),$$

My questions is this: has anyone ever proven the Littlewood--Richardson rule by producing an algorithm for rewriting a single semistandard tableau of shape $\nu\setminus\lambda$ and weight $\mu$ as a pair of tableaux of the form $${\rm SStd}(\tau,\mu) \times {\rm Latt}(\nu\setminus \lambda,\tau)$$ where the tableau in $${\rm SStd}(\tau,\mu)$$ identifies a composition factor $S(\tau)$ of $M(\mu)$ and the tableau in $${\rm Latt}(\nu\setminus \lambda,\tau)$$ identifies a homomorphism from that particular copy of $S(\tau)$ to $S(\nu\setminus\lambda)$.

Of course, numerically this is obvious, (by the Young and LR rules!), but I was wondering if there exists an explicit bijection (as $\tau$ ranges over all partitions more dominant that $\mu$) in the literature.