Littlewood-Richardson coefficients in terms of Specht modules

Question

Littlewood-Richardson coefficients $c_{\nu\mu}^{\lambda}$ (where $\nu$,$\mu$ and $\lambda$ are integer partitions such that $|\nu| + |\mu| = |\lambda|$) are well-known coefficients appearing in various contexts.

In terms of Schur functions, which form a basis of the symmetric functions, the coefficients are the multiplicative constants, meaning that

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

This is the context in which the rule was first stated. As Schur functions are the characters of irreducible representations of $\text{GL}_n$, it means that if we denote $V_{\lambda}$ for the irreducible representation of $\text{GL}_n$ associated to $\lambda$, then:

$$ V_{\nu} \otimes V_{\mu} \cong \displaystyle \bigoplus_{\lambda} (V_{\lambda})^{\oplus c_{\nu\mu}^{\lambda}} $$

The transition between those two contexts is relatively easy.

There is also another context closely related to the two above, which is the representations the symmetric group. Via the Schur-Weyl duality, the above results implies that if $S^{\lambda}$ is the Specht module associated to $\lambda$ (which are the irreducible representations of $\mathfrak{S}_{|\lambda|}$), then

$$ \left( S^{\nu} \otimes S^{\mu} \right) \big\uparrow_{\mathfrak{S}_{|\nu|} \times \mathfrak{S}_{|\mu|}}^{\mathfrak{S}_{|\nu|+|\mu|}} \cong \bigoplus_{\lambda} \left( S^{\lambda} \right)^{\oplus c_{\nu\mu}^{\lambda}} $$

Using Schur's lemma, the latter means that

$$ \text{dim} \left( \text{Hom}_{\mathfrak{S}_{|\nu|+|\mu|}} \left( S^{\lambda}, \left( S^{\nu} \otimes S^{\mu} \right) \big\uparrow_{\mathfrak{S}_{|\nu|} \times \mathfrak{S}_{|\mu|}}^{\mathfrak{S}_{|\nu|+|\mu|}} \right) \right) = c_{\nu\mu}^{\lambda} $$

My question is: is there any construction of the Littlewood-Richardson coefficients which is stated only in terms of the symmetric group and that gives the last equality? For example, Specht modules is the span of certain elements of the symmetric group algebra, called polytabloids. Can we calculate the Littlewood-Richardson coefficients using those?

Answer 1

The answer is Yes. You don't need $GL(n, \mathbb C)$ or symmetric functions to define/understand LR-coefficients. There are numerous references to choose from, which wary how much Young tableau combinatorics you want to see.

1. For more combinatorial explanations, I recommend two similarly titled papers by Kerov and by Zelevinsky. Roughly, you want to take pairs of SYTs of shapes $\mu \circ \nu$, apply the jeu-de-taquin and count the number of times a fixed $A\in$SYT$(\lambda)$ appears. When $A$ is lexicographically smallest, these gives one of the standard combinatorial interpretation for $c^\lambda_{\mu,\nu}$.

2. For a more algebraic interpretation, I recommend a concise paper by Donin which defines LR-coefficients in terms of dimension of certain spaces of homomorphisms between $S_n$-modules related to Zelevinsky's "pictures".

There are many more references, generalizations, etc. Note that in both cases you need to start with the Frobenius reciprocity formula since it's a bit easier to think about decomposition of large reduced $S_n$-modules.

Answer 2

Gordon James gave a fairly simple proof entirely within the representation theory of the symmetric groups:

James, G. D. A characteristic-free approach to the representation theory of $𝔖_n$. J. Algebra 46 (1977), no. 2, 430–450.