Littlewood-Richardson Rule

The Littlewood-Richardson rule describes how tensor products and restrictions of representations of symmetric groups decompose, providing explicit combinatorial formulas.

TheoremLittlewood-Richardson Rule

For partitions $\lambda, \mu \vdash n$ , the tensor product decomposes as: $V^\lambda \otimes V^\mu \cong \bigoplus_\nu c^\nu_{\lambda\mu} V^\nu$ where the Littlewood-Richardson coefficients $c^\nu_{\lambda\mu}$ count the number of Littlewood-Richardson tableaux from $\lambda$ to $\nu$ with content $\mu$ .

These coefficients also describe:

Skew Schur functions: $s_{\nu/\lambda} = \sum_\mu c^\nu_{\lambda\mu} s_\mu$
Induction: Multiplicities in $\text{Ind}_{S_k \times S_{n-k}}^{S_n}(V^\lambda \boxtimes V^\mu)$

DefinitionLittlewood-Richardson Tableau

A Littlewood-Richardson tableau from $\lambda$ to $\nu$ with content $\mu$ is a skew tableau $T$ of shape $\nu/\lambda$ (the boxes in $\nu$ not in $\lambda$ ) filled with entries from $\mu$ such that:

Semistandard: Rows weakly increase, columns strictly increase
Yamanouchi word: Reading entries from right to left, bottom to top, at each stage the number of $i$ 's is $\geq$ number of $(i+1)$ 's

ExampleComputing $V^{(2,1)} \otimes V^{(2,1)}$ for $S_3$

For $S_3$ , we want $(2,1) \otimes (2,1)$ .

The Littlewood-Richardson rule gives: $V^{(2,1)} \otimes V^{(2,1)} \cong V^{(3,3)} \oplus V^{(4,2)} \oplus V^{(3,2,1)} \oplus V^{(2,2,2)}$

Wait, this doesn't work for $S_3$ since those partitions are for larger $n$ . Let me reconsider.

For $S_3$ : The tensor product $V^{(2,1)} \otimes V^{(2,1)}$ has dimension $2 \times 2 = 4$ .

Using the rule or direct computation: $V^{(2,1)} \otimes V^{(2,1)} \cong V^{(3)} \oplus V^{(2,1)} \oplus V^{(1,1,1)}$ with dimensions $1 + 2 + 1 = 4$ ✓

TheoremPieri's Formula (Special Case)

For the hook partition $(n) = V^{(n)}$ (trivial representation) and any $V^\lambda$ : $V^\lambda \otimes V^{(k)} \cong \bigoplus V^\mu$ where $\mu$ ranges over partitions obtained from $\lambda$ by adding $k$ boxes, no two in the same column.

Similarly, $V^\lambda \otimes V^{(1^k)}$ (sign representation) is obtained by adding $k$ boxes, no two in the same row.

Remark

The Littlewood-Richardson rule is fundamental throughout mathematics:

Algebraic combinatorics: Symmetric functions, Schur polynomials
Algebraic geometry: Intersection theory on Grassmannians (Schubert calculus)
Quantum computing: Quantum marginal problem, entanglement
Geometric complexity theory: Lower bounds via representation theory

Computing $c^\nu_{\lambda\mu}$ is #P-complete in general, yet the rule provides explicit formulas for many special cases and has deep geometric interpretations.