In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur in many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of finite-dimensional representations of general linear groups, or in the decomposition of certain induced representations in the representation theory of the symmetric group, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric polynomials. Littlewood–Richardson coefficients depend on three partitions, say λ , μ , ν {\displaystyle \lambda ,\mu ,\nu } , of which λ {\displaystyle \lambda } and μ {\displaystyle \mu } describe the Schur functions being multiplied, and ν {\displaystyle \nu } gives the Schur function of which this is the coefficient in the linear combination; in other words they are the coefficients c λ , μ ν {\displaystyle c_{\lambda ,\mu }^{\nu }} such that
s λ s μ = ∑ ν c λ , μ ν s ν . {\displaystyle s_{\lambda }s_{\mu }=\sum _{\nu }c_{\lambda ,\mu }^{\nu }s_{\nu }.}
The Littlewood–Richardson rule states that c λ , μ ν {\displaystyle c_{\lambda ,\mu }^{\nu }} is equal to the number of Littlewood–Richardson tableaux of skew shape ν / λ {\displaystyle \nu /\lambda } and of weight μ {\displaystyle \mu } .
History
The Littlewood–Richardson rule was first stated by D. E. Littlewood and A. R. Richardson (1934, theorem III p. 119) but though they claimed it as a theorem they only proved it in some fairly simple special cases. Robinson (1938) claimed to complete their proof, but his argument had gaps, though it was so obscurely written that these gaps were not noticed for some time, and his argument is reproduced in the book (Littlewood 1950). Some of the gaps were later filled by Macdonald (1995). The first rigorous proofs of the rule were given four decades after it was found, by Schützenberger (1977) and Thomas (1974), after the necessary combinatorial theory was developed by C. Schensted (1961), Schützenberger (1963), and Knuth (1970) in their work on the Robinson–Schensted correspondence. There are now several short proofs of the rule, such as (Gasharov 1998), and (Stembridge 2002) using Bender-Knuth involutions. Littelmann (1994) used the Littelmann path model to generalize the Littlewood–Richardson rule to other semisimple Lie groups. The Littlewood–Richardson rule is notorious for the number of errors that appeared prior to its complete, published proof. Several published attempts to prove it are incomplete, and it is particularly difficult to avoid errors when doing hand calculations with it: even the original example in D. E. Littlewood and A. R. Richardson (1934) contains an error.
Littlewood–Richardson tableaux
A Littlewood–Richardson tableau is a skew semistandard tableau with the additional property that the sequence obtained by concatenating its reversed rows is a lattice word (or lattice permutation), which means that in every initial part of the sequence any number i {\displaystyle i} occurs at least as often as the number i + 1 {\displaystyle i+1} . Another equivalent (though not quite obviously so) characterization is that the tableau itself, and any tableau obtained from it by removing some number of its leftmost columns, has a weakly decreasing weight. Many other combinatorial notions have been found that turn out to be in bijection with Littlewood–Richardson tableaux, and can therefore also be used to define the Littlewood–Richardson coefficients.
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