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Why should the Littlewood-Richardson rule be true? (English) Zbl 1300.20053

The aim of this paper is to give a proof of the Littlewood-Richardson (LR) rule, which gives a description of tensor products of two irreducible (finite-dimensional) polynomial representations of the complex general linear group \(\text{GL}_n(\mathbb C)\), and to show that the two types of conditions used to define LR tableaux are the appropriate ones for the validity of the rule.
The authors first give a summary of the occurrences and applications of the LR coefficients in various fields (representation theory, combinatorics, geometry, sums of Hermitian matrices, extensions of Abelian groups) and define the following mathematical objects necessary to formulate the LR Rule [W. Fulton, Young tableaux. With applications to representation theory and geometry. Lond. Math. Soc. Student Texts 35. Cambridge: Cambridge University Press (1997; Zbl 0878.14034)]:
\(\bullet\) a skew tableau with the shape \(F-D\) and content \(E=\{\mu_1,\mu_2\ldots,\mu_m\}\), where \(F,D\) are Young diagrams with \(D\subset F\) and the entries of this tableau are taken from \(\{1,2,\ldots,m\}\) with \(\mu_j\) of them being \(j\) for \(1\leq j\leq m\);
\(\bullet\) an LR (skew) tableau, i.e., a skew tableau which is semistandard and satisfies the Yamanouchi word condition;
\(\bullet\) an LR coefficient \(c^F_{D,E}\), which is the number of LR tableaux of shape \(F-D\) and content \(E\);
\(\bullet\) an irreducible polynomial representation \(\rho^D_n\) of \(\text{GL}_n(\mathbb C)\), where \(D\) is the Young diagram with at most \(n\) rows which labels this representation.
Then, instead of using the combinatorics of partitions and symmetric functions as in previous proofs, they apply representation theory in a substantial way to prove the LR rule: The multiplicity of \(\rho^F_n\) in the tensor product \(\rho^D_n\otimes\rho^E_n\) is given by the LR coefficient \(c^F_{D,E}\).

MSC:

20G05 Representation theory for linear algebraic groups
05E10 Combinatorial aspects of representation theory
05E15 Combinatorial aspects of groups and algebras (MSC2010)

Citations:

Zbl 0878.14034
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