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Multiplicity-free products of Schur functions. (English) Zbl 0990.05130
The Littlewood–Richardson coefficients are defined as the structure constants for multiplication of Schur functions: $$s_{\mu}s_{\nu}=\sum_{\lambda}c(\lambda;\mu,\nu)s_{\lambda}$$. There are several well-known examples for which the product of two Schur functions is multiplicity-free, i.e., for which $$c(\lambda;\mu,\nu)\leq 1$$. First and foremost there are the Pieri rules which state that $$c(\lambda;\mu,(k))=1$$ if $$\lambda/\mu$$ is a horizontal $$k$$-strip and zero otherwise, and $$c(\lambda;\mu,(1^k))=1$$ if $$\lambda/\mu$$ is a vertical $$k$$-strip and zero otherwise. Another important example is provided by $$c(\lambda;\mu,\nu)\leq 1$$ if $$\mu$$ and $$\nu$$ are partitions of rectangular shape.
In the paper under review the following neat result is proved: $$c(\lambda;\mu,\nu)\leq 1$$ if and only if (i) $$\mu$$ or $$\nu$$ is of the form $$(k)$$ or $$(1^k)$$, (ii) $$\mu$$ or $$\mu'$$ is of the form $$(2^k)$$ and $$\nu$$ is of the form $$(a^m b^n)$$, $$a>b>0$$ (or vice versa), (iii) $$\mu$$ is of the form $$(a^m)$$ and $$\nu$$ or $$\nu'$$ is of the form $$(b^n d)$$ or $$(b d^n)$$, $$b>d>0$$ (or vice versa), and (iv) $$\mu$$ is of the form $$(a^m)$$ and $$\nu$$ is of the form $$(b^n)$$. Here “vice versa” refers to reversing the roles of $$\mu$$ and $$\nu$$, which follows from the obvious symmetry $$c(\lambda;\mu,\nu)= c(\lambda;\nu,\mu)$$. The first statement of the theorem is of course nothing but a weak form of the Pieri rules and the final statement corresponds to the above-mentioned result for rectangular shapes. The occurrence of the conjugates of $$\mu$$ and $$\nu$$ in the above is due to the fact that $$c(\lambda;\mu,\nu)=c(\lambda';\nu',\mu')$$.
An analogous (and more general) result is proved by restricting the partitions $$\lambda,\mu$$ and $$\nu$$ to having at most $$n$$ parts. This latter result classifies the multiplicity-free tensor products of the irreducible representations of $$\text{SL}(n,C)$$.

##### MSC:
 05E05 Symmetric functions and generalizations 20G05 Representation theory for linear algebraic groups 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
##### Keywords:
Schur function; multiplicity-free; tensor product; Pieri rules
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