Multiplicity-free products of Schur functions.

*(English)*Zbl 0990.05130The 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 \(\text{SL}(n,C)\).

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 \(\text{SL}(n,C)\).

Reviewer: S.Ole Warnaar (Melbourne)