The universal form of the Littlewood-Richardson rule
Let F be a free module of rank n over any commutative ring R and let \(L_{\lambda}F\) be the GL(F)-irreducible homogeneous polynomial representations parametrized by the partition \(\lambda =(\lambda_ 1,\lambda_ 2,...)\) of n. Given any two partitions \(\lambda\), \(\mu\) of n, one has \(L_{\lambda}F\otimes L_{\mu}F\cong \sum c(\lambda,\mu;\nu)L_{\nu}F\), \(c(\lambda,\mu;\nu)\in {\mathbb{N}}\), in the case when \(R={\mathbb{C}}\), where the coefficients \(c(\lambda,\mu;\nu)\) can be described by the Littlewood-Richardson rule. In the present paper, the author proves that there exists a universal filtration for \(L_{\lambda}F\otimes L_{\mu}F\) such that its associated graded module is isomorphic to \(\sum c(\lambda,\mu;\nu)L_{\nu}F\). Furthermore, he gives the coefficients \(c(\lambda,\mu;\nu)\) an interpretation in his description of the associated graded module \(\sum L_{\nu}F\otimes C_{\nu}\), where \(C_{\nu}\) is a suitable universally free R-module of \(U^+(\lambda_ 1,R)\)-invariants, \(U^+(\lambda_ 1,R)\) denotes the upper-unitriangular matrices of order \(\lambda_ 1\) with entries in R.
irreducible homogeneous polynomial representations
scientific article; zbMATH DE number 4077556