On the Littlewood-Richardson polynomials
The authors continue their study of semi-invariants of quivers. Let \(Q\) be a quiver, \(\alpha\) a dimension vector, \(\text{GL}(\alpha)\) the corresponding product of general linear groups. Write \(SI(Q,\alpha)\) for the algebra of polynomial invariants of the derived subgroup \(\text{SL}(\alpha)\) of \(\text{GL}(\alpha)\), acting on the space of \(\alpha\)-dimensional representations of \(Q\) over an algebraically closed field. Under the assumption that \(Q\) has no oriented cycles, a spanning set of \(SI(Q,\alpha)\) was given by \textit{H. Derksen} and \textit{J. Weyman} [J. Am. Math. Soc. 13, No. 3, 467-479 (2000; Zbl 0993.16011)]. Here Schofield's double quiver construction is applied, in order to derive from this generators of \(SI(Q,\alpha)\) for arbitrary quivers, recovering this way results of \textit{L. Le Bruyn} and \textit{C. Procesi} [Trans. Am. Math. Soc. 317, No. 2, 585-598 (1990; Zbl 0693.16018)] and \textit{S. Donkin} [Comment. Math. Helv. 69, No. 1, 137-141 (1994; Zbl 0816.16015)] on generating \(\text{GL}(\alpha)\)-invariants, and results of \textit{A. Schofield} and \textit{M. Van den Bergh} [Indag. Math., New Ser. 12, No. 1, 125-138 (2001; Zbl 1004.16012)] and \textit{M. Domokos} and \textit{A. N. Zubkov} [Transform. Groups 6, No. 1, 9-24 (2001; Zbl 0984.16023)] on generating semi-invariants. The second part of the paper concentrates on the subalgebra \(SI(Q,\alpha,\sigma)\) spanned by those relative \(\text{GL}(\alpha)\)-invariants whose weight is a scalar multiple of some fixed weight \(\sigma\) (where \(Q\) has no oriented cycles). This is a graded algebra, and its projective spectrum provides a moduli space for certain representations of \(Q\). The authors prove that for all sufficiently large \(n\), the algebra \(SI(Q,\alpha,\sigma)\) has a homogeneous system of parameters consisting of elements of degree \(n\). The Poincaré series of \(SI(Q,\alpha,\sigma)\) can be written as a rational function \(P(t)/(1-t)^d\), where \(d\) is the Krull dimension of \(SI(Q,\alpha,\sigma)\), and \(P(t)\) has degree smaller than \(d\). As a corollary, for any three partitions \(\lambda,\mu,\nu\), the function assigning to a natural number \(n\) the Littlewood-Richardson coefficient \(c^{n\nu}_{n\lambda,n\mu}\) is a polynomial in \(n\). (Also submitted to MR).
scientific article; zbMATH DE number 1868002