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On the Littlewood-Richardson polynomials. (English) Zbl 1018.16012

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 H. Derksen and 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 L. Le Bruyn and C. Procesi [Trans. Am. Math. Soc. 317, No. 2, 585-598 (1990; Zbl 0693.16018)] and S. Donkin [Comment. Math. Helv. 69, No. 1, 137-141 (1994; Zbl 0816.16015)] on generating \(\text{GL}(\alpha)\)-invariants, and results of A. Schofield and M. Van den Bergh [Indag. Math., New Ser. 12, No. 1, 125-138 (2001; Zbl 1004.16012)] and M. Domokos and 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).

MSC:

16G20 Representations of quivers and partially ordered sets
16R30 Trace rings and invariant theory (associative rings and algebras)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14L24 Geometric invariant theory
15A72 Vector and tensor algebra, theory of invariants
20G05 Representation theory for linear algebraic groups
13A50 Actions of groups on commutative rings; invariant theory
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
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References:

[1] Boutot, J.-F., Singularités rationelles et quotients par les groups reductifs, Invent. Math., 88, 65-68 (1987) · Zbl 0619.14029
[2] Donkin, S., Polynomial invariants of representations of quivers, Comment. Math. Helv., 69, 137-141 (1994) · Zbl 0816.16015
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