A conjecture of Mallows and Sloane with the universal denominator of Hilbert series (Q6595251)

The paper under review addresses a conjecture of Mallows and Sloane concerning the universal denominator of Hilbert series for invariant rings of finite linear groups. The authors extend the understanding of the conjecture by introducing constraints on the degrees of homogeneous systems of parameters (h.s.o.p.) of such rings, based on the universal denominator, as defined by Derksen. This work provides criteria for determining when the conjecture holds and presents counterexamples, including a generalization of Stanley's well-known counterexample.\N\NThe authors first recall foundational concepts in invariant theory, such as rings of invariants of finite linear groups, the Hilbert series, and the concept of h.s.o.p., underlining the significance of the Cohen-Macaulay property in non-modular cases. They present Molien's formula and derive explicit expressions for Hilbert series in terms of basic invariants, setting the stage for the conjecture of Mallows and Sloane.\N\NThe core contribution is twofold. First, the authors impose a new constraint on the degrees of h.s.o.p., derived from the universal denominator of the Hilbert series. This result links the structure of invariant rings to the degrees of their generators and explains how Stanley's counterexample violates the conjecture. They further demonstrate that, under these constraints, the conjecture holds for pseudoreflection groups and certain abelian groups.\N\NSecond, the authors provide a detailed analysis of the conjecture for specific cases, including minimal faithful representations of cyclic groups and symmetric groups. They introduce techniques for decomposing abelian group representations into simpler components, allowing for a systematic study of the conjecture's validity. Also, they establish sufficient conditions for the conjecture to hold for representations of cyclic groups, extending the applicability of the universal denominator framework.\N\NThe paper employs some tools from modular invariant theory, including the study of universal denominators, combinatorial methods, and representation-theoretic techniques. The authors derive explicit criteria for the existence of h.s.o.p. with optimal degrees and provide new examples and counterexamples to illustrate their findings. These contributions not only clarify the scope of the conjecture but also offer a pathway for future exploration of related algebraic problems.\N\NOn the whole, the paper presents some advancements in the study of Hilbert series and invariant theory, particularly in the context of finite linear groups.