Free extensions and Lefschetz properties, with an application to rings of relative coinvariants

Abstract

Graded Artinian algebras can be regarded as algebraic analogues of cohomology rings (in even degrees) of compact topological manifolds. In this analogy, a free extension of a base ring with a fibre ring corresponds to a fibre bundle over a manifold. If the manifold is Kähler, then its cohomology ring satisfies the strong Lefschetz property, which means multiplication by a linear form has the largest possible Jordan type. In this paper, we study the behaviour of strong Lefschetz and Jordan type with respect to free extensions, using relative coinvariant rings of finite groups as prototypical models. We show that if V is a vector space and if the subgroup W of the general linear group Gl(V) is a non-modular finite reflection group and is a non-parabolic reflection subgroup, then the relative coinvariant ring cannot have a linear element of strong Lefschetz Jordan type. However, we give examples where these rings , some with non-unimodal Hilbert functions, nevertheless have (non-homogeneous) elements of strong Lefschetz Jordan type. Some of these examples are related to open combinatorial questions proposed and partially solved by G. Almkvist.

COMMUNICATED BY:

• P. Semrl

Keywords:

• Artinian algebra
• coinvariant
• free extension
• Hilbert function
• invariant
• Jordan type
• Lefschetz property
• reflection group
• tensor product

2010 Mathematics Subject Classifications:

• Primary: 13E10
• Secondary: 13A50
• 13D40
• 13H10
• 14C05
• 14L35
• 20F55

Acknowledgements

This paper began with discussions after a talk Chris McDaniel gave at Northeastern University in Spring 2017 and his presentation of Example 3.7. Shujian Chen in a Junior-Senior thesis project supervised by the third author found with Chris further examples of relative covariants having non-unimodular Hilbert functions; Pedro Marques joined. The authors would like to thank Junzo Watanabe, Uwe Nagel, and Alexandra Secealanu for helpful comments/examples, Donald King and Gordana Todorov for their comments, Emre Sen and Ivan Martino for discussions, and Richard Stanley and Stephanie van Willigenburg for their helpful responses to questions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Reproofs for the standard graded case were essentially the same as J. Briançon's, see also [45, Theorem 2.7].

2 In [13], T. Harima and J. Watanabe show the following: assume that , that V,W are A,B modules with symmetric unimodal Hilbert functions; then has the SLP property as module if and only if both V,W are SLP modules.

3 The Hilbert polynomial is sometimes termed ‘Poincaré polynomial’ or ‘Poincaré series’ by topologists or topology-influenced writers [24,46]; but ‘Poincaré series’ has a very different meaning to commutative algebraists, so we use ‘Hilbert polynomial’ in this paper.

4 Indeed if is not unimodal, then there must be indices a<b<c such that , which implies that the the map has rank at most , and thus cannot have full rank.

5 Proposition 2.22 was first established by T. Harima and J. Watanabe in their paper [15,16] for standard grading; they extended is to non-standard grading over in [13, Proposition 6.1] (also [17, Proposition 4.12]) where , but the proof in large enough characteristic p can be shown similarly.

6 This has the same Hilbert functions as the (different) Example 3.7.

7 See Proposition 4.26 in [17]

8 is not to be confused with the invariant coinvariant ring defined as subring of K-invariant coinvariants . See [25] for more on the invariant coinvariant ring.

9 For a generalization to see [9, Example 2.16].

Additional information

Funding

Pedro Macias Marques was partially supported during the work by CIMA – Centro de Investigação em Matemática e Aplicações, Universidade de Évora, project PEst-OE/MAT/UI0117/2014 (Fundação para a Ciência e a Tecnologia).