Rings of invariants for modular representations of elementary abelian <Emphasis Type="Italic">p</Emphasis>-groups

We initiate a study of the rings of invariants of modular representations of elementary abelian p-groups. With a few notable exceptions, the modular representation theory of an elementary abelian p-group is wild. However, for a given dimension, it is possible to parameterise the representations. We describe parameterisations for modular representations of dimension two and of dimension three. We compute the ring of invariants for all two-dimensional representations; these rings are generated by two algebraically independent elements. We compute the ring of invariants of the symmetric square of a two-dimensional representation; these rings are hypersurfaces. We compute the ring of invariants for all three-dimensional representations of rank at most three; these rings are complete intersections with embedding dimension at most five. We conjecture that the ring of invariants for any three-dimensional representation of an elementary abelian p-group is a complete intersection.     

On the coinvariants of modular representations of cyclic groups of prime order

We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we give a reduced Gröbner basis for the Hilbert ideal and the corresponding monomial basis for the coinvariants. We also describe the decomposition of the coinvariants as a module over the group ring. For one family of representations, we are able to describe the coinvariants despite the fact that an explicit generating set for the invariants is not known. In all cases our results confirm the conjecture of Harm Derksen and Gregor Kemper on degree bounds for generators of the Hilbert ideal. As an incidental result, we identify the coefficients of the monomials appearing in the orbit product of a terminal variable for the three-dimensional indecomposable representation.     

S.A.G.B.I. bases for rings of formal modular seminvariants

We use the theory of S.A.G.B.I. bases to construct a generating set for the ring of invariants for the four and five dimensional indecomposable modular representations of a cyclic group of prime order. We observe that for the four dimensional representation the ring of invariants is generated in degrees less than or equal to 2p–3, and for the five dimensional representation the ring of invariants is generated in degrees less than or equal to 2p–2. Received: January 22, 1997     

The relative trace ideal and the depth of modular rings of invariants

We prove that for a modular representation, the depth of the ring of invariants is the sum of the dimension of the fixed point space of the p-Sylow subgroup and the grade of the relative trace ideal. We also determine which of the Dickson invariants lie in the radical of the relative trace ideal and we describe how to use the Dickson invariants to compute the grade of the relative trace ideal.     

Frobenius splitting and invariant theory

We consider varieties over an algebraically closed field k of characteristicp>0. Given a linear representation of a reductive group, we prove that the ring of invariants is F-regular provided the associated projective quotient is Frobenius-split, the twisting sheaves are Cohen-Macaulay (C-M), and a mild technical condition is met. As an example of how this can be used, we show that the ring of invariants (under the adjoint action of SL (3)) ofg copies ofM ₃ is C-M. (HereM ₃ denotes the vector space of 3×3 matrices over k andp>3.) The method of proof involves an induction, and is potentially of wide applicability. As a corollary we obtain that the moduli space of rank 3 and degree 0 bundles on a smooth projective curve of genusg is C-M.     

The Cohen–Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non-Cohen–Macaulay actually imply that no graded separating algebra is Cohen–Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen–Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen–Macaulay graded separating algebra implies the group is generated by bireections. Additionally, we give an example which shows that Cohen–Macaulay separating algebras can occur when the ring of invariants is not Cohen–Macaulay.     

The multiplicity and the Cohen-Macaulayness of extended Rees algebras of equimultiple ideals

Let I be an equimultiple ideal of Noetherian local ring A. This paper gives some multiplicity formulas of the extended Rees algebras T=A[It,t^-1]. In the case A generalized Cohen-Macaulay, we determine when T is Cohen-Macaulay and as an immediate consequence we obtain e.g., some criteria for the Cohen-Macaulayness of Rees algebra R(I) over a Cohen-Macaulay ring in terms of reduction numbers and ideals.     

Cohen-Macaulay types of subgroup lattices of finite abelianp-groups

For a minimal free resolution of a Stanley-Reisner ring constructed from the order complex of a modular lattice. T. Hibi showed that its last Betti number (called the Cohen-Macaulay type) is computed by means of the Möbius function of the given modular lattice. Using this result, we consider the Stanley-Reisner ring of the subgroup lattice of a finite abelianp-group associated with a given partition, and show that its Cohen-Macaulay type is a polynomial inp with integer coefficients.     

On arithmetic Macaulayfication of Noetherian rings

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

     

Explicit and Canonical Dress Induction

We prove a strengthened statement of Dress's induction theorem for the Green ring of modular representations of a finite group using the theory of the multiplicity module of an indecomposable modular representation. Moreover, we construct an integral canonical induction formula for the Green ring for indecomposable representations with normal vertex. Presented by J. Carlson Mathematics Subject Classifications (2000) 20C20, 19A22. Robert Boltje: Research supported by the NSF, DMS-0200592 and 0128969. Burkhard Külshammer: Research supported by the DAAD.     

Doset Hibi rings with an application to invariant theory

We define the concept of a doset Hibi ring and a generalized doset Hibi ring which are subrings of a Hibi ring and are normal a?ne semigroup rings. We apply the theory of (generalized) doset Hibi rings to analyze the rings of absolute orthogonal invariants and absolute special orthogonal invariants and show that these rings are normal and Cohen-Macaulay and has rational singularities if the characteristic of the base field is zero and is F-rational otherwise. We also state criteria of Gorenstein property of these rings.     

LENGTH ONE IDEAL EXTENSIONS AND THEIR ASSOCIATED GRADED RINGS

Let (R. m) be a d-dimensional Cohen-Macaulay local ring. Given m-primary ideals J ? I of R such that I is contained in the integral closure of J and λ(I/J)= I, we compare depth G(J) and depth G(J). For example, if J has reduction number one, JI = I², and μ(J)≤ d + 1, we prove that depth G(I)≥d – 1. If, in addition, μ(I)= d + 1, we show that I has reduction number one, and hence G(I) is Cohen-Macaulay. These results, besides leading to statements comparing depths of associated graded rings along a composition series, make visible the possibility of studying powers of an ideal by using reductions that are not minimal reductions.     

Cohen-Macaulay and Gorenstein property of Rees algebras of non-singular equimultiple prime ideals

We give criteria for the Cohen-Macaulay and Gorenstein property of Rees algebras of height 2 non-singular equimultiple prime ideals in terms of explicite representations of the associated graded rings. As consequences, we show that in general, the Cohen-Macaulay resp. Gorenstein property of such Rees algebras does not imply the Cohen-Macaulay resp. Gorenstein property of the base ring and that these properties depend upon the characteristic. Dedicated to the memrory of Professor Lê Van Thiêm Professor Lê Van Thiêm was the first directorof Hanoi Institute of Mathematics     

Galois ring extensions and localized modular rings of invariants of p-groups

We apply recent results on Galois-ring extensions and trace surjective algebras to analyze dehomogenized modular invariant rings of finite p-groups, as well as related localizations. We describe criteria for the dehomogenized invariant ring to be polynomial or at least regular and we show that for regular affine algebras with possibly non-linear action by a p-group, the singular locus of the invariant ring is contained in the variety of the transfer ideal. If V is the regular module of an arbitrary finite p-group, or V is any faithful representation of a cyclic p-group, we show that there is a suitable invariant linear form, inverting which renders the ring of invariants into a “localized polynomial ring” with dehomogenization being a polynomial ring. This is in surprising contrast to the fact that for a faithful representation of a cyclic group of order larger than p, the ring of invariants itself cannot be a polynomial ring by a result of Serre. Our results here generalize observations made by Richman [R] and by Campbell and Chuai [CCH].     

Classifying thick subcategories of the stable category of Cohen-Macaulay modules

Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher-dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p-group due to Benson, Carlson and Rickard, we consider classifying thick subcategories of the stable category of Cohen-Macaulay modules over a Gorenstein local ring. The main result of this paper yields a complete classification of the thick subcategories of the stable category of Cohen-Macaulay modules over a local hypersurface in terms of specialization-closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus.     

Some algebraic invariants related to mixed product ideals

We compute some algebraic invariants (e.g. depth, Castelnuovo-Mumford regularity) for a special class of monomial ideals, namely the ideals of mixed products. As a consequence, we characterize the Cohen-Macaulay ideals of mixed products. Received: 25 October 2007     

Mayer’s transfer operator and representations of GL<Subscript>2</Subscript>

In this paper we relate Mayer’s transfer operator for the geodesic flow of the modular surface to the representation theory of the semigroup of invertible 2×2-matrices with non-negative entries. It turns out that similarly to the case of the Kac-Baker model (see Hilgert et al., Convex Cones, and Semigroups, Oxford University Press, London, 1989 and Hilgert and Mayer, Commun. Math. Phys. 232:19–58, 2002) from statistical mechanics which is related to Howe’s oscillator semigroup one has to introduce an additional multiplication operator to obtain a self-adjoint Hilbert space operator of trace class with the correct spectrum from the natural operators provided by the representation theory. In the present case the representations naturally live on weighted Bergman spaces, but can also be realized on weighted L ²-spaces. Using the representation theory of Ol’shanskiĭ semigroups the semigroup representations can be analytically extended to the simply connected covering of SL(2,ℝ) where they can be identified as holomorphic discrete series representations. To Karl Heinrich Hofmann on the occasion of his 75th birthday.