The Balmer spectrum of rational equivariant cohomology theories

The category of rational G-equivariant cohomology theories for a compact Lie group G is the homotopy category of rational G-spectra and therefore tensor-triangulated. We show that its Balmer spectrum is the set of conjugacy classes of closed subgroups of G, with the topology corresponding to the topological poset of [7]. This is used to classify the collections of subgroups arising as the geometric isotropy of finite G-spectra. The ingredients for this classification are (i) the algebraic model of free spectra of the author and B. Shipley [14], (ii) the Localization Theorem of Borel–Hsiang–Quillen [21] and (iii) tom Dieck's calculation of the rational Burnside ring [4].     

Shephard–Todd–Chevalley Theorem for Skew Polynomial Rings

We prove the following generalization of the classical Shephard–Todd–Chevalley Theorem. Let G be a finite group of graded algebra automorphisms of a skew polynomial ring \(A:=k_{p_{ij}}[x_1,\cdots,x_n]\). Then the fixed subring A ^G has finite global dimension if and only if G is generated by quasi-reflections. In this case the fixed subring A ^G is isomorphic to a skew polynomial ring with possibly different p _ij ’s. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings.     

The linear index of a group representation

This paper introduces a generalization of Polya's cycle index of a permutation representation. Let G denote a finite group of order n, and V a complex module affording a representation of G. For any class function χ of G, we construct a complex polynomial in n variables called the χ-linear index of G, denoted L(G, V; χ). In the case that V arises from a permutation representation. L(G, V; χ) can be converted to the usual cycle index and conversely. Basic properties of the linear index are discussed, including linear indices of sums and products of modules. By fixing a module V, we get the following map from any set of class functions of G to the ring of complex polynomials in n variables: assign to χ the polynomial L(G, V; χ). Consider the ring of complex generalized characters of G. Conditions are given when the mapping for this ring is or is not one-to-one.     

Sur la catégorie Db,G(X) pour G réductif fini

In this Note, we are interested in the G-equivariant derived category of a smooth projective scheme over an algebraically closed field k, on which a reductive finite group G is acting. We compare the G-equivariant derived category of X with the derived category of the quotient by giving a descent criterion. The result generalizes a theorem of Lønsted in G-equivariant K-theory on curves (K. Lønsted, J. Math. Kyoto Univ. 23 (4) (1983) 775–793). We also give an equivariant version of Be??linson's equivalence of categories (Funct. Anal. Appl. 12 (1979) 214–216) and treat the exemple of the projective line. To cite this article: S. Térouanne, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Rational torus-equivariant stable homotopy I: Calculating groups of stable maps

We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show that it is of finite injective dimension. It can be used as a model for rational G-spectra in the sense that there is a homology theory

     

Dunkl Operators for Complex Reflection Groups

Dunkl operators for complex reflection groups are defined inthis paper. These commuting operators give rise to a parameterizedfamily of deformations of the polynomial De Rham complex. Thisleads to the study of the polynomial ring as a module over the‘rational Cherednik algebra’, and a natural contravariantform on this module. In the case of the imprimitive complexreflection groups G(m, p, N), the set of singular parametersin the parameterized family of these structures is describedexplicitly, using the theory of non-symmetric Jack polynomials.2000 Mathematical Subject Classification: 20F55 (primary), 52C35,05E05, 33C08 (secondary).     

Structure theorems over polynomial rings

Given a polynomial ring R over a field k and a finite group G, we consider a finitely generated graded RG-module S. We regard S as a kG-module and show that various conditions on S are equivalent, such as only containing finitely many isomorphism classes of indecomposable summands or satisfying a structure theorem in the sense of [D. Karagueuzian, P. Symonds, The module structure of a group action on a polynomial ring: A finiteness theorem, preprint, http://www.ma.umist.ac.uk/pas/preprints].     

Fusion Procedure for Wreath Products of Finite Groups by the Symmetric Group

Let G be a finite group. A complete system of pairwise orthogonal idempotents is constructed for the wreath product of G by the symmetric group by means of a fusion procedure, that is by consecutive evaluations of a rational function with values in the group ring. This complete system of idempotents is indexed by standard Young multi-tableaux. Associated to the wreath product of G by the symmetric group, a Baxterized form for the Artin generators of the symmetric group is defined and appears in the rational function used in the fusion procedure.     

Constructing Witt-Burnside rings

The Witt-Burnside ring of a profinite group G over a commutative ring A generalizes both the Burnside ring of virtual G-sets and the rings of universal and p-typical Witt vectors over A. The Witt-Burnside ring of G over the monoid ring Z[M], where M is a commutative monoid, is proved isomorphic to the Grothendieck ring of a category whose objects are almost finite G-sets equipped with a map to M that is constant on G-orbits. In particular, if A is a commutative ring and A_× denotes the set A as a monoid under multiplication, then the Witt-Burnside ring of G over Z[A_×] is isomorphic to Graham's ring of “virtual G-strings with coefficients in A.” This result forms the basis for a new construction of Witt-Burnside rings and provides an important missing link between the constructions of Dress and Siebeneicher [Adv. in Math. 70 (1988) 87-132] and Graham [Adv. in Math. 99 (1993) 248-263]. With this approach the usual truncation, Frobenius, Verschiebung, and Teichmüller maps readily generalize to maps between Witt-Burnside rings.     

Polynomial invariants for a semisimple and cosemisimple Hopf algebra of finite dimension

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. The coefficients of polynomial invariants are integers if is a finite Galois extension of Q, and A is a scalar extension of some finite-dimensional semisimple Hopf algebra over Q. Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of A, and recognize the difference between the representation category and the representation ring of A. Actually, by computing and comparing polynomial invariants, we find new examples of pairs of Hopf algebras whose representation rings are isomorphic, but whose representation categories are distinct.     

A Representation Stability Theorem for VI-modules

Let VI be the category whose objects are the finite dimensional vector spaces over a finite field of order q and whose morphisms are the injective linear maps. A VI-module over a ring is a functor from the category VI to the category of modules over the ring. A VI-module gives rise to a sequence of representations of the finite general linear groups. We prove that the sequence obtained from any finitely generated VI-module over an algebraically closed field of characteristic zero is representation stable - in particular, the multiplicities which appear in the irreducible decompositions eventually stabilize. We deduce as a consequence that the dimension of the representations in the sequence {V _n } obtained from a finitely generated VI-module V over a field of characteristic zero is eventually a polynomial in q ⁿ . Our results are analogs of corresponding results on representation stability and polynomial growth of dimension for FI-modules (which give rise to sequences of representations of the symmetric groups) proved by Church, Ellenberg, and Farb.     

Remarks on Hilbert series of graded modules over polynomial rings

In this article we discuss a result on formal Laurent series and some of its implications for Hilbert series of finitely generated graded modules over standard-graded polynomial rings: For any integer Laurent function of polynomial type with non-negative values the associated formal Laurent series can be written as a sum of rational functions of the form ${\frac{Q_j(t)}{(1-t)^j}}$ , where the numerators are Laurent polynomials with non–negative integer coefficients. Hence any such series is the Hilbert series of some finitely generated graded module over a suitable polynomial ring ${\mathbb{F}[X_1 , \ldots , X_n]}$ . We give two further applications, namely an investigation of the maximal depth of a module with a given Hilbert series and a characterization of Laurent polynomials which may occur as numerator in the presentation of a Hilbert series as a rational function with a power of (1 ? t) as denominator.     

System-theoretic and algebraic aspects of the rings of stable and proper stable rational functions

A new approach in transfer-function methods for solving a variety of control-theoretic problems is to work with fractional representations over the ring of stable or proper stable rational functions of various transfer matrices. The rings of stable and proper stable rational functions are well known to be Euclidean domains. These rings differ critically from the polynomial ring by the nonuniqueness of the remainders obtained. This major difficulty in extending the idea of polynomial models of Fuhrmann to the rings of stable and causal stable rational functions is circumvented by choosing a remainder in a special form. A natural realization theory is thus developed for matrix fraction representations of transfer-function matrices over these rings. As an application of the new theory developed, linear matrix equations (QX+RY=T and QX+YR=T) over the rings of stable and causal stable rational functions are reduced to finite sets of linear equations over the base field.     

On the string topology category of compact Lie groups

We answer a question of Blumberg, Cohen and Teleman, showing that the Chas–Sullivan loop homology is the Hochschild cohomology of any object in the rational string topology category of a compact, simply connected, Lie group G. Moreover, we show that the answer follows from the classification of the localizing subcategories of the derived category of chains on the based loops of G, which we achieve using the stratification machinery of Benson, Iyengar and Krause. For integral coefficients we get similar results for G a simply-connected special unitary group.     

The transvection free groups with polynomial rings of invariants

Let G be a finite linear group containing no transvections. This paper proves that the ring of invariants of G is polynomial if and only if the pointwise stabilizer in G of any subspace is generated by pseudoreflections. Kemper and Malle used the classification of finite irreducible groups generated by pseudoreflections to prove the irreducible case in arbitrary characteristic. We extend their result to the reducible case.     

Morphic groups

A group G is called morphic if every endomorphism α:G→G for which Gα is normal in G satisfies G/Gα≅ker(α). This concept originated in a 1976 paper of Gertrude Ehrlich characterizing when the endomorphism ring of a module is unit regular. The concept has been extensively studied in module and ring theory, and this paper investigates the idea in the category of groups. After developing their basic properties, we characterize the morphic groups among the dihedral groups and the groups whose normal subgroups form a finite chain. We investigate when a direct product of morphic groups is again morphic, prove that a finite nilpotent group is morphic if and only if its Sylow subgroups are morphic, and present some results for the case where a p-group is morphic.     

The complexity of cover graph recognition for some varieties of finite lattices

We address the following decision problem: Instance: an undirected graphG. Problem: IsG a cover graph of a lattice? We prove that this problem is NP-complete. This extends results of Brightwell [5] and Ne?et?il and Rödl [12]. On the other hand, it follows from Alvarez theorem [2] that recognizing cover graphs of modular or distributive lattices is in P. An important tool in the proof of the first result is the following statement which may be of independent interest: Given an integerl, l?3, there exists an algorithm which for a graphG withn vertices yields, in time polynomial inn, a graphH with the number of vertices polynomial inn, and satisfying girth(H)?l and χ(H)=χ(G).