a general theory of canonical induction formulae abstract so far there exist three independent constructions of two different canonical versions of brauer s induction theorem for complex characters due to v snaith p symonds and the author canonical in this context means functorial with respect to restrictions along group homomorphisms in this article we axiomatize the situation in which the above canonical induction formulae are constructed mackey functors and related structures arise in this way naturally as a convenient language this approach allows us to construct canonical induction formulae for arbitrary mackey functors in particular we obtain canonical induction formulae for the brauer character ring the group of projective characters the ring of trivial source modules and the ring of linear source modules in most cases it is not difficult to construct such formulae over the rational numbers a much more subtle question is whether the constructed formula comes from a canonical induction formula defined over the integers we give a sufficient condition in the general framework of mackey functors for a canonical induction formula to be integral as an application we show how canonical induction formulae allow the construction of functorial maps on representation rings in terms of functorial maps on subrings as for example the span of linear characters in the case of the canonical brauer induction formula this will be used in a subsequent article in the case of adams operations and chern classes

on a theorem of benson and curtis iwahori conjectured 5 that the hecke algebra over q of a chevalley group g over a finite field f with respect to a bore1 subgroup is isomorphic to the group algebra over q of the weyl group the analogous statement with q replaced by its algebraic closure was proved by tits 3 ex 27 p 561 using a deformation argument in 11 benson and curtis proved using a case by case analysis that iwahori s conjecture is true over q whenever g is simple of type ze e with the help of springer they proved that when g is of type e or e there exists irreducible representations of the hecke algebra whose character involve 4 and hence iwahori s conjecture over q is false in general the main result of this paper is an explicit construction of an isomorphism of the hecke algebra over q q with the group algebra over q q of the weyl group the construction makes essential use of the polynomials p w of 8 at some point in the proof i have to appeal to the interpretation of p l in terms of multiplicities in verma modules 8 i 51 which has recently been established by brylinski and kashiwara 4 and independently by belinson and bernstein

simple modules over green biset functors we present three examples of green biset functors for which their simple modules can be parametrized these are particular cases of a conjecture by serge bouc classifying the simple modules over a green biset functor a that generalizes the classification of simple biset functors we also prove this conjecture under certain hypothesis for a

foncteurs d ensembles munis d une double action resume sommaire 1 introduction 2 notations generalites 3 ensembles munis d une double action 3 1 notations 3 2 formule de mackey 3 3 decomposition d un morphisme 3 4 sous categories 3 4 1 ensembles p q libres 3 4 2 groupes reductifs 4 foncteurs simples 5 une theorie de vortex et de source 6 une seconde theorie de vortex et source 7 le foncteur de burnside 7 1 un homomorphisme deb g dans end g 7 2 les sous foncteurs du foncteur de burnside en caracteristique 0 7 2 1 premier exemple 7 2 2 les constantesmg n 7 2 3 lesb groupes 7 2 4 les sous foncteurs deb 7 2 5 exemples deb groupes 8 exemples de foncteurs simples 8 1 le cas des foncteurs de mackey globaux 8 2 les foncteurssh k 8 3 certains foncteurs simples en caracteristique 0 9 l algebreegg end g egg 9 1 une base naturelle 9 2 multiplication 9 3 semi simplicite 10 le foncteur des representations complexes 10 1 caracteres 10 2 sous foncteurs et decomposition 10 3 semi simplicite

the functor of p permutation modules for abelian groups let k be a field of characteristic p where p is a prime number let pp k g be the grothendieck group of p permutation kg modules where g is a finite group and let cpp k g be pp k g tensored with the field of complex numbers c in this article we find all the composition factors of the biset functor cpp k restricted to the category of abelian groups

on the rational homotopy type of function spaces the main result of this paper is the construction of a minimal model for the function space f x y of continuous functions from a finite type finite dimensional space x to a finite type nilpotent space y in terms of minimal models for x and y for the component containing the constant map 7r f x y 0 q 7r y 0 h x q in positive dimensions when x is formal there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for y and the coproduct of h x q we also give a version of the main result for the space of cross sections of a fibration

diagonal p permutation functors abstract let k be an algebraically closed field of positive characteristic p and let f be an algebraically closed field of characteristic 0 we consider the f linear category f p p k of finite groups in which the set of morphisms from g to h is the f linear extension f t h g of the grothendieck group t h g of p permutation k h k g bimodules with twisted diagonal vertices the f linear functors from f p p k to f mod are called diagonal p permutation functors they form an abelian category f p p k we study in particular the functor f t sending a finite group g to the grothendieck group f t g of p permutation kg modules and show that f t is a semisimple object of f p p k equal to the direct sum of specific simple functors parametrized by isomorphism classes of pairs p s of a finite p group p and a generator s of a p subgroup acting faithfully on p this leads to a precise description of the evaluations of these simple functors in particular we show that the simple functor indexed by the trivial pair 1 1 is isomorphic to the functor sending a finite group g to f k 0 k g where k 0 k g is the grothendieck group of projective kg modules

the primitive idempotents of the p permutation ring let g be a finite group let p be a prime number and let k be a field of characteristic 0 and k be a field of characteristic p both large enough in this note we state explicit formulae for the primitive idempotents of the ring of p permutation kg modules tensored with k

representations of affine hecke algebras and based rings of affine weyl groups it is known that an interesting part of the study of the representation theory of p adic groups can be reduced to the study of the representation theory of affine hecke algebras b v let w s be an extended affine weyl group and hk qo the corresponding hecke algebra over a field k with a nonzero parameter qo k when k is the complex numbers field and qo is not a root of unity a classification of simple representations of hk qo was established in kl2 deligne langlands lusztig classification for affine type a a classification of simple representations of hk qo was obtained in am for any qo and arbitrary sufficiently large k when k is algebraically closed and has positive characteristic the representations of hk qo were studied by vign ras as part of her study of modular representations of p adic groups v in this paper we shall verify a conjecture of lusztig l6 7 a by means of the based ring of an extended affine weyl group theorem 3 3 the conjecture says that if the parameter qo is not a root of the corresponding poincar polynomial then the classification established in kl2 remains valid the restriction is necessary for the classification see remark 3 4 a

a study of a simple p permutation functor abstract in this article we are going to give a description of a simple p permutation functor s 1 c we can prove the existence of an isomorphism between this functor and the functor which sends a group to the image of the cartan homomorphism this fact allows us to link the evaluation of s 1 c with the image of the cartan matrix of comack k g and gives us an explicit formula for the dimension of this evaluation

a remark on the dade group and the burnside group abstract the object of this note is to show that the formula for tensor induction of relative syzygies in the dade group stated in s bouc j reine angew math 523 2000 113 can be viewed as a special case of a functorial homomorphism from the dual b of the burnside group to the subgroup d of the dade group generated by relative syzygies it follows that there exists a short exact sequence of functors 0 r q b d d tors 0 where r q is the functor of rational representations this may be viewed as an improvement from q to z of theorem d of invent math 139 2000 275

correspondence functors and finiteness conditions abstract we investigate the representation theory of finite sets the correspondence functors are the functors from the category of finite sets and correspondences to the category of k modules where k is a commutative ring they have various specific properties which do not hold for other types of functors in particular if k is a field and if f is a correspondence functor then f is finitely generated if and only if the dimension of f x grows exponentially in terms of the cardinality of the finite set x moreover in such a case f has actually finite length also if k is noetherian then any subfunctor of a finitely generated functor is finitely generated

on p permutation equivalences between rickard equivalences and isotypies broue and rickard defined in their landmark papers from 1990 and 1996 the notions of an isotypy and a splendid equivalence between p blocks of finite groups here we define a notion of equivalence which we call a p permutation equivalence that implies an isotypy and is implied by a splendid equivalence moreover we study properties of p permutation equivalences

splendid equivalences derived categories and permutation modules we introduce the notion of a splendid equivalence between blocks of finite groups which is a strengthening of the notion of an equivalence of derived categories such an equivalence is always induced by a two sided tilting complex whose terms are summands of permutation modules a feature that is compatible with the belief that these complexes should be related to the adic cohomology of deligne lusztig varities in the case of reductive groups over finite fields we show that a splendid equivalence has good consequences over and above those that follow from a derived equivalence in particular it gives for principal blocks a structural explanation of brou s concept of an isotypy a compatible family of perfect isometries we give several examples including examples of morita equivalences that are splendid in an unexpectedly non trivial way

automorphisms of spaces with finite fundamental group let x be a finite cw complex with finite fundamental group we show that the group aut x of homotopy classes of self homotopy equivalences of x is commensurable to an arithmetic group if in addition x is an oriented manifold then the subgroup autt x of homotopy classes of tangential homotopy equivalences is commensurable to an arithmetic group moreover if x is a smooth manifold of dimension 5 then the subgroup diff x of aut x the elements of which are represented by diffeomorphisms is also commensurable to an arithmetic group

tensor product of correspondence functors abstract as part of the study of correspondence functors the present paper investigates their tensor product and proves some of its main properties in particular the correspondence functor associated to a finite lattice has the structure of a commutative algebra in the tensor category of all correspondence functors and a converse also holds

a new proof for classification of irreducible modules of a hecke algebra of type an 1 abstract in this paper we give a new proof for the classification of irreducible modules of a hecke algebra of type a n 1 which was obtained by dipper and james in 1986

borel smith functions and the dade group we show that there is an exact sequence of biset functors over p groups 0 cb jb d 0 where cb is the biset functor for the group of borel smith functions b is the dual of the burnside ring functor d is the functor for the subgroup of the dade group generated by relative syzygies and the natural transformation is the transformation recently introduced by the first author in s bouc a remark on the dade group and the burnside group j algebra 279 2004 180 190 we also show that the kernel of mod 2 reduction of is naturally equivalent to the functor b of units of the burnside ring and obtain exact sequences involving the torsion part of d mod 2 reduction of cb and b

mackey functors induction from restriction functors and coinduction from transfer functors abstract boltje s plus constructions extend two well known constructions on mackey functors the fixed point functor and the fixed quotient functor in this paper we show that the plus constructions are induction and coinduction functors of general module theory as an application we construct simple mackey functors from simple restriction functors and simple transfer functors we also give new proofs for the classification theorem for simple mackey functors and semisimplicity theorem of mackey functors

on fibred biset functors with fibres of order prime and four abstract this note has two purposes first to present a counterexample to a conjecture parametrizing the simple modules over green biset functors appearing in an author s previous article this parametrization fails for the monomial burnside ring over a cyclic group of order four second to classify the simple modules for the monomial burnside ring over a group of prime order for which the above mentioned parametrization holds

continuous cohomology and real homotopy type various aspects of homotopy theory in the category of simplicial spaces are developed topics covered include continuous cohomology continuous de rham cohomology the kan extension condition the homotopy relation fibrations the serre spectral sequence real homotopy type and its relation to graded commutative differential algebras over the reals

the ideals of the slice burnside p biset functor abstract let g be a finite group and k be a field of characteristic zero our purpose is to investigate the ideals of the slice burnside functor k it turns out that they are the subfunctors f of k such that for any finite group g the evaluation f g is an ideal of the algebra k g this allows for a determination of the full lattice of ideals of the slice burnside p biset functor k p

maximal subgroups of direct products we determine all maximal subgroups of the direct product g n of n copies of a group g if g is finite we show that the number of maximal subgroups of g n is a quadratic function of n if g is perfect but grows exponentially otherwise we deduce a theorem of wiegold about the growth behaviour of the number of generators of g n

the de rham homotopy theory and differential graded category this paper is a generalization of moriya in j pure appl algebra 214 4 422 439 2010 we develop the de rham homotopy theory of not necessarily nilpotent spaces we use two algebraic objects closed dg categories and equivariant dg algebras we see these two objects correspond in a certain way proposition 3 3 4 theorem 3 4 5 we prove an equivalence between the homotopy category of schematic homotopy types toen in selecta math n s 12 1 39 135 2006 and a homotopy category of closed dg categories theorem 1 0 1 we give a description of homotopy invariants of spaces in terms of minimal models theorem 1 0 2 the minimal model in this context behaves much like the sullivan s minimal model we also provide some examples we prove an equivalence between fiberwise rationalizations bousfield and kan in lecture notes in mathematics vol 304 springer berlin 1972 and closed dg categories with subsidiary data theorem 1 0 4

rational l s category and its applications let s be a 1 connected cw complex of finite type and put cat0 s lusternik schnirelmann category of the localization sq this invariant is char acterized in terms of the minimal model of s it is shown that if is the fibre inclusion of a fibration it is also shown that if dim h s q oo then either dim it s q oo or the groups nk s q grow exponentially with k introduction in this paper topological space means a pointed path connected normal space s1 such that the inclusion of the basepoint is a cofibration and the rational singular homology is finite dimensional in each degree all vector spaces and algebras have q as ground field homology and cohomology of spaces is singular with rational coefficients we shall be dealing with category of spaces cat s in the sense of lusternik and schnirelmann 18 a recent survey is given in 15 but our definition differs by one from that of 15 so that for us spheres have category one not two our purpose is to provide a computationally useful characterization of the rational category cat0 5 of space 5 and to apply this in a variety of situations the definition of cat0 5 is sketched below and given precisely in 4 it satisfies cat0 s cauls when 5 is 1 connected cat0 5 coincides with the category of the localization sq which has been studied by toomer in 26 and 27 and lemaire and sigrist in 17 the characterization problem was posed by berstein 26 most of the applications depend on the following fundamental mapping theorem which we derive as a consequence of our characterization

the functor of rational representations for p groups abstract let k be a field of characteristic p 0 let c p k be the category whose objects are the finite p groups morphisms are the k linear combinations of bisets and composition of morphisms is obtained by k linear extension from the usual product of bisets let f p k denote the category of k linear functors from c p k to the category of k vector spaces this paper investigates the structure of the functor kr q mapping a p group p to kr q p k z r q p as an object of f p k where r q p is the grothendieck group of the category of finite dimensional q p modules the main result is an explicit description of the lattice of all subfunctors of kr q in particular for p odd it is shown that kr q is a uniserial object in f p k for p 2 the lattice of subfunctors of kr q can be described as the lattice of closed subsets of a graph whose vertices are the 2 groups of normal 2 rank 1 in both cases a composition series of kr q is obtained which leads to a formula giving the dimension of the evaluations sq k p of the simple objects sq k of f p k associated to p groups q of normal p rank 1 different from cp this formula can be phrased in terms of rational representations of p but also in terms of the geometry of the lattice of subgroups of p using the notions of basic subgroups and origins for example if p 2 the dimension of s 1 k p is equal to the number of absolutely irreducible q p modules