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

isomorphic rings of monomial representations abstract let d g and d g be the rings of monomial representations of finite groups g and g of odd order we prove that the isomorphy d g d g implies the isomorphy b g b g of the respective burnside rings moreover we prove that the isomorphy d g d g with finite z groups g and g i e groups which have only cyclic sylow subgroups implies the isomorphy g g in particular a group g of squarefree order is uniquely determined up to isomorphy by the ring d g

harnack estimates and extinction profile for weak solutions of certain singular parabolic equations we establish an intrinsic harnack estimate for nonnegative weak solutions of the singular equation 1 1 below for m in the optimal range n 2 n 1 intrinsic means that due to the singularity the space time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself consequences are sharp supestimates on the solutions and decay rates as t approaches the extinction time analogous results are shown for p laplacian type equations

the diagonalization of cubic matrices this paper is devoted to analysing the problem of the diagonalization of cubic matrices we extend the familiar algebraic approach which is based on the cardano formulae we rewrite the complex roots of the associated resolvent secular equation in terms of transcendental functions and we derive the diagonalizing matrix

fibred permutation sets and the idempotents and units of monomial burnside rings we study the units of monomial burnside rings and the idempotents of monomial burnside algebras introducing a tenduction map we realise the unit group and the torsion unit group as a mackey functor

non negative solutions of the evolution p laplacian equation initial traces and cauchy problem when 1 p 2 from the introduction we continue here the investigation initiated earlier by us trans amer math soc 314 1989 no 1 187 224 concerning the solvability of the cauchy problem and the existence of initial traces for nonnegative weak solutions of the nonlinear evolution equation 1 ut div du p 2 du 0 in st rn 0 t 0 2 both in terms of results and techniques the main difference stems from the fact that unlike solutions in the case p 2 solutions of 1 are not in general locally bounded

a harnack inequality for weighted degenerate parabolic equations abstract in this paper we generalize the recent result of dibenedetto gianazza vespri on the harnack inequality for degenerate parabolic equations to the case of a weighted p laplacian type operator in the spatial part the weight is assumed to belong to the suitable muckenhoupt class

crossed burnside rings and bouc s construction of green functors abstract we use a formula for primitive idempotents of the crossed burnside ring given by f oda and t yoshida and a result from the theory of green functors obtained by s bouc to prove an induction theorem for the drinfel d double for a finite group over the complex field we obtain artin s induction theorem for group algebras as a corollary of the theorem

boundary behavior of the fast diffusion equation the fast diffusion equation avm vt 0 m 1 is a degenerate nonlinear parabolic equation of which the existence of a unique continuous weak solution has been established in this paper we are going to obtain a lipschitz growth rate of the solution at the boundary of nl and estimate that in terms of the various data

non abelian geometric phase for general three dimensional quantum systems adiabatic u 2 geometric phases are studied for arbitrary quantum systems with a three dimensional hilbert space necessary and sufficient conditions for the occurrence of the non abelian geometrical phases are obtained without actually solving the full eigenvalue problem for the instantaneous hamiltonian the parameter space of such systems which has the structure of is explicitly constructed the results of this article are applicable for arbitrary multipole interaction hamiltonians and their linear combinations for spin j 1 systems in particular it is shown that the nuclear quadrupole hamiltonian does actually lead to non abelian geometric phases for j 1 this system being bosonic is time reversal invariant therefore it cannot support abelian adiabatic geometrical phases

stability of non isolated asymptotic profiles for fast diffusion the stability of asymptotic profiles of solutions to the cauchy dirichlet problem for fast diffusion equation fde for short is discussed the main result of the present paper is the stability of any asymptotic profiles of least energy it is noteworthy that this result can cover non isolated profiles e g those for thin annular domain cases the method of proof is based on the ojasiewicz simon inequality which is usually used to prove the convergence of solutions to prescribed limits as well as a uniform extinction estimate for solutions to fde besides local minimizers of an energy functional associated with this issue are characterized furthermore the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the ojasiewicz simon inequality in a different way

cayley klein parameters and evolution of two and three level systems and squeezed states in this paper the time behavior of quantum states ruled by hamiltonians linear in the su 2 su 1 1 and su 3 generators in terms of the cayley klein parameters originally introduced in classical mechanics is analyzed also pointed out is the link between the cayley klein parameters and the wei norman ordering functions exploited in the context of the schrodinger representation

evolution of su 2 and su 1 1 states a further mathematical analysis the problem of the evolution of su 2 and su 1 1 states is analyzed from a unified point of view a generalized rabi matrix for a time dependent su 2 su 1 1 coherence preserving hamiltonian is also derived

parabolic approach to nonlinear elliptic eigenvalue problems we consider the asymptotic profiles of the nonlinear parabolic flows ut um to show the geometric properties of the following elliptic nonlinear eigenvalue problems posed in a strictly convex domain for different values of p in the classical case p 1 we establish that log f is a strictly concave function when 0 p 1 the function is strictly concave while for there is at least a solution such that is strictly convex moreover eventual convexity properties for the parabolic flows are proved

alternative descriptions in quaternionic quantum mechanics we characterize the quasianti hermitian quaternionic operators in quaternionic quantum mechanics by means of their spectra moreover we state a necessary and sufficient condition for a set of quasianti hermitian quaternionic operators to be anti hermitian with respect to a uniquely defined positive scalar product in a infinite dimensional right quaternionic hilbert space according to such results we obtain two alternative descriptions of a quantum optical physical system in the realm of quaternionic quantum mechanics while no alternative can exist in complex quantum mechanics and we discuss some differences between them

some remarks on g functors and the brauer morphism keywords finite group g functor restriction maps transfer maps mackey s formula frobenius reciprocity brauer morphism defect base nilpotent ideal burnside ring idempotent formula induction formula symmetric g functor character ring green ring reference ctg article 1988 003 record created on 2008 12 16 modified on 2017 05 12

wei norman kolokolov dual approach for arbitrary spin s in a time dependent magnetic field a comprehensive theory describing spin dynamics in a time dependent magnetic field is constructed on the basis of a maurer cartan mc form selected following the exponential wei norman ordering technique we demonstrate that the purely diffusive su 2 problem described by a mc form yields the fokker planck equation for the probability density function the fokker planck like operator is observed to correspond to the casimir operator the case of a multichannel coulomb model is considered for illustration

crossed burnside rings i the fundamental theorem abstract let g be a finite group and s a finite g monoid a crossed g set over s is a finite g set equipped with a g map into s called a weight function a crossed burnside ring x g s is the grothendieck ring of the category of crossed g sets with respect to disjoint unions and tensor products in this paper we prove the fundamental theorem of crossed burnside rings and an idempotent formula at characteristic 0

the units of a partial burnside ring relative to the young subgroups of a symmetric group the unit group of a partial burnside ring relative to the young subgroups of the symmetric group sn on n letters is included in the image by the tom dieck homomorphism as a consequence of this fact the alternating character n of sn is expressed explicitly as a z linear combinations of permutation characters associated with finite left sn sets sn y for the young subgroups y

complex positive maps and quaternionic unitary evolution the complex projection of any n dimensional quaternionic unitary dynamics defines a one parameter positive semigroup dynamics we show that the converse is also true i e that any one parameter positive semigroup dynamics of complex density matrices with maximal rank can be obtained as the complex projection of suitable quaternionic unitary dynamics

the crossed burnside rings iii the dress construction for a tambara functor abstract we discuss the dress construction for a tambara functor associated to a finite commutative monoid with action of a finite group

induction formulae for mackey functors with applications to representations of the twisted quantum double of a finite group abstract in the theory of canonical induction formulae for mackey functors boltje 4 demonstrated that the plus constructions together with the mark morphism are useful for the study of canonical versions of induction theorems analogous to those in representation theory of finite groups in this paper we present a short exact sequence for the plus constructions derived from cauchy frobenius lemma and apply it to the proof of boltje s integrality result for canonical induction formulae the methods appearing in boltje s theory combined with the dress construction for mackey functors are applicable to induction theorems on representations of the twisted quantum double of a finite group as a sequel to such a research we describe canonical versions of two induction theorems whose origins are artin s induction theorem and brauer s induction theorem on c characters of a finite group

holder regularity and uniqueness theorem on weak solutions to the degenerate keller segel system abstract in this paper we present local holder estimates for the degenerate keller segel system k s m below in the range of m 1 and q 2 before a blow up of solutions to deal with difficulties caused by the degeneracy of the operator we find uniform estimates depending on sup norm of the density function and modified the energy estimates and intrinsic scales considered in porous medium equation as its application the uniqueness of weak solution to k s m is also showed for the case q max 1 m 2 2 in the class of holder continuous functions by proving l 1 contraction in this class

crossed burnside rings ii the dress construction of a green functor abstract we present some results about the dress construction of green functor associated to a g monoid in particular we see that the crossed burnside rings the representation rings of quantum double and hochschild cohomology rings are constructed by the dress construction of some green functors

global positivity estimates and harnack inequalities for the fast diffusion equation abstract we investigate local and global properties of positive solutions to the fast diffusion equation u t u m in the range d 2 d m 1 corresponding to general nonnegative initial data for the cauchy problem posed in the whole euclidean space r d we prove sharp local positivity estimates weak harnack inequalities and elliptic harnack inequalities we use them to derive sharp global positivity estimates and a global harnack principle for the mixed initial and boundary value problem posed in a bounded domain of r d with homogeneous dirichlet condition we prove weak and elliptic harnack inequalities our work shows that these fast diffusion flows have regularity properties comparable and in some senses better than the linear heat flow

positivity local smoothing and harnack inequalities for very fast diffusion equations abstract we investigate qualitative properties of local solutions u t x 0 to the fast diffusion equation t u u m m with m 1 corresponding to general nonnegative initial data our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form 0 t with r d they combine into forms of new harnack inequalities that are typical of fast diffusion equations such results are new for low m in the so called very fast diffusion range precisely for all m m c d 2 d the boundedness statements are true even for m 0 while the positivity ones cannot be true in that range

lie algebraic methods and solutions of linear partial differential equations in this paper an algebraic method to obtain the solution of linear partial differential equations of the evolution type is discussed the proposed method exploits the lie differential operators and their matrix realization to reduce the equation to an easily solvable generalized matrix form some applications to problems of specific interest are also discussed

multiple burnside rings and brauer induction formulae abstract let g be a finite group and suppose that g is an operator group of a finite group a define s g a h h s g and z 1 h a where s g is the set of subgroups of g and z 1 h a is the set of crossed homomorphisms from h to a we view g as an operator group of the opposite group a of a and make s g a into a left a g set the ring g a is defined to be a commutative ring consisting of all formal z linear combinations of a g orbits in s g a idempotent formulae for q z g a not only imply a generalization of dress induction theorem but bring in the case where z 1 g a is the set of linear c characters of g boltje s explicit formula for brauer s induction theorem and its hyperelementary version

on the isomorphism problem for the ring of monomial representations of a finite group in this paper we are concerned with the problem of finding properties of a finite group g in the ring d g of monomial representations of g we determine the conductors of the primitive idempotents of q zd g where c is a primitive g th root of unity and prove a structure theorem for the torsion units of d g using these results we show that an abelian group g is uniquely determined by the ring d g we state an explicit formula for the primitive idempotents of z p zd g where z p is a localization of z we get further results for nilpotent and p nilpotent groups and we obtain properties of sylow subgroups of g from d g

the cauchy problem for ut δu m when 0 m 1 this paper deals with the cauchy problem for the nonlinear diffusion equation u t u u m 1 0 on 0 x rn u 0 u0 when 0 m 1 fast diffusion case we prove that there exists a global time solution for any locally integrable function u0 hence no growth condition at infinity for u0 is required moreover the solution is shown to be unique in that class behavior at infinity of the solution and l loc regularizing effects are also examined when m max n 2 n 0 1

multiplicative induction and units for the ring of monomial representations abstract let g be a finite group and let a be a finite abelian g group for each subgroup h of g h a denotes the ring of monomial representations of h with coefficients in a which is a generalization of the burnside ring h of h we research the multiplicative induction map h a g a derived from the tensor induction map h g and also research the unit group of g a the results are explained in terms of the first cohomology groups h 1 k a for k g we see that tensor induction for 1 cocycles plays a crucial role in a description of multiplicative induction the unit group of g a is identified as a finitely generated abelian group we especially study the group of torsion units of g a and study the unit group of g as well

group cohomology and gauge equivalence of some twisted quantum doubles the purpose of the present paper is to investigate relationships that exist between certain quasi hopf algebras namely twisted quantum doubles of a finite abelian group and their module categories this is closely related to aspects of group cohomology concerning the relationship of the bar complex to some other complexes defined and studied by eilenberg and maclane mac52 em50a and em50b although our results are purely homological and algebraic in nature the motivation for studying these questions derives mainly from connections with conformal field theory cft and vertex operator algebras dw90 dpr92 ms89 dl93 we will thus review background motivation although it is hardly necessary for an understanding of our main results and then give a more detailed overview of the paper it is not necessary for the reader to appreciate what a vertex operator algebra voa is cf bor86 flm88 fhl93 but only to understand this voas have a representation theory hence a linear module category and it is important to understand the nature of this category it is expected that the module category v mod for a voa v has the structure of a braided monoidal category js93 roughly this means that there is a notion of tensor product of v modules and that the tensor product of three modules is associative by an isomorphism that depends on the particular modules moreover tensor product of modules is commutative again via an isomorphism that depends on the particular modules particularly nice voas called rational have the additional property that v mod is a semisimple category so that v has only a finite number of isomorphism classes of simple modules and every object in v mod is a direct sum of simple objects the simplest voas of all from this perspective are called holomorphic they are rational and have but one simple module namely the adjoint module v itself thus if v is a holomorphic voa then v mod is equivalent as a semi simple braided monoidal category to the category of vector spaces

holder regularity and harnack inequality for degenerate parabolic equations related to caffarelli kohn nirenberg inequalities abstract this paper deals with the following parabolic equations 1 u t div x p u p 2 u f x t u in t 0 t where p 2 0 r n is a regular bounded domain containing the origin and f is a function with suitable hypothesis that we will precise later in order to obtain some weak harnack inequality

holder estimates for singular non local parabolic equations abstract in this paper we establish local holder estimate for non negative solutions of the singular equation m p below for m in the range of exponents n 2 n 2 1 since we have trouble in finding the local energy inequality of v directly we use the fact that the operator can be thought as the normal derivative of some extension v of v to the upper half space caffarelli and silvestre 2007 5 i e v is regarded as boundary value of v the solution of some local extension problem therefore the local holder estimate of v can be obtained by the same regularity of v in addition it enables us to describe the behavior of solution of non local fast diffusion equation near their extinction time

the table of marks the kostka matrix and the character table of the symmetric group abstract in this article we introduce a short proof of t k c where t is the table of marks with respect to the young subgroups k is the kostka matrix and c is the character table of the symmetric group s n as a corollary of the result we could determine the explicit elements of the unit group of the generalized burnside ring with respect to the young subgroups of s n

the representation ring of the quantum double of a finite group abstract we present some results about the representation ring of the quantum double of a finite group over fields of arbitrary characteristic we give a direct sum decomposition of this representation ring into ideals involving green rings of subgroups given characters of such green rings we construct characters of the representation ring of the quantum double and show that all characters arise in this fashion we prove that the characters thereby obtained from brauer characters separate modules of the quantum double up to composition factors

local smoothing effects positivity and harnack inequalities for the fast p laplacian equation we study qualitative and quantitative properties of local weak solutions of the fast p laplacian equation tu pu with 1 p 2 our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of rn 0 t we combine these lower and upper bounds in different forms of intrinsic harnack inequalities which are new in the very fast diffusion range that is when 1 p 2n n 1 the boundedness results may be also extended to the limit case p 1 while the positivity estimates cannot we prove the existence as well as sharp asymptotic estimates for the so called large solutions for any 1 p 2 and point out their main properties we also prove a new local energy inequality for suitable norms of the gradients of the solutions as a consequence we prove that bounded local weak solutions are indeed local strong solutions more precisely tu lloc2

global representation rings abstract in this paper we combine the concepts of burnside ring and character ring in what we call the global representation ring of a finite group we compute all ring homomorphisms from this ring to the complex numbers we compute its spectrum its connected components and determine the primitive idempotents

an induction theorem for the unit groups of burnside rings of 2 groups abstract let g be a 2 group and b g denote the group of units of the burnside ring of g for each subquotient h k of g there is a generalized induction map from b h k to b g defined as the composition of inflation and multiplicative induction maps we prove that the product of generalized induction maps b h k b g is surjective when the product is taken over the set of all subquotients that are isomorphic to the trivial group or a dihedral 2 group of order 2 n with n 4 as an application we give an algebraic proof for a theorem by tornehave the unit group for the burnside ring of a 2 group aarhus universitet preprint series 1983 84 41 may 1984 which states that tom dieck s exponential map from the real representation ring of g to b g is surjective we also give a sufficient condition for the surjectivity of the exponential map from the burnside ring of g to b g

the species and idempotents of the green algebra of a finite group with a cyclic sylow subgroup we study the green algebra a kg generated by the isomorphism classes of indecomposable kg modules where k is a field with chark p 0 and g a finite group with a cyclic sylow p subgroup more precisely we discuss the homomorphisms a kg c called species and state a formula for the idempotents of a kg