counting characters of upper triangular groups abstract working over a finite field of order q let u n q be the group of upper triangular n n matrices with all diagonal entries equal to 1 it is known that all complex irreducible characters of u n q have degrees that are powers of q and that all powers q e occur for 0 e n where the upper bound n is defined by the formulas 2 m m m 1 and 2 m 1 m 2 it has been conjectured that the number of irreducible characters having degree q e is some polynomial in q with integer coefficients depending on n and e in this paper we construct explicit polynomials for the cases where e is one of n n 1 and 1 also a list of polynomials is presented that appear to give the correct character counts for n 9 several related results concerning pattern groups are also included

superinduction for pattern groups abstract it is well known that the representation theory of the finite group of unipotent upper triangular matrices u n over a finite field is a wild problem by instead considering approximately irreducible representations supercharacters one obtains a rich combinatorial theory analogous to that of the symmetric group where we replace partition combinatorics with set partitions this paper studies diaconis isaacs concept of superinduction in pattern groups while superinduction shares many desirable properties with usual induction it no longer takes characters to characters we begin by finding sufficient conditions guaranteeing that superinduction is in fact induction it turns out for two natural embeddings of u m in u n superinduction is induction we conclude with an explicit combinatorial algorithm for computing this induction analogous to the pieri formulas for the symmetric group

character degrees and nilpotence class of finite p groups an approach via pro p groups let s be a finite set of powers of p containing 1 it is known that for some choices of s if p is a finite p group whose set of character degrees is s then the nilpotence class of p is bounded by some integer that depends on s while for some other choices of s such an integer does not exist the sets of the first type are called class bounding sets the problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set s is class bounding if and only if p s in this article we provide a new approach to this problem our main result shows the relevance of certain p adic space groups in this problem with its help we are able to prove some results that provide new class bounding sets we also show that there exist non class bounding sets s such that p s

stability of large amplitude ekman hartmann boundary layers in mhd the case of ill prepared data in this paper we study an incompressible highly rotating fluid submitted to a high magnetic field between two planes with a dirichlet boundary condition we investigate the nonlinear stability of ekman hartmann boundary layers under a spectral assumption for general initial data this means that the data can be chosen as an arbitrary but smooth enough three dimensional divergence free vector field independent of the small parameter

counting conjugacy classes for unitriangular matrices abstract let g n be the subgroup of gl n q consisting of upper unitriangular matrices a longstanding conjecture attributed to g higman states that the number of conjugacy classes of g n is given by a polynomial in q with integer coefficients recent results show that the resolution of this conjecture is linked to the classification of the canonical primitive connected matrices of g n in this paper we calculate the number of canonical primitive connected matrices a of g n satisfying that a the graph associated to a has two pivot lines and there are at most two edges connecting the pivot lines we prove that this number is a polynomial in q and it is connected with the combinatory of catalan numbers

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

supercharacters of unipotent groups defined by involutions abstract we construct supercharacter theories of finite unipotent groups in the orthogonal symplectic and unitary types our method utilizes group actions in a manner analogous to that of diaconis and isaacs in their construction of supercharacters of algebra groups the resulting supercharacter theories agree with those of andre and neto in the case of the unipotent orthogonal and symplectic groups and generalize to a large collection of subgroups in the unitary group case we describe the supercharacters and superclasses in terms of labeled set partitions and calculate the supercharacter table

on polynomial hamiltonian planar vector fields abstract we study the dynamical behaviour of polynomial hamiltonian planar vector fields particularly we analyze the structure of finite and infinite critical points and we obtain the best upper bound of the number of centers and of the number of saddles that a system of this type can exhibit depending on its degree

the exact number of conjugacy classes of the sylow p subgroups of gl n q modulo q 1 13 abstract let g be a finite p group of order p n a well known result of p hall determines the number of conjugacy classes of g r g modulo p 2 1 p 1 namely he proved the existence of a non negative constant k such that r g n p 2 1 p e k p 2 1 p 1 we denote by g n the group of the upper unitriangular matrices over f q the finite field with q p t elements in a vera lopez j m arregi and f j vera lopez on the number of conjugacy classes of the sylow p subgroups of gl n q bull austral math soc 53 1996 431 439 the number r g n is given modulo q 1 5 in this paper we introduce the concept of primitive canonical matrix the knowledge of the number of primitive canonical matrices with connected graph of size less than or equal to n should be sufficient to determine the number of all canonical matrices of size n moreover we give explicitly the polynomial formulas i i n i 0 12 depending only on n and not on q such that r g n i mu i n q 1 i k n q q 1 13 n n

on the u module structure of the unipotent specht modules of finite general linear groups gordon james investigated the specht modules of the symmetric groups for each partition of n there is a specht module defined in terms of the intersection of the kernels of certain homomorphisms following the philosophy that gln q is a q analog to sn gordon james defined the unipotent specht modules over a field f for gln q if the characteristic of f is coprime to q then the dimension of the unipotent specht module is independent of f indeed one expects the representation theory for gln q to translate into that for sn by setting q equal 1 in the sense of the conjecture by richard dipper and gordon james we have the analogous concept of standard basis this conjecture was proved by marco brandt richard dipper gordon james and sinead lyle for the case of a two part partition but the proof is rather combinational hence it seems the method there will not work for an arbitrary partition our new strategy is to investigate the fu module structure of the unipotent specht module the advantage of restriction to fu module is that fu is semisimple since by general assumption the characteristic of the field is coprime to q gordon james definiert spechtmoduln fur symmetrische gruppen fur jede partition von n gibt es einen spechtmodul von sn der als schnitt von kernen gewisser homomorphismen definiert ist in der tat gibt es eine basis von den spechtmoduln die durch standard tableaux indiziert ist diese basis heist standardbasis in der philosophie dass gln q ein q analog zu sn ist definiert gordon james den unipotenten spechtmoduln uber einem korper f fur gln q die dimension von den unipotenten spechtmoduln ist unabhangig von f falls die charakteristik von f koprim zu q ist wir hoffen die darstellungstheorie von gln q zu der von sn ubersetzen zu konnen im sinne der vermutung von richard dipper und gordon james haben wir den analogen begriff der standardbasis marco brandt richard dipper gordon james und sinead lyle zeigen diese vermutung fur den fall dass partition eine 2 teile partition ist da der beweis sehr kombinatorisch ist scheint es unwahrscheinlich dass diese methoden fur beliebige partition funktionieren dies motiviert uns neue methoden zu finden unsere neue strategie ist untersuchung der fu modul struktur von den unipotenten spechtmoduln der vorteil der einschrankung zu fu moduln ist dass fu halbeinfach ist wenn die charakteristik von f koprim zu q ist

characters of the borel and sylow subgroups of classical groups abstract let g be a classical group over the field with q p d elements we prove that if p is an odd prime then the irreducible characters of the sylow p subgroup of g have degrees that are powers of q moreover we prove that if p is odd and g is not the unitary group then the borel subgroup of g is an m group and its character degrees as well as the sizes of the conjugacy classes can be written in the form q r q 1 s 2 k

a supercharacter analogue for normality diaconis and isaacs define in 8 diaconis and isaacs 2008 a supercharacter theory for algebra groups over a finite field by constructing certain unions of conjugacy classes called superclasses and certain reducible characters called supercharacters this work investigates the properties of algebra subgroups h g which are unions of some set of the superclasses of g we call such subgroups supernormal after giving a few useful equivalent formulations of this definition we show that products of supernormal subgroups are supernormal and that all normal pattern subgroups are supernormal we then classify the set of supernormal subgroups of un q the group of unipotent upper triangular matrices over the finite field fq and provide a formula for the number of such subgroups when q is prime following this we give supercharacter analogues for clifford s theorem and mackey s method of little groups specifically we show that a supercharacter restricted to a supernormal subgroup decomposes as a sum of supercharacters with the same degree and multiplicity we then describe how the supercharacters of an algebra group of the form un uh ua where ua is supernormal and a2 0 are parametrized by uh orbits of the supercharacters of ua and the supercharacters of the stabilizer subgroups of these orbits

supercharacter formulas for pattern groups c andre and n yan introduced the idea of a supercharacter theory to give a tractable substitute for character theory in wild groups such as the unipotent uppertriangular group u n f q in this theory superclasses are certain unions of conjugacy classes and supercharacters are a set of characters which are constant on superclasses this paper gives a character formula for a supercharacter evaluated at a superclass for pattern groups and more generally for algebra groups

combinatorial methods of character enumeration for the unitriangular group abstract let ut n q denote the group of unipotent n n upper triangular matrices over a field with q elements the degrees of the complex irreducible characters of ut n q are precisely the integers q e with 0 e n 2 n 1 2 and it has been conjectured that the number of irreducible characters of ut n q with degree q e is a polynomial in q 1 with nonnegative integer coefficients depending on n and e we confirm this conjecture when e 8 and n is arbitrary by a computer calculation in particular we describe an algorithm which allows us to derive explicit bivariate polynomials in n and q giving the number of irreducible characters of ut n q with degree q e when n 2 e and e 8 when divided by q n e 2 and written in terms of the variables n 2 e 1 and q 1 these functions are actually bivariate polynomials with nonnegative integer coefficients suggesting an even stronger conjecture concerning such character counts as an application of these calculations we are able to show that all irreducible characters of ut n q with degree q 8 are kirillov functions we also discuss some related results concerning the problem of counting the irreducible constituents of individual supercharacters of ut n q

classification of global phase portraits and bifurcation diagrams of hamiltonian systems with rational potential in this paper we study the global dynamics of the hamiltonian systems x hy x y y hx x y where the hamiltonian function h has the particular form h x y y2 2 p x q x h x y y2 2 p x q x p x q x r x p x q x r x are polynomials in particular h is the sum of the kinetic and a rational potential energies firstly we provide the normal forms by a suitable symplectic change of variables then the global topological classification of the phase portraits of these systems having canonical forms in the poincare disk in the cases where degree p 0 1 2degree p 0 1 2 and degree q 0 1 2degree q 0 1 2 are studied as a function of the parameters that define each polynomial we use a blow up technique for finite equilibrium points and the poincare compactification for the infinite equilibrium points finally we show some applications

irreducible characters of groups associated with finite nilpotent algebras with involution abstract an algebra group is a group of the form p 1 j where j is a finite dimensional nilpotent associative algebra a theorem of z halasi asserts that in the case where j is defined over a finite field f every irreducible character of p is induced from a linear character of an algebra subgroup of p if j is a nilpotent algebra with involution then naturally defines a group automorphism of p 1 j and we may consider the fixed point subgroup c p assuming that f has odd characteristic p we show that every irreducible character of c p is induced from a linear character of a subgroup of the form c q where q is a invariant algebra subgroup of p as a particular case the result holds for the sylow p subgroups of the finite classical groups of lie type

counting irreducible representations of large degree of the upper triangular groups abstract let u n q be the upper triangular group of degree n over the finite field f q with q elements in this paper we present constructions of large degree complex irreducible representations of u n q where n 7 and then determine the number of irreducible representations of largest second largest and third largest degrees

reduction for characters of finite algebra groups abstract let j be a finite dimensional nilpotent algebra over a finite field f q we formulate a procedure for analysing characters of the group 1 j in particular we study characters of the group u n q of unipotent triangular n n matrices over f q using our procedure we compute the number of irreducible characters of u n q of each degree for n 13 also we explain and generalise a phenomenon concerning the group u 13 2 discovered by isaacs and karagueuzian

supercharacters and superclasses for algebra groups we study certain sums of irreducible characters and compatible unions of conjugacy classes in finite algebra groups these groups generalize the unimodular upper triangular groups over a finite field and the supercharacter theory we develop extends results of carlos andre and ning yan that were originally proved in the upper triangular case this theory sometimes allows explicit computations in situations where it would be impractical to work with the full character table we discuss connections with the kirillov orbit method and with gelfand pairs and we give conditions for a supercharacter or a superclass to be an ordinary irreducible character or conjugacy class respectively we also show that products of supercharacters are positive integer combinations of supercharacters

stability of large ekman boundary layers in rotating fluids the aim of this paper is to investigate the stability of ekman boundary layers for rotating fluids when the ekman number and the rossby number go to zero more precisely we prove that spectral stability implies linear and nonlinear stabilities of approximate solutions in particular we replace the smallness condition obtained with energy methods in 5 by a weaker spectral condition which is sharp

conjugacy classes in maximal parabolic subgroups of general linear groups abstract we compute conjugacy classes in maximal parabolic subgroups of the general linear group this computation proceeds by reducing to a matrix problem such problems involve finding normal forms for matrices under a specified set of row and column operations we solve the relevant matrix problem in small dimensional cases this gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6 in particular this includes every maximal parabolic subgroup of gln k for n

ekman layers of rotating fluids the case of general initial data in this paper we describe the weak solutions of the navier stokes equations with a large coriolis term as the rossby and the ekman numbers go to zero in a special domain with various boundary conditions this work extends the results in the case of well prepared initial data in 19 30 2000 john wiley sons inc

conjugacy classes in unitriangular matrices let gn gn q be the group of the upper unitriangular matrices of size n n over fq the finite field with q pt elements higman has conjectured that for each n the number of conjugacy classes of elements of gn is a polynomial expression in q in this paper we develop the algorithm given in j algebra 177 1995 899 introducing new ideas and theoretical properties which lead us to get the conjugacy vector of gn for n 13 these vectors for n 5 resp for n 6 7 8 were given in j algebra 152 1 1992 1 resp j algebra 177 1995 899 in particular we conclude that for these values the number of conjugacy classes of gn is a polynomial in q with integral coefficients and degree n n 6 12 thus higman s conjecture holds for n 13 on the other hand for each positive integer n we find explicitly a set of q n n 6 12 different conjugacy classes

characters of the sylow p subgroups of the chevalley groups d4 pn abstract let u q be a sylow p subgroup of the chevalley groups d 4 q where q is a power of a prime p we describe a construction of all complex irreducible characters of u q and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters furthermore we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in q 1 with nonnegative integer coefficients

on the irreducible characters of a sylow 2 subgroup of the finite symplectic group in characteristic 2 abstract we show that if p m is a sylow 2 subgroup of the finite symplectic group sp 2 m q where q is a power of 2 then p m has irreducible complex characters of degree 2 t q m m 1 2 where t is any integer satisfying 0 t m 2 and that q m m 1 2 is the largest possible degree of an irreducible complex character of p m we include related results on the degrees of the irreducible characters of a sylow 2 subgroup of the split special orthogonal group so 2 m q when q is a power of 2

irreducible characters of large degree of sylow p subgroups of classical groups abstract if g is a sylow p subgroup of a classical group defined over a finite field of characteristic p 2 and g is the associated lie algebra we show that the irreducible characters of g with sufficiently large degree p f can be parameterized by the orbits of g on the dual space of g of size p 2 f we use this correspondence to count the number of irreducible characters of g of maximum degree

boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems abstract in this paper we study viscous perturbations of quasilinear hyperbolic systems in several dimensions as the viscosity goes to zero the boundary is noncharacteristic for the hyperbolic system we in particular describe the boundary layer which arises near the boundary and give a sufficient condition for the convergence of the solution to the solution of some mixed hyperbolic problem with some nonlinear maximal dissipative boundary conditions a counterexample is given when this condition is not satisfied and the solution blows up as the viscosity goes to 0

representations of mclain groups abstract basic modules of mclain groups m m r are defined and investigated these are possibly infinite dimensional analogues of andre s supercharacters of u n q the ring r needs not be finite or commutative and the field underlying our representations is essentially arbitrary we deal with all characteristics prime or zero on an equal basis the set totally ordered by is allowed to be infinite we show that distinct basic modules are disjoint determine the dimension of the endomorphism algebra of a basic module find when a basic module is irreducible and exhibit a full decomposition of a basic module as direct sum of irreducible submodules including their multiplicities several examples of this decomposition are presented and a criterion for a basic module to be multiplicity free is given in general not every irreducible module of a mclain group is a constituent of a basic module

hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields we provide normal forms and the global phase portraits in the poincare disk for all hamiltonian nilpotent centers of linear plus cubic homogeneous planar polynomial vector fields

phase portraits of separable hamiltonian systems abstract we study a generalization of potential hamiltonian systems h x y y 2 f x with one degree of freedom namely those with hamiltonian functions of type h x y f x g y which will be denoted by x h we present an algorithm to obtain the phase portrait including the behaviour at infinity of x h when f and g are arbitrary polynomials indeed from the graphs of the one variable functions f and g we are able to give the full description on the poincare disk therefore extending the well known method to obtain the phase portrait of potential systems in the finite plane the fact that the phase portraits can be fully described in terms of the two one variable real functions f and g allows as well a complete study of the bifurcation diagrams in complete families of vector fields the algorithm can be applied to study separable hamiltonian systems with one degree of freedom which include a vast amount of examples in physical applications

on products of matrices of a fixed order abstract we study products of matrices of fixed orders we show that if g is an upper triangular matrix finite or infinite over a field of q elements then g can be expressed as a product of at most four triangular matrices whose orders are divisors of q 1 this result can be applied to the general linear and to the vershik kerov group we also present some facts about conjugacy of elements of orders dividing q 1

some algorithms for the calculation of conjugacy classes in the sylow p subgroups of gl n q abstract the aim of this paper is to fix theoretical bases and develop algorithms for the symbolic calculation of the conjugacy classes in the sylow p subgroup of gl n q formed by the upper unitriangular matrices q pt being arbitrary our main method is to go back along a central series whose factor groups correspond to the entries in the upper diagonals according to a suitable order by linearizing the problem we are able to construct representatives of the conjugacy classes for small values of n n 8 we find the conjugacy vector and the number of conjugacy classes as polynomials in q of degree depending on n furthermore for n 7 we observe that the distinction between zero and non zero values in the entries suffices to construct a complete set of representatives of the conjugacy classes

unitriangular actions on quadratic forms and character degrees we determine canonical representatives and generating functions of orbit sizes for sesquilinear and quadratic forms under unitriangular action these results are used to control how far character degrees and class sizes of unipotent subgroups of classical groups are from being powers of the order of the underlying field

conjugacy in groups of upper triangular matrices abstract we show that in the group u n f 2 a sylow 2 subgroup of gl n f 2 there are elements not conjugate to their inverses if n 13 it follows that there are irreducible characters of this group that are not real valued and that a conjecture of kirillov is not correct as stated we give a partial explanation of why these group elements do not exist if n 12 and an example to show that analogous phenomena can occur for odd primes

supercharacters symmetric functions in noncommuting variables and related hopf algebras we identify two seemingly disparate structures supercharacters a useful way of doing fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field and the ring of symmetric functions in noncommuting variables each is a hopf algebra and the two are isomorphic as such this allows developments in each to be transferred the identification suggests a rich class of examples for the emerging field of combinatorial hopf algebras

combinatorics related to higman s conjecture i parallelogramic digraphs and dispositions abstract in this paper we introduce a kind of directed graphs digraphs arranged in shifted rows of different lengths which arise in a natural way related to problems of finding the number of certain families of canonical primitive connected cellular matrices of the p sylow g n of g l n q formed by the upper unitriangular matrices over the finite field with q elements higman s conjecture states that the number of conjugacy classes of g n is a polynomial in q we associate a number which we call the counter to each directed graph which gives additional information about the polynomial structure of the number of conjugacy classes we focus on a family of digraphs which we call parallelogramic digraphs in which we have n rows of length k each one shifted one place to the right with respect to the previous one we give explicit formulas for their counters for n up to 5 we prove also that the counters satisfy recurrence equations for fixed k when we vary n proving thus a fact that was empirically observed by r h harding and a p heinz and proved by p sun for k up to 5 when n 1 this number multiplied by q 1 n k 1 corresponds to the cardinality of the family of canonical cellular n k n k matrices over the field f q with n pivot lines of length k and exactly k 1 links connecting the pilots of the lines we indicate other kinds of digraphs related to higman s conjecture that establish lines of future research on this topic

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

polynomial properties in unitriangular matrices let gn gn q be the group of the upper unitriangular matrices of size n over fq the finite field of q pt elements g higman has conjectured that for each n the number of conjugacy classes of elements of gn is a polynomial expression in q in this paper we prove that the number of conjugacy classes of gn of cardinality qs with s n 3 is a polynomial in q 1 with non negative integral coefficients fs q 1 of degree less than or equal to the integer part of 2s 1 in addition fs q 1 depends only on s and not on n we determine these polynomials arguing with the methods we gave previously 1995 j algebra177 899 925 in fact the coefficients of these polynomials are obtained by certain generating functions

on the characters of the unit group of dn algebras in this paper we examine the structure and the characters of the unit group of some special algebras called dn algebras and we prove that the irreducible characters are induced from linear characters of the unit group of some dn subalgebras this extends a result of b szegedy

belitskii s canonical forms of upper triangular nilpotent matrices under upper triangular similarity abstract all indecomposable canonical forms are determined for upper triangular nilpotent matrices of size less than or equal to 7 under upper triangular similarity via belitskii s algorithm furthermore we show that there exists an indecomposable canonical form of upper triangular nilpotent n n matrix which admits at least n 2 2 parameters for n 8

conjugacy classes in sylow p subgroups of gl n q taking s to consist of lower unitriangular matrices higman notes that for i 3 ic g j

calculating conjugacy classes in sylow p subgroups of finite chevalley groups abstract in s m goodwin on the conjugacy classes in maximal unipotent subgroups of simple algebraic groups transform groups 11 1 2006 51 76 8 the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a sylow p subgroup u q of a finite chevalley group g q valid when q is a power of a good prime for g q in this paper we develop this algorithm and discuss an implementation in the computer algebra language gap using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in u q when g q is of rank at most 6 in these cases we observe that the number of conjugacy classes of u q is given by a polynomial in q with integer coefficients

upper triangular similarity of upper triangular matrices abstract we consider the following equivalence relation in the set of all complex upper triangular n n matrices a and b are called u similar if there exists an invertible upper triangular matrix s such that a s 1 bs if a b are u similar then they must have the same diagonal and the same jordan form it is known that for n 6 there are infinitely many mutually non u similar nilpotent upper triangular matrices with the same jordan form we introduce an appropriate generalization of the jordan block called an irreducible matrix and we prove that each upper triangular matrix is u similar to a generalized direct sum of irreducible blocks where the location and the order of the blocks is fixed and each block is determined uniquely up to u similarity

upper bounds for the number of conjugacy classes of a finite group abstract for a finite group g let k g denote the number of conjugacy classes of g we prove that a simple group of lie type of untwisted rank l over the field of q elements has at most 6 q l conjugacy classes using this estimate we show that for completely reducible subgroups g of gl n q we have k g q 10 n confirming a conjecture of kovacs and robinson for finite groups g with f g a p group we prove that k g cp a where p a is the order of a sylow p subgroup of g and c is a constant for groups with o p g 1 we obtain that k g g p this latter result confirms a conjecture of iranzo navarro and monasor we also improve various earlier results concerning conjugacy classes of permutation groups and linear groups as a by product we show that any finite group g has a soluble subgroup s and a nilpotent subgroup n such that k g s and k g n 3

a supercharacter theory for involutive algebra groups if j is a finite dimensional nilpotent algebra over a finite field k the algebra group p 1 j admits a standard supercharacter theory as defined in 16 if j is endowed with an involution then naturally defines a group automorphism of p 1 j and we may consider the fixed point subgroup cp x p x x 1 assuming that k has odd characteristic p we use the standard supercharacter theory for p to construct a supercharacter theory for cp in particular we obtain a supercharacter theory for the sylow p subgroups of the finite classical groups of lie type and thus extend in a uniform way the construction given by andre and neto in 7 8 for the special case of the symplectic and orthogonal groups

computing in unitriangular matrices over finite fields abstract let g n p be the sylow p subgroup of sl n p formed by the upper unitriangular matrices the aim of this paper is to describe algorithms for the computation of the number of conjugacy classes the conjugacy vector of g n p the character rational or real of the elements of g n p the cardinality of the centralizer of each matrix of g n p the conjugacy vector of the normal subset n corresponding to a pivot disposition and the character inert or ramification of each entry of any matrix of g n p for p 2 by using these algorithms we have proved that kirillov s conjecture every matrix of g n 2 is conjugate to its inverse holds for n 12 but for n 13 there exists a unique pair of inverse conjugacy classes not conjugate a representative pair of these conjugacy classes is given in j algebra 202 1998 704 for n 14 we give the complete list of the canonical matrices of the 22 counterexamples to kirillov s conjecture for n 14 we have proved that a and a 5 are conjugate and for n 25 we have found a matrix a g 25 2 such that a and a 5 are not conjugate in addition for n 32 we have found a matrix a g 32 2 such that a and a 1 are conjugate but a and a 5 are not conjugate so isaacs conjecture every real matrix in g n 2 m is actually rational is not true for p 3 isaacs and kereguezian give in ibid a matrix a g 20 3 such that a and a 4 are not conjugate in this paper for every odd prime p we obtain a matrix a g n p with n 6 p 1 such that a and a 1 p are not conjugate consequently there exist irreducible characters of g 6p 1 p that are not q p valued where p is a primitive p th root of unity besides for p 3 and n 13 we have computed the conjugacy vector of g n 3 and verified that every matrix a g n 3 is conjugate to a 4 thus for all n 13 the character values of g n 3 always lie in q 3

bifurcation diagrams for hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields as a natural continuation of the work done in 7 we provide the bifurcation diagrams for the global phase portraits in the poincare disk of all the hamiltonian linear type centers of linear plus cubic homogeneous planar polynomial vector fields

hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields abstract we provide normal forms and the global phase portraits in the poincare disk for all the hamiltonian linear type centers of linear plus cubic homogeneous planar polynomial vector fields

on the number of conjugacy classes in a finite group in the following g will denote a finite group and we will use the standard notation of the theory of groups we say that g is a d group if g contains a unique conjugacy class of maximal rc subgroups in addition a d group g is said to be a lx group if all epimorphic images of subgroups of g are d groups for example if g is n separable then g is a lx group in particular all rr solvable groups are l groups in this work we get new results relative to the conjugacy classes of a finite group g let n be a normal subgroup of g rt a set of prime numbers r g resp r g the number of conjugacy classes of elements resp n elements of g rc g the set of all different primes dividing igil g the set of all n elements of g and g g n in this paper we analyze the number r g of conjugacy classes of n elements of g through the local analysis of the number r gn of conjugacy classes of n elements of g which intersect the coset gn our aims are threefold to obtain upper and lower bounds of the number r g of conjugacy classes of rc elements of g in terms of the numbers r g n r n and g l where g denotes the derived subgroup of g to get the residue class of r g modulo the best number given in terms of the primes dividing gi and finally to analyze the conjugacy vector d ic g l ic g l of g assuming that g is the disjoint union of the classes cl g i 1 r r g and lcg g1 3 3 ic g l the results obtained are useful both for the calculation of the conjugacy vector of a finite group and for the classification of finite groups according to the number of conjugacy classes see examples 11 13 in section 4 moreover they enable us to obtain the following inequalities

iterative character constructions for algebra groups abstract we construct a family of orthogonal characters of an algebra group which decompose the supercharacters defined by diaconis and isaacs 2008 6 like supercharacters these characters are given by nonnegative integer linear combinations of kirillov functions and are induced from linear supercharacters of certain algebra subgroups we derive a formula for these characters and give a condition for their irreducibility generalizing a theorem of otto 2010 20 we also show that each such character has the same number of kirillov functions and irreducible characters as constituents in proving these results we observe as an application how a recent computation by evseev 2010 7 implies that every irreducible character of the unitriangular group ut n q of unipotent n n upper triangular matrices over a finite field with q elements is a kirillov function if and only if n 12 as a further application we discuss some more general conditions showing that kirillov functions are characters and describe some results related to counting the irreducible constituents of supercharacters

on the characters and commutators of finite algebra groups let a be a finite dimensional algebra over a finite field and let j a be the jacobson radical of a we prove a new commutator identity in the group g 1 j a and using this identity we show that every irreducible character of g is induced from a linear character of some subgroup of the form h 1 u where u is a subalgebra of j a this answers a question of i m isaacs

an analog of nilpotence arising from supercharacter theory abstract the goal of this paper is to generalize several group theoretic concepts such as the center and commutator subgroup central series and ultimately nilpotence to a supercharacter theoretic setting and to use these concepts to show that there can be a strong connection between the structure of a group and the structure of its supercharacter theories we then use these concepts to show that the upper and lower annihilator series of j can be described in terms of certain central series for the algebra group g 1 j defined by s when s is the algebra group supercharacter theory defined by diaconis isaacs

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

irreducible characters of sylow subgroups of symplectic and unitary groups abstract we construct canonical families of irreducible characters of borel and sylow p subgroups of sp 2 n q and u 2 n q 2 where q is a power of a prime p

involutions and characters of upper triangular matrix groups we study the realizability over r of representations of the group u n of upper triangular n x n matrices over f 2 we prove that all the representations of u n are realizable over r if n 12 but that if n 13 u n has representations not realizable over r this theorem is a variation on a result that can be obtained by combining work of j arregi and a vera lopez and of the authors but the proof of the theorem in this paper is much more natural