representations of the symmetric group are reducible over simply transitive subgroups if the characteristic of f is zero this problem has been solved by saxl 16 an important feature of saxl s result is that a groupg as in the problem above is either 2 transitive or fixes a point i e is contained in some n 1 the caseg n 1 can then be settled using the branching rule and induction on the other hand an explicit list of k transitive groups for k 2 is available which can be used to complete the proof in characteristic zero see 16 for more details from now on we assume that p 0 in this case the problem is important for determining maximal subgroups of finite classical groups 1 13 however the situation is nowmore complicated for example to determine the pairs g d as above withg an one needs the mullineux conjecture 7 2 ifg is intransitive then up to a conjugation it is contained in a standard young subgroup of the form n k k the irreducible restrictions from n to n 1 have been described in 14 see also 11 6 in 11 it is also

an equilibrium existence theorem bewley 14 theorem l proved an infinite dimensional equilibrium existence theorem which is a significant extension of the classical finite dimensional theorem of arrow and debreu 11 the assumptions on technology and preferences are natural and applicable in a wide variety of cases the proof is based on a limit argument that makes direct use of the existence of equilibrium in the finite dimensional case this paper establishes the existence of equilibrium under assumptions which are essentially the same as those given by bewley with the additional assumption that the preference orderings of consumers are representable by real valued utility functions this approach is related to the welfare approach of negishi 9 and arrow and hahn 2 chap 51 in the finite dimensional case and simplifies the approach originally adopted by bewley 5 in addition to making clear the role played by each of the assumptions in establishing the existence of equilibrium this approach has the merit of constructing directly a certain real valued function that is maximised at an equilibrium a result that provides a powerful tool in the analysis of qualitative properties of an equilibrium a model of resource allocation in continuous time over an infinite horizon that may be viewed as an application of the model that follows is given in 181

regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

irreducible restrictions of representations of symmetric and alternating groups in small characteristics abstract building on reduction theorems and dimension bounds for symmetric groups obtained in our earlier work we classify the irreducible restrictions of representations of the symmetric and alternating groups to proper subgroups such a classification is known when the characteristic of the ground field is greater than 3 but the small characteristics cases require a substantially more delicate analysis and new ideas our results fit into the aschbacher scott program on maximal subgroups of finite classical groups

representations of the alternating group which are irreducible over subgroups let f be an algebraically closed field of characteristic p geq 0 and a n be the alternating group on n letters the main goal of this paper is to describe the pairs g e where e is an irreducible fa n module and g 3 the case p 0 has been treated by j saxl the problem is important for the classification of maximal subgroups in finite classical groups 2000 mathematical subject classification 20c20 20c30 20b35 20b20

local continuity and asymptotic behaviour of degenerate parabolic systems abstract we study the local holder continuity and the asymptotic behaviour of solution u u 1 u k of the degenerate system u t i m u m 1 u i for m 1 and i 1 k which describes the population densities of k species whose diffusions are determined by their total population density u u 1 u k for the local holder continuity we adopt the intrinsic scaling and iteration arguments of degiorgi moser and dibenedetto under some regularity conditions we also prove that the population density of i th species with the population m i converges in c s to the function m i m b m x t as t where b m is the barenblatt profile of the standard porous medium equation with l 1 mass m m 1 m k as a consequence of asymptotic behaviour it is shown that each density becomes a concave function after a finite time

a proof of the mullineux conjecture a partition of a positive integer n is a sequence 1 2 m 0 of integers such that i n for a positive integer p a partition 1 2 m or its young diagram is called p regular if it does not have p or more equal parts i e if there does not exist t m p 1 with t t 1 t p 1 let f be a field of characteristic p 0 it is well known that irreducible representations of the symmetric group sn over f are naturally parametrized by pregular partitions of n cf for example 9 12 if is such a partition we denote the corresponding irreducible module by d let sgnn be the one dimensional sign representation of sn over f i e sgnn f as a vector space and g f sgn g f for any g sn f f here sgn g is just the sign of the permutation g it is clear that for any irreducible d the tensor product d sgnn is also irreducible the problem usually called the problem of mullineux is to find the p regular partition such that d sgnn d put d sgnn dn in this way a bijection bn on the set pn of p regular partitions of n is defined for each positive integer n and the problem is

representations of the symmetric group which are irreducible over subgroups let f be an algebraically closed field of characteristic p and n be the symmetric group on n letters in this paper we classify all pairs g d where d is an irreducible f n module of dimension greater than 1 and g is a proper subgroup of n such that the restriction d g is irreducible provided p 3

irreducible restrictions of representations of symmetric groups in small characteristics reduction theorems we study irreducible restrictions of modules over symmetric groups to subgroups we get reduction results which substantially restrict the classes of subgroups and modules for which this is possible such results are known when the characteristic of the ground field is greater than 3 but the small characteristics cases require a substantially more delicate analysis and new ideas this work fits into the aschbacher scott program on maximal subgroups of finite classical groups

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

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

representations of the general linear groups which are irreducible over subgroups we classify all triples g v h such that sl n q leq g leq gl n q v is a representation of g of dimension greater than one over an algebraically closed field bbb f of characteristic prime to q and h is a proper subgroup of g such that the restriction v downarrow h is irreducible this problem is a natural part of the aschbacher scott program on maximal subgroups in finite classical groups

smooth solution for the porous medium equation in a bounded domain abstract in this paper we show the short time existence of the smooth solution for the porous medium equations in a smooth bounded domain 0 1 u t u m m 1 with zero boundary condition on the fixed boundary the flux u m is nonzero while the gradient of pressure u m 1 is zero as a consequence the parabolic equation above becomes degenerate the proof is based on a definition of weighted space c s 2 corresponding to the given degeneracy and the schauder estimates in its linearized equation 0 2 w t x n a i j x d i j w in c s 2 for 0 m 1 m 1

systems of partial differential equations in porous medium abstract we investigate systems of degenerate parabolic equations idealizing reactive solute transport in porous media taking advantage of the inherent structure of the system that allows to deduce a scalar generalized porous medium equation for the sum of the solute concentrations we show existence of a unique weak solution to the coupled system and derive regularity estimates we also prove that the system supports solutions propagating with finite speed thus giving rise to free boundaries and interaction of compactly supported initial concentrations of different species

on 2 modular representations of the symmetric groups abstract let g gl n denote the general linear group of invertible n n matrices with entries in an algebraically closed field f of characteristic 2 we prove the existence of certain composition factors in the restriction to gl n k gl k of some polynomial g modules as a consequence we resolve the final case of the problem addressed by jantzen and seitz of when an irreducible modular representation of the symmetric group s n remains irreducible upon restriction to the young subgroup s n k s k

on restrictions of modular spin representations of symmetric and alternating groups let f be an algebraically closed field of characteristic p and h be an almost simple group or a central extension of an almost simple group an important problem in representation theory is to classify the subgroups g of h and fh modules v such that the restriction v g is irreducible for example this problem is a natural part of the program of describing maximal subgroups in finite classical groups in this paper we investigate the case of the problem where h is the schur s double cover n or s n

representations of the alternating group which are irreducible over subgroups ii nsf dms 1161094 dms 0901241 dms 1201374 humboldt foundation simons foundation 204181 simons foundation fellowship 305247

asymptotics of the fast diffusion equation via entropy estimates we consider non negative solutions of the fast diffusion equation ut um with m 0 1 in the euclidean space mathbb r d d 3 and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t for m mc d 2 d or as t approaches the extinction time when m mc for a class of initial data we prove that the solution converges with a polynomial rate to a self similar solution for t large enough if m mc or close enough to the extinction time if m mc such results are new in the range m mc where previous approaches fail in the range mc m 1 we improve on known results

on restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups i 1994 on restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups ii communications in algebra vol 22 no 15 pp 6175 6208