zeta regularized products if k is a sequence of nonzero complex numbers then we define the zeta regularized product of these numbers k l k to be exp z 0 where z s k 0 l k s we assume that z s has analytic continuation to a neighborhood of the origin if k is the sequence of positive eigenvalues of the laplacian on a manifold then the zeta regularized product is known as det the determinant of the laplacian and k k is known as the functional determinant the purpose of this paper is to discuss properties of the determinant and functional determinant for general sequences of complex numbers we discuss asymptotic expansions of the functional determinant as and its relationship to the weierstrass product

good filtrations and strong f regularity of the ring of u p invariants abstract let k be an algebraically closed field of positive characteristic g a reductive group over k and v a finite dimensional g module let p be a parabolic subgroup of g and u p its unipotent radical we prove that if s sym v has a good filtration then s u p is strongly f regular

on relations on minors of generic symmetric matrices let r be a commutative ring with 1 and fix an integer n b 1 suppose x are variables with 1 d i j n we denote by s r x the polynomial ring over r with n n 1 2 variables x assume x x when 1 d i j n then x is the generic n x n symmetric matrix with entries in s for a positive integer p such that 1 dp n let j be the ideal of s defined by all p minors of x the rings s j have been the classical objects of intensive study for instance by the first and the second fundamental theorems 2 111 it is well known that when an orthogonal group acts on a certain polynomial ring in a certain way the ring of invariants is described in the form of s j furthermore it has been proved that when r is cohen macaulay s j is also cohen macaulay and the equality depth j proj dim s j n p 1 n p 2 2 holds 7 in s jbzetiak pragacz and weyman constructed the minimal free resolution of s j for any n and p when r contains the rational number field the minimal free resolution means the resolution such that its entries of boundary maps do not have constant terms but over an arbitrary commutative ring r minimal free resolutions have not been obtained in general in the cases where p 1 and p n we can construct the minimal free resolutions of s j from the koszul complex further when p n 1 minimal free resolutions of s j were explicitly constructed in 3 41 in this article we prove that the relation module of minors of x is generated by degree 0 and 1 relations on minors and explicitly describe those relations moreover by using this result we prove the existence of minimal free resolutions of s j when p n 2 the basic idea is the same as that of 6 but we use plethysm formulas in place of cauchy s formula used in 6 388 0021 8693 89 3 00

new good quasi cyclic ternary and quaternary linear codes let n k d q codes be linear codes of length n dimension k and minimum hamming distance d over gf q the following quasi cyclic codes are constructed in this paper 44 11 20 3 55 11 26 3 66 11 32 3 48 12 21 3 60 12 28 3 56 13 24 3 65 13 29 3 56 14 23 3 60 15 23 3 64 16 25 3 36 9 19 4 90 9 55 4 99 9 61 4 30 10 14 4 50 10 27 4 55 10 30 4 33 11 15 4 44 11 22 4 55 11 29 4 36 12 16 4 48 12 23 4 60 12 31 4 all of these codes have established or exceed the respective lower bounds on the minimum distance given by brouwer

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

optimal quaternary linear codes of dimension five let d sub q n k be the maximum possible minimum hamming distance of a q ary n k d code for given values of n and k it is proved that d sub 4 33 5 22 d sub 4 49 5 34 d sub 4 131 5 96 d sub 4 142 5 104 d sub 4 147 5 108 d sub 4 152 5 112 d sub 4 158 5 116 d sub 4 176 5 spl ges 129 d sub 4 180 5 spl ges 132 d sub 4 190 5 spl ges 140 d sub 4 195 5 144 d sub 4 200 5 148 d sub 4 205 5 152 d sub 4 216 5 160 d sub 4 227 5 168 d sub 4 232 5 172 d sub 4 237 5 176 d sub 4 240 5 178 d sub 4 242 5 180 and d sub 4 247 5 184 a survey of the results of recent work on bounds for quaternary linear codes in dimensions four and five is made and a table with lower and upper bounds for d sub 4 n 5 is presented

consta abelian codes over galois rings we study n length consta abelian codes a generalization of the well known abelian codes and constacyclic codes over galois rings of characteristic p sup a where n and p are coprime a twisted discrete fourier transform dft is used to generalize transform domain results of abelian and constacyclic codes to consta abelian codes further we characterize consta abelian codes invariant under two kinds of monomials whose underlying permutations are effected by i multiplying the coordinates with a unit in the appropriate mixed radix representation of the coordinate positions and ii shifting the coordinates by t positions all the codes studied here belong to the class of quasi twisted codes which are known to contain some good codes we show that the dual of a consta abelian code invariant under the two monomials is also a consta abelian code closed under both monomials

new quasi twisted quaternary linear codes let n k d sub q codes be linear codes of length n dimension k and minimum hamming distance d over gf q in this article 31 new quaternary q 4 codes are constructed which improve the respective lower bounds on minimum distance in brouwer s tables

the genus and a regularization of an s1 equivariant euler class we show that a new multiplicative genus in the sense of hirzebruch can be obtained by generalizing a calculation due to atiyah and witten we introduce this as the genus compute its value for some examples and highlight some of its interesting properties we also indicate a connection with the study of multiple zeta values which gives an algebraic interpretation for our proposed regularization procedure

some ternary and quaternary codes and associated sphere packings tables of good ternary and quaternary codes are presented and they are used in the construction of dense sphere packings results include 1 tables of the best ternary and quaternary constacyclic codes including cyclic codes up to block length 50 2 a class of optimal n 2 codes over gf q 3 the u v w mod 2u v mod u construction a new ternary code construction technique that can be used to construct the ternary reed muller codes and others and 4 tables of linear ternary and quaternary codes obtained by modifying and combining various codes here and in the literature packings are generated in even dimensions up to 100 using these codes and a sphere packing construction in dimensions 36 and 60 new record densities appear to have been achieved

the frobenius morphism in invariant theory abstract let r be the homogeneous coordinate ring of the grassmannian g gr 2 n defined over an algebraically closed field of characteristic p 0 in this paper we give a completely characteristic free description of the decomposition of r considered as a graded r p module into indecomposables frobenius summands as a corollary we obtain a similar decomposition for the frobenius pushforward of the structure sheaf of g and we obtain in particular that this pushforward is almost never a tilting bundle on the other hand we show that r provides a noncommutative resolution for r p when p n 2 generalizing a result known to be true for toric varieties in both the invariant theory and the geometric setting we observe that if the characteristic is not too small the frobenius summands do not depend on the characteristic in a suitable sense in the geometric setting this is an explicit version of a general result by bezrukavnikov and mirkovic on frobenius decompositions for partial flag varieties we are hopeful that it is an instance of a more general p uniformity principle

rings of invariants of 2 2 matrices in positive characteristic abstract we prove that the rings of invariants of 2 2 matrices over an infinite field are cohen macaulay this result generalizes the similar theorem of mehta and ramadas in odd characteristics our approach is more elementary and it uses only some standard facts from the theory of modules with good filtrations and the theory of determinantal rings

good filtrations of symmetric algebras and strong f regularity of invariant subrings in this paper we prove that a linear action of a reductive group on a polynomial ring with good filtrations over a field of characteristic p 0 yields a strongly f regular in particular cohen macaulay invariant subring the strongly f regular property of some known examples of invariant subrings such as the coordinate rings of schubert varieties in grassmannians are recovered a similar result over a field of characteristic zero is also proved

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

hankel determinantal rings have rational singularities abstract hankel determinantal rings i e determinantal rings defined by minors of hankel matrices of indeterminates arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves they may also be viewed as linear specializations of generic determinantal rings we prove that over fields of characteristic zero hankel determinantal rings have rational singularities in the case of positive prime characteristic we prove that they are f pure independent of the characteristic we give a complete description of the divisor class groups of these rings and show that each divisor class group element is the class of a maximal cohen macaulay module

new optimal quaternary linear codes of dimension 5 in this correspondence new optimal quaternary linear codes of dimension 5 are presented which extend previously known results these codes belong to the class of quasi twisted qt codes and have been obtained using a greedy local search algorithm other codes are also given which provide a lower bound on the maximum possible minimum distance the generator polynomials for these codes are tabulated and the minimum distances of known qt codes are given

addendum to on the variation in the cohomology of the symplectic form of the reduced phase space here we have used the notation rlx for the inner product of the vector field x with the form a and ix m x j m for the x component of a in an earlier paper 4 it was shown that the push forward j dm of the liouville measure dm on m under the momentum mapping j is a piecewise polynomial measure on moreover in case x has isolated isolated zeros on m an explicit formula for the integral

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

generating invariant rings of quivers in arbitrary characteristic abstract it is well known that the ring of polynomial invariants of a reductive group is finitely generated however it is difficult to give strong upper bounds on the degrees of the generators especially over fields of positive characteristic in this paper we make use of the theory of good filtrations along with recent results on the null cone to provide polynomial bounds for matrix semi invariants in arbitrary characteristic and consequently for matrix invariants our results generalize to invariants and semi invariants of quivers

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal