Cylindric Partitions
Ira
M.
Gessel;
C.
Krattenthaler
429-479
Abstract: A new object is introduced into the theory of partitions that generalizes plane partitions: cylindric partitions. We obtain the generating function for cylindric partitions of a given shape that satisfy certain row bounds as a sum of determinants of $q$-binomial coefficients. In some special cases these determinants can be evaluated. Extending an idea of Burge (J. Combin. Theory Ser. A 63 (1993), 210-222), we count cylindric partitions in two different ways to obtain several known and new summation and transformation formulas for basic hypergeometric series for the affine root system $\widetilde A_{r}$. In particular, we provide new and elementary proofs for two $\widetilde A_{r}$ basic hypergeometric summation formulas of Milne (Discrete Math. 99 (1992), 199-246).
Coherent functors, with application to torsion in the Picard group
David
B.
Jaffe
481-527
Abstract: Let $A$ be a commutative noetherian ring. We investigate a class of functors from $\langle \langle$commutative $A$-algebras$\rangle \rangle$ to $\langle \langle$sets$\rangle \rangle$, which we call coherent. When such a functor $F$ in fact takes its values in $\langle \langle$abelian groups$\rangle \rangle$, we show that there are only finitely many prime numbers $p$ such that ${}_pF(A)$ is infinite, and that none of these primes are invertible in $A$. This (and related statements) yield information about torsion in $\operatorname {Pic} (A)$. For example, if $A$ is of finite type over $\mathbb {Z}$, we prove that the torsion in $\operatorname {Pic} (A)$ is supported at a finite set of primes, and if ${}_p\operatorname {Pic} (A)$ is infinite, then the prime $p$ is not invertible in $A$. These results use the (already known) fact that if such an $A$ is normal, then $\operatorname {Pic} (A)$ is finitely generated. We obtain a parallel result for a reduced scheme $X$ of finite type over $\mathbb {Z}$. We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field.
The generalized Berg theorem and BDF-theorem
Huaxin
Lin
529-545
Abstract: Let $A$ be a separable simple $AF$-algebra with finitely many extreme traces. We give a necessary and sufficient condition for an essentially normal element $x\in M(A)$, i.e., $\pi (x)$ is normal ($\pi : M(A)\to M(A)/A$ is the quotient map), having the form $y+a$ for some normal element $y\in M(A)$ and $a\in A.$ We also show that a normal element $x\in M(A)$ can be quasi-diagonalized if and only if the Fredholm index $ind(\lambda -x)=0$ for all $\lambda \not\in sp(\pi (x)).$ In the case that $A$ is a simple $C^*$-algebra of real rank zero, with stable rank one and with continuous scale, $K_1(A)=0,$ and $K_0(A)$ has countable rank, we show that a normal element $x\in M(A)$ with zero Fredholm index can be written as \begin{equation*}x=\sum _{n=1}^{\infty }\lambda _n(e_n-e_{n-1})+a, \end{equation*} where $\{e_n\}$ is an (increasing) approximate identity for $A$ consisting of projections, $\{\lambda _n\}$ is a bounded sequence of numbers and $a\in A$ with $\|a\|<\epsilon$ for any given $\epsilon >0.$
Jet Cohomology of Isolated Hypersurface Singularities and Spectral Sequences
Xiao
Er
Jian
547-577
Abstract: We study jet cohomology of isolated hypersurface singularities defined by partial differential forms and prove formulas to compute jet cohomology groups by linear algebra.
On Matroids Representable over $GF(3)$ and Other Fields
Geoff
Whittle
579-603
Abstract: The matroids that are representable over $GF(3)$ and some other fields depend on the choice of field. This paper gives matrix characterisations of the classes that arise. These characterisations are analogues of the characterisation of regular matroids as the ones that can be represented over the rationals by a totally-unimodular matrix. Some consequences of the theory are as follows. A matroid is representable over $GF(3)$ and $GF(5)$ if and only if it is representable over $GF(3)$ and the rationals, and this holds if and only if it is representable over $GF(p)$ for all odd primes $p$. A matroid is representable over $GF(3)$ and the complex numbers if and only if it is representable over $GF(3)$ and $GF(7)$. A matroid is representable over $GF(3)$, $GF(4)$ and $GF(5)$ if and only if it is representable over every field except possibly $GF(2)$. If a matroid is representable over $GF(p)$ for all odd primes $p$, then it is representable over the rationals.
Congruences, Trees, and $p$-adic Integers
Wolfgang
M.
Schmidt;
C.
L.
Stewart
605-639
Abstract: Let $f$ be a polynomial in one variable with integer coefficients, and $p$ a prime. A solution of the congruence $f(x) \equiv 0 (\text {mod} \,p)$ may branch out into several solutions modulo $p^{2}$, or it may be extended to just one solution, or it may not extend to any solution. Again, a solution modulo $p^{2}$ may or may not be extendable to solutions modulo $p^{3}$, etc. In this way one obtains the ``solution tree'' $T = T(f)$ of congruences modulo $p^{\lambda }$ for $\lambda = 1,2,\ldots$. We will deal with the following questions: What is the structure of such solution trees? How many ``isomorphism classes'' are there of trees $T(f)$ when $f$ ranges through polynomials of bounded degree and height? We will also give bounds for the number of solutions of congruences $f(x) \equiv 0 (\text {mod} \,p^{\lambda })$ in terms of $p, \lambda$ and the degree of $f$.
Locally conformal Kähler structures in quaternionic geometry
Liviu
Ornea;
Paolo
Piccinni
641-655
Abstract: We consider compact locally conformal quaternion Kähler manifolds $M$. This structure defines on $M$ a canonical foliation, which we assume to have compact leaves. We prove that the local quaternion Kähler metrics are Ricci-flat and allow us to project $M$ over a quaternion Kähler orbifold $N$ with fibers conformally flat 4-dimensional real Hopf manifolds. This fibration was known for the subclass of locally conformal hyperkähler manifolds; in this case we make some observations on the fibers' structure and obtain restrictions on the Betti numbers. In the homogeneous case $N$ is shown to be a manifold and this allows a classification. Examples of locally conformal quaternion Kähler manifolds (some with a global complex structure, some locally conformal hyperkähler) are the Hopf manifolds quotients of $\mathbb H^n-\{0\}$ by the diagonal action of appropriately chosen discrete subgroups of $CO^+(4)$.
Tangential flatness and global rigidity of higher rank lattice actions
Nantian
Qian
657-673
Abstract: We establish the continuous tangential flatness for orientable weakly Cartan actions of higher rank lattices. As a corollary, we obtain the global rigidity of Anosov Cartan actions.
Confluence of cycles for hypergeometric functions on $Z_{2,n+1}$
Yoshishige
Haraoka
675-712
Abstract: The hypergeometric function of general type, which is a generalization of the classical confluent hypergeometric functions, admits an integral representation derived from a character of a linear abelian group. For the hypergeometric function on the space of $2\times (n+1)$ matrices, a basis of cycles for the integral is constructed by a limit process, which is called a process of confluence. The determinant of the period matrix is explicitly evaluated to show the independence of the cycles.
A Lie theoretic Galois theory for the spectral curves of an integrable system. II
Andrew
McDaniel;
Lawrence
Smolinsky
713-746
Abstract: In the study of integrable systems of ODE's arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. Many of these systems are algebraically completely integrable in that they linearize on the Jacobian of a spectral curve. In an earlier paper the authors gave a classification of the spectral curves in terms of the Weyl group and arranged the spectral curves in a hierarchy. This paper examines the Jacobians of the spectral curves, again exploiting the Weyl group action. A hierarchy of Jacobians will give a basis of comparison for flows from various representations. A construction of V. Kanev is generalized and the Jacobians of the spectral curves are analyzed for abelian subvarieties. Prym-Tjurin varieties are studied using the group ring of the Weyl group $W$ and the Hecke algebra of double cosets of a parabolic subgroup of $W.$ For each algebra a subtorus is identified that agrees with Kanev's Prym-Tjurin variety when his is defined. The example of the periodic Toda lattice is pursued.
Symmetric powers of complete modules over a two-dimensional regular local ring
Daniel
Katz;
Vijay
Kodiyalam
747-762
Abstract: Let $(R,m)$ be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free $R$-module $A$, write $A_{n}$ for the $n$th symmetric power of $A$, mod torsion. We study the modules $A_{n}$, $n \geq 1$, when $A$ is complete (i.e., integrally closed). In particular, we show that $B\cdot A = A_{2}$, for any minimal reduction $B \subseteq A$ and that the ring $\oplus _{n \geq 1} A_{n}$ is Cohen-Macaulay.
Enriched $P$-Partitions
John
R.
Stembridge
763-788
Abstract: An (ordinary) $P$-partition is an order-preserving map from a partially ordered set to a chain, with special rules specifying where equal values may occur. Examples include number-theoretic partitions (ordered and unordered, strict or unrestricted), plane partitions, and the semistandardtableaux associated with Schur's $S$-functions. In this paper, we introduce and develop a theory of enriched $P$-partitions; like ordinary $P$-partitions, these are order-preserving maps from posets to chains, but with different rules governing the occurrence of equal values. The principal examples of enriched $P$-partitions given here are the tableaux associated with Schur's $Q$-functions. In a sequel to this paper, further applications related to commutation monoids and reduced words in Coxeter groups will be presented.
Limiting subhessians, limiting subjets and their calculus
Alexander
D.
Ioffe;
Jean-Paul
Penot
789-807
Abstract: We study calculus rules for limiting subjets of order two. These subjets are obtained as limits of sequences of subjets, a subjet of a function $f$ at some point $x$ being the Taylor expansion of a twice differentiable function which minorizes $f$ and coincides with $f$ at $x$. These calculus rules are deduced from approximate (or fuzzy) calculus rules for subjets of order two. In turn, these rules are consequences of delicate results of Crandall-Ishii-Lions. We point out the similarities and the differences with the case of first order limiting subdifferentials.
Stiefel-Whitney classes and the conormal cycle of a singular variety
Joseph
H. G.
Fu;
Clint
McCrory
809-835
Abstract: A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety $X$ is given by means of the conormal cycle of an embedding of $X$ in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. We also show that, for a complex analytic variety, these classes are the mod 2 reductions of the Chern-MacPherson classes.
Congruences between Modular Forms, Cyclic Isogenies of Modular Elliptic Curves, and Integrality of $p$-adic $L$-Functions
Shu-Leung
Tang
837-856
Abstract: Let $\Gamma$ be a congruence subgroup of type $(N_1,N_2)$ and of level $N$. We study congruences between weight 2 normalized newforms $f$ and Eisenstein series $E$ on $\Gamma$ modulo a prime $\gp$ above a rational prime $p$. Assume that $p\nmid 6N$, $E$ is a common eigenfunction for all Hecke operators and $f$ is ordinary at $\gp$. We show that the abelian variety associated to $f$ and the cuspidal subgroup associated to $E$ intersect non-trivially in their $p$-torsion points. Let $A$ be a modular elliptic curve over $\Q$ with good ordinary reduction at $p$. We apply the above result to show that an isogeny of degree divisible by $p$ from the optimal curve $A_1$ in the $\Q$-isogeny class of elliptic curves containing $A$ to $A$ extends to an étale morphism of Néron models over $\Z _p$ if $p>7$. We use this to show that $p$-adic distributions associated to the $p$-adic $L$-functions of $A$ are $\Z _p$-valued.