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Recently, Stanley [21] has defined a symmetric function generalization of the chromatic polynomial of a graph. Independently, Chung and Graham [4] have defined a digraph polynomial called the cover polynomial which is closely related to the chromatic polynomial of a graph (in fact, as we shall see, the cover polynomial of a certain digraph associated to a poset P coincides with the chromatic polynomial of the incomparability graph of P) and also to rook polynomials. The starting point for the present paper is the suggestion in Chung and Graham's paper that the cover polynomial might generalize to a symmetric function in much the same way that the chromatic polynomial does. This is indeed the case, and in this paper we shall study this symmetric function generalization of the cover polynomial. As one would expect, there are a number of generalizations and analogues of known results about the cover polynomial, the rook polynomial, and Stanley's chromatic symmetric function. In addition, however, we will obtain some unexpected dividends, such as a combinatorial reciprocity theorem that answers a question of Chung and Graham and ties together a number of known results that previously seemed unrelated ([2, Theorem 7.3], [19, Chapter 7, Theorem 2], [21, Theorem 4.2], [23, Theorem 3.2]), and a new symmetric function basis that appears to be a natural counterpart of the polynomial basis ( x+n d )n=0, ..., d . One reason for these unexpected results is that this topic lies at the intersection of several branches of combinatorics; their interaction naturally gives rise to new connections and ideas. We shall indicate several directions for further research in this potentially very rich area of study.

This is the first in a series of articles on singular perturbation series in quantum mechanics. In these papers we will compute transition amplitudes non-perturbatively for several simple quantum mechanical systems. This paper treats the behavior of a harmonic oscillator as a function of the spring constant. All the systems to be considered exhibit the same phenomenon: there is, in general, no power series expansion, even of an asymptotic nature, in powers of the perturbation parameter. For reasonable initial and final states there are, however, expansions of a more complicated type. In the case of the harmonic oscillator there is a log-power series expansion. This example is sufficiently simple and explicit that one can see how the logarithmic terms arise mathematically. How they arise physically is rather more mysterious. One point however is clear; the standard techniques of perturbation theory, whether expressed in terms of Dyson series, Feynman diagrams, successive approximation, or path integrals, are all structured so as to produce only power series and thus fail to detect logarithmic terms.

In this paper we consider the Birman–Wenzl algebras over an arbitrary field and prove that they are cellular in the sense of Graham and Lehrer. Furthermore, we determine for which parameters the Birman–Wenzl algebras are quasi-hereditary. So the general theory of cellular algebras and quasi-hereditary algebras applies to Birman–Wenzl algebras. As a consequence, we can determine all irreducible representations of the Birman–Wenzl algebras by linear algebra methods. We prove also that the new Hecke algebras induced from Birman–Wenzl algebras are Frobenius over a field (but not always cellular).

In this note, we show explicitly how to obtain the structure of a Lie bialgebra on the Virasoro algebra (with or without a central extension), on the Witt algebra, and on many other Lie algebras. Previously, V. G. Drinfel′d (in a fundamental paper (1983, Soviet Math. Dokl . 27, No. 1, 68-71)), introduced the notion of what later (1987, in "Proceedings, International Congress of Mathematicians, 1986, Berkeley, CA," pp. 798-820, Amer. Math. Soc., Providence, RI) came to be known as a triangular, coboundary Lie bialgebra G associated to a solution r ∈ G ⊗ G of the Classical Yang-Baxter Equation; and together with A. A. Belavin he showed (1982, Functional Anal. Appl . 16, No. 3, 159-180) that all invertible solutions r of that equation could be obtained as inverses of 2-cocycles. E. Witten (1988, Comm. Math. Phys . 114, 1-53) considered the case of such invertible r for diff( S 1 ), the Lie algebra of smooth vector fields on the circle. E. Beggs and S. Majid (1990, Ann. Inst. H. Poincare Phys. Theor . 53, No. 1, 15 34) considered the invertible case further (in a topological framework) and found solutions for the complexification of diff( S 1 ) but not for the real Lie algebra diff( S 1 ). In this note, we exhibit a Lie bialgebra structure corresponding to a non-invertible solution r . The method by which we obtain such a structure works for both the real and the complex case of diff( S 1 ), the Virasoro algebra (with or without a central extension), the Witt algebra, and many other Lie algebras. We present an explicit expression for the diagonal in all these cases. In fact, we show, more generally, how to obtain the structure of a triangular, coboundary Lie bialgebra on any Lie algebra containing linearly independent elements a and b satisfying [ a , b ] = k · b for some non-zero k ∈ K . Depending on the particular Lie algebra under consideration, the flexibility provided by allowing k to vary enables us to obtain different Lie bialgebra structures (some locally finite, others not) on the same underlying Lie algebra.

A combinatorial structure is said to be asymmetric when its automorphism group reduces to the identity. Asymmetric structures may be enumerated with asymmetry indicator series. As with classical cycle indicator series, which count unlabelled structures or types, formal power series operations such as addition, multiplication. and substitution of asymmetry indicator series correspond to combinatorial operations performed directly on the appropriate combinatorial structures. In this paper, the coefficients of asymmetry indicator series are computed directly by applying Mobius inversion techniques on lattices related to the group action. Substitution is shown to be well behaved in this context. In addition, the connections between asymmetry indicator series and cycle indicator series are explained.

This paper introduces and analyzes a class of directionally reinforced random walks. The work is motivated by an elementary model for time and space correlations in ocean surface wave fields. We develop some basic properties of these walks. For instance, we investigate recurrence properties and give conditions under which the limiting continuous versions of the walks are Gaussian diffusion processes.

The weight lattice of a crystallographic root system is partially ordered by the rule that λ > μ if λ − μ is a nonnegative integer linear combination of positive roots. In this paper, we study the subposet formed by the dominant weights. In particular, we prove that λ covers μ in this partial order only if λ − μ belongs to a distinguished subset of the positive roots. Also, if the root system is irreducible, we prove that the Mobius function of the partial order takes on only the values {0, ±1, ±2}.

Barbasch and Vogan gave a beautiful rule for restricting and inducing Kazhdan–Lusztig representations of Weyl groups. In this paper we show that this rule implies and generalizes the Littlewood–Richardson rule for decomposing outer products of representations of the symmetric groups. A new recursive rule for computing characters of arbitrary Coxeter groups follows. Another application is a generalization of Garsia–Remmel's algorithm for decomposing certain tensor products of symmetric groups representations.

Let R denote a commutative (and associative) ring with 1 and let A denote a finitely generated commutative R -algebra. Let G denote a finite group of R -algebra automorphisms of A . In the case that R is a field of characteristic 0, Noether constructed a finite set of R -algebra generators of the invariants of G . This paper proves that the same construction produces a set of generators of the invariants of G when | G |! is invertible in R . Generators of the invariants of G are also explicitly described in the case that G is solvable and | G | is invertible in R .

/quad A regular polytope is locally toroidal if its minimal sections which are not of spherical type are toroids. The locally toroidal polytopes of rank 4 have been extensively discussed, and their classification is now nearly complete. In this paper, the locally toroidal polytopes of higher rank are investigated. Again, an almost complete classification of these regular polytopes is obtained; as well as sporadic examples, there are several infinite families.

In this paper we show that for a wide class of totally unrectifiable 1-sets in the plane (or even a Hilbert space) satisfying a mild measure-theoretic flatness condition almost everywhere, at sufficiently small scales, the lower spherical density is bounded above by 1 2 at almost every point, thereby affirming Besicovitch's conjecture, which states that for all totally unrectifiable 1-sets in the plane (or possibly even in R n , or a Hilbert space), the lower spherical density is bounded above by 1 2 at almost every point.

Let R be a commutative ring, V a finitely generated free R-module and G⩽GLR(V) a finite group acting naturally on the graded symmetric algebra A=Sym(V). Let β(AG) denote the minimal number m, such that the ring AG of invariants can be generated by finitely many elements of degree at most m. Furthermore, let H◁G be a normal subgroup such that the index |G : H| is invertible in R. In this paper we prove the inequality β(A G )⩽β(A H )·|G : H|. For H=1 and |G| invertible in R we obtain Noether's bound β(AG)⩽|G|, which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order, |G|!, is invertible in R.

Let F denote a field of characteristic different from two. In this paper we describe the mod2 cohomology of a Galois group GF (called the W-group of F) which is known to essentially characterize the Witt ring WF of anisotropic quadratic modules over F. We show that H*(GF, F2) contains the mod2 Galois cohomology of F and that its structure will reflect important properties of the field. We construct a space XF endowed with an action of an elementary abelian group E such that the computation of the cohomology of GF reduces to calculating the equivariant cohomology H*E(XF, F2). For the case of a field which is not formally real this amounts to computing the cohomology of an explicit Euclidean space form, an object which is interesting in its own right. We provide a number of examples and a substantial combinatorial computation for the cohomology of the universal W-groups.

In this paper we construct a tensor (or monoidal) category for any two-sided cell in a finite or affine Weyl group. The isomorphism classes of simple objects of this category correspond to the elements of the two-sided cell; the tensor product arises by a certain truncation procedure from the standard convolution of perverse sheaves on a flag manifold, which imitates geometrically the definition of the ring J given in G. Lusztig ( J. Algebra 109 , 1987, 536–548). Similarly, we construct, for any left cell, a tensor category in which the isomorphism classes of simple objects correspond to the elements in the left cell intersected with its inverse (a right cell). This contains as a special case the category of equivariant perverse sheaves on an affine grassmannian, which, as a consequence of G. Lusztig ( Asterisque 101 – 102 (1983), 208–209) is a tensor category for convolution (without truncation), closely related to the tensor category of representations of the group dual, in the sense of Langlands, to the group which gives rise to the affine grassmannian. But our construction can be specialized in other ways, and it gives a (conjectural) realization of the tensor category of representations of the reductive quotient of the centralizer of any unipotent element of the Langlands dual group.

Strong exact Borel subalgebras and strong Δ -subalgebras are shown to exist for quasi-hereditary algebras which possess exact Borel subalgebras and Δ -subalgebras. This implies that the algebras associated with blocks of category O have strong exact Borel subalgebras and strong Δ -subalgebras. The structure of these subalgebras is shown to be closely related to abstract Kazhdan–Lusztig theory. The main technical tool in this paper is a construction which has an exact Borel subalgebras (of a given quasi-hereditary algebra) as input and a strong exact Borel subalgebra as output. From this result, Morita invariance of the existence of exact Borel subalgebras is derived.

We introduce the concept of a bounded below set in a lattice. This can be used to give a generalization of Rota's broken circuit theorem to any finite lattice. We then show how this result can be used to compute and combinatorially explain the M\"obius function in various examples including non-crossing set partitions, shuffle posets, and integer partitions in dominance order. Next we present a generalization of Stanley's theorem that the characteristic polynomial of a semimodular supersolvable lattice factors over the integers. We also give some applications of this second main theorem, including the Tamari lattices., Comment: 29 pages, 6 figures, Latex, see related papers at http://www.math.msu.edu/~sagan

In this paper we derive a q -analog of Holman′s U ( n ) generalization of the Pfaff-Saalschutz or balanced 3 F 2 summation theorem. This provides a multivariable generalization of Jackson′s classical q -analog of the balanced 3 F 2 summation theorem. We obtain our main result from our new general q -difference equations for "balanced" basic hypergeometric series in U ( n ) and "both" q -analog of Holman′s U ( n ) generalization of the Vandermonde summation theorem. Even the classical case of our proof appears to be new. The q -analog of U ( n ) Pfaff-Saalschutz is invariant under inversion of the base q and also multivariable reversal of series. Corresponding theorems for ordinary series are deduced by taking the limit as q → 1 of all these results.

In this paper we present a self-contained and detailed exposition of the new renormalization group technique proposed in [1, 2]. Its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator, the partition function, or correlation functions. We apply this renormalization transformation to a Hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field, and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing.

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a k-open subgroup of the fixed point group of θ, and Gk (resp. Hk) the set of k-rational points of G (resp. H). The variety Gk/Hk is called a symmetric k-variety. These varieties occur in many problems in representation theory, geometry, and singularity theory. Over the last few decades the representation theory of these varieties has been extensively studied for k=R and C. As most of the work in these two cases was completed, the study of the representation theory over other fields, like local fields and finite fields, began. The representations of a homogeneous space usually depend heavily on the fine structure of the homogeneous space, like the restricted root systems with Weyl groups, etc. Thus it is essential to study first this structure and the related geometry. In this paper we give a characterization of the isomorphy classes of these symmetric k-varieties together with their fine structure of restricted root systems and also a classification of this fine structure for the real numbers, p-adic numbers, finite fields and number fields.

The study of the action of the Steenrod algebra on the mod $p$ cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of Steenrod operations on the cohomology of Grassmannians, both in the Borel and the Schubert picture. We consider integral lifts of Steenrod operations, which lie in a certain Hopf algebra of differential operators. The latter has been considered recently as a realization of the Landweber-Novikov algebra in complex cobordism theory; it also has connections with the action of the Virasoro algebra on the boson Fock space. Our formulas for Steenrod operations are based on combinatorial methods which have not been used before in this area, namely Hammond operators and the combinatorics of Schur functions. We also discuss several applications of our formulas to the geometry of Grassmannians.

Very little is known about the existence of curves, and families of curves, whose Jacobians are acted on by large rings of endomorphisms. In this paper, we show the existence of curves X with an injection K↪Hom(Jac(X), Jac(X))⊗ZQ, where K is a subfield of even index at most 10 in a primitive cyclotomic field Q(ζp), or a subfield of index 2 in Q(ζpq) or Q(ζpα). This result generalizes previous work of Brumer, Mestre, and Tautz, Top, and Verberkmoes. Our curves are constructed as branched covers of the projective line, and the endomorphisms arise as quotients of double coset algebras of the Galois groups of these coverings. In certain cases, we show the existence of continuous families of Jacobians admitting the desired endomorphism algebras. At the end, we raise some questions about upper bounds for endomorphism algebras of the type described here and ask about a “non-Galois version of Hurwitz's theorem.”

A self-affine tile in R n is a set T of positive measure with A ( T )=∪ d ∈ D ( T + d ), where A is an expanding n × n real matrix with |det( A )|= m an integer, and D ={ d , d 2 , ..., d m }⊆ R n is a set of m digits. It is known that self-affine tiles always give tilings of R n by translation. This paper extends known characterizations of digit sets D yielding self-affine tiles. It proves several results about the structure of tilings of R n possible using such tiles, and gives examples showing the possible relations between self-replicating tilings and general tilings, which clarify results of Kenyon on self-replicating tilings.

The object of this paper is to obtain a set of inequalities relating the face numbers of different orbit types of a simplicial polytope P with a finite solvable group G of linear symmetries. It is assumed that (1) for each subgroup H of G, the fixed point set PH is a subpolytope of P, and (2) the toric variety X(P) associated to P is nonsingular. The action of G on P induces an action on X(P), and we describe a set of Smith-type inequalities between the Betti numbers of X(P)H, where H ranges through the set of subgroups of G. By relating each X(P)H with X(PH), we then express these inequalities in terms of the face numbers of the different orbit types of P and the rank of fixed point sets of certain compact tori. This rank is determined explicitly when G is abelian. Moreover, assumption (2) is removed for a polytope of dimension 2.

In this paper we classify the 3-dimensional vector spaces (nets) of 4 × 4 and of 5 × 5 alternating matrices over an algebraically closed field. We apply the second classification to check our "linear syzygy conjectures" for modules over the polynomial ring of dimension 5.

As a mathematical model for time and space correlations in ocean surface wave fields, Mauldin, Monticino, and von Weizsacker (1996,Adv. in Math.117, 239–252), introduced directionally reinforced random walks and analyzed their recurrence properties. This paper develops limiting properties for these random walks inRd. In particular, it is shown that the limit distribution is a Brownian motion if the time between changes of directions has a finite second moment, and is a stable process if the time between changes of directions is in a domain of attraction of a stable law. The latter gives an affirmative answer to an open question posed by Mauldinet al.. The strong law of large numbers and strong approximations are also discussed.

Ever since Viggo Brun's pioneering work, number theorists have developed increasingly sophisticated refinements of the sieve of Eratosthenes to attack problems such as the twin prime conjecture and Goldbach's conjecture. Ever since Gian-Carlo Rota's pioneering work, combinatorialists have found more and more areas of combinatorics where sieve methods (or Mobius inversion) are applicable. Unfortunately, these two developments have proceeded largely independently of each other even though they are closely related. This paper begins the process of bridging the gap between them by showing that much of the theory behind the number-theoretic refinements carries over readily to many combinatorial settings. The hope is that this will result in new approaches to and more powerful tools for sieve problems in combinatorics such as the computation of chromatic polynomials, the enumeration of permutations with restricted position, and the enumeration of regions in hyperplane arrangements.

We consider algebras with one binary operation $\cdot$ and one generator ({\it monogenic}) and satisfying the left distributive law $a\cdot (b\cdot c)=(a\cdot b)\cdot (a\cdot c)$. One can define a sequence of finite left-distributive algebras $A_n$, and then take a limit to get an infinite monogenic left-distributive algebra~$A_\infty$. Results of Laver and Steel assuming a strong large cardinal axiom imply that $A_\infty$ is free; it is open whether the freeness of $A_\infty$ can be proved without the large cardinal assumption, or even in Peano arithmetic. The main result of this paper is the equivalence of this problem with the existence of a certain algebra of increasing functions on natural numbers, called an {\it embedding algebra}. Using this and results of the first author, we conclude that the freeness of $A_\infty$ is unprovable in primitive recursive arithmetic.

In this paper we extend the work of Fomin and Greene on noncommutative Schur functions by defining noncommutative analogs of Schubert polynomials. If the variables satisfy certain relations (essentially the same as those needed in the theory of noncommutative Schur functions), we prove a Pieri-type formula and a Cauchy identity for our noncommutative polynomials. Our results imply the conjecture of Fomin and Kirillov concerning the expansion of an arbitrary Grothendieck polynomial in the basis of Schubert polynomials; we also present a combinatorial interpretation for the coefficients of the expansion. We conclude with some open problems related to it.

This paper introduces a class of geometric objects called Crofton k -densities which are the analogue of closed differential forms. In R n , Crofton densities can be represented by means of a generalization of the Radon transform. This transform maps functions on the space of ( n - k )-planes in R n into Crofton k -densities. We write a system of PDE′s satisfied by Crofton k -densities. Crofton k -denisites can be considered as Lagrangians, and we can write multidimensional Euler-Lagrange equations for them. The system of PDE′s for Crofton densities coincides with the lower order terms of the Euler operator. So Euler-Lagrange equations for Lagrangians corresponding to Crofton densities have more simple form than in general. In addition Crofton k -densities can be characterized by the fact that their extremals include all k -dimensional planes. In particular Crofton 1-densities can be considered as Lagrangians, and their extremals are straight lines. We present a Crofton 1-density as a "nonlocal differential" of a function in R n . The Poincare lemma is valid for Crofton 1-densities that satisfy some growth conditions. In this work we connect two integral geometries, classical integral geometry of Poincare and Chern and integral geometry of Radon transform.