Showing papers in "Transactions of the American Mathematical Society in 1990"
TL;DR: In this article, a two-variable Laurent polynomial invariant of regular isotopy for unoriented knots and links is proposed, denoted LK for a link K, and it satisfies the axioms: 1. Lo0=1.
Abstract: This paper studies a two-variable Laurent polynomial invariant of regular isotopy for classical unoriented knots and links. This invariant is denoted LK for a link K, and it satisfies the axioms: 1. Regularly isotopic links receive the same polynomial. 2. Lo0=1. 3 L _ = aL, L_? = a-'L. 4. L ) Small diagrams indicate otherwise identical parts of larger diagrams. Regular isotopy is the equivalence relation generated by the Reidemeister moves of type II and type III. Invariants of ambient isotopy are obtained from L by writhe-normalization.
663 citations
TL;DR: In this article, Martin et al. developed fundamental results on the existence and behavior of solutions to semilinear functional differential equations in a Banach space setting, including differential inequalities, invariant sets, and Lyapunov functions.
Abstract: FUNCTIONAL DIFFERENTIAL EQUATIONS AND REACTION-DIFFUSION SYSTEMS R. H. MARTIN, JR. AND H. L. SMITH ABSTRACT. Several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a Banach space setting. The ideas are applied to reaction-diffusion systems that have time delays in the nonlinear reaction terms. The techniques presented here include differential inequalities, invariant sets, and Lyapunov functions, and therefore they provide for a wide range of applicability. The results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays. Several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a Banach space setting. The ideas are applied to reaction-diffusion systems that have time delays in the nonlinear reaction terms. The techniques presented here include differential inequalities, invariant sets, and Lyapunov functions, and therefore they provide for a wide range of applicability. The results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays. Suppose Q is a bounded domain in RN with aQ smooth and A is the Laplacian operator on Q. Also, let m be a positive integer, z a positive number, and f = (fi)m a continuous, bounded function from [0, xc] x Q x C([-z, 0])m into Rm where C([-T, 0]) is the space of continuous functions from [-z, 0] into R. The purpose of this paper is to apply abstract results for semilinear functional differential equations in Banach spaces to reactiondiffusion systems with time delays having the form a tu(x, t) = diAu' (x, t) + ?i(t, x, ut(x, *)), t>a, xeQ, i=1,...,m, (RDD) ai(x)ui(x, t) + au i(x, t) = ,81(x, t), t > a , x EaQ u'(x , a + 0) = X'(x , 0) 5 -T 0, di > 0, and ca:Q * [0,o ) is C' and ,i:Q x [0, oo) R is C2. Here an is the outward normal derivative on aQ and if di = 0 it is assumed that no boundary conditions are specified for this i. Also, tu'(x, t) denotes the partial with respect to t, whereas ut(x, *) denotes the member of C([-z, 0]) defined by 0 -u(x, t + 0) = (u'(x, t + 0))M. Our techniques provide basic existence criteria, but the main point is that they can also be effectively applied to obtain estimates for solutions, especially Received by the editors October 7, 1987 and, in revised form, October 7, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 35R10, 34K30.
503 citations
259 citations
TL;DR: In this article, a characterization of a class of classical Lorentz spaces on which the Hardy Littlewood maximal operator is bounded is given by determining the weights for which Hardy's inequality holds for nonincreasing functions.
Abstract: A characterization is given of a class of classical Lorentz spaces on which the Hardy Littlewood maximal operator is bounded. This is done by determining the weights for which Hardy's inequality holds for nonincreasing functions. An alternate characterization, valid for nondecreasing weights, is also derived.
239 citations
TL;DR: In this paper, the invariant theory of representations of a quiver is discussed and a trace satisfying a formal Cayley-Hamilton identity is given, which is then connected to a trace of the quiver.
Abstract: We discuss the invariant theory of the variety of representations of a quiver and present generators and relations. We connect this theory of algebras with a trace satisfying a formal CayleyHamilton identity
234 citations
TL;DR: In this paper, the stability of property (RD) with respect to some constructions of groups such as subgroups, over-groups of finite index, semidirect and amalgamated products was studied.
Abstract: Let r be a group. We associate to any length-function L on r the space H'{' (r) of rapidly decreasing functions on r (with respect to L), which coincides with the space of smooth functions on the k-dimensional torus when r = Zk. We say that r has property (RD) if there exists a length-function L on r such that H,{,(r) is contained in the reduced C·-algebra C;(r) of r. We study the stability of property (RD) with respect to some constructions of groups such as subgroups, over-groups of finite index, semidirect and amalgamated products. Finally, we show that the following groups have property (RD): (1) Finitely generated groups of polynomial growth; (2) Discrete cocompact subgroups of the group of all isometries of any hyperbolic space.
227 citations
TL;DR: In this article, the authors consider supersolutions of the Hamilton-Jacobi equation up to the boundary of a bounded open subset of a real-valued function, where the gradient of the unknown function u is the gradient gradient of u. The main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system.
Abstract: In the present paper we consider Hamilton-Jacobi equations of the form H(x, u, Vu) = 0, x E Q, where Q is a bounded open subset of Rn H is a given continuous real-valued function of (x, s, p) E Q x R x Rn and Vu is the gradient of the unknown function u . We are interested in particular solutions of the above equation which are required to be supersolutions, in a suitable weak sense, of the same equation up to the boundary of Q. This requirement plays the role of a boundary condition. The main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system, as well from related questions in constrained geodesics.
226 citations
190 citations
TL;DR: In this article, it was shown that the classification of irreducible modules in this category can be reduced to the class of torsion free modules, which are defined as modules that decompose into direct sums of finite dimensional 4-weight spaces.
Abstract: Let g denote a reductive Lie algebra over an algebraically closed field of characteristic zero, and let 4 denote a Cartan subalgebra of g. In this paper we study finitely generated g-modules that decompose into direct sums of finite dimensional 4-weight spaces. We show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules, those we call torsion free modules. We also show that if g is a simple Lie algebra that admits a torsion free module, then g is of type A or C.
165 citations
TL;DR: In this article, the enveloping algebra of sl(2, C) was studied for different f, and it was shown how they are similar to (and different from) U(sl(2)), the envelope algebra for sl(3, C).
Abstract: Fix f E C[X]. Define R = C[A, B, H] subject to the relations HA-AH = A, HB-BH =-B, AB-BA = f(H). We study these algebras (for different f) and in particular show how they are similar to (and different from) U(sl(2)), the enveloping algebra of sl(2, C). There is a notion of highest weight modules and a category a' for such R. For each n > 0, if f (x) = (x + 1)n+l _ Xn+1 , then R has precisely n simple modules in each finite dimension, and every finite-dimensional R-module is semisimple.
151 citations
TL;DR: In this article, large deviation estimates for continuous maps of compact metric spaces are proved for attractors in differentiable dynamics, rate of escape problems, and shift spaces, and applied to shift spaces.
Abstract: We prove some large deviation estimates for continuous maps of compact metric spaces and apply them to attractors in differentiable dynamics, rate of escape problems, and to shift spaces.
TL;DR: In this article, it was shown that if u is a Holder continuous solution to this equation, then the measure,u satisfies the growth property j[B(x, r)] 1 is given by the structure of the differential operator.
Abstract: This paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form divA(x, u, Vu) + B(x, u, Vu), where A and B are Borel measurable, are solutions to the equation divA(x, u, Vu) + B(x, u, Vu) = ,u for some nonnegative Radon measure ,i. Among other things, it is shown that if u is a Holder continuous solution to this equation, then the measure ,u satisfies the growth property j[B(x, r)] 1 is given by the structure of the differential operator. Conversely, if u is assumed to satisfy this growth condition, then it is shown that u satisfies a Harnack-type inequality, thus proving that u is locally bounded. Under -the additional assumption that A is strongly monotonic, it is shown that u is Holder continuous.
TL;DR: The Leray functor as discussed by the authors was introduced to define the cohomological Conley index of an isolated invariant set of a homeomorphism on a locally compact metric space.
Abstract: We introduce the Leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological Conley index of an isolated invariant set of a homeomorphism on a locally compact metric space. We prove the homotopy and additivity properties for this index and compute the index in some examples. As one of applications we prove the existence of nonconstant, bounded solutions of the Euler approximation of a certain system of ordinary differential equations.
TL;DR: In this article, it was shown that the image of the unit disk under a Holder continuous conformal mapping is a p-Poincare domain and that the imbedding of the Sobolev space is compact.
Abstract: A domain D c Rd of finite volume is said to be a p-Poincare domain if there is a constant Mp(D) so that fU UDII dx d , then D is a p-Poincare domain. This answers a question of Axler and Shields regarding the image of the unit disk under a Holder continuous conformal mapping. We also consider geometric conditions which imply that the imbedding of the Sobolev space W" "P(D) -+ Lp(D) is compact, and prove that this is the case for a Holder domain D.
TL;DR: In this article, an ultra-filter approach to "central" sets (sets which are combinatorially rich) is developed, which enables us to show that for any partition of the positive integers one cell is both additively and multiplicatively central.
Abstract: Applying ideas from topological dynamics in compact metric spaces to the Stone-Cech compactification of a discrete semigroup, several new proofs of old results and some new results in Ramsey Theory are obtained. In particular, two ultrafilter proofs of van der Waerden's Theorem are given. An ultrafilter approach to "central" sets (sets which are combinatorially rich) is developed. This enables us to show that for any partition of the positive integers one cell is both additively and multiplicatively central. Also, a fortuitous answer to a question of Ellis is obtained.
TL;DR: In this article, it was shown that for generic bounded polynomial vector fields with isolated critical points, the sum of the indices at all their critical points is (-1) and that there are exactly seventeen different behaviors at infinity for bounded cubic polynomials in the plane.
Abstract: We prove that, for generic bounded polynomial vector fields in R\" with isolated critical points, the sum of the indices at all their critical points is (-1)\" . We characterize the local phase portrait of the isolated critical points at infinity for any bounded polynomial vector field in R\" . We apply this characterization to show that there are exactly seventeen different behaviours at infinity for bounded cubic polynomial vector fields in the plane. 0. Introduction Let X : U —> Rk be a vector field where U is an open set of Rk . Let y(t) = y(t, x) be the integral curve of X such that y(0) = x. Let Ix be its maximal interval of definition. We shall say that X is a bounded vector field if for all x e U, there exists some compact set K c U such that y(t) G K for each t g Ix n (0, +00). In § 1 we introduce the stereographic compactification of X, s(X). We then use the index formula of Bendixson and the Poincaré-Hopf theorem to prove the following result: Proposition A. Let X be a bounded polynomial vector field in the plane. If all the critical points of s(X) are isolated, then the sum of the indices at all those critical points is 1. In §2 we use the Poincaré compactification of X, p(X), to characterize the local phase portrait of the isolated critical points at infinity for bounded polynomial vector fields X = (P, Q) in the plane. The degree n of X is defined by n = max{degreeP, degree Q}. We denote by ix(q) the index of Y at a critical point q of X. We then prove the following theorem: Theorem B. Let X be a bounded polynomial vector field in the plane. If q is an isolated infinite critical point of X, then (a) The local phase portrait of p(X) at q is described in Figure 2.2 (resp. Figure 2.4) when the degree of X is even (resp. odd). Received by the editors July 12, 1988. The contents of this paper have been presented to the meeting \"Qualitative Theory of Differential Equations\" in Szeged, Hungary, August 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 34C05; Secondary 58F14.
Journal Article•
TL;DR: The existence of weak solutions for the Navier-Stokes equations for the infinite cylinder with initial data in L p is considered in this paper, where the existence theorem gives a new method of constructing global solutions.
Abstract: The existence of weak solutions for the Navier-Stokes equations for the infinite cylinder with initial data in L p is considered in this paper. We study the case of initial data in L p (R n ), 2 n) is bridged. The existence theorem gives a new method of constructing global solutions. The cases p=n are treated at the end of the paper
TL;DR: In this article, the group algebra of the symmetric group and properties of the irreducible characters are used to derive combinatorial properties of embeddings of rooted maps in orientable surfaces of arbitrary genus.
Abstract: The group algebra of the symmetric group and properties of the irreducible characters are used to derive combinatorial properties of embeddings of rooted maps in orientable surfaces of arbitrary genus. In particular, we show that there exists, for each genus, a correspondence between the set of rooted quadrangulations and a set of rooted maps of all lower genera with a distinguished subset of vertices.
TL;DR: In this article, the authors consider the case where G is a connected, simply connected solvable Lie group and K C Aut(G) is a compact, connected group and determine a moduli space for the associated K-spherical functions.
Abstract: Let G be a locally compact group, and let K be a compact subgroup of Aut(G) , the group of automorphisms of G. There is a natural action of K on the convolution algebra L (G), and we denote by LK(G) the subalgebra of those elements in L (G) that are invariant under this action. The pair (K, G) is called a Gelfand pair if LI(G) is commutative. In this paper we consider the case where G is a connected, simply connected solvable Lie group and K C Aut(G) is a compact, connected group. We characterize such Gelfand pairs (K, G), and determine a moduli space for the associated K-spherical functions.
TL;DR: In this paper, the authors apply the method used by Hile and Protter [2] to a variety of second-order elliptic problems, in particular, to all constant coefficient problems, where the Laplacian is replaced by a more general operator in a Hilbert space.
Abstract: Let Q C IR"m be a bounded open set, AQ its boundary and A the Laplacian on Rm Consider the elliptic differential equation: (1) -Au = Au in Q; u = 0 on AQ It is known that the eigenvalues, )i, of (1) satisfy n M n (2) i=l n+1l 1 provided that 'n+l > An In this paper we abstract the method used by Hile and Protter [2] to establish (2) and apply the method to a variety of second-order elliptic problems, in particular, to all constant coefficient problems We then consider a variety of higher-order problems and establish an extension of (2) for problem (1) where the Laplacian is replaced by a more general operator in a Hilbert space
TL;DR: In this paper, the authors characterize the pairs of nonnegative functions (u, v) for which MB+ applies LP(v) in LP(u) or in weakLp(u).
Abstract: Let M+ be the maximal operator defined by g MB f(x) = sup ( f(t)lg(t) dt) (f+g(t) dt) where g is a positive locally integrable function on R. We characterize the pairs of nonnegative functions (u, v) for which MB+ applies LP(v) in LP(u) or in weakLp(u) . Our results generalize Sawyer's (case g = 1 ) but our proofs are different and we do not use Hardy's inequalities, which makes the proofs of the inequalities self-contained.
TL;DR: In this paper, the authors give operators on the flag variety of a compact Lie group and a complex-oriented generalized cohomology theory, which satisfy braid relations only for the two classical cases, thereby giving a topological interpretation of a theorem proved by Gutkin.
Abstract: Let X be the flag variety of a compact Lie group and let h* be a complex-oriented generalized cohomology theory. We introduce operators on h*(X) which generalize operators introduced by Bernstein, Gel'fand, and Gel'fand for rational cohomology and by Demazure for K-theory. Using the Becker-Gottlieb transfer, we give a formula for these operators, which enables us to prove that they satisfy braid relations only for the two classical cases, thereby giving a topological interpretation of a theorem proved by the authors and extended by Gutkin. One of the central issues in Lie theory is the geometry of the flag variety associated to a compact Lie group G. An important problem concerning the flag variety is the Schubert calculus, which studies the ring structure of the cohomology of the flag variety. Work initiated by Borel, Bott and Kostant, which culminated in a paper by Bernstein, Gel'fand and Gel'fand [BGG], gave a complete solution to the problem. Demazure studied the same problem for K-theory. Moreover, these techniques have been generalized to the Kac-Moody situation by Kac-Peterson, Kostant-Kumar, and others. This work has focussed on algebro-geometric properties of the flag variety. Here, on the other hand we study the flag variety from the point of view of algebraic topology. As a consequence, not only do we recover the classical results described above, but we extend these results to a certain class of cohomology theories-those which are complex-oriented. Examples of complex-oriented theories include ordinary cohomology, K-theory, complex cobordism, and elliptic cohomology. Since the context we have chosen in very general, the proofs are universal and are often simpler than the classical arguments. In the work of BGG, a crucial role is played by operators Ai associated to each simple reflection si of the Weyl group of G (defined by Demazure in K-theory). These operators Ai satisfy the braid relations, which are the relations between pairs of simple reflections. In this paper, we generalize the A, to give operators D, acting on h*(G/T) for any complex-oriented theory h*. We prove that braid relations are satisfied only for cohomology theories with the formal group law of rational cohomology or of K-theory (Theorem Received by the editors June 21, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 55N20, 57T15. The second author was supported by an NSF graduate fellowship. (3 1990 American Mathematical Society 0002-9947/90 $1.00 + $.25 perpage
TL;DR: In this article, the enveloping algebra of sl(2, C) was studied for different f, and it was shown how they are similar to (and different from) U(sl(2)), the envelope algebra for sl(3, C).
Abstract: Fix f E C[X]. Define R = C[A, B, H] subject to the relations HA-AH = A, HB-BH =-B, AB-BA = f(H). We study these algebras (for different f) and in particular show how they are similar to (and different from) U(sl(2)), the enveloping algebra of sl(2, C). There is a notion of highest weight modules and a category a' for such R. For each n > 0, if f (x) = (x + 1)n+l _ Xn+1 , then R has precisely n simple modules in each finite dimension, and every finite-dimensional R-module is semisimple.
TL;DR: The notion of pseudouniform convergence of a sequence of functions and optimal convergence of continuous functions to a bounded Baire class 1 function is introduced in this article, where the authors show their essential equivalence.
Abstract: We study in this paper various ordinal ranks of (bounded) Baire class 1 functions and we show their essential equivalence. This leads to a natural classification of the class of bounded Baire class 1 functions B_1 in a transfinite hierarchy B^ξ_1 ξ < ω_1) of "small" Baire classes, for which (for example) an analysis similar to the Hausdorff-Kuratowski analysis of Δ^0_2 sets via transfinite differences of closed sets can be carried out. The notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a Baire class 1 function ƒ are introduced and used in this study.
TL;DR: In this article, it was shown that the rings R all of whose left modules are direct sums of finitely generated modules satisfy the following finiteness condition: for each positive integer n there are only finitely many isomorphism types of (a) indecomposable left R-modules of length n; (b) finitely presented indecompositionable right R -modules having minimal projective resolutions with n relations.
Abstract: It is shown that the rings R all of whose left modules are direct sums of finitely generated modules satisfy the following finiteness condition: For each positive integer n there are only finitely many isomorphism types of (a) indecomposable left R-modules of length n; (b) finitely presented indecomposable right R-modules of length n; (c) indecomposable right R-modules having minimal projective resolutions with n relations. Moreover, our techniques yield a very elementary proof for the fact that the presence of the above decomposability hypothesis for both left and right R-modules entails finite representation type.
TL;DR: In this article, the authors examined differentiability properties of the convex envelope conv E of a given function E: Rn __ (-cc, cc] in terms of properties of E. They showed that 0 < a < 1 can be obtained under general conditions.
Abstract: We examine differentiability properties of the convex envelope conv E of a given function E: Rn __ (-cc, cc] in terms of properties of E. It is shown that C1 as well as optimal C1'a regularity results, 0 < a < 1 can be obtained under general conditions.
TL;DR: Theorem 3.1 (c) was proved in this paper for locally compact groups, where T is an isometric isomorphism from the Banach algebra LUC(G1) * onto LUC (G2)*, equipped with Arens multiplication.
Abstract: Let G1, G2 be locally compact groups. We prove in this paper that if T is an isometric isomorphism from the Banach algebra LUC(G1) * (the continuous dual of the Banach space of left uniformly continuous functions on G1, equipped with Arens multiplication) onto LUC(G2)*I then T maps M(G1) onto M(G2) and LI(G,) onto L'(G2). We also prove that any isometric isomorphism from LI(G,)** (second conjugate algebra of L' (GI)) onto Ll (G2) maps L' (G1) onto L' (G2) . 0. INTRODUCTION AND PRELIMINARIES Let G1, G2 be locally compact groups. Let M(Gi), i = 1, 2, be the Banach algebra of regular Borel measures on Gi . A well-known result of B. E. Johnson [10] asserts that if T is an isometric isomorphism from M(G1) onto M(G2), then T maps Ll (G,) onto L (G2) (and hence G1 and G2 must be isomorphic by Wendel's theorem [21]). In this paper we prove (Theorem 3.1 (c)), among other things, that if T is an isometric isomorphism from Ll(G1)** onto L1(G2)**, then T maps L1(G1) onto Ll (G2). This answers affirmatively a question raised in [4]. Theorem 3.1 (c) was proved for abelian locally compact groups by Lau and Losert in 13], and for compact and discrete groups by Ghahramani and Lau in [4]. Let G be a locally compact group. Let C(G) denote the space of bounded continuous complex-valued functions on G with the sup norm topology, and LUC(G) denote the closed subspace of bounded left uniformly continuous functions on G, i.e. all f E C(G) such that the map x -?lxf from G into C(G) is continuous, where (lxf)(y) = f(xy), x, y E G. Then LUC(G)* is a Banach algebra with the Arens multiplication defined by (nm, f) = (n, m1f), n, m E LUC(G)*, f E LUC(G), where mlf(x) = (m, lxf) x E G. Furthermore, M(G) may be identified with a closed subspace of LUC(G)* by the natural embedding (,u, f) = f f(x) du(x), f E LUC(G), ,u E M(G) . It was Received by the editors November 1, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 43A20.
Abstract: Local approximation order to smooth complex valued functions by a finite dimensional space #7B-H, spanned by certain products of exponentials by polynomials, is investigated. The results obtained, together with a suitable quasi-interpolation scheme, are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline. The analysis of a typical #7B-H is based here on the identification of its dual with a certain space #7B-P of multivariate polynomials. This point of view allows us to solve a class of multivariate interpolation problems by the polynomials from #7B-P
TL;DR: In this paper, a new chromatic skein invariant for a special class of dichromatic links was proposed, which is a mutual extension of both the two-variable Alexander polynomial and the twisted Alexander invariant.
Abstract: We investigate the skein theory of oriented dichromatic links in S3. We define a new chromatic skein invariant for a special class of dichromatic links. This invariant generalizes both the two-variable Alexander polynomial and the twisted Alexander polynomial. Alternatively, one may view this new 1 2 invariant as an invariant of oriented monochromatic links in S x D , and as such it is the exact analog of the twisted Alexander polynomial. We discuss basic properties of this new invariant and applications to link interchangeability. For the full class of dichromatic links we show that there does not exist a chromatic skein invariant which is a mutual extension of both the two-variable Alexander polynomial and the twisted Alexander polynomial.
TL;DR: In this paper, the authors apply the theory of Hall-Littlewood functions to prove several basic hypergeometric series identities, including some previously known generalizations of the Rogers-Ramanujan identities due to G. E. Andrews and D. M. Proctor.
Abstract: We apply the theory of Hall-Littlewood functions to prove several multiple basic hypergeometric series identities, including some previously known generalizations of the Rogers-Ramanujan identities due to G. E. Andrews and D. M. Bressoud. The techniques involve the adaptation of a method due to I. G. Macdonald for calculating partial fraction expansions of certain types of symmetric formal power series. Macdonald originally used this method to prove a pair of generating function identities for plane partitions conjectured by MacMahon and Bender-Knuth. We show that this method can also be used to prove another pair of plane partition identities recently obtained by R. A. Proctor. 0. Introduction Many identities from the theory of symmetric functions can be viewed as generalizations of standard results from the theory of partitions and/or basic hypergeometric series. See Chapter I of [M], for example. The sense of genralization typically derives from the fact that when the variables xx, x2, ... in a symmetric function identity are specialized to be powers of a single variable q , one usually obtains combinatorial information about partitions (linear or plane), or else a well-known fact about basic hypergeometric series, i.e., qseries. However, most of the results that have been obtained in this way do not extend very deeply into the theory of ^-series. The main purpose of this paper is to present methods for deriving some nontrivial ^-series identities via the theory of Hall-Littlewood symmetric functions. For example, we will obtain the Rogers-Ramanujan identities oo r oo EQ TT,, 5n-l,-l,, 5/1-4.-1 ^(,-«)(,-,v.(,-o-B\"-' ' (1~* ' ■ oo „'-(''+1) °° EQ TT/, 5/1-2. —1,, 5/i-3, — l -—-T-T.-7A = i[(la ) i1\"« ) r=oil-Q)il~Q )'\"il-Q) »=1 as a consequence of these methods. Received by the editors January 23, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 05A19, 05A17, 05A30, 11P68. Partially supported by an NSF Postdoctoral Research Fellowship. The author was also supported by the Institute for Mathematics and its Applications with funds provided by the NSF. ©1990 American Mathematical Society 0002-9947/90 $1.00+ $.25 per page 469 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 470 J. R. STEMBRIDGE The most crucial part of the method we present was suggested by Macdonald's derivation of a partial fraction expansion involving Hall-Littlewood functions (see Example III.5.5 in [M]). Macdonald [M] used this expansion to prove a pair of plane partition conjectures due to MacMahon (Example 1.5.17), and Bender and Knuth (Example 1.5.19), and to calculate the Hecke series for the group of symplectic similitudes over a local field (§V.5). By specializing the variables in Macdonald's expansion, one obtains an interesting pair of apparently new multiple (1 + Qlar), r=\\ oo (0.4) (íUaUí/fl)«, = £(-1W*>(i Q2r+i/a r=0 Also, if we replace q with q and z with —z/q , the special case a = q\" of (0.2) becomes the following well-known ^-analogue of the Binomial Theorem: (0.5) (-z;q)n = J2zq^ r=0 where [\"] = (q)„/(q)r(q)n_r denotes the ^-binomial coefficient. A partition of r is a weakly decreasing sequence X = (Xx, X2, ...) of nonnegative integers such that |A| = ¿f,X¡ = r. The number of nonzero terms of X is called the length and denoted by f(X). The conjugate X' is the partition whose zth term X'¡ is defined to be the number of terms > z in A. In most cases, we License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 472 J. R. STEMBRIDGE will follow the partition notation of [M], although one important exception will be the parameter \"«:=£(*<) This quantity would be n(X') in the notation of [M]. It will be convenient to extend the notation (z)n to partitions X by defining iz)x = iz)il-i2i*h2-xf ' and to extend the n . Finally, if x = (xx, ... , xn) is an zz-tuple of variables and a e Z\" (or even R\"), we will use the notation xa as an abbreviation for x\"' ■ ■ -x°\". 1. Basic techniques For each partition X = (Xx, ... ,Xf of length at most zz, let Pfx) = Px(x; q) denote the Hall-Littlewood function indexed by X in the variables x = (xx, ... , Xn), i.e., 11 X;-X¡ (1.1) Pfx;q)= £ w weSJX where Sn/X denotes a set of left coset representatives for the subgroup of the symmetric group 5 consisting of those permutations of (xx, ... , xf) that fix xk [M, III]. Let px > p2 > ■ ■ ■ > Pj > 0 denote the distinct integers occurring among (Xx, ... , Xn), and let m¡ denote the multiplicity of p¡ in X. The /-tuple (m) = (mx, ... , mf is thus a composition of zz. The distinct permutations of xÁ may be indexed by (ordered) partitions of the indeterminate-set x into blocks of sizes mx, ... , m¡,or equivalen tly, by functions f:x —>■ {1, 2, ...,/} with \\f~ (i)\\ = m¡. We will refer to such a map / as a function of type (m). Using fij as an abbreviation for the product of the variables in fi~ (i), the definition (1.1) may be rewritten in the form (1-2) ^;<7) = £/f'---./rv */= II ir^rf f(x,) HI.(2.1)]. From this definition it is also clear that Pfx ; 0) is the Schur function sfx). The principal source from which we intend to derive ^-series identities develops from the following pair of Hall-Littlewood function expansions: