Green Functions Associated to Complex Reflection Groups

It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the fundamental expansion of a Macdonald symmetric function and for the plethysm of two Schur functions, while the Schur expansions of these expressions are still elusive. Based on work of Egge, Loehr and Warrington, Garsia and Remmel provided a method to obtain the Schur expansion from the fundamental expansion by replacing each quasisymmetric function by a Schur function (not necessarily indexed by a partition) and using straightening rules to obtain the Schur expansion. Here we provide a new method that only involves the coefficients of the quasisymmetric functions indexed by partitions and the quasi-Kostka matrix. As an application, we identify the lexicographically largest term in the Schur expansion of the plethysm of two Schur functions. We provide the Schur expansion of $s_w[s_h](x,y)$ for $w=2,3,4$ using novel symmetric chain decompositions of Young's lattice for partitions in a $w\times h$ box. For $w=4$, this is the first known combinatorial expression for the coefficient of $s_{\lambda}$ in $s_{w}[s_{h}]$ for two-row partitions $\lambda$, and for $w=3$ the combinatorial expression is new.

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space. In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.

Using the vertex operator representations for symplectic and orthogonal Schur functions, we define two families of symmetric functions and show thatthey are the skew symplectic and skew orthogonal Schur polynomials defined implicitly by Koike and Terada and satisfy the general branching rules. Furthermore, we derive the Jacobi-Trudi identities and Gelfand-Tsetlin patterns for these symmetric functions. Additionally, the vertex operator method yields their Cauchy-type identities. This demonstrates that vertex operator representations serve not only as a tool for studying symmetric functions but also offers unified realizations for skew Schur functions of types A, C, and D.

We derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor T? of Young symmetry type ? where ? is an integer partition. The branchings and fixing a vector vi, a symmetric tensor gij = gji and an antisymmetric tensor fij = ?fji, respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function s? ? {?} by the basic M series of complete symmetric functions and the L = M?1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains ?-generalized Newell?Littlewood formulae and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for and , showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine and in some instances non-reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2-cocycles. The algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters. Modification rules are beyond the scope of the present paper, but are briefly discussed.

Let $h_\lambda$, $e_\lambda$, and $m_\lambda$ denote the homogeneous symmetric function, the elementary symmetric function and the monomial symmetric function associated with the partition $\lambda$ respectively. We give combinatorial interpretations for the coefficients that arise in expanding $m_\lambda$ in terms of homogeneous symmetric functions and the elementary symmetric functions. Such coefficients are interpreted in terms of certain classes of bi-brick permutations. The theory of Lyndon words is shown to play an important role in our interpretations.

Starting with a family of Markov chains corresponding to a trellis (or ladder) model we show how to find symmetric functions in the eigenvalues of the transition matrix. Schur functions arise via minors of powers of the transition matrix. We give a path model for finding the Schur functions in terms of the transition probabilities of the trellis. Limit theorems for powers of the transition matrix yield limit theorems for the exterior powers of the transition matrix which in turn yield limit theorems for corresponding families of symmetric functions. We present a theorem on Ces`aro convergence of powers of a matrix that is the exterior power of a stochastic matrix.

Symmetric Green's functions for certain hyperbolic problems

Green's function and position correlation function for a charged oscillator in a heat bath and a magnetic field

Extending the elementary and complete homogeneous symmetric functions, we introduce the truncated homogeneous symmetric function in (1) for any integer partition λ, and show that the transition matrix from to the power sum symmetric functions is given by where and z are nonsingular diagonal matrices. Consequently, forms a basis of the ring Λ of symmetric functions. In addition, we show that the generating function satisfies where ω is the involution of Λ sending each elementary symmetric function to the complete homogeneous symmetric function .

PREFACE CLASSICAL AND QUANTUM SCATTERING Experiments with Nuclear Particles Theories and Experiments Reactions Channels Conservation Laws Kinematics of Nuclear Reactions Cross Sections, Center of Mass and Laboratory Frames Classical Scattering The Classical Cross Section Example: Rutherford scattering Orbiting, rainbow and glory scattering Stationary scattering of a plane wave Appendix 1.A - Systems of units Appendix 1.B - Useful constants and conversion factors Exercises References THE PARTIAL-WAVE EXPANSION METHOD The Scattering Wave Function Radial Equation Free Particle in Spherical Coordinates Phase Shifts Scattering Amplitude and Cross Sections Integral Formulae for the Phase-Shifts Hard Sphere Scattering Resonances Scattering from a Square-Well Low Energy Scattering: Scattering Length Scattering Length for Nucleon-Nucleon Scattering The Effective Range Formula Effective Range for Nucleon-Nucleon Scattering Coulomb Scattering An Illustration: a - a Scattering Appendix 2.A - Absolute Phase Shifts and Levinson Theorem Exercises References FORMAL SCATTERING THEORY Introduction: Green's Functions Free Particle's Green's Functions Scattering Amplitude Born Approximation Transition and Scattering Matrices The Two-Potential Formula Distorted Wave Born Approximation Partial-Wave Expansion of the S-Matrix Partial-Wave Free Particle's Green's Functions Collision of Particles with Spin Collisions of Identical Particles Scattering of Clusters of Identical Fermions Imaginary Potentials: Absorption Cross Section Appendix 3.A - Analytical Properties of the S-Matrix Exercises References COMPOUND NUCLEUS REACTIONS Introduction The Nucleon-Nucleon Interaction The Nucleus as a Strongly Absorbing Medium Mean Free Path of a Nucleon in Nuclei Fermi Gas Model Formal Theory of the Optical Potential Empirical Optical Potential Compound Nucleus Formation R-Matrix Average of the Cross Sections Level Densities in Nuclei Compound Nucleus Decay: The Weisskopf-Ewing Theory Reciprocity Theorem The Hauser-Feshbach Theory Appendix 4.A - The Shell Model Exercises References FUSION AND FISSION Introduction The Liquid Drop Model General Considerations on Fusion Reactions The One Dimensional WKB Approximation Connection Formulas in WKB The Three-Dimensional WKB Approximation Heavy Ion Fusion Reactions Sub-Barrier Fusion Superheavy Elements Occurrence of Fission Mass Distribution of the Fragments Neutrons Emitted in Fission Cross Sections for Fission Energy Distribution in Fission Isomeric Fission The Nuclear Reactor Appendix 5.A - The Nilsson Model Exercises References DIRECT REACTIONS Introduction Level Width and Fermi's Golden Rule Direct Reactions: A Simple Approach Direct Reactions: Detailed Calculations Applications of the Shell Model Direct Reactions as Probe of the Shell Model Nuclear Vibrations Photonuclear Reactions - Giant Resonances Coulomb Excitation Electromagnetic Transition Probabilities for Nuclear Vibrations Nuclear Excitation in the Deformed Potential Model Appendix 6.A - Multipole Moments and the Electromagnetic Interaction Exercises References NUCLEAR REACTIONS IN THE COSMOS Cosmic Rays Stellar Evolution: Hydrogen and CNO Cycles White Dwarfs and Neutron Stars Synthesis of Heavier Elements Supernovae Explosions Thermonuclear Cross Sections and Reaction Rates Reaction Networks Models for Astrophysical Nuclear Cross Sections Slow and Rapid Capture Processes Tests of the Solar Models Indirect Methods for Nuclear Astrophysics Reactions Exercises References HIGH ENGERGY COLLISIONS Introduction Nucleons as Billiard Balls Applications of the Classical Model The Eikonal Wavefunction Elastic Scattering Coulomb Amplitude and Coulomb Eikonal Phase Total Reaction Cross Sections Scattering of Particles with Spin The Optical Limit of Glauber Theory Pauli Blocking of Nucleon-Nucleon Scattering Glauber Theory of Multiple Scattering Coulomb Excitation Inelastic Scattering Charge-Exchange Reactions Exercises References RELAVISTIC COLLISIONS Unpacking the Nucleus The Boltzmann-Uehling-Uhlenbeck Equation Wigner Function Numerical Treatment of Transport Equations Structure of Hadrons Quantum Chromodynamics The Quark-Gluon Plasma Exercises References

Lam and Pylyavskyy introduced loop symmetric functions as a generalization of symmetric functions. They defined loop Schur functions as generating functions over semistandard tableaux with respect to a `colored weight,' and they proved a Jacobi--Trudi-style determinantal formula for these generating functions. We prove that loop Schur functions can be expressed as a ratio of `loop alternants,' extending the analogy with Schur functions. As an application, we give a new proof of the loop version of the Murnaghan--Nakayama rule.

We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K K -theoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. Our expansions extend to flagged Grothendieck polynomials.

The Jack polynomials J (x; ) form a remarkable class of symmetric polynomials. They are indexed by a partition and depend on a parameter . One of their properties is that several classical families of symmetric functions can be obtained by specializing , e.g., the monomial symmetric functions m ( =∞), the elementary functions e ′ ( = 0), the Schur functions s ( = 1) and nally the two classes of zonal polynomials ( = 2, = 1=2). The Jack polynomials can be de ned in various ways, e.g.: a) as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b) as simultaneous eigenfunctions of certain di erential operators (the Sekiguchi–Debiard operators). Recently Opdam, [O], constructed a similar family F (x; ) of nonsymmetric polynomials. The index runs now through all compositions ∈ Nn. They are de ned in a completely similar fashion, e.g., the Sekiguchi–Debiard operators are being replaced by the Cherednik di erential-re ection operators (see Sect. 3). It is becoming more and more clear that these polynomials are as important as their symmetric counterparts. The purpose of this paper is to add to the existing characterizations of Jack polynomials two further ones: c) a recursion formula among the F together with two formulas to obtain J from them. d) combinatorial formulas of both J and F in terms of certain generalized tableaux.

We introduce certain generalisations of the characters of the classical Lie groups, extending the recently defined factorial characters of Foley and King. In this extension, the factorial powers are replaced with an arbitrary sequence of polynomials, as in Sergeev–Veselov's generalised Schur functions and Okada's generalised Schur P- and Q-functions. We also offer a similar generalisation for the rational Schur functions. We derive Littlewood-type identities for our generalisations. These identities allow us to give new (unflagged) Jacobi–Trudi identities for the Foley–King factorial characters and for rational versions of the factorial Schur functions. We also propose an extension of the original Macdonald's ninth variation of Schur functions to the case of symplectic and orthogonal characters, which helps us prove Nägelsbach–Kostka identities.

In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over the integers to a finite field $\mathbb{F}_q$, what is the probability that the Schur function $s_\lambda$ maps to zero? We show that this probability is always at least $1/q$ and is asymptotically $1/q$. Moreover, we give a complete classification of all shapes that can achieve probability $1/q$. In addition, we identify certain families of shapes where the corresponding Schur functions being sent to zero are independent events, and we look into the probability that a Schur functions is mapped to nonzero values in $\mathbb{F}_q$.