Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

/pdf/orthogonal-polynomials-of-several-variables-2cwkynqk1v.pdf

Orthogonal Polynomials of Several Variables

https://hal.archives-ouvertes.fr/hal-00017721/document

Noncommutative Symmetrical Functions

https://www-irma.u-strasbg.fr/~loday/PAPERS/98LodayRonco(PBT).pdf

Hopf Algebra of the Planar Binary Trees

Schubert polynomials were introduced by Bernstein et al. and Demazure, and were extensively developed by Lascoux, Schutzenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polynomial {\mathfrak S}_w in terms of the reduced decompositions of the permutation w. Using this result, a variation of Schensted's correspondence due to Edelman and Greene allows one to associate in a natural way a certain set {\cal M}_w of tableaux with w, each tableau contributing a single term to {\mathfrak S}_w. This correspondence leads to many problems and conjectures, whose interrelation is investigated. In Section 2 we consider permutations with no decreasing subsequence of length three (or 321-avoiding permutations). We show for such permutations that {\mathfrak S}_w is a flag skew Schur function. In Section 3 we use this result to obtain some interesting properties of the rational function s_{\lambda/\mu}(1,q,q^2,\cdots), where s_{\lambda/\mu} denotes a skew Schur function.

/pdf/some-combinatorial-properties-of-schubert-polynomials-26mw5ek8iu.pdf

Some Combinatorial Properties of Schubert Polynomials

We define the notion of connectivity set for elements of any finitely generated Coxeter group. Then we define an order related to this new statistic and show that the poset is graded and each interval is a shellable lattice. This implies that any interval is Cohen-Macauley. We also give a Galois connection between intervals in this poset and a boolean poset. This allows us to compute the Mobius function for any interval.

Posets related to the connectivity set of Coxeter groups

We give a type free formula for the expansion of $k$-Schur functions indexed by fundamental coweights within the affine nilCoxeter algebra. Explicit combinatorics are developed in affine type $C$.

/pdf/expansions-of-k-schur-functions-in-the-affine-nilcoxeter-2ppzklno3r.pdf

Expansions of k-Schur Functions in the Affine nilCoxeter Algebra

We give a type free formula for the expansion of k-Schur functions indexed by fundamental coweights within the affine nilCoxeter algebra. Explicit combinatorics are developed in affine type C.

Expansions of $k$-Schur functions in the affine nilCoxeter algebra

FPSAC 2013 Extended Abstract. We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions according to a signed combinatorial formula.

pdf/the-immaculate-basis-of-the-non-commutative-symmetric-1qzktciuak.pdf

The immaculate basis of the non-commutative symmetric functions

The space $M_{\mu/i,j}$ spanned by all partial derivatives of the lattice polynomial $\Delta_{\mu/i,j}(X;Y)$ is investigated in math.CO/9809126 and many conjectures are given. Here, we prove all these conjectures for the $Y$-free component $M_{\mu/i,j}^0$ of $M_{\mu/i,j}$. In particular, we give an explicit bases for $M_{\mu/i,j}^0$ which allow us to prove directly the central {\sl four term recurrence} for these spaces.

Lattice Diagram polynomials in one set of variables

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