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

Given a finite graded poset with labeled Hasse diagram, we construct a quasi- symmetric generating function for (saturated) chains whose labels have fixed descents. This is a common generalization of a generating function for the flag f-vector defined by Ehrenborg and of a symmetric function associated to certain edge-labeled posets which arose in the theory of Schubert polynomials. We show this construction gives a Hopf morphism from an incidence algebra of edge-labeled posets to the Hopf algebra of quasi-symmetric functions.

Hopf Algebras and Edge-Labeled Posets

We show that the Grothendieck bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function basis for this bialgebra.

pdf/grothendieck-bialgebras-partition-lattices-and-symmetric-30hc23j4f2.pdf

Grothendieck bialgebras, Partition lattices and symmetric functions in noncommutative variables

The immaculate basis of the non-commutative symmetric functions was recently introduced by the first and third authors to lift certain structures in the symmetric functions to the dual Hopf algebras of the non-commutative and quasi-symmetric functions. It was shown that immaculate basis satisfies a positive, multiplicity free right Pieri rule. It was conjectured that the left Pieri rule may contain signs but that it would be multiplicity free. Similarly, it was also conjectured that the dual quasi-symmetric basis would also satisfy a signed multiplicity free Pieri rule. We prove these two conjectures here.

The Pieri rule for dual immaculate quasi-symmetric functions

We give an LU-decomposition of the supercharacter table of the group of n × n unipotent upper triangular matrices over &#x1d53d;q, into a lower-triangular matrix with entries in ℤ[q] and an upper-triangular matrix with entries in ℤ[q-1]. To this end, we introduce a q deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommuting variables. The decomposition is obtained from the transition matrices between the supercharacter basis, the q-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group Sn given by the transition matrices between Schur functions, monomials and power-sums. We deduce some combinatorial results associated to this decomposition. In particular, we compute the determinant of the supercharacter table.

A supercharacter table decomposition via power-sum symmetric functions

Structure constants for the multiplication of Schubert polynomials by Schur symmetric polynomials are related to the enumeration of chains in a new partial order on S∞, the Grassmannian Bruhat order. Here we present a monoid M related to this order. We develop a notion of reduced sequences for M and show that M is analogous to the nil-Coxeter monoid for the weak order on S∞.

A Monoid for the Grassmannian Bruhat Order

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