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.

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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 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.
 Nous introduisons une nouvelle base des fonctions syméetriques non commutatives dont les images commutatives sont des fonctions de Schur. Nous construisons la base duale des fonctions quasi-symétriques qui s’expriment de façon positive en fonction de la base fondamentale et décomposent les fonctions de Schur.

Immaculate basis of the non-commutative symmetric functions

q and q,t-analogs of non-commutative symmetric functions

In this paper, a Hopf monoid is an algebraic structure built on objects in the category of Joyal’s vector species. There are two Fock functors, K and K, that map a Hopf monoid H to graded Hopf algebras K(H) and K(H), respectively. There is a natural Hopf monoid structure on linear orders L, and the two Fock functors are related by K(H) = K(H × L). Unlike the functor K, the functor K applied to H may not preserve the antipode of H. In view of the relation between K and K, one may consider instead of H the larger Hopf monoid L×H and study the antipode of L×H. One of the main results in this paper provides a cancellation free and multiplicity free formula for the antipode of L ×H. As a consequence, we obtain a new antipode formula for the Hopf algebra H = K(H). We explore the case when H is commutative and cocommutative, and obtain new antipode formulas that, although not cancellation free, they can be used to obtain an antipode formula for K(H) in some cases. We also recover many well-known identities in the literature involving antipodes of certain Hopf algebras. In our study of commutative and cocommutative Hopf monoids, hypergraphs and acyclic orientations play a central role. We obtain polynomials analogous to the chromatic polynomial of a graph, and also identities parallel to Stanley’s (−1)-color theorem. An important consequence of our notion of acyclic orientation of hypergraphs is a geometric interpretation for the antipode formula for hypergraphs. This interpretation, which differs from the recent work of Aguiar and Ardila as the Hopf structures involved are different, appears in subsequent work by the authors. Introduction Computing antipode formulas in any graded Hopf algebra is a classical yet difficult problem. Recently, numerous results in this direction have been provided for various families of Hopf algebras [1, 4, 7, 10, 12, 15, 16]. Some motivations to find such formulas lie in their geometric interpretation [1], their use in quantum field theories [14], or their role in deriving combinatorial invariants of the discrete objects in play. A key example of this is the Hopf algebra of graphs G as given in [16], where the authors derive the antipode formula and use it to obtain the celebrated Stanley’s (−1)-color theorem: the chromatic polynomial of a graph evaluated at −1 is, up to a sign, the number of acyclic orientations of the graph. A remarkable result in [1] shows that the antipode formula of a graph as given in [16] is encoded in the f -vector of the graphical zonotope corresponding to the underlying graph. Manuscript received 23rd February 2018, revised 23rd November 2018, accepted 7th December 2018.

/pdf/the-antipode-of-linearized-hopf-monoids-3xmld7rb35.pdf

The antipode of linearized Hopf monoids

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n\ge0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n\ge0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim(A_n)=r^nn!$ where $r = \dim(A_1)$. In the case $r=1$ we give a conjectural classification. We then investigate a quantum version of the main theorem. We conclude with some open problems and a categorification of the construction. This paper is a full version of the summary arXiv: 0710.3744.

Combinatorial Hopf algebras and Towers of Algebras - Dimension, Quantization, and Functoriality

We present complete simplicial fan realizations of any spherical subword complex of type $$A_n$$An for $$n\le 3$$n≤3. This provides complete simplicial fan realizations of simplicial multi-associahedra $$\varDelta _{2k+4,k}$$Δ2k+4,k, whose facets are in correspondence with $$k$$k-triangulations of a convex $$(2k+4)$$(2k+4)-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. We also present fan realizations of two previously unknown cases of subword complexes of type $$A_4$$A4, namely the multi-associahedra $$\varDelta _{9,2}$$Δ9,2 and $$\varDelta _{11,3}$$Δ11,3.

Fan Realizations of Type $$A$$A Subword Complexes and Multi-associahedra of Rank 3

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