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European Mathematical Society Publishing House
2024-03-28 22:21:05
3
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=AIHPD&vol=2&iss=3&update_since=2024-03-28
Annales de l’Institut Henri Poincaré D
Ann. Inst. Henri Poincaré Comb. Phys. Interact.
AIHPD
2308-5827
2308-5835
General
Combinatorics
Quantum theory
10.4171/AIHPD
http://www.ems-ph.org/doi/10.4171/AIHPD
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
2
2015
3
A solution to the combinatorial puzzle of Mayer’s virial expansion
Stephen James
Tate
Imperial College London, LONDON, UNITED KINGDOM
Virial expansion, cluster expansion, two-connected graph, involution, Tonks gas, hard-core gas
Mayer’s second theorem in the context of a classical gasmodel allows us to write the coe fficients of the virial expansion of pressure in terms of weighted two-connected graphs. Labelle, Leroux and Ducharme studied the graph weights arising from the one-dimensional hardcore gas model and noticed that the sum of these weights over all two-connected graphs with $n$ vertices is $–n(n–2)!$. Th is paper addresses the question of achieving a purely combinatorial proof of this observation and extends the proof of Bernardi for the connected graph case.
Statistical mechanics, structure of matter
Combinatorics
229
262
10.4171/AIHPD/18
http://www.ems-ph.org/doi/10.4171/AIHPD/18
Clustering properties of rectangular Macdonald polynomials
Charles
Dunkl
University of Virginia, CHARLOTTESVILLE, UNITED STATES
Jean-Gabriel
Luque
Université de Rouen, SAINT-ÉTIENNE-DU-ROUVRAY CEDEX, FRANCE
Fractional quantum Hall effect, clustering properties, Macdonald polynomials, Hecke algebras, multivariate polynomials
The clustering properties of Jack polynomials are relevant in the theoretical study of the fractional Hall states. In this context, some factorization properties have been conjectured for the $(q,t)$-deformed problem involving Macdonald polynomials (which are also the quantum eigenfunctions of a familly of commuting di fference operators with signifi cance in the relativistic Ruijsenaars–Schneider model). The present paper is devoted to the proof of this formula. To this aim we use four families of Jack/Macdonald polynomials: symmetric homogeneous, nonsymmetric homogeneous, shifted symmetric and shifted nonsymmetric.
Combinatorics
Group theory and generalizations
Special functions
Quantum theory
263
307
10.4171/AIHPD/19
http://www.ems-ph.org/doi/10.4171/AIHPD/19
Veldkamp-space aspects of a sequence of nested binary Segre varieties
Metod
Saniga
Vienna University of Technology, WIEN, AUSTRIA
Hans
Havlicek
TU Wien, WIEN, AUSTRIA
Frédéric
Holweck
Université de Bourgogne Franche-Comté, BELFORT CEDEX, FRANCE
Michel
Planat
Institut FEMTO-ST, BESANÇON, FRANCE
Petr
Pracna
National Information Centre for European Research, PRAGUE 6, CZECH REPUBLIC
Binary Segre varietes, Veldkamp spaces, hyperbolic quadrics
Let $S_{(N)} \equiv \operatorname{PG}(1,\,2) \times \operatorname{PG}(1,\,2) \times \cdots \times \operatorname{PG}(1,\,2)$ be a Segre variety that is an $N$-fold direct product of projective lines of size three. Given two geometric hyperplanes $H'$ and $H''$ of $S_{(N)}$, let us call the triple $\{H', H'', \overline{H' \Delta H''}\}$ the Veldkamp line of $S_{(N)}$. We shall demonstrate, for the sequence $2 \leq N \leq 4$, that the properties of geometric hyperplanes of $S_{(N)}$ are fully encoded in the properties of Veldkamp {\it lines} of $S_{(N-1)}$. Using this property, a complete classification of all types of geometric hyperplanes of $S_{(4)}$ is provided. Employing the fact that, for $2 \leq N \leq 4$, the (ordinary part of) Veldkamp space of $S_{(N)}$ is $\operatorname{PG}(2^N-1,2)$, we shall further describe which types of geometric hyperplanes of $S_{(N)}$ lie on a certain hyperbolic quadric $\mathcal{Q}_0^+(2^N-1,2) \subset \operatorname{PG}(2^N-1,2)$ that contains the $S_{(N)}$ and is invariant under its stabilizer group; in the $N=4$ case we shall also single out those of them that correspond, via the Lagrangian Grassmannian of type $LG(4,8)$, to the set of 2295 maximal subspaces of the symplectic polar space $\mathcal{W}(7,2)$.
Geometry
Linear and multilinear algebra; matrix theory
Group theory and generalizations
Quantum theory
309
333
10.4171/AIHPD/20
http://www.ems-ph.org/doi/10.4171/AIHPD/20