- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 12:05:23
9
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=CMH&vol=79&iss=1&update_since=2024-03-29
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
79
2004
1
Pieri-type formulas for the non-symmetric Jack polynomials
P.
Forrester
University of Melbourne, PARKVILLE, VIC, AUSTRALIA
D.
McAnally
University of Melbourne, PARKVILLE, VIC, AUSTRALIA
Jack polynomials, Pieri-type formulas, recurrence relations
In the theory of symmetric Jack polynomials the coefficients in the expansion of the $p$th elementary symmetric function $e_p(z)$ times a Jack polynomial expressed as a series in Jack polynomials are known explicitly. Here analogues of this result for the non-symmetric Jack polynomials $E_\eta(z)$ are explored. Necessary conditions for non-zero coefficients in the expansion of $e_p(z) E_\eta(z)$ as a series in non-symmetric Jack polynomials are given. A known expansion formula for $z_i E_\eta(z)$ is rederived by an induction procedure, and this expansion is used to deduce the corresponding result for the expansion of $\prod_{j=1, \, j\ne i}^N z_j \, E_\eta(z)$, and consequently the expansion of $e_{N-1}(z) E_\eta(z)$. In the general $p$ case the coefficients for special terms in the expansion are presented.
Special functions
General
1
24
10.1007/s00014-003-0789-2
http://www.ems-ph.org/doi/10.1007/s00014-003-0789-2
Formes différentielles abéliennes, bornes de Castelnuovo et géométrie des tissus
Alain
Hénaut
Université de Bordeaux I et C.N.R.S., TALENCE CEDEX, FRANCE
Web geometry, analytic algebraic geometry, Abelian differentials
A $d$-web ${\Cal W}(d)$ is given by $d$ complex analytic foliations of codimension $n$ in $({\sumbbb C}^N,0)$ such that the leaves are in general position. We are interested in the geometry of such configurations. A complex $({\Cal A}^{\bullet},\delta)$ of ${\sumbbb C}$-vector spaces is defined in which ${\Cal A}^0$ corresponds to functions and ${\Cal A}^p$ to $p$-forms of the web ${\Cal W}(d)$ for $1\leq p\leq n$. If $N=kn$ with $k\geq 2$, it is proved that $r_p:=\dim_{\,\sumbbb C}{\Cal A}^p$ is a finite analytic invariant of ${\Cal W}(d)$ with an optimal upper bound $\pi_{p}(d,k,n)$ for $0\leq p\leq n$. These bounds generalize the Castelnuovos ones for genus of curves in ${\sumbbb P}^{k}$ with degree $d$. Some characterization of the the space $H^0(V_n,\omega^p_{V_n})$ of abelian differentials to an algebraic variety $V_n$ in ${\sumbbb P}^{n+k-1}$ of pure dimension $n$ with degree $d$ is given. Moreover, using duality and Abels theorem, we investigate how for suitable $V_n$ the natural complex $\bigr(H^0(V_n,\omega^{\bullet}_{V_n}),d\,\bigr)$ and the abelian relation complex $({\Cal A}^{\bullet},\delta)$ of the linear web associated to $V_n$ in $({\sumbbb C}^{kn},0)$ are related.
Differential geometry
Algebraic geometry
Several complex variables and analytic spaces
General
25
57
10.1007/s00014-003-0787-4
http://www.ems-ph.org/doi/10.1007/s00014-003-0787-4
Commutator length of symplectomorphisms
Michael
Entov
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Commutator length, quasimorphism, symplectic manifold, Hamiltonian symplectomorphism, quantum cohomology, Floer homology
Each element $x$ of the commutator subgroup $[G, G]$ of a group $G$ can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of $x$. The commutator length of $G$ is defined as the supremum of commutator lengths of elements of $[G, G]$. We show that for certain closed symplectic manifolds $(M,\omega)$, including complex projective spaces and Grassmannians, the universal cover $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ of the group of Hamiltonian symplectomorphisms of $(M,\omega)$ has infinite commutator length. In particular, we present explicit examples of elements in $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ that have arbitrarily large commutator length -- the estimate on their commutator length depends on the multiplicative structure of the quantum cohomology of $(M,\omega)$. By a different method we also show that in the case $c_1 (M) = 0$ the group $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ and the universal cover ${\widetilde{\Symp}}_0\, (M,\omega)$ of the identity component of the group of symplectomorphisms of $(M,\omega)$ have infinite commutator length.
Differential geometry
General
58
104
10.1007/s00014-001-0799-0
http://www.ems-ph.org/doi/10.1007/s00014-001-0799-0
Structure in the classical knot concordance group
Tim
Cochran
Rice University, HOUSTON, UNITED STATES
Kent
Orr
Indiana University, BLOOMINGTON, UNITED STATES
Peter
Teichner
University of California, BERKELEY, UNITED STATES
Knot concordance, von Neumann signatures, Blanchfield pairing, Casson-Gordon invariants
We provide new information about the structure of the abelian group of topological concordance classes of knots in $S^3$. One consequence is that there is a subgroup of infinite rank consisting entirely of knots with vanishing Casson-Gordon invariants but whose non-triviality is detected by von Neumann signatures.
Manifolds and cell complexes
General
105
123
10.1007/s00014-001-0793-6
http://www.ems-ph.org/doi/10.1007/s00014-001-0793-6
A geometric construction of the Conway potential function
David
Cimasoni
Université de Genève, GENÈVE 4, SWITZERLAND
Conway potential function, Conway polynomial, colored link, C-complex
We give a geometric construction of the multivariable Conway potential function for colored links. In the case of a single color, it is Kauffman's definition of the Conway polynomial in terms of a Seifert matrix.
Manifolds and cell complexes
General
124
146
10.1007/s00014-003-0777-6
http://www.ems-ph.org/doi/10.1007/s00014-003-0777-6
Noether's problem for dihedral 2-groups
HUAH
CHU
NATIONAL TAIWAN UNIVERSITY, TAIPEI, TAIWAN
SHOU-JEN
HU
TAMKANG UNIVERSITY, TAIPEI, TAIWAN
MING-CHANG
KANG
NATIONAL TAIWAN UNIVERSITY, TAIPEI, TAIWAN
Rationality, Noether's problem, generic Galois extension, generic polynomials, dihedral groups
Let $K$ be any field and $G$ be a finite group. Let $G$ act on the rational function field $K(x_g: \, g \in G)$ by $K$-automorphisms defined by $g \cdot x_h= x _{gh}$ for any $g, \, h \in G$. Denote by $K(G)$ the fixed field $K(x_g: \, g \in G)^G$. Noethers problem asks whether $K(G)$ is rational (= purely transcendental) over $K$. We shall prove that $K(G)$ is rational over $K$ if $G$ is the dihedral group (resp. quasi-dihedral group, modular group) of order 16. Our result will imply the existence of the generic Galois extension and the existence of the generic polynomial of the corresponding group.
Field theory and polynomials
Number theory
Commutative rings and algebras
Algebraic geometry
147
159
10.1007/s00014-003-0783-8
http://www.ems-ph.org/doi/10.1007/s00014-003-0783-8
Auslander-Reiten theory over topological spaces
Peter
Jørgensen
University of Newcastle, NEWCASTLE UPON TYNE, UNITED KINGDOM
Auslander-Reiten triangle, Auslander-Reiten quiver, cochain differential graded algebra, Poincaré duality, topological space, sphere
Auslander-Reiten triangles and quivers are introduced into algebraic topology. It is proved that the existence of Auslander-Reiten triangles characterizes Poincaré duality spaces, and that the Auslander-Reiten quiver is a weak homotopy invariant. The theory is applied to spheres whose Auslander-Reiten triangles and quivers are computed. The Auslander-Reiten quiver over the $d$-dimensional sphere turns out to consist of $d-1$ copies of ${\mathbb Z} A_{\infty}$. Hence the quiver is a sufficiently sensitive invariant to tell spheres of different dimension apart.
Algebraic topology
Associative rings and algebras
General
160
182
10.1007/s00014-001-0795-4
http://www.ems-ph.org/doi/10.1007/s00014-001-0795-4
A new critical pair theorem applied to sum-free sets in Abelian groups
Yahya ould
Hamidoune
Université Pierre et Marie Curie, PARIS, FRANCE
Alain
Plagne
Ecole Polytechnique, PALAISEAU CEDEX, FRANCE
Additive number theory, Vosper theorem, $(k,l)$-free sets, structure theorem
We shall prove a new generalization of Vosper critical pair theorem to finite Abelian groups. We next apply this new tool to the theory of $(k,l)$-free sets in finite Abelian groups. In particular, in most cases, we describe the structure of maximal $(k,l)$-free sets and determine the maximal cardinality of such a set. This result allows us for instance to give precisions on an old result of Yap: we are able to describe completely the maximal sum-free sets with cardinality at least one third of that of the ambient group.
Number theory
Group theory and generalizations
General
183
207
10.1007/s00014-003-0786-5
http://www.ems-ph.org/doi/10.1007/s00014-003-0786-5
Quaternions, octonions and the forms of the exceptional simple classical Lie superalgebras
Alberto
Elduque
Universidad de Zaragoza, ZARAGOZA, SPAIN
Lie superalgebra, exceptional, forms, quaternions, octonions, Tits construction
The forms of the exceptional simple classical Lie superalgebras are determined over arbitrary fields of characteristic $\ne 2,3$.
Nonassociative rings and algebras
General
208
228
10.1007/s00014-003-0790-9
http://www.ems-ph.org/doi/10.1007/s00014-003-0790-9