Triality for Homogeneous Polynomials

Through the triality of SO(8,C), we study three interrelated homogeneous basis of the ring of invariant polynomials of Lie algebras, which give the basis of three Hitchin fibrations, and identify the explicit automorphisms that relate them.

A generalization of Kruskal–Katona’s theorem

Let K be a field, E the exterior algebra of a finite dimensional K-vector space, and F a finitely generated graded free E-module with homogeneous basis g1, . . ., gr such that deg g1 ≤ deg g2 ≤ · · · ≤ deg gr. We characterize the Hilbert functions of graded E–modules of the type F/M, with M graded submodule of F. The existence of a unique lexicographic submodule of F with the same Hilbert function as M plays a crucial role.

Stability properties of Plethysm: new approach with combinatorial proofs (Extended abstract)

International audience Plethysm coefficients are important structural constants in the theory of symmetric functions and in the representations theory of symmetric groups and general linear groups. In 1950, Foulkes observed stability properties: some sequences of plethysm coefficients are eventually constants. Such stability properties were proven by Brion with geometric techniques and by Thibon and Carré by means of vertex operators. In this paper we present a newapproach to prove such stability properties. This new proofs are purely combinatorial and follow the same scheme. We decompose plethysm coefficients in terms of other plethysm coefficients (related to the complete homogeneous basis of symmetric functions). We show that these other plethysm coefficients count integer points in polytopes and we prove stability for them by exhibiting bijections between the corresponding sets of integer points of each polytope. Les coefficients du pléthysme sont des constantes de structure importantes de la théorie des fonctions symétriques, ainsi que de la théorie de la représentation des groupes symétriques et des groupes généraux linéaires. En 1950, Foulkes a observé pour ces coefficients de phénomènes de stabilité: certaines suites de coefficients du pléthysme sont stationnaires. De telles propriétés ont été démontrées par Brion, au moyen de techniques géométriques, et par Thibon et Carré, au moyen d’opérateurs vertex. Dans ce travail, nous présentons une nouvelle approche, purement combinatoire, pour démontrer des propriétés de stabilité de ce type. Nous décomposons les coefficients du pléthysme comme somme alternées de coefficients de pléthysme d’un autre type (liés à la base des fonctions symétriques sommes complètes), qui comptent les points entiers dans des polytopes. Nous démontrons la stabilité des suites de ces coefficients en exhibant des bijections entres les ensembles de points entiers des polytopes correspondants.

A Schur-Like Basis of NSym Defined by a Pieri Rule

Recent research on the algebra of non-commutative symmetric functions and the dual algebra of quasi-symmetric functions has explored some natural analogues of the Schur basis of the algebra of symmetric functions. We introduce a new basis of the algebra of non-commutative symmetric functions using a right Pieri rule. The commutative image of an element of this basis indexed by a partition equals the element of the Schur basis indexed by the same partition and the commutative image is $0$ otherwise. We establish a rule for right-multiplying an arbitrary element of this basis by an arbitrary element of the ribbon basis, and a Murnaghan-Nakayama-like rule for this new basis. Elements of this new basis indexed by compositions of the form $(1^n, m, 1^r)$ are evaluated in terms of the complete homogeneous basis and the elementary basis.

Nonhomogeneous Parking Functions and Noncrossing Partitions

For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in terms of the (reduced) type of $k$-divisible noncrossing partitions. Our work extends Haiman's notion of a parking function symmetric function.

Perfectly Homogeneous Bases in Banach Spaces

A bounded basis {xn} of a Banach space X is called perfectly homogeneous if every bounded block basic sequence {yn} of {xn} is equivalent to {xn}. By a result of M. Zippin [4], a basis in a Banach space is perfectly homogeneous if and only if it is equivalent to the unit vector basis of c0 or lp, 1 ≤ p < + ∞. A basis {xn} of a Banach space X is called symmetric, if every permutation {xσ(n)} of {xn} is a basis of X, equivalent to the basis {xn}. It is clear that every perfectly homogeneous basis is symmetric.

Hjelmslev Planes Derived from Modular Lattices

In several papers, W. Klingenberg has elaborated the connections between Hjelmslev planes and a class of rings, called H-rings (4; 5; 6), which are rings of coordinates for the corresponding Hjelmslev planes. Certain homomorphic images of valuation rings are examples of H-rings. In these examples, the lattice of (right) ideals of the ring, say R,is a chain, and the coordinatization of the corresponding Hjelmslev plane yields a natural embedding of the plane in the lattice L(R3) of (right) submodules of the module R3. Now, L(R3) is a modular lattice with a homogeneous basis of order 3 given by the submodules a1 = (1, 0, 0)R, a2 = (0, 1, 0)R, a2 = (0, 0, 1)R, and the sublattices L(N, ai) of elements less than or equal to ai are chains. Forgetting about the ring, we find ourselves in the situation of a problem suggested by Skornyakov (7, Problem 23, p. 166), namely, to study modular lattices with a homogeneous basis of chains. Baer (2) and Inaba (3) investigated lattices of this kind with Desarguesian properties and assuming that the chains L(N, ai) were finite. Representations of the lattices by means of certain rings can be found in both articles.

Some Examples of Complemented Modular Lattices

Let L be a complemented, χ-complete modular lattice. A theorem of Amemiya and Halperin (see [l], Theorem 4.3) asserts that if the intervals [O, a] and [O, b], a, bεL, are upper χ-continuous then [O, a∪b] is also upper χ-continuous. Roughly speaking, in L upper χ-continuity is additive. The following question arises naturally: is χ-completeness an additive property of complemented modular lattices? It follows from Corollary 1 to Theorem 1 below that the answer to this question is in the negative.A complemented modular lattice is called a Von Neumann geometry or continuous geometry if it is complete and continuous. In particular a complete Boolean algebra is a Von Neumann geometry. In any case in a Von Neumann geometry the set of elements which possess a unique complement form a complete Boolean algebra. This Boolean algebra is called the centre of the Von Neumann geometry. Theorem 2 shows that any complete Boolean algebra can be the centre of a Von Neumann geometry with a homogeneous basis of order n (see [3] Part II, definition 3.2 for the definition of a homogeneous basis), n being any fixed natural integer.