Regular Embeddings Research Articles

Cohomology of regular embeddings

We give in this paper an explicit description of the rational cohomology ring of a “complete symmetric variety” or regular compactification of an algebraic symmetric variety. The motivation for this computation comes from the desire to describe an explicit Schubert calculus for these varieties, which have been considered since the work of Chasles and Schubert in several special cases. The technique we use is the result of our attempts to understand better the work of Jurkiewicz and Danilov in the case of torus embeddings. The combinatorial presentation of the cohomology ring of a smooth torus embedding finds a natural explanation and proof using the language of equivariant cohomology. The key point is that the Reisner-Stanley algebra of the torus embedding coincides with its equivariant cohomology ring. We consider a class of spaces, regular embeddings, that contains both “complete symmetric varieties” and smooth torus embeddings. By suitably generalizing the notion of Reisner-Stanley algebra, we are able to compute the rational equivariant cohomology ring of these spaces in an explicit way.

Vertex operator representations of basic affine superalgebras

Using the basic construction of the affine superalgebra OSp(M‖N)(1) in terms of vertex operators, vertex operator representations of a large class of untwisted and twisted affine basic superalgebras of the unitary and orthosymplectic series are found. The construction uses two methods: first, the regular embeddings of semisimple superalgebras (untwisted case) and, second, the folding method developed in a previous paper [Frappat et al., Nucl. Phys. B 305, 164 (1988)] to construct twisted vertex operators for Lie algebras with fermionic elementary fields (twisted case).

Regular embeddings of coloured digraphs

A particular kind of 2-cell embeddings, called regular, for arc-coloured digraphs is introduced, and a method for constructing both orientable and non-orientable regular embeddings is presented. Furthermore, by using combinatorial concepts and the Euler-Poincare formula, we derive upper bounds for both the orientable and non-orientable genera of such arc-coloured digraphs.

The extension of regular and rational embeddings

The extension of regular and rational embeddings

Regular embeddings of a graph

In this paper we study embeddings of a graph G in Euclidean space R that are 'regular' in the following sense: given any two distinct vertices u and v of G, the distance between the corresponding points in R equals a if u and v are adjacent, and equals β otherwise. It is shown that for any given value of s — (β2 — a2)/β29 the minimum dimension of a Euclidean space in which G is regularly embeddable is determined by the characteristic polynomials of G and (7.

SU(3) c × U(1) em real fermions in regular embeddings of color and charge

The reality under the SU(3) c × U(1) em subgroup of complex fermion representations of SU( N) is studied for regular embeddings of color and charge: Q = diag( a, a, a, b, b − 1, c 1, c 2,… c N−5 ) for a fundamental representation. Also, complex spinor representations of O(4 n + 2) are studied. The SU(5) charge assignment ( − 1 3 , − 1 3 , − 1 3 ,1,0,0,…,0 ) is always real in SU(3) c × U(1) em. We find that the spinor representations of SU(2 n + 1) with n ⩾ 3 can allow more general charge assignments.

Remarks on the requirement matrix of school timetable problems and regular embeddings of graphs

Some properties and extensions of a construction given by Papoulias and Panayiotopoulos are discussed. A graph-theoretical interpretation is mentioned.

Regular Embeddings which Preserve Lattice Structure

Regular Embeddings which Preserve Lattice Structure

Regular embeddings which preserve lattice structure

A lattice L is said to be regularly embedded in a complete lattice K if there exists an isomorphic mapping of L into K which preserves all least upper and greatest lower bounds that exist in L. With the MacNeille completion by cuts every lattice can be regularly embedded in a complete lattice. And if the lattice is a Boolean algebra, then its completion by cuts is also a Boolean algebra. In general, however, this embedding does not preserve lattice structure; there are examples of distributive lattices whose completions by cuts are not even modular. On the other hand every lattice L can be embedded in the complete lattice of its ideals which satisfies every identity that holds in L. This embedding preserves greatest lower bounds but does not preserve infinite least upper bounds. Recently A. Horn has asked whether or not any distributive lattice can be regularly embedded in some complete distributive lattice. Here this question is answered in the negative. We show with an example that there exists a distributive lattice which cannot be regularly embedded in a complete modular lattice. What appears to be most important in the example is the lack of certain continuity laws. If we impose such a law on a lattice, then a structure preserving regular embedding is possible. Following [2] let us define a subset A of a lattice L to be a complete ideal if for every element aEL and every subset SCA for which US exists in L we have