Complementary Algorithms For Tableaux

We study four operations defined on pairs of tableaux. Algorithms for the first three involve the familiar procedures of jeu de taquin, row insertion, and column insertion. The fourth operation, hopscotch, is new, although specialised versions have appeared previously. Like the other three operations, this new operation may be computed with a set of local rules in a growth diagram, and it preserves Knuth equivalence class. Each of these four operations gives rise to an a priori distinct theory of dual equivalence. We show that these four theories coincide. The four operations are linked via the involutive tableau operations of complementation and conjugation.Comment: 29 pages, 52 .eps files for figures, JCTA, to appea

Gröbner basis structure of finite sets of points

We study the relationship between certain Gröbner bases for zero-dimensional radical ideals, and the varieties defined by the ideals. Such a variety is a finite set of points in an affine n-dimensional space. We are interested in monomial orders that “eliminate ” one variable, say z. Eliminating z corresponds to projecting points in n-space to (n − 1)-space by discarding the z-coordinate. We show that knowing a minimal Gröbner basis under an elimination order immediately reveals some of the geometric structure of the corresponding variety, and knowing the variety makes available information concerning the basis. These relationships can be used to decompose polynomial systems into smaller systems