Let U − q (g) be the negative part of the quantized universal enveloping algebra constructed from a Cartan matrix associated to a complex semisimple Lie algebra g. Let λ be a dominant integral weight and V (λ) the irreducible U − q (g)-module with highest weight λ. There is on the one hand the canonical basis for U − q (g) [4, 12, 13] and on the other the standard monomial theoretic basis for t...

w x Hodge and Pedoe 5 constructed bases of the homogeneous coordinate rings of the Schubert varieties in the Grassmannian, consisting of certain products of Plucker coordinates called standard monomials. Lakshmibai, ̈ w x Musili, and Seshadri 6 have generalized this theory extensively, giving standard monomial bases of the spaces of global sections of certain line bundles over unions of Schubert...

Abstract. In this paper we study standard bases for submodules of K[[t1, . . . , tm]][x1, . . . , xn] s respectively of their localisation with respect to a t-local monomial ordering. The main step is to prove the existence of a division with remainder generalising and combining the division theorems of Grauert and Mora. Everything else then translates naturally. Setting either m = 0 or n = 0 w...

The monomial initial ideals of a graded polynomial ideal are in bijection with the vertices of a convex polytope known as the state polytope of the ideal. The Grr obner fan of the ideal is the normal fan of its state polytope. In this paper we present a software system called TiGERS (Toric Grr obner bases Enumeration by Reverse Search) for computing the Grr obner fan of a toric ideal by enumera...

Egecioglu, ii. and J.B. Remmel, Brick tabloids and the connection matrices between bases of symmetric functions, Discrete Applied Mathematics 34 (1991) 107-120. Let H, denote the space of symmetric functions, homogeneous of degree n. In this paper we introduce a new set of combinatorial objects called I-brick tabloids and its variants, which we use *to give combinatorial interpretations of the ...

We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around the coordinate hyperplanes and trivial monodromy around all other hyperplanes. In the case where the local system is equivariant for the symmetric group, we ...

We study the relationship between certain Gröbner bases for zerodimensional 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...

We combine the theory of Pommaret bases with a (slight generalisation of a) recent construction by Sköldberg based on discrete Morse theory. This combination allows us the explicit determination of a (generally non-minimal) free resolution for a graded polynomial module with the computation of only one Pommaret basis. If only the Betti numbers are needed, one can considerably simplify the compu...

For uniform meshes it is shown that any linear fc-step formula can be formulated so that only k values need to be saved between steps. By saving additional m values it is possible to construct local polynomial approximations of degree k + m — 1, which can be used as predictor formulas. Different polynomial bases lead to different equivalent forms of multistep formulas. In particular, local mono...

In this paper we introduce a new type of monomial division called Janet-like, since its properties are similar to those of Janet division. We show that the former division improves the latter one. This means that a Janet divisor is always a Janet-like divisor but the converse is generally not true. Though Janet-like division is not involutive, it preserves all algorithmic merits of Janet divisi...