Since it is well-known that the Vandermonde matrix is ill-conditioned, while the interpolation itself is not unstable in function space, this paper surveys the choices of other new bases. These bases are data-dependent and are categorized into discretely l2-orthonormal and continuously L2-orthonormal bases. The first one construct a unitary Gramian matrix in the space l2(X) while the late...

Classically, Gröbner bases are computed by first prescribing a fixed monomial order. Moss Sweedler suggested an alternative in the mid 1980s and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on K(x, y) that are suitable for this framework. We then perform such computations for i...

Monomial representations and operations for Grr obner bases computations are investigated from an implementation point of view. The technique of vectorized monomial operations is introduced and it is shown how it expedites computations of Grr obner bases. Furthermore, a rank-based monomial representation and comparison technique is examined and it is concluded that this technique does not yield...

In this paper, we develop the Gr6bner-Shirshov basis theory for the representations of associative algebras by introducing the notion of Gr6bner-Shirshov pairs. Our result can be applied to solve the reduction problem in representation theory and to construct monomial bases of representations of associative algebras. As an illustration, we give an explicit construction of Gr6bner-Shirshov pairs...

in this article, we introduce monomial irreducible representations of the special linear lie algebra $sln$. we will show that this kind of representations have bases for which the action of the chevalley generators of the lie algebra on the basis elements can be given by a simple formula.

We describe a recursive algorithm that produces an integral basis for the centre of the Hecke algebra of type A consisting of linear combinations of monomial symmetric polynomials of Jucys–Murphy elements. We also discuss the existence of integral bases for the centre of the Hecke algebra that consist solely of monomial symmetric polynomials of Jucys–Murphy elements. Finally, for n = 3, we show...

In this article, we introduce monomial irreducible representations of the special linear Lie algebra $sln$. We will show that this kind of representations have bases for which the action of the Chevalley generators of the Lie algebra on the basis elements can be given by a simple formula.

We consider two types of quotients of the integrable modules of sl2. These spaces of coinvariants have dimensions described in terms of the Verlinde algebra of level-k. We describe monomial bases for the spaces of coinvariants, which leads to a fermionic description of these spaces. For k = 1, we give the explicit formulas for the characters. We also present recursion relations satisfied by the...