We construct a monomial basis of the positive part U of the quantized enveloping algebra associated to a finite–dimensional simple Lie algebra. As an application we give a simple proof of the existence and uniqueness of the canonical basis of U. 0. Introduction In [L1], Lusztig showed that the positive part U of the quantized enveloping algebra associated to a finite–dimensional simple Lie alge...

Let F be a field and let G be a finite graph with a total ordering on its edge set. Richard Stanley noted that the Stanley-Reisner ring F (G) of the broken circuit complex of G is CohenMacaulay. Jason Brown gave an explicit description of a homogeneous system of parameters for F (G) in terms of fundamental cocircuits in G. So F (G) modulo this hsop is a finite dimensional vector space. We conje...

This paper addresses the problem of eecient construction of monomial bases for the coordinate rings of zero-dimensional varieties. Existing approaches rely on Grr ob-ner bases methods { in contrast, we make use of recent developments in sparse elimination techniques which allow us to strongly exploit the structural sparseness of the problem at hand. This is done by establishing certain properti...

Using the Beilinson-Lusztig-MacPherson construction of the quantized enveloping algebra of gln and its associated monomial basis, we investigate q-Schur algebras Sq(n, r) as “little quantum groups”. We give a presentation for Sq(n, r) and obtain a new basis for the integral q-Schur algebra Sq(n, r), which consists of certain monomials in the original generators. Finally, when n > r, we interpre...

We construct linear operators factorizing the three bases of symmetric polynomials: monomial symmetric functions mλ(x), elementary symmetric polynomials Eλ(x), and Schur functions sλ(x), into products of univariate polynomials.

We construct linear operators factorizing the three bases of symmetric polynomials: monomial symmetric functions mλ(x), elementary symmetric polynomials Eλ(x), and Schur functions sλ(x), into products of univariate polynomials.

We investigate Gröbner bases of contraction ideals under monomial homomorphisms. As an application, we generalize the result of Aoki–Hibi–Ohsugi– Takemura and Ohsugi–Hibi for toric ideals of nested configurations.

A new O(n) algorithm is given for evaluating univariate polynomials of degree n in the P olya basis. Since the Lagrange, Bernstein, and monomial bases are all special instances of the P olya basis, this technique leads to e cient evaluation algorithms for these special bases. For the monomial basis, this algorithm is shown to be equivalent to Horner's rule. 1 P olya basis functions Let nk(t) be...

In the ring of symmetric functions the inverse Kostka matrix appears as the transition matrix from the bases given by monomial symmetric functions to the Schur bases. We present both a combinatorial characterization and a recurrent formula for the entries of the inverse Kostka matrix which are different from the results obtained by Egecioglu and Remmel [ER] in 1990. An application to the topolo...