Tight monomials and the monomial basis property

More Tight Monomials in Quantized Enveloping Algebras

On Tight Monomials in Quantized Enveloping Algebras

In this paper, the author studies when some monomials are in the canonical basis of the quantized enveloping algebra corresponding to a simply laced semisimple finite dimensional complex Lie algebra. 0. Introduction To any graph Γ, Lusztig has associated in [L1] and [L2] an algebra U− over Z[v, v−1] provided with a canonical basis B. In the case that Γ is the Dynkin graph of a simply laced semi...

A Complete and Tight Average-Case Analysis of Learning Monomials

We advocate to analyze the average complexity of learning problems. An appropriate framework for this purpose is introduced. Based on it we consider the problem of learning monomials and the special case of learning monotone monomials in the limit and for on-line predictions in two variants: from positive data only, and from positive and negative examples. The well-known Wholist algorithm is co...

Minimal Gröbner bases and the predictable leading monomial property

We focus on Gröbner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the " predictable leading monomial (PLM) property " that is shared by minimal Gröbner bases of modules in F[x] q , no matter what positional term order is used. The PLM property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable deg...

Monomial Ideals, Almost Complete Intersections and the Weak Lefschetz Property

has maximal rank, i.e. it is injective or surjective. In this case, the linear form L is called a Lefschetz element of A. (We will often abuse notation and say that the corresponding ideal has the WLP.) The Lefschetz elements of A form a Zariski open, possibly empty, subset of (A)1. Part of the great interest in the WLP stems from the fact that its presence puts severe constraints on the possib...