In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of t-minors of a generic pluri-circulant matrix is a transversal monomial ideal . Using a Gröbner basis for this ideal, it is sho...

We characterize componentwise linear monomial ideals with minimal Taylor resolution and consider the lower bound for the Betti numbers of componentwise linear ideals. INTRODUCTION Let S = K[x1, . . . ,xn] denote the polynomial ring in n variables over a field K with each degxi = 1. Let I be a monomial ideal of S and G(I) = {u1, . . . ,us} its unique minimal system of monomial generators. The Ta...

We show that all monomial ideals in the polynomial ring in at most 3 variables are pretty clean and that an arbitrary monomial ideal I is pretty clean if and only if its polarization I p is clean. This yields a new characterization of pretty clean monomial ideals in terms of the arithmetic degree, and it also implies that a multicomplex is shellable if and only the simplicial complex correspond...

Let Λ be a finite-dimensional (D, A)-stacked monomial algebra. In this paper, we give necessary and sufficient conditions for the variety of a simple Λ-module to be nontrivial. This is then used to give structural information on the algebra Λ, as it is shown that if the variety of every simple module is nontrivial, then Λ is a D-Koszul monomial algebra. We also provide examples of (D, A)-stacke...

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...

The algebra Sym of symmetric functions is a proper subalgebra of QSym: for example, M11 and M12 +M21 are symmetric, but M12 is not. As an algebra, QSym is generated by those monomial symmetric functions corresponding to Lyndon words in the positive integers [11, 6]. The subalgebra of QSym ⊂ QSym generated by all Lyndon words other than M1 has the vector space basis consisting of all monomial sy...

Monomial ideals form an important link between commutative algebra and combinatorics. In this chapter, we demonstrate how to implement algorithms in Macaulay 2 for studying and using monomial ideals. We illustrate these methods with examples from combinatorics, integer programming, and algebraic geometry.

Let K be a field of characteristic 0 containing all roots of unity. We classified all the Hopf structures on monomial K-coalgebras, or, in dual version, on monomial K-algebras.

In analogy to the disjoint cycle decomposition in permutation groups, Ore and Specht define a of elements full monomial group exploit this describe conjugacy classes centralisers group. We generalise their results wreath products whose base need not be finite top acts faithfully on set. parameterise such explicitly. For products, our approach yields efficient algorithms for finding conjugating ...