Related papers: Minors of symmetric and exterior powers
We define symmetric and exterior powers of categories, fitting into categorified Koszul complexes. We discuss examples and calculate the effect of these power operations on the categorical characters of matrix 2-representations.
We give criteria for graded ideals to have the property that all their powers are componentwise linear. Typical examples to which our criteria can be applied include the vertex cover ideals of certain finite graphs.
We present a close relationship between matching number, covering numbers and their fractional versions in combinatorial optimization and ordinary powers, integral closures of powers, and symbolic powers of monomial ideals. This…
This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial…
We establish characteristic-free criteria for the componentwise linearity of graded ideals. As applications, we classify the componentwise linear ideals among the Gorenstein ideals, the standard determinantal ideals, and the ideals…
We characterize monomial ideals which are intersections of monomial prime ideals and study classes of ideals with this property, among them polymatroidal ideals.
The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Discrete 1-matrix, 2-matrix, ``conformal'' (multicomponent) and Kontsevich models are considered in some detail, together with the…
This paper is about the local geometry of a real surfaces. It introduces machinery for studying families of subsets which are determined by conditions which are similar to base conditions, but also involve positivity/non-negativity. The…
We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic…
In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the $n$-th symbolic power of a given prime ideal consists of the…
We give a description of the minimal primes of the ideal generated by the 2 x 2 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent m x m minors of an m x n generic matrix when the…
In this paper, we study various properties of matroidal ideals.
Symbolic powers are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph…
An ideal I in a polynomial ring S has linear powers if all the powers I^k of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required…
One studies certain degenerations of the generic square matrix over a field $k$ along with its main related structures, such as the determinant of the matrix, the ideal generated by its partial derivatives, the polar map defined by these…
This work is about symbolic powers of codimension two perfect ideals in a standard polynomial ring over a field, where the entries of the corresponding presentation matrix are general linear forms. The main contribution of the present…
In this paper we obtain some statements concerning ideals of polynomials and apply these results in a number of different situations. Among other results, we present new characterizations of $\mathcal{L}_{\infty}$-spaces, Coincidence…
This paper investigates the symbolic powers of toric ideals. We first describe them in terms of the kernel of certain linear maps derived from the lattice structure of the toric ideal. Furthermore, we apply our results to show that symbolic…
The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Determinantal formulas, relation to conformal field models and the theory of Generalized Kontsevich model are discussed in some…
We study a class of determinantal ideals arising from conditional independence (CI) statements with hidden variables. Such CI statements translate into determinantal conditions on a matrix whose entries represent the probabilities of events…