Related papers: Relations between the minors of a generic matrix
A set of polynomials G in a polynomial ring S over a field is said to be a universal Groebner basis, if G is a Groebner basis with respect to every term order on S. Twenty years ago Bernstein, Sturmfels, and Zelevinsky proved that the set…
Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph theoretical terms the primitive, the minimal, the indispensable and the fundamental binomials of the toric ideal $I_G$.
A standard tool for classifying the complexity of equivalence relations on $\omega$ is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce…
Let $R$ be a local principal ideal ring of length two, for example, the ring $R=\Z/p^2\Z$ with $p$ prime. In this paper we develop a theory of normal forms for similarity classes in the matrix rings $M_n(R)$ by interpreting them in terms of…
The purpose of this article is to study determinants of matrices which are known as generalized Pascal triangles (see [1]). We present a factorization by expressing such a matrix as a product of a unipotent lower triangular matrix, a…
Minimal representations of a real reductive group G are the "smallest" irreducible unitary representations of G. The author suggests a program of global analysis built on minimal representations from the philosophy: small representation of…
Despite the popularity of low-rank matrix completion, the majority of its theory has been developed under the assumption of random observation patterns, whereas very little is known about the practically relevant case of non-random…
For positive integers $n,n'$, we give a combinatorial characterization for the set of quadratic inequalities on minors that are valid for all $n\times n'$ totally nonnegative matrices. This is obtained as a consequence from our earlier…
We study relations between two log minimal models of a fixed lc pair. For any two log minimal models of an lc pair constructed with log MMP, we prove that there are small birational models of the log minimal models which can be connected by…
Permutation matrices play a key role in matching and assignment problems across the fields, especially in computer vision and robotics. However, memory for explicitly representing permutation matrices grows quadratically with the size of…
We obtain minimal dimension matrix representations for each indecomposable five-dimensional Lie algebra over $\R$ and justify in each case that they are minimal. In each case a matrix Lie group is given whose matrix Lie algebra provides the…
In this paper, a model is proposed to solve the gauge hierarchy problem. Beyond the standard model, we introduce an extra scalar field that non-minimally couples to gravity. The fundamental scale is set at weak scale and Planck scale…
Consider the affine space consisting of pairs of matrices $(A,B)$ of fixed size, and its closed subvariety given by the rank conditions $\operatorname{rank} A \leq a$, $\operatorname{rank} B \leq b$ and $\operatorname{rank} (A\cdot B) \leq…
Let $R$ be a ring with unity. The upper ideal relation graph $\Gamma_U(R)$ of the ring $R$ is a simple undirected graph whose vertex set is the set of all non-unit elements of $R$ and two distinct vertices $x, y$ are adjacent if and only if…
We investigate the space of syzygies of the apolar ideals $\det_n^\perp$ and ${\rm perm}_n^\perp$ of the determinant $\det_n$ and permanent ${\rm perm}_n$ polynomials. Shafiei had proved that these ideals are generated by quadrics and…
The main theorem of the paper allows to generalize a class of identities among the quantum minors for quantum linear groups to similar identities but with the row labels of the quantum minors involved permuted.
Gauge theories describe the interactions of the fundamental building blocks of nature with great success. The Standard Model achieves a partial unification of the electromagnetic and weak interactions, and it also acomodates the strong…
We determine sufficient conditions for certain classes of $(n+k) \times n$ matrices $E$ to have all order-$n$ minors to be nonzero. For a special class of $(n+1) \times n$ matrices $E,$ we give the formula for the order-$n$ minors. As an…
Over a field of characteristic $0$, we construct a minimal set of generators of the defining ideals of closures of nilpotent conjugacy class in the set of $n \times n$ matrices. This modifies a conjecture of Weyman and provides a complete…
In this chapter, we discuss normal generators for mapping class groups of surfaces. Especially, we focus on the relation between normal generation of a mapping class with its asymptotic translation lengths on the Teichm\"uller space and the…