Related papers: On certain spaces of lattice diagram determinants

Let $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ be the sets of quadratic forms and symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$, respectively. The orbits of $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ under a…

We investigate categorical and amalgamation properties of the functor Idc assigning to every partially ordered abelian group G its semilattice of compact ideals Idc G. Our main result is the following. Theorem 1. Every diagram of finite…

Matrix polynomials given in an orthogonal basis are considered. Following the ideas of Mackey et al. "Vector spaces of Linearizations for Matrix Polynomials" (2006), the vec- tor spaces, called M1(P), M2(P) and DM(P), of potential…

Given a square-free monomial ideal $I$ in a polynomial ring $R$ over a field $\mathbb{K}$, one can associate it with its LCM-lattice and its hypergraph. In this short note, we establish the connection between the LCM-lattice and the…

We extend the existing skew polynomial representations of matrix algebras which are direct sum of matrix spaces over division rings. In this representation, the sum-rank distance between two tuples of matrices is captured by a weight…

For a nonempty compact set D of R we determine the maximal possible dimension of a subspace X of polynomial functions of degree at most m which possesses a positive bases (where positivity is understood on D). The exact value of this…

Young's partition lattice $L(m,n)$ consists of unordered partitions having $m$ parts where each part is at most $n$. Using methods from complex algebraic geometry, R. Stanley proved that $L(m,n)$ is rank-symmetric, unimodal, and strongly…

Given an ideal $I$ in a polynomial ring $K[x_1,\dots,x_n]$ over a field $K$, we present a complete algorithm to compute the binomial part of $I$, i.e., the subideal ${\rm Bin}(I)$ of $I$ generated by all monomials and binomials in $I$. This…

A set of polynomials $M$ is called a {\it submodule} of $\mathbb{C} [x_1, \dots, x_n ]$ if $M$ is a translation invariant linear subspace of $\mathbb{C} [x_1, \dots, x_n ]$. We present a description of the submodules of $\mathbb{C} [x,y]$…

Let $G$ be a graph and let $m_{ij}(G)$, $i,j\ge 1$, be the number of edges $uv$ of $G$ such that $\{d_v(G), d_u(G)\} = \{i,j\}$. The {\em $M$-polynomial} of $G$ is introduced with $\displaystyle{M(G;x,y) = \sum_{i\le j} m_{ij}(G)x^iy^j}$.…

Let $\mathbb{D}$ be a division ring and $\mathbb{F}$ be a subfield of the center of $\mathbb{D}$ over which $\mathbb{D}$ has finite dimension $d$. Let $n,p,r$ be positive integers and $\mathcal{V}$ be an affine subspace of the…

Fill each box in a Young diagram with the number of paths from the bottom of its column to the end of its row, using steps north and east. Then, any square sub-matrix of this array starting on the south-east boundary has determinant one. We…

In recent work of Cachazo, Guevara, Mizera and the author, a generalization of the biadjoint scattering amplitude $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ was introduced as an integral over the moduli space of $n$ points in $\mathbb{CP}^{k-1}$,…

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda + \sum_{k = 1}^d [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are linear forms in…

In a recent work, Coronel et al. initiated the study of the relation between the diversity-multiplexing tradeoff (DMT) performance of a multiuser multiple-input multiple-output (MU-MIMO) lattice code and the rate of the decay of the…

This paper treats certain integral lattices with respect to ternary quadratic forms, which are obtained from the data of a non-zero element and a maximal lattice in a quaternary quadratic space. Such a lattice can be described by means of…

We study the \emph{picture space} $X^d(G)$ of all embeddings of a finite graph $G$ as point-and-line arrangements in an arbitrary-dimensional projective space, continuing previous work on the planar case. The picture space admits a natural…

In this article, we consider the monomial complete intersection algebra $\mathbb{K}[x,y]/\langle x^d,y^q\rangle$ in two variables. For elements $l_1,\ldots,l_{d+q-2k}$ of degree $1$, we give a formula of the deteminant of linear map from…

Multilevel lattice codes, such as those associated to Constructions $C$, $\overline{D}$, D and D', have relevant applications in communications. In this paper, we investigate some properties of lattices obtained via Constructions D and D'…

This paper studies the differential lattice, defined to be a lattice $L$ equipped with a map $d:L\to L$ that satisfies a lattice analog of the Leibniz rule for a derivation. Isomorphic differential lattices are studied and classifications…