Related papers: $2$-Modular Matrices

It is a well-known result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank $n$. This follows from a double-exponential bound on the maximal denominator in an Egyptian…

We compute all fusion algebras with symmetric rational $S$-matrix up to dimension 12. Only two of them may be used as $S$-matrices in a modular datum: the $S$-matrices of the quantum doubles of $\mathbb{Z}/2\mathbb{Z}$ and $S_3$. Almost all…

Every m by n matrix A with rank r has exactly r independent rows and r independent columns. The fact has become the most fundamental theorem in linear algebra such that we may favor it in an unconscious way. The sole aim of this paper is to…

We pursue a classification of low-rank super-modular categories parallel to that of modular categories. We classify all super-modular categories up to rank=$6$, and spin modular categories up to rank=$11$. In particular, we show that, up to…

We prove that any odd-dimensional modular category of rank at most 23 is pointed. We also show that an odd-dimensional modular category of rank 25 is either pointed, perfect, or equivalent to $\operatorname{Rep}(D^\omega(\mathbb Z_7\rtimes…

A real symmetric matrix $M$ is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size $d$. The smallest such $d$ is called the (complex) completely positive semidefinite…

Let R be a Noetherian commutative ring of dimension n >2 and let A=R[T,T^{-1}]. Assume that the height of the Jacobson radical of R is atleast 2. Let P be a projective A-module of rank n=dim A - 1 with trivial determinant. We define an…

A set of arithmetical sequences $$ a_1\, (\bmod{ \,\, m_1}) \quad, \quad a_2 \, (\bmod{\,\, m_2}) \quad, \quad \dots \quad , \quad a_k \, (\bmod{\,\,m_k}) \quad \quad , $$ with $$ m_1 \leq m_2 \leq \dots \leq m_k \quad \quad , $$ is called…

Let $A \in \mathbb{Z}^{m \times n}$ be an integral matrix and $a$, $b$, $c \in \mathbb{Z}$ satisfy $a \geq b \geq c \geq 0$. The question is to recognize whether $A$ is $\{a,b,c\}$-modular, i.e., whether the set of $n \times n$…

We show for each positive integer $a$ that, if $\mathcal{M}$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $c$ such that either every rank-$r$ matroid in $\mathcal{M}$ can…

We introduce the tree-decomposition-based parameter totally $\Delta$-modular treewidth (TDM-treewidth) for matrices with two nonzero entries per row. We show how to solve integer programs whose matrices have bounded TDM-treewidth in…

We start with observing that the only connected finite dimensional algebras with finitely many isomorphism classes of indecomposable bimodules are the quotients of the path algebras of uniformly oriented $A_n$-quivers modulo the radical…

This is part of an ongoing project to find a general algebraic framework for semiring theory. The structure theory of semirings is quite challenging, largely because of the lack of negation, and such basic properties such as unique…

For an algebraic number $\alpha$ of degree $n$, let $\mathcal{M}_{\alpha}$ be the $\mathbb{Z}$-module generated by $1,\alpha ,\ldots ,\alpha^{n-1}$; then $\mathbb{Z}_{\alpha}:=\{\xi\in\mathbb{Q} (\alpha ):\,…

Let A be a regular domain of dimension d containing an infinite field and let n be an integer with 2n\geq d+3. For a stably free A-module P of rank n, we prove that (i) P has a unimodular element if and only if the euler class of P is zero…

We show that, if $k$ and $\ell$ are positive integers and $r$ is sufficiently large, then the number of rank-$k$ flats in a rank-$r$ matroid $M$ with no $U_{2,\ell+2}$-minor is less than or equal to number of rank-$k$ flats in a rank-$r$…

We consider the problem of determining which matrices are permutable to be supmodular. We show that for small dimensions any matrix is permutable by a universal permutation or by a pair of permutations, while for higher dimensions no…

We study the 'bad science matrix problem': among all matrices $A\in\mathbb{R}^{n\times n}$ whose rows have unit $\ell_2$-norm, determine the maximum of $\beta(A)=\frac{1}{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty$. Steinerberger [1]…

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. We determine the maximal dimension for an affine subspace of $n$ by $n$ symmetric (or alternating) matrices with entries in $\mathbb{K}$ and with…

All Lie algebras and representations will be assumed to be finite dimensional over the complex numbers. Let $V(m)$ be the irreducible $\sl(2)$-module with highest weight $m\geq 1$ and consider the perfect Lie algebra $\g=\sl(2)\ltimes…