Related papers: On unit weighing matrices with small weight

Let ${\mathcal W}_n$ be the Lie algebra of polynomial vector fields. We classify simple weight ${\mathcal W}_n$-modules $M$ with finite weight multiplicities. We prove that every such nontrivial module $M$ is either a tensor module or the…

We study the decomposition matrices for the unipotent $\ell$-blocks of finite special unitary groups SU$_n(q)$ for unitary primes $\ell$ larger than $n$. Up to very few unknown entries, we give a complete solution for $n=2,\ldots,10$. We…

We give explicit structure of the graded ring of modular forms with respect to Gamma(N) (N=1,2,3,4,5,6,7,8,9,10,12,16,18) and for some other congruence groups. We also study the modular forms of half-integer weight for certain groups.

The long run behaviour of linear dynamical systems is often studied by looking at eventual properties of matrices and recurrences that underlie the system. A basic problem that lies at the core of many questions in this setting is the…

We derive weighted summation identities involving the second order recurrence sequence $\{w_n\} =\{ w_n(a,b; p, q)\}$ defined by $w_0 = a,\,w_1 = b;\,w_n = pw_{n - 1} - qw_{n - 2}\, (n \ge 2)$, where $a$, $b$, $p$ and $q$ are arbitrary…

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study the largest minimum weight $d(n,k)$ among all binary linear complementary dual $[n,k]$ codes. We…

A matrix $T \in \M_n(\C)$ is \emph{UECSM} if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we…

We count with a smooth weight the number of $2 \times 2$ integer matrices with a fixed characteristic polynomial with a main term and an error term using bounds for sums of Weyl sums for quadratic roots.

We use the Poincar\'e series method to compute gravity partition functions associated to SU(N) level 1 WZW models with arbitrarily large numbers of modular invariants. The result is an average over these invariants, with the weights being…

We explore the set of unitary matrices characterized by a given structure in the context of their applications in the field of Quantum Information. In the first part of the Thesis we focus on classification of special classes of unitary…

Some mutually quasi-unbiased weighing matrices are constructed from binary codes satisfying certain conditions. Motivated by this, in this note, we study binary codes satisfying the conditions. The weight distributions of binary codes…

We present a method to calculate integrals over monomials of matrix elements with invariant measures in terms of Wick contractions. The method gives exact results for monomials of low order. For higher--order monomials, it leads to an error…

In these notes we focus a bit on the complex case for some families of matrices and equivalences between them.

An $n\times n$ complex matrix $A$ is called coninvolutory if $\bar AA=I_n$ and skew-coninvolutory if $\bar AA=-I_n$ (which implies that $n$ is even). We prove that each matrix of size $n\times n$ with $n>1$ is a sum of 5 coninvolutory…

We study a coarse moduli space of irreducible representations of the group of unipotent matrices of order $\mathbb{4}$ over the ring of integers which have finite weight. All such representations are known to be monomial. To describe a…

The decomposition matrix of a finite group in prime characteristic p records the multiplicities of its p-modular irreducible representations as composition factors of the reductions modulo p of its irreducible representations in…

We investigate the variety of commuting matrices. We classify its components for any number of matrices of size at most 7. We prove that starting from quadruples of size 8 matrices, this scheme has generically nonreduced components, while…

In this paper we describe an algorithm for generating all the possible $PIW(m,n,k)$ - integer $m\times n$ Weighing matrices of weight $k$ up to Hadamard equivalence. Our method is efficient on a personal computer for small size matrices, up…

We provide a classification method of weighing matrices based on a classification of self-orthogonal codes. Using this method, we classify weighing matrices of orders up to 15 and order 17, by revising some known classification. In…

We prove that for any fixed unitary matrix $U$, any abelian self-adjoint algebra of matrices that is invariant under conjugation by $U$ can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by…