Related papers: On certain lattices associated with generic divisi…

In this paper we investigate some new problems in additive combinatorics. Our problems mainly involve permutations (or circular permutations) $n$ distinct numbers (or elements of an additive abelian group) $a_1,\ldots,a_n$ with adjacent…

We construct all solvable Lie algebras with a specific n-dimensional nilradical n_{n,3} which contains the previously studied filiform nilpotent algebra n_{n-2,1} as a subalgebra but not as an ideal. Rather surprisingly it turns out that…

In this paper we consider arbitrary intervals in the left weak order on the symmetric group $S_n$. We show that the Lehmer codes of permutations in an interval form a distributive lattice under the product order. Furthermore, the…

We show that a finite graded lattice of rank n is supersolvable if and only if it has an EL-labeling where the labels along any maximal chain form a permutation. We call such a labeling an S_n EL-labeling and we consider finite graded…

Given $n \times n$ matrices, $A_1, \dots, A_k$, consider the linear operator $L(A_1,\dots,A_k) \, \colon \; \operatorname{M}_n \to \operatorname{M}_n$ given by \[ L(A_1,\dots,A_k)(A_{k+1})= \sum_{\sigma\in S_{k+1}}…

The Dinitz conjecture states that, for each $n$ and for every collection of $n$-element sets $S_{ij}$, an $n\times n$ partial latin square can be found with the $(i,j)$\<th entry taken from $S_{ij}$. The analogous statement for $(n-1)\times…

We construct $N$-complexes of non completely antisymmetric irreducible tensor fields on $\mathbb R^D$ which generalize the usual complex $(N=2)$ of differential forms. Although, for $N\geq 3$, the generalized cohomology of these…

In this article we construct a large family of $R$-matrices for various extensions of small quantum groups by grouplike elements. The extensions are in correspondence to lattices between root and weight lattice and admit $R$-matrices in…

Let K be an arbitrary (commutative) field with at least three elements. It was recently proven that an affine subspace of M_n(K) consisting only of non-singular matrices must have a dimension lesser than or equal to n(n-1)/2. Here, we…

The structure of the multiplicative group $M_n = ({\mathbb Z}/n{\mathbb Z})^\times$ encodes a great deal of arithmetic information about the integer $n$ (examples include $\phi(n)$, the Carmichael function $\lambda(n)$, and the number…

We introduce the notion of universal odd generalized Poisson superalgebra associated to an associative algebra A, by generalizing a construction made in [5]. By making use of this notion we give a complete classification of simple linearly…

The space K^n of all n-dimensional { Lie} algebras has a natural non-Hausdorff topology k^n, which has characteristic limits, called transitions, A -> B, between distinct Lie algebras A and B. The entity of these transitions are the natural…

We describe an $N=2$ supersymmetric Poisson vertex algebra structure of $N=1$ (resp. $N=0$) classical $W$-algebra associated with $\mathfrak{sl}(n+1|n)$ and the odd (resp. even) principal nilpotent element. This $N=2$ supersymmetric…

A concise study of ternary and cubic algebras with $Z_3$ grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, $S_3$, and its abelian subgroup…

Let $G = GL(V)$ for an N-dimensional vector space $V$ over an algebraically closed field k, and $G^{\theta}$ the fixed point subgroup of $G$ under an involution $\theta$ on $G$. In the case where $G^{\theta} = O(V)$, the generalized…

We study orbits of semigroups of $\text{SL}(2,\mathbb{Z})$, and demonstrate reciprocity obstructions: we show that certain such orbits avoid squares, but not as a consequence of obstructions inherited from an algebraic set, and not as a…

In this paper we present short algebraic proofs of the Linear Conway--Gordon--Sachs and the Linear van Kampen--Flores theorems in the spirit of the Radon theorem on convex hulls. {\bf Theorem.} {\it Take any $n+3$ general position points in…

The classical Stern sequence of positive integers was extended to a polynomial sequence $S_n(\lambda)$ by Klav\v{z}ar et. al. by defining $S_0(\lambda) = 0$, $S_1(\lambda) = 1$, and $$S_{2n}(\lambda) = \lambda S_n(\lambda),\quad…

We consider the following question: Which real sequences (a(n)) that satisfy a linear recurrence with constant coefficients are positive for sufficiently large n? We show that the answer is negative for both (a(n)) and (-a(n)), if the…

For a tuple of square complex-valued $N\times N$ matrices $A_1,\dots,A_n$ the determinant of their linear combination $x_1A_1+\cdots +x_nA_n$, which is called \textit{a pencil}, is a homogeneous polynomial of degree $N$ in $\C[x_1,...x_n]$.…