Related papers: Tridiagonal pairs of shape (1,2,1)

If for a vector space V of dimension g over a characteristic zero field we denote by $\wedge^iV$ its alternating powers, and by $V^\vee$ its linear dual, then there are natural Poincar\'e isomorphisms: $\wedge^i V^\vee \cong \wedge^{g-i}…

Veldkamp polygons are certain graphs $\Gamma=(V,E)$ such that for each $v\in V$, $\Gamma_v$ is endowed with a symmetric anti-reflexive relation $\equiv_v$. These relations are all trivial if and only if $\Gamma$ is a thick generalized…

For an arbitrary field $\mathbb{K}$ and a family of inner products in a $\mathbb{K}$-vector space $V$ of arbitrary dimension, we study necessary and sufficient conditions in order to have an orthogonal basis relative to all the inner…

Let $n,n'$ be positive integers and let $V$ be an $(n+n')$-dimensional vector space over a finite field $\mathbb{F}$ equipped with a non-degenerate alternating, hermitian or quadratic form. We estimate the proportion of pairs $(U, U')$,…

We consider surjective endomorphisms f of degree > 1 on the projective n-space with n = 3, and f^{-1}-stable hypersurfaces V. We show that V is a hyperplane (i.e., deg(V) = 1) but with four possible exceptions; it is conjectured that deg(V)…

Let $V$ be a braided vector space of diagonal type. Let $\mathfrak B(V)$, $\mathfrak L^-(V)$ and $\mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial…

We consider quadrangles of perimeter $2$ in the plane with marked directed edge. To such quadrangle $Q$ a two-dimensional plane $\Pi\in\mathbb{R}^4$ with orthonormal base is corresponded. Orthogonal plane $\Pi^\bot$ defines a plane…

We construct a constrained trivariate extension of the univariate normalized B-basis of the vector space of trigonometric polynomials of arbitrary (finite) order n defined on any compact interval [0,\alpha], where \alpha is a fixed (shape)…

We study linear relations between face numbers of levels in arrangements. Let $V = \{ v_1, \ldots, v_n \} \subset \mathbf{R}^{r}$ be a vector configuration in general position, and let $\mathcal{A}(V)$ be polar dual arrangement of…

Given an alternating trilinear form T on the V=F^n let L_T denote the set of T-singular lines in the projective space PV. These are all lines <a,b> for which the linear form T(a,b,.) is identically zero. Amongst the immense profusion of…

Let $D$ be a division ring, $\mathcal V$ and $ \mathcal W$ vector spaces over $D$, and ${\mathcal L(\mathcal V,\mathcal W)}$ the ${\mathcal L(\mathcal W)}$-${\mathcal L(\mathcal V)}$ bimodule of all linear transformations from $\mathcal V$…

Let $\mathbb{K}$ denote an algebraically closed field. Let $V$ denote a vector space over $\mathbb{K}$ with finite positive dimension. By a Leonard triple on $V$ we mean an ordered triple of linear transformations in ${\rm End}(V)$ such…

Let $V$ be an infinite-dimensional vector space over a field. In a previous article, we have shown that every endomorphism of $V$ splits into the sum of four square-zero ones but also into the sum of four idempotent ones. Here, we study…

Let $\mathbb V$ be an arbitrary linear space and $f:\mathbb V \times \ldots \times \mathbb V \to \mathbb V$ an $n$-linear map. It is proved that, for each choice of a basis ${\mathcal B}$ of $\mathbb V$, the $n$-linear map $f$ induces a…

Let $\mathcal{A}=(A_{1},...,A_{n},...)$ be a finite or infinite sequence of $2\times2$ matrices with entries in an integral domain. We show that, except for a very special case, $\mathcal{A}$ is (simultaneously) triangularizable if and only…

Let $V$ be a vector space with countable dimension over a field, and let $u$ be an endomorphism of it which is locally finite, i.e. $(u^k(x))_{k \geq 0}$ is linearly dependent for all $x$ in $V$. We give several necessary and sufficient…

In the free probability theory of Voiculescu two of the most frequently used *-distributions are those of a Haar unitary and of a circular element. We define an $R$-diagonal pair as a generalization of these distributions by the requirement…

Let $U$ and $V$ be finite-dimensional vector spaces over a (commutative) field $\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space $\mathcal{L}(U,V)$ of all linear operators from $U$ to $V$. A map $F : \mathcal{S} \rightarrow…

The aim of this paper is to present some results concerning the $\rho_*$-orthogonality in real normed spaces and its preservation by linear operators. Among other things, we prove that if $T\,: X \longrightarrow Y$ is a nonzero linear $(I,…

Given an $n$-dimensional vector space $V$ over a field $\mathbb K$, let $2\leq k < n$. There is a natural correspondence between the alternating $k$-linear forms $\varphi$ of $V$ and the linear functionals $f$ of $\bigwedge^kV$. Let…