Finite-dimensional Vector Space Research Articles

A note on bireflectional elements of an algebra

A classical theorem of Wonenburger, Djokovič, Hoffmann and Paige [2, 3, 7] states that an element of the general linear group of a finite-dimensional vector space is the product of two involutions if and only if it is similar to its inverse. We give a very elementary proof of this result when the underlying field F is algebraically closed with characteristic other than 2. In that situation, the result is generalized to the group of invertibles of any finite-dimensional algebra over F .

Polyhedral compactifications, I

Abstract In this work we describe horofunction compactifications of metric spaces and finite-dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, by ultrapowers of the spaces at hand. The polyhedral compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.

Classification of Consistent Systems of Handlebody Group Representations

Abstract The classifying spaces of handlebody groups form a modular operad. Algebras over the handlebody operad yield systems of representations of handlebody groups that are compatible with gluing. We prove that algebras over the modular operad of handlebodies with values in an arbitrary symmetric monoidal bicategory $\mathcal{M}$ (we introduce for these the name ansular functor) are equivalent to self-dual balanced braided algebras in $\mathcal{M}$. After specialization to a linear framework, this proves that consistent systems of handlebody group representations on finite-dimensional vector spaces are equivalent to ribbon Grothendieck-Verdier categories in the sense of Boyarchenko-Drinfeld. Additionally, it produces a concrete formula for the vector space assigned to an arbitrary handlebody in terms of a generalization of Lyubashenko’s coend. Our main result can be used to obtain an ansular functor from vertex operator algebras subject to mild finiteness conditions. This includes examples of vertex operator algebras whose representation category has a non-exact monoidal product.

Realizations of unisingular representations by hyperelliptic Jacobians

A representation of a finite group G on a finite dimensional vector space V is called unisingular if every has 1 as an eigenvalue in its action on V. In this paper we show that certain unisingular representations can be realized as mod 2 representations of hyperelliptic Jacobians over Q. We additionally identify new unisingular representations of the symmetric and alternating groups.

Sums of two square-zero selfadjoint or skew-selfadjoint endomorphisms

Let V be a finite-dimensional vector space over a field F, equipped with a symmetric or alternating non-degenerate bilinear form b. When the characteristic of F is not 2, we characterize the endomorphisms u of V that split into u=a1+a2 for some pair (a1,a2) of b-selfadjoint (respectively, b-skew-selfadjoint) endomorphisms of V such that (a1)2=(a2)2=0. In the characteristic 2 case, we obtain a similar classification for the endomorphisms of V that split into the sum of two square-zero b-alternating endomorphisms of V when b is alternating (an endomorphism v is called b-alternating whenever b(x,v(x))=0 for all x∈V).Finally, if the field F is equipped with a non-identity involution, we characterize the pairs (h,u) in which h is a Hermitian form on a finite-dimensional space over F, and u is the sum of two square-zero h-selfadjoint endomorphisms.

A lower bound for the dimension of tetrahedral splines in large degree

Splines are piecewise polynomial functions which are continuously differentiable to some order r. For a fixed integer d the space of splines of degree at most d is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable attention has been given to this problem in the bivariate setting, the literature on trivariate splines is less conclusive. In particular, the dimension of generic trivariate splines is not known even in large degree when r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}. In this paper we use a bound we previously derived for splines on vertex stars to compute a new lower bound on the dimension of trivariate splines in large enough degree. We illustrate in several examples that our formula gives the exact dimension of the spline space in large enough degree if vertex positions are generic. In contrast, for splines continuously differentiable of order r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}, every lower bound in the literature diverges (often significantly) in large degree from the dimension of the spline space in these examples. We derive the bound using commutative and homological algebra.

Stable lattices in p-adic representations III. Indecomposable reduction and holonomy

Let G be a group, V a finite dimensional vector space over a complete discrete valuation field K, and ρ:G⟶GLK(V) an irreducible representation with bounded image (in the metric topology). First, we show that there are many stable lattices with indecomposable reduction, in the sense that their simplicial convex hull in the Bruhat-Tits building of GLK(V) is the set S(ρ) of stable norms.Then we investigate the reductions more closely, in case K has finite residue field and the simplicial set S(ρ) has dimension 2. When the reduction type (defined below) is (2,1), we show that there exists a stable lattice with indecomposable reduction that has at most 2 composition series (uniserial or biserial). When the reduction type is (3), we show that the number of v∈S(ρ)0 with semisimple reduction is bounded above by a constant times |S(ρ)0|.To prove these last two statements, we prove and apply an index theorem (of Gauss-Bonnet type).

Semi-projective representations and twisted representation groups

A semi-projective representation is a homomorphism of a finite group into the group of semi-projective transformations of a finite dimensional vector space over a field. Schur’s concept of a representation group for projective representations is extended to semi-projective representations under the assumption that the field is algebraically closed. A computer algorithm is given that produces, for a given finite group, all such twisted representation groups under trivial or conjugation actions on the field of complex numbers.

Subset sums and block designs in a finite vector space

In this paper we settle the question of whether a finite-dimensional vector space {{mathcal {V}}} over {mathbb {F}}_p, with p an odd prime, and the family of all the k-sets of elements of {mathcal {V}} summing up to a given element x, form a 1-(v,k,lambda _1) or a 2-(v,k,lambda _2) block design, and, in either case, we find a closed form for lambda _i and characterize the automorphism group. The question is discussed also in the case where the elements of the k-sets are required to be all nonzero, as the two cases happen to be intrinsically inseparable. The “twin case” p=2, which has strict connections with coding theory, was completely discussed in a recent paper by G. Falcone and the present author.

Circular bidiagonal pairs

A square matrix is said to be circular bidiagonal whenever (i) each nonzero entry is on the diagonal, or the subdiagonal, or in the top-right corner; (ii) each subdiagonal entry is nonzero, and the entry in the top-right corner is nonzero. Let F denote a field, and let V denote a nonzero finite-dimensional vector space over F. We consider an ordered pair of F-linear maps A:V→V and A⁎:V→V that satisfy the following two conditions:•there exists a basis for V with respect to which the matrix representing A is circular bidiagonal and the matrix representing A⁎ is diagonal;•there exists a basis for V with respect to which the matrix representing A⁎ is circular bidiagonal and the matrix representing A is diagonal. We call such a pair a circular bidiagonal pair on V. We classify the circular bidiagonal pairs up to affine equivalence. There are two infinite families of solutions, which we describe in detail.

Lagrangian Grassmannians, CKP Hierarchy and Hyperdeterminantal Relations

This work concerns the relation between the geometry of Lagrangian Grassmannians and the CKP integrable hierarchy. The Lagrange map from the Lagrangian Grassmannian of maximal isotropic (Lagrangian) subspaces of a finite dimensional symplectic vector space $V\oplus V^*$ into the projectivization of the exterior space $\Lambda V$ is defined by restricting the Pl\"ucker map on the full Grassmannian to the Lagrangian sub-Grassmannian and composing it with projection to the subspace of symmetric elements under dualization $V \leftrightarrow V^*$. In terms of the affine coordinate matrix on the big cell, this reduces to the principal minors map, whose image is cut out by the $2 \times 2 \times 2$ quartic {\em hyperdeterminantal} relations. To apply this to the CKP hierarchy, the Lagrangian Grassmannian framework is extended to infinite dimensions, with $V\oplus V^*$ replaced by a polarized Hilbert space $ {\mathcal H} ={\mathcal H}_+\oplus {\mathcal H}_-$, with symplectic form $\omega$. The image of the Plucker map in the fermionic Fock space ${\mathcal F}= \Lambda^{\infty/2}{\mathcal H}$ is identified and the infinite dimensional Lagrangian map is defined. The linear constraints defining reduction to the CKP hierarchy are expressed as a fermionic null condition and the infinite analogue of the hyperdeterminantal relations is deduced. A multiparametric family of such relations is shown to be satisfied by the evaluation of the $\tau$-function at translates of a point in the space of odd flow variables along the cubic lattices generated by power sums in the parameters.

Stationary probability measures on projective spaces 1: block-Lyapunov dominated systems

Given a finite-dimensional real vector space V, a probability measure μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} on PGL(V)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {PGL}}(V)$$\\end{document} and a μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-invariant subspace W, under a block-Lyapunov contraction assumption, we prove existence and uniqueness of lifts to P(V)\\P(W)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P(V){\\setminus } P(W)$$\\end{document} of stationary probability measures on the quotient P(V/W). In the other direction, i.e. block-Lyapunov expansion, we prove that stationary measures on P(V/W) have lifts if any only if the group generated by the support of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} stabilizes a subspace W′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W'$$\\end{document} not contained in W and exhibiting a faster growth than on W∩W′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W \\cap W'$$\\end{document}. These refine the description of stationary probability measures on projective spaces as given by Furstenberg, Kifer and Hennion, and under the same assumptions, extend corresponding results by Aoun, Benoist, Bruère, Guivarc’h, and others.

Cayley subspace sum graph of vector spaces

Let $\mathbb{V}$ be a finite dimensional vector space over the field $\mathbb{F}$. Let $S(\mathbb{V})$ be the set of all subspaces of $\mathbb{V}$ and $\mathbb{A}\subseteq S^*(\mathbb{V})=S(\mathbb{V})\backslash\{0\}.$ In this paper, we define the Cayley subspace sum graph of $\mathbb{V},$ denoted by Cay$(S^*(\mathbb{V}),\mathbb{A}), $ as the simple undirected graph with vertex set $S^*(\mathbb{V})$ and two distinct vertices $X$ and $Y$ are adjacent if $X+Z=Y$ or $Y+Z=X$ for some $Z\in \mathbb{A}$. Having defined the Cayley subspace sum graph, we study about the connectedness, diameter and girth of several classes of Cayley subspace sum graphs Cay$(S^*(\mathbb{V}), \mathbb{A})$ for a finite dimensional vector space $\mathbb{V}$ and $\mathbb{A}\subseteq S^*(\mathbb{V})=S(\mathbb{V})\backslash\{0\}.$

Tridiagonal pairs, alternating elements, and distance-regular graphs

The positive part Uq+ of Uq(slˆ2) has a presentation with two generators W0, W1 and two relations called the q-Serre relations. The algebra Uq+ contains some elements, said to be alternating. There are four kinds of alternating elements, denoted {W−k}k∈N, {Wk+1}k∈N, {Gk+1}k∈N, {G˜k+1}k∈N. The alternating elements of each kind mutually commute. A tridiagonal pair is an ordered pair of diagonalizable linear maps A,A⁎ on a nonzero, finite-dimensional vector space V, that each act in a (block) tridiagonal fashion on the eigenspaces of the other one. Let A, A⁎ denote a tridiagonal pair on V. Associated with this pair are six well-known direct sum decompositions of V; these are the eigenspace decompositions of A and A⁎, along with four decompositions of V that are often called split. In our main results, we assume that A, A⁎ has q-Serre type. Under this assumption A, A⁎ satisfy the q-Serre relations, and V becomes an irreducible Uq+-module on which W0=A and W1=A⁎. We describe how the alternating elements of Uq+ act on the above six decompositions of V. We show that for each decomposition, every alternating element acts in either a (block) diagonal, (block) upper bidiagonal, (block) lower bidiagonal, or (block) tridiagonal fashion. We investigate two special cases in detail. In the first case the eigenspaces of A and A⁎ all have dimension one. In the second case A and A⁎ are obtained by adjusting the adjacency matrix and a dual adjacency matrix of a distance-regular graph that has classical parameters and is formally self-dual.

Persistence and the Sheaf-Function Correspondence

Abstract The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M. When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $\mathbf {k}$ -vector spaces on M. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.

Inverse theorem for certain directional Gowers uniformity norms

Let G be a finite-dimensional vector space over a prime field Fp with some subspaces H1,...,Hk. Let f : G ? C be a function. Generalizing the notion of Gowers uniformity norms, Austin introduced directional Gowers uniformity norms of f over (H1,...,Hk) as ||f||2kU(H1,...,Hk) = E x?G,h1?H1,...,hk?Hk ? ?h1... ??hk f(x) where ??uf(x) : = f(x + u)?f(x) is the discrete multiplicative derivative. Suppose that G is a direct sum of subspaces G = U1 ? U2 ?...? Uk. In this paper we prove the inverse theorem for the norm ||?||U(U1,...,Uk,G,...,G), with ? copies of G in the subscript, which is the simplest interesting unknown case of the inverse problem for the directional Gowers uniformity norms. Namely, writing ||? ||U for the norm above, we show that if f : G ? C is a function bounded by 1 in magnitude and obeying ||f||U ? c, provided ? < p, one can find a polynomial ?: G ? Fp of degree at most k + ? ? 1 and functions gi : ?j?[k]\{i} Uj ? {z ? C: |z| ? 1} for i ? [k] such that |Ex?G f(x)??(x) ? i? [k] gi(x1,... ,xi?1, xi+1,..., xk)| ? (exp(Op,k,?(1))(Op,k,?(c?1)))?1. The proof relies on an approximation theorem for the cuboid-counting function that is proved using the inverse theorem for Freiman multi-homomorphisms.

Fourier expansions of vector-valued automorphic functions with non-unitary twists

We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.

Categorifications of rational Hilbert series and characters of FSop modules

We introduce a method for associating a chain complex to a module over a combinatorial category, such that if the complex is exact then the module has a rational Hilbert series. We prove homology--vanishing theorems for these complexes for several combinatorial categories including: the category of finite sets and injections, the opposite of the category of finite sets and surjections, and the category of finite dimensional vector spaces over a finite field and injections. Our main applications are to modules over the opposite of the category of finite sets and surjections, known as $FS^{op}$ modules. We obtain many constraints on the sequence of symmetric group representations underlying a finitely generated $FS^{op}$ module. In particular, we describe its character in terms of functions that we call character exponentials. Our results have new consequences for the character of the homology of the moduli space of stable marked curves, and for the equivariant Kazhdan-Luzstig polynomial of the braid matroid.

Annihilating Entanglement Between Cones

Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform the k-fold maximal tensor product of a cone {textsf{C}}_1 into the k-fold minimal tensor product of a cone {textsf{C}}_2, and the pair ({textsf{C}}_1,{textsf{C}}_2) is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that ({textsf{C}}_1,{textsf{C}}_2) is resilient if either {textsf{C}}_1 or {textsf{C}}_2 is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.

Poles of finite-dimensional representations of Yangians

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $Y_{\hbar}(\mathfrak{g})$ be the Yangian of $\mathfrak{g}$. In this paper, we study the sets of poles of the rational currents defining the action of $Y_{\hbar}(\mathfrak{g})$ on an arbitrary finite-dimensional vector space $V$. Using a weak, rational version of Frenkel and Hernandez' Baxter polynomiality, we obtain a uniform description of these sets in terms of the Drinfeld polynomials encoding the composition factors of $V$ and the inverse of the $q$-Cartan matrix of $\mathfrak{g}$. We then apply this description to obtain a concrete set of sufficient conditions for the cyclicity and simplicity of the tensor product of any two irreducible representations, and to classify the finite-dimensional irreducible representations of the Yangian double.