Finite-dimensional Complex Vector Space Research Articles

TQFTs and quantum computing

Quantum computing is captured in the formalism of the monoidal subcategory of VectC generated by C2 — in particular, quantum circuits are diagrams in VectC — while topological quantum field theories, in the sense of Atiyah, are diagrams in VectC indexed by cobordisms. We initiate a program to formalize this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a higher category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable higher categorical structure which we call FVectC. We realize quantum circuits as images of cobordisms under monoidal functors from these modified cobordisms to FVectC, which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain category.

On the normalizer of the reflexive cover of a unital algebra of linear transformations

Given a unital algebra A of linear transformations on a finite-dimensional complex vector space V, in this paper we study the set Col(A) consisting of those invertible linear transformations S on V which map every subspace M∈Lat(A) to a subspace SM∈Lat(A). We show that Col(A) is the normalizer of the group of invertible linear transformations in the reflexive cover of A. For the unital algebra (A) which is generated by a linear transformation A, we give the complete description of Col(A). By using the primary decomposition of A, we first reduce the problem of characterizing Col(A) to the problem of characterizing the group Col(N) of a given nilpotent linear transformation N. While Col(N) always contains all invertible linear transformations of the commutant (N)′ of N, it is always contained in the reflexive cover of (N)′. We prove that Col(N) is a proper subgroup of (AlgLat(N)′)−1 if and only if at least two Jordan blocks in the Jordan decomposition of N are of dimension 2 or more. We also determine the group Col(J2⊕J2).

Open Subsets in a Stein Space with Singularities

Serre proved that a domain $Y$ in $\mathbb{C}^n$ is Stein if and only if $H^i(Y,\mathcal{O}_Y) = 0$ for all $i \gt 0$. Laufer showed that if $Y$ is an open subset of a Stein manifold of dimension $n$ and $H^i(Y,\mathcal{O}_Y)$ is a finite dimensional complex vector space for every $i \gt 0$, then $Y$ is Stein. Vâjâitu generalized these theorems to singular Stein space of dimension $n$. In this paper, we consider singular Stein spaces $X$ with arbitrary dimension and give necessary and sufficient conditions for an open subset $Y$ in $X$ to be Stein. We show that if $Y$ is an open subset of a reduced Stein space $X$ with arbitrary dimension and singularities, then $Y$ is Stein if and only if $H^i(Y,\mathcal{O}_Y)$ is a finite dimensional complex vector space for every $i \gt 0$. Without cohomology condition, if $X-Y$ is a closed subspace of $X$, then we show that the geometric condition of the boundary $X-Y$ determines the Steinness of $Y$. More precisely, we show that if $X$ is normal and the boundary $X-Y$ is the support of an effective $\mathbb{Q}$-Cartier divisor, or $X-Y$ is of pure codimension $1$ and does not contain any singular points of $X$, then $Y$ is Stein.

Doubly transitive lines I: Higman pairs and roux

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. In doing so, we make fundamental connections with both discrete geometry and algebraic combinatorics. In particular, we show that doubly transitive lines are necessarily optimal packings in complex projective space, and we introduce a fruitful generalization of regular abelian distance-regular antipodal covers of the complete graph.

Orthogonal projections on hyperplanes intertwined with unitaries

Fix a point in a finite-dimensional complex vector space and consider the sequence of iterates of this point under the composition of a unitary map with the orthogonal projection on the hyperplane orthogonal to the starting point. We prove that, generically, the series of the squared norms of these iterates sums to the dimension of the underlying space. This leads us to construct a (device-dependent) dimension witness for quantum systems which involves the probabilities of obtaining certain strings of outcomes in a sequential yes–no measurement. The exact formula for this series in non-generic cases is provided as well as its analogue in the real case.

Refinement of Novikov–Betti Numbers and of Novikov Homology Provided by an Angle Valued Map

To a pair (X, f), where X is a compact ANR and f : X → 𝕊1 is a continuous angle valued map, a field κ, and a nonnegative integer r, one assigns a finite configuration of complex numbers z with multiplicities δfr (z) and a finite configuration of free κ[t−1, t]-modules $$ {\hat{\delta}}_r^f $$ of rank $$ {\delta}_r^f $$ (z) indexed by the same numbers z. This is in analogy with the configuration of eigenvalues and of generalized eigenspaces of a linear operator in a finite-dimensional complex vector space. The configuration $$ {\delta}_r^f $$ refines the Novikov–Betti number in dimension r, and the configuration $$ {\hat{\delta}}_r^f $$ refines the Novikov homology in dimension r associated with the cohomology class defined by f. In the case of the field κ = C, the configuration $$ {\hat{\delta}}_r^f $$ provides by “von-Neumann completion” of a configuration $$ {\hat{\hat{\delta}}}_r^f $$ of mutually orthogonal closed Hilbert submodules of the L2-homology of the infinite cyclic cover of X determined by the map f, which is an L∞(𝕊1)-Hilbert module.

Independent multistructures, flag structures, and sets of linearly independent polar elements

Let V be a finite dimensional complex vector space, and for each positive integer n let Sn(V) denote the vector space whose elements are the complex-valued symmetric n-linear functions on V×V×⋯×V (n-copies). If f∈S1(V), that is, f is a linear functional on V, then we define the symmetric n-linear function fn by fn(x1,x2,…,xn)=∏i=1nf(xi) for all x1,x2,…,xn∈V. It is well known that the elements fn, known as polar elements, span Sn(V). Our focus is the simple generation of sets of linearly independent polar elements of given order n. A basis for Sn(V) consisting entirely of polar elements is a polar basis. We introduce several new structures that are closely connected to the problem of generating independent sets of polar elements. In particular, we introduce the notion of independent multistructure of order n, n≥0, and depth p, p≥1, and show how such structures generate sets of linearly independent polar elements of cardinality (n+pp). When p=dim⁡(V) we obtain a basis for Sn(V) consisting entirely of polar elements. We also introduce the dual concept of flag structure of order n and depth p. Both of these concepts evolved from a consideration of subspace filters which were defined previously in [6].

Iteration of the exterior power on representation rings

In this note we investigate the idea of Michael Atiyah of using, as a possible approach to the Theorem of Feit–Thompson on the solvability of finite groups of odd order, the iterations of the transformation which replaces a representation of a finite group G on a finite dimensional complex vector space E by the difference between the associated representation of G on the sum of exterior powers of E and the trivial representation. We show that G has odd order if and only if the above operation extends to virtual representations and we then express it in terms of the exponential and the Adams operations in the complexified representation ring. We show the relevance of the idea in a concrete example by exhibiting convergence to a non-trivial character.

Re-parameterizing and reducing families of normal operators

We present a new proof of results of Kurdyka & Paunescu, and of Rainer, about real-analytic multi-parameters generalizations of classical results by Rellich and Kato about the reduction in families of univariate deformations of normal operators over real or complex vector spaces of finite dimensions. Given a real analytic family of normal operators over a finite dimensional real or complex vector space, there exists a locally finite composition of blowings-up with smooth centers re-parameterizing the given family such that at each point of the source space of the re-parameterizing mapping, there exists a neighbourhood of any given point over which exists a real analytic orthonormal frame in which the pull back of the operator is in reduced form at every point of the neighbourhood. A free by-product of our proof is the local real analyticity of the eigen-values, which in all prior works was a prerequisite step to get local regular reducing bases.

On relationships between two linear subspaces and two orthogonal projectors

AbstractSum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of them-dimensional complex column vector space 𝔺mwith respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

A refinement of Betti numbers and homology in the presence of a continuous function, I

We propose a refinement of the Betti numbers and of the homology with coefficients in a field of a compact ANR in the presence of a continuous real valued function. The refinement of Betti numbers consists of finite configurations of points with multiplicities in the complex plane whose total cardinality are the Betti numbers and the refinement of homology consists of configurations of vector spaces indexed by points in complex plane, with the same support as the first, whose direct sum is isomorphic to the homology. When the homology is equipped with a scalar product these vector spaces are canonically realized as mutually orthogonal subspaces of the homology. The assignments above are in analogy with the collections of eigenvalues and generalized eigenspaces of a linear map in a finite dimensional complex vector space. A number of remarkable properties of the above configurations are discussed.

Subspace filters and polar bases for spaces of symmetric multilinear functions

Let V be a finite dimensional complex vector space, and for each positive integer n let denote the vector space whose elements are the complex valued symmetric n-linear functions on (n-copies). If , that is, f is a linear functional on V, then we define the symmetric n-linear function by for all . It is well known that the elements , known as polar elements, span . A basis for consisting entirely of polar elements is a polar basis. We present a surprisingly simple recursive algorithm for the generation of polar bases. Explicit stand alone formulas for polar bases are also presented. One such explicit basis is so natural that it should probably be called the canonical polar basis for . We introduce the notion of subspace filter and show that each subspace filter of order n provides polar bases for all of the spaces simultaneously, and we extend this result to the entire symmetric algebra .

Cherednik and Hecke Algebras of Varieties with a Finite Group Action

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on the set of complex reflections in $G$, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety $X$ with a finite group $G$ of automorphisms. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero-Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case $\pi_2(X)\otimes \Bbb Q=0$) a flat deformation of the orbifold fundamental group of the orbifold $X/G$, which we call the Hecke algebra of $X/G$. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples.

Tensor diagrams and cluster algebras

The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. We show that when V is 3-dimensional, each of these rings carries a natural cluster algebra structure (typically, many of them) whose cluster variables include Weyl's generators. We describe and explore these cluster structures using the combinatorial machinery of tensor diagrams. A key role is played by the web bases introduced by G. Kuperberg.

Classification of regular dicritical foliations

In this paper we give complete analytic invariants for the set of germs of holomorphic foliations in$(\mathbb{C}^{2},0)$that become regular after a single blow-up. Some of the invariants describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in a finite dimensional complex vector space. Such singularities admit separatrices tangentially to any direction at the origin. When enough separatrices are leaves of a radial foliaton (a condition that can always be attained if the multiplicity of the germ at the origin is at most four) we are able to describe and realize all the analytical invariants geometrically and provide analytic normal forms. As a consequence, we prove that any two such germs sharing the same transverse invariants are conjugated by a very particular type of birational transformation. We also provide explicit examples of universal equisingular unfoldings of foliations that cannot be produced by unfolding functions. With these at hand we are able to explicitly parametrize families of analytically distinct foliations that share the same transverse invariants.

Nonnormal operators in physics, a singular-vectors approach: illustration in polarization optics.

The singular-vectors analysis of a general nonnormal operator defined on a finite-dimensional complex vector space is given in the frame of a pure operatorial ("nonmatrix," "coordinate-free") approach, performed in a Dirac language. The general results are applied in the field of polarization optics, where the nonnormal operators are widespread as operators of various polarization devices. Two nonnormal polarization devices representative for the class of nonnormal and even pathological operators-the standard two-layer elliptical ideal polarizer (singular operator) and the three-layer ambidextrous ideal polarizer (singular and defective operator)-are analyzed in detail. It is pointed out that the unitary polar component of the operator exists and preserves, in such pathological case too, its role of converting the input singular basis of the operator in its output singular basis. It is shown that for any nonnormal ideal polarizer a complementary one exists, so that the tandem of their operators uniquely determines their (common) unitary polar component.

Cherednik and Hecke Algebras of Varieties with a Finite Group Action

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on the set of complex reflections in $G$, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety $X$ with a finite group $G$ of automorphisms. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero-Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case $\pi_2(X)\otimes \Bbb Q=0$) a flat deformation of the orbifold fundamental group of the orbifold $X/G$, which we call the Hecke algebra of $X/G$. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples.

BASIC VECTOR VALUED SIEGEL MODULAR FORMS OF GENUS TWO

We consider the space $[\Gamma, rr_{0}, v^{r}, \varrho]$ of all vector-valued holomorphic modular forms $f\colon \mathbb{H}_{n} \to \mathcal{Z}$ of transformation type \begin{equation*} f(MZ)=v^{r}(M)\det(CZ+D)^{r_{0}r}\varrho(CZ+D)f(Z). \end{equation*} $\varrho\colon \mathrm{GL}(n, \mathbb{C}) \to \mathrm{GL}(\mathcal{Z})$ is a rational representation on a finite dimensional complex vector space $\mathcal{Z}$. These spaces can be collected in a graded $A(\Gamma)$-module \begin{equation*} \mathcal{M} = \mathcal{M}_{\Gamma}(r_{0}, v, \varrho) := \bigoplus_{r \in \mathbb{Z}}[\Gamma, rr_{0}, v^{r}, \varrho]. \end{equation*} We treat in this paper some special cases in genus 2. The first one is essentially due to Wieber. Here the starting weight is $1/2$, the starting multiplier system is the multiplier system $v_{\Theta}$ and for $\varrho$ we take the second symmetric power of standard representation. Thus we consider a variant of this case and a new example. In this final case the starting weight is $1/2$, the starting multiplier system is the theta multiplier system $v_{\vartheta}$ and for $\varrho$ we take the standard representation. In all these cases we will determine the structure of $\mathcal{M}$.

Finite complex reflection arrangements are K(pi,1)

Let V be a finite dimensional complex vector space and W ⊆ GL(V ) be a finite complex reflection group. Let V reg be the complement in V of the reflecting hyperplanes. We prove that V reg is a K(π, 1) space. This was predicted by a classical conjecture, originally stated by Brieskorn for complexified real reflection groups. The complexified real case follows from a theorem of Deligne and, after contributions by Nakamura and OrlikSolomon, only six exceptional cases remained open. In addition to solving these six cases, our approach is applicable to most previously known cases, including complexified real groups for which we obtain a new proof, based on new geometric objects. We also address a number of questions about π1(W\V ), the braid group of W . This includes a description of periodic elements in terms of a braid analog of Springer’s theory of regular elements.

Linear semigroups with coarsely dense orbits

Let S be a finitely generated abelian semigroup of invertible linear operators on a finite-dimensional real or complex vector space V. We show that every coarsely dense orbit of S is actually dense in V. More generally, if the orbit contains a coarsely dense subset of some open cone C in V, then the closure of the orbit contains the closure of C. In the complex case the orbit is then actually dense in V. For the real case we give precise information about the possible cases for the closure of the orbit.