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In this paper, the 2-category $\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$ of (weak) representations of an arbitrary (weak) 2-group $\mathbb{G}$ on (some version of) Kapranov and Voevodsky's 2-category of (complex) 2-vector spaces is studied. In particular, the set of equivalence classes of representations is computed in terms of the invariants $\pi_0(\mathbb{G})$, $\pi_1(\mathbb{G})$ and $[\alpha]\in H^3(\pi_0(\mathbb{G}),\pi_1(\mathbb{G}))$ classifying $\mathbb{G}$. Also the categories of morphisms (up to equivalence) and the composition functors are determined explicitly. As a consequence, we obtain the the {\it monoidal} category of linear representations (more generally, the category of $[z]$-projective representations, for any given cohomology class $[z]\in H^2(\pi_0(\mathbb{G}),\mathbb{C}^*)) of the first homotopy group $\pi_0(\mathbb{G})$ as well as its category of representations on finite sets both live in $\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$, the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of morphisms between suitable 1-dimensional representations) and the second as a subcategory of the homotopy category of $\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$.

Annie

The main purpose of this note is to establish the continuity of seminorms on finite-dimensional vector spaces over the real or complex numbers.

In this paper, we introduce a graph structure called linear dependence graph of a finite dimensional vector space over a finite field. Some basic properties of the graph like connectedness, completeness, planarity, clique number, chromatic number etc. have been studied. It is shown that two vector spaces are isomorphic if and only if their corresponding linear dependence graphs are isomorphic. Also adjacency spectrum, Laplacian spectrum and distance spectrum of the linear dependence graph have been studied.

Given $n+1$ unit vectors in $\mathbf{R}^n$ or $\mathbf{C}^n,$ consider the absolute values of the determinants of the vectors taken $n$ at a time. By taking a geometric perspective, we show that the minimum of these determinants is maximized when the vectors point from the origin to the vertices of a regular simplex inscribed in the unit sphere in $\mathbf{R}^n,$ even in the complex case. We also discuss variations on this problem and a few connections to other problems.

An introduction to linear algebra from the pure math perspective.

Let $V$ be a finite dimensional vector space over a local field $F$. Let $\chi: F^\times \rightarrow \mathbb C^\times$ be an arbitrary character of $F^\times$. We determine the structure of the natural representation of $\mathrm{GL}(V)$ on the space $\mathcal{S}^*(V)^\chi$ of $\chi$-invariant distributions on $V$.

We consider a class of measures called autophage which was introduced and studied by Szekely for measures on the real line. We show that the autophage measures on finite-dimensional vector spaces over real or p-adic field are infinitely divisible without idempotent factors and are absolutely continuous with bounded continuous density. We also show that certain semistable measures on such vector spaces are absolutely continuous.

We consider a class of measures called autophage which was introduced and studied by Szekely for measures on the real line. We show that the autophage measures on finite-dimensional vector spaces over real or p-adic field are infinitely divisible without idempotent factors and are absolutely continuous with bounded continuous density. We also show that certain semistable measures on such vector spaces are absolutely continuous.

We consider a class of measures called autophage which was introduced and studied by Szekely for measures on the real line. We show that the autophage measures on finite-dimensional vector spaces over real or p-adic field are infinitely divisible without idempotent factors and are absolutely continuous with bounded continuous density. We also show that certain semistable measures on such vector spaces are absolutely continuous.

We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence.

We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence.

Book Source: Digital Library of India Item 2015.5812 dc.contributor.author: Halmos Paul R dc.date.accessioned: 2015-06-19T15:24:04Z dc.date.available: 2015-06-19T15:24:04Z dc.date.digitalpublicationdate: 2006-01-03 dc.date.citation: 1948 dc.identifier.barcode: 99999990174335 dc.identifier.origpath: /data1/upload/0028/645 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/5812 dc.description.scanningcentre: C-DAC, Noida dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 212 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: Princeton University Press Prienceton London dc.rights: Out_of_copyright dc.source.library: Ratan Tata Library, University Delhi dc.subject.classification: Natural Sciences dc.subject.classification: Mathematics dc.subject.keywords: Algebraic dc.title: Finite Dimensional Vector Spaces

Source: Digital Library of India Scanning Centre: C-DAC, Noida Source Library: Ratan Tata Library, University Delhi Date Accessioned: 7/10/2015 20:53 The Digital Library of India was a project under the auspices of the Government of India.

We give a general formula for the equivariant complex $K$-theory $K_G^*(V)$ of a finite dimensional real linear space $V$ equipped with a linear action of a compact group $G$ in terms of the representation theory of a certain double cover of $G$. Using this general formula, we give explicit computations in various interesting special cases. In particular, as an application we obtain explicit formulas for the $K$-theory of $C_r^*(GL(n,\RR))$, the reduced group C*-algebra of $GL(n,\RR)$.

The paper deals with various centering problems for probability measures on finite dimensional vector spaces. We show that for every such measure there exists a vector $h$ satisfying $\mu*\delta(h)=S(\mu*\delta (h))$ for each symmetry $S$ of $\mu$, generalizing thus Jurek's result obtained for full measures. An explicit form of the $h$ is given for infinitely divisible $\mu$. The main result of the paper consists in the analysis of quasi-decomposable (operator-semistable and operator-stable) measures and finding conditions for the existence of a `universal centering' of such a measure to a strictly quasi-decomposable one.

In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezout's theorem, we obtain the sufficient and necessary conditions on general curves where sharp $L^p-L^r$ extension estimates hold. Our main result can be considered as a nice generalization of works by Mochenhaupt and Tao and Iosevich and Koh. As an application of our sharp extension estimates, we also study the Falconer distance problems in two dimensions.

It is shown that, for any field F \subseteq R, any ordered vector space structure of F^n with Riesz interpolation is given by an inductive limit of sequence with finite stages (F^n,\F_{>= 0}^n) (where n does not change). This relates to a conjecture of Effros and Shen, since disproven, which is given by the same statement, except with F replaced by the integers, Z. Indeed, it shows that although Effros and Shen's conjecture is false, it is true after tensoring with Q.