Integral Representation Theory Research Articles

Local exterior square and Asai L-functions for GL(n) in odd characteristic

Let $F$ be a non-archimedean local field of odd characteristic $p > 0$. In this paper, we consider local exterior square $L$-functions $L(s,\pi,\wedge^2)$, Bump-Friedberg $L$-functions $L(s,\pi,BF)$, and Asai $L$-functions $L(s,\pi,As)$ of an irreducible admissible representation $\pi$ of $GL_m(F)$. In particular, we establish that those $L$-functions, via the theory of integral representations, are equal to their corresponding Artin $L$-functions $L(s,\wedge^2(\phi(\pi)))$, $L(s+1/2,\phi(\pi))L(s,\wedge^2(\phi(\pi)))$, and $L(s,As(\phi(\pi)))$ of the associated Langlands parameter $\phi(\pi)$ under the local Langlands correspondence. These are achieved by proving the identity for irreducible supercuspidal representations, exploiting the local to global argument due to Henniart and Lomeli.

Integration with respect to Baire vector measures with applications to the spectral theory

Let X be a completely regular Hausdorff space and C_b(X) the space of all bounded continuous scalar functions on X , equipped with the strict topology \beta_\sigma . We develop a general integral representation theory of (\beta_\sigma,\xi) -continuous linear operators from C_b(X) to a locally convex Hausdorff space (E,\xi) . We present equivalent conditions for a (\beta_\sigma,\xi) -continuous operator T\colon C_b(X) \to E to be weakly compact. If (\mathcal{A},\xi) is a sequentially complete unital locally convex algebra, we establish an integral representation of a (\beta_\sigma,\xi) -continuous unital algebra homomorphism T\colon C_b(X) \to \mathcal{A} . As an application, we develop spectral theory for operators T\colon C_b(X) \to \mathcal{L}_s(\mathcal{Y}) , where \mathcal{L}_s(\mathcal{Y}) denotes the algebra of bounded linear operators of a Banach space \mathcal{Y} into itself, equipped with the topology \tau_s of simple convergence.

An integral theory of dominant dimension of Noetherian algebras

Dominant dimension is introduced into integral representation theory, extending the classical theory of dominant dimension of Artinian algebras to projective Noetherian algebras (that is, algebras which are finitely generated projective as modules over a commutative Noetherian ring). This new homological invariant is based on relative homological algebra introduced by Hochschild in the 1950s. Amongst the properties established here are a relative version of the Morita-Tachikawa correspondence and a relative version of Mueller's characterization of dominant dimension. The behaviour of relative dominant dimension of projective Noetherian algebras under change of ground ring is clarified and we explain how to use this property to determine the relative dominant dimension of projective Noetherian algebras. In particular, we determine the relative dominant dimension of Schur algebras and quantized Schur algebras.

A survey of semisimple algebras in algebraic combinatorics

This is a survey of semisimple algebras of current interest in algebraic combinatorics, with a focus on questions which we feel will be new and interesting to experts in group algebras, integral representation theory, and computational algebra. The algebras arise primarily in two families: coherent algebras and subconstituent (aka. Terwilliger) algebras. Coherent algebras are subalgebras of full matrix algebras having a basis of 01-matrices satisfying the conditions that it be transpose-closed, sum to the all 1’s matrix, and contain a subset \(\Delta \) that sums to the identity matrix. The special case when \(\Delta \) is a singleton is the important case of an adjacency algebra of a finite association scheme. A Terwilliger algebra is a semisimple extension of a coherent algebra by a set of diagonal 01-matrices determined canonically from its basis elements and a choice of row. We will survey the current state of knowledge of the complex, real, rational, modular, and integral representation theory of these semisimple algebras, indicate their connections with other areas of mathematics, and present several open questions.

Flexibility of Analysis Serving Computational Polynomial Algebra or Arithmetics

Fundamental concepts in algebraic geometry such as algebraic cycles (with respect to intersection problems) or residues (with respect to division or interpolation ones) gain in flexibility once they are interpreted within the frame of currents in complex geometry ; such interpretation may even be transposed to the frame of arithmetics thanks to recent developments in analytic geometry over algebraically closed fields equipped with an ultrametric (instead of archimedean) absolute value. This survey intends to revisit from such currential interpretation (transposed if possible to the arithmetic setting, for example on Berkovich analytic spaces in place of complex spaces) improper intersection theory, together with the notions of local or global multiplicity, as well as multivariate residue theory with its articulations with Cauchy integral representation theory and Lagrange interpolation formula.

Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem

One of the most beautiful results in the integral representation theory of finite groups is a theorem of A. Weiss that detects a permutation R-lattice for the finite p-group G in terms of the restriction to a normal subgroup N and the N-fixed points of the lattice, where R is a finite extension of the p-adic integers. Using techniques from relative homological algebra, we generalize Weiss' Theorem to the class of infinitely generated pseudocompact lattices for a finite p-group, allowing R to be any complete discrete valuation ring in mixed characteristic. A related theorem of Cliff and Weiss is also generalized to this class of modules. The existence of the permutation cover of a pseudocompact module is proved as a special case of a more general result. The permutation cover is explicitly described.

Differential principal factors and Pólya property of pure metacyclic fields

Barrucand and Cohn’s theory of principal factorizations in pure cubic fields [Formula: see text] and their Galois closures [Formula: see text] with [Formula: see text] types is generalized to pure quintic fields [Formula: see text] and pure metacyclic fields [Formula: see text] with [Formula: see text] possible types. The classification is based on the Galois cohomology of the unit group [Formula: see text], viewed as a module over the automorphism group [Formula: see text] of [Formula: see text] over the cyclotomic field [Formula: see text], by making use of theorems by Hasse and Iwasawa on the Herbrand quotient of the unit norm index [Formula: see text] by the number [Formula: see text] of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different [Formula: see text]. The precise structure of the group of differential principal factors is determined with the aid of kernels of norm homomorphisms and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units [Formula: see text]. Generalizing criteria for the Pólya property of Galois closures [Formula: see text] of pure cubic fields [Formula: see text] by Leriche and Zantema, we prove that pure metacyclic fields [Formula: see text] of only one type cannot be Pólya fields. All theoretical results are underpinned by extensive numerical verifications of the [Formula: see text] possible types and their statistical distribution in the range [Formula: see text] of [Formula: see text] normalized radicands.

CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS

AbstractWe give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result, we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.

Elliptic and K-theoretic stable envelopes and Newton polytopes

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $$K$$ -theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the $$K$$ -theoretic stable envelopes and our elliptic stable envelopes. We show that the $$K$$ -theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the $${\mathfrak {gl}}_2$$ case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the $$K$$ -theoretic stable envelopes.

Tables of Pure Quintic Fields

By making use of our generalization of Barrucand and Cohn’s theory of principal factorizations in pure cubic fields and their Galois closures with 3 possible types to pure quintic fields and their pure metacyclic normal fields with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields N having normalized radicands in the range 2≤D3. Our classification is based on the Galois cohomology of the unit group UN, viewed as a module over the automorphism group Gal(N/K) of N over the cyclotomic field K=Q(ξ5), by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index (Uk:NN/K(UN)) by the number #(PN/K/PK) of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different DN/K. The precise structure of the F5-vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units (UN:U0). The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.

Holomorphy of adjoint L functions for quasisplit A_2

We study the poles of the twisted adjoint L function of a generic cuspidal automorphic representation of GL(3) or a quasisplit unitary group using a method pioneered by Ginzburg and Jiang and based on the theory of integral representations.

Lensless two-color ghost imaging from the perspective of coherent-mode representation**Project supported by the National Natural Science Foundation of China (Grant Nos. 61771067, 61631014, 61471051, and 61401036) and the Youth Research and Innovation Program of Beijing University of Posts and Telecommunications, China (Grant Nos. 2015RC12 and 2017RC10).

The coherent-mode representation theory is firstly used to analyze lensless two-color ghost imaging. A quite complicated expression about the point-spread function (PSF) needs to be given to analyze which wavelength has a stronger affect on imaging quality when the usual integral representation theory is used to ghost imaging. Unlike this theory, the coherent-mode representation theory shows that imaging quality depends crucially on the distribution of the decomposition coefficients of the object imaged in a two-color ghost imaging. The analytical expression of the decomposition coefficients of the object is unconcerned with the wavelength of the light used in the reference arm, but has relevance with the wavelength in the object arm. In other words, imaging quality of two-color ghost imaging depends primarily on the wavelength of the light illuminating the object. Our simulation results also demonstrate this conclusion.

The Bartle–Dunford–Schwartz and the Dinculeanu–Singer theorems revisited

Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. By the classical Bartle–Dunford–Schwartz theorem, any operator S∈L(C(Ω),Y) admits an integral representation with respect to a Y⁎⁎-valued measure. By the Dinculeanu–Singer theorem, each operator U∈L(C(Ω,X),Y) admits an integral representation with respect to an L(X,Y⁎⁎)-valued measure. We establish an integral representation for any operator S∈L(C(Ω),L(X,Y)) with respect to an L(X,Y⁎⁎)-valued measure. This far-reaching extension of the Bartle–Dunford–Schwartz theorem serves as a departure point for a general integral representation theory developed in the present paper. In particular, it is an efficient tool that enables us to give an alternative simple proof to the Dinculeanu–Singer theorem. The latter theorem is proved in a more general context of X-valued continuous functions with p-compact range. Among others, useful formulas which connect different vector measures are deduced.

Integral Representation of Continuous Operators with Respect to Strict Topologies

Let ,X, be a completely regular Hausdorff space and ,mathcal {B}o, be the sigma -algebra of Borel sets in X. Let C_b(X) (resp. B(mathcal {B}o)) be the space of all bounded continuous (resp. bounded mathcal {B}o-measurable) scalar functions on X, equipped with the natural strict topology beta . We develop a general integral representation theory of (beta ,xi )-continuous operators from C_b(X) to a lcHs (E,xi ) with respect to the representing Borel measure taking values in the bidual E''_xi of (E,xi ). It is shown that every (beta ,xi )-continuous operator T:C_b(X)rightarrow E possesses a (beta ,xi _mathcal {E})-continuous extension {hat{T}}:B(mathcal {B}o)rightarrow E''_xi , where xi _mathcal {E} stands for the natural topology on E''_xi . If, in particular, X is a k-space and (E,xi ) is quasicomplete, we present equivalent conditions for a (beta ,xi )-continuous operator T:C_b(X)rightarrow E to be weakly compact. As an application, we have shown that if X is a k-space and a quasicomplete lcHs (E,xi ) contains no isomorphic copy of c_0, then every (beta ,xi )-continuous operator T:C_b(X)rightarrow E is weakly compact.

Rational Conjugacy of Torsion Units in Integral Group Rings of Non-Solvable Groups

AbstractWe introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus conjecture for the group PSL(2, 19). We also prove the Zassenhaus conjecture for PSL(2, 23). In a second application we show that there are no normalized units of order 6 in the integral group rings of M10 and PGL(2, 9). This completes the proof of a theorem of Kimmerle and Konovalov that shows that the prime graph question has an affirmative answer for all groups having an order divisible by at most three different primes.

A Riesz representation theory for completely regular Hausdorff spaces and its applications

Abstract Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb (X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · || F ) -continuous operators T : Cb (X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · || F )-continuous operators T : Cb (X, E) → F. As an application, we study (β, || · || F )-continuous weakly compact and unconditionally converging operators T : Cb (X, E) → F. In particular, we establish the relationship between these operators and the corresponding Borel operator measures given by the Riesz representation theorem. We obtain that if X is a k-spaceand E is reflexive, then (Cb (X, E), β) has the V property of Pełczynski.

On Šilov boundary for function spaces

In this paper, we develop the notion of Šilov boundary for closed, point separating subspaces of spaces of continuous functions on a compact Hausdorff space. We give an example to show that the set of peak points need not be a boundary. We give a proof of Bishop's theorem that describes the Choquet boundary in the case of closed, point separating, subalgebras of continuous functions that do not contain the constant function. For closed point separating subspaces, using results of R.R. Phelps on integral representation theory for subspaces, we show that the closure of the Choquet boundary as a bundle is the Šilov boundary bundle similar in the sense to the one considered by D. Blecher [5].

Differential operators for a scale of Poisson type kernels in the unit disc

In this paper, we introduce a scale of differential operators which is shown to correspond canonically to a certain scale of solution kernels generalizing the classical Poisson kernel for the unit disc. The scale of kernels studied is very natural and appears in many places in mathematical analysis, such as in the theory of integral representations of biharmonic functions in the unit disc.

The study of a generalized integral of the Temlyakov type in the unit bicircle

The theory of integral representations of analytic functions of many complex variables is an important branch of the multidimensional complex analysis. An essential contribution to this theory was done by A. A. Temlyakov, who obtained integral representations for functions of two complex variables [1, 2]. The functions were assumed to be analytic in the class of parametrically defined bounded convex complete bicircular domains. L. A. Aizenberg [3] considered an arbitrary function summable in the sense of Lebesgue at the boundary of the defining domain as the density in the Temlyakov integrals. On the base of Temlyakov integral representations he introduced the notion of Temlyakov type integrals. This paper is dedicated to the study of properties of a generalized integral of the Temlyakov type

Some Remarks on the Tensor Product of Algebras and Applications II

We apply the G-algebra theory to the tensor product of algebras. These considerations are applied to extend the results of Alghamdi and Khammash [1], Khammash [4] and Külshammer [5, Proposition 1.2] on the tensor product of group algebras and modules over an algebraically closed field to lattices over a complete discrete valuation ring. This places these results in the standard integral finite group modular representation theory of G-algebras as pioneered by Puig (cf. [8]). We also study some aspects of covering homomorphisms and the Green correspondence in this context (cf. [8, Sections 20 and 25]).