自由能展开式中点群的不可约表示的基函数

一个群的基函数的选择并非唯一，不同的基函数对应不同的表示矩阵。即使相同的表示矩阵，基函数也可以有不同的选择。在相变的宏观唯象理论中， 自由能展开式可由同一个表示矩阵的基函数来构造，因而给出不可约表示的基函数表就非常有意义。现有文献给出的32点群不可约表示的基函数只写到二次幂，而且有些文献中的同一个不可约表示所选取的不同的基函数却对应不同的表示矩阵，这样在构造群变换不变式时，就会出错。该文将基函数表写至三次幂, 这有助于准确、迅速地写出到六次幂群变换不变的自由能表达式。由新的基函数表发现十八种点群有三次幂的群操作不变式, 高温相属这些点群铁电体, 发生的本征铁电相变为一级相变。     

Representations of the Quantum Algebra Uq(un, 1)

The main aim of the paper is to study infinite-dimensional representations of the real form U _q (u _{n, 1}) of the quantized universal enveloping algebra U _q (gl _{n + 1}). We investigate the principal series of representations of U _q (u _{n, 1}) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U _q (u _{n, 1}) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U _q (u _{n, 1}) has finite-dimensional irreducible *-representations.     

Binary polyhedral groups and Euclidean diagrams

Recently, J. McKay [7] has observed that the irreducible complex representations of the binary polyhedral groups can be arranged in order to form the vertices of a Euclidean diagram in such a way that the tensor product of any irreducible representation M with the standard two-dimensional representation is the direct sum of the irreducible representations which are the neighbors of M in the diagram, and he asked for an explanation. In this note, we will show that any self-dual two-dimensional representation gives rise to a generalized Euclidean diagram, and that this in fact can be used to give a proof of the classification theorem of the binary polyhedral groups which at the same time furnishes a list of the irreducible representations and also gives the minimal splitting field.     

Characters and the structure of finite semigroups

It is shown that properties of the structure of a finite semigroup S such as the order of the set ofJ of S and the existence of normal subsemigroups may be deduced from the knowledge of the characters of the irreducible representations of S. The character table of the full transformation semigroup T₄ of a four-element set is given.     

Reduction of discontinuity for derivations on Frechet algebras

We consider strictly irreducible representations with whichthe discontinuity of a derivation on a (locally multiplicativelyconvex) Fréchet algebra must be associated. Only thosestrictly irreducible representations which are compatible withthe topology of the algebra are considered. The main resultsshow that when consideration is fixed upon each seminorm, theexceptional set of primitive ideals supporting the discontinuitymust be a finite set, with each ideal being the kernel of somefinite-dimensional irreducible representation. This result isthe best possible, as can be seen by considering the radicalFréchet algebra constructed by Charles Read with identityadjoined which has a derivation with separating ideal that isthe entire algebra, and one could take (countable) Fréchetproducts of his counterexample. It is also proved that derivationson commutative Fréchet algebras, the structure spacesof which are compact metric in the weak* topology, have onlyfinitely many such exceptional points overall.     

Free group representations from vector-valued multiplicative functions,I

Let Γ denote a noncommutative free group and let Ω stand for its boundary. We construct a large class of unitary representations of Γ. This class contains many previously studied representations, and is closed under several natural operations. Each of the constructed representations is in fact a representation of Γ ⋉_λ C(Ω). We prove here that each of them is irreducible as a representation of Γ ⋉_λ C(Ω). Actually, as will be shown in further work, each of them is irreducible as a representation of Γ, or is the direct sum of exactly two irreducible, inequivalent Γ-representations. This research was supported by the Italian CNR.     

Constructing irreducible representations of discrete groups

The decomposition of unitary representations of a discrete group obtained by induction from a subgroup involves commensurators. In particular Mackey has shown that quasi-regular representations are irreducible if and only if the corresponding subgroups are self-commensurizing. The purpose of this work is to describe general constructions of pairs of groups Γ₀ with Γ its own commensurator in Γ. These constructions are then applied to groups of isometries of hyperbolic spaces and to lattices in algebraic groups.     

Genuine Representations of the Metaplectic Group

This paper determines the –correspondence for the dual pairs (O(p, q), Sp(2n, R)) when p+q=2n+1. As a consequence, there is a natural bijection between genuine irreducible representations of the metaplectic group Mp(2n, R) and irreducible representations of SO(p, q) with p+q=2n+1.     

Modular irreducible representations of the symmetric group as linear codes

We describe a particularly easy way of evaluating the modular irreducible matrix representations of the symmetric group. It shows that Specht’s approach to the ordinary irreducible representations, along Specht polynomials, can be unified with Clausen’s approach to the modular irreducible representations using symmetrized standard bideterminants. The unified method, using symmetrized Specht polynomials, is very easy to explain, and it follows directly from Clausen’s theorem by replacing the indeterminate x_ij of the letter place algebra by x_jⁱ.Our approach is implemented in SYMMETRICA. It was used in order to obtain computational results on code theoretic properties of the p-modular irreducible representation [λ]_p corresponding to a p-regular partition λ via embedding it into representation spaces obtained from ordinary irreducible representations. The first embedding is into the permutation representation induced from the column group of a standard Young tableau of shape λ. The second embedding is the embedding of [λ]_p into the space of , the p-modular representation obtained from the ordinary irreducible representation [λ] by reducing the coefficients modulo p.We include a few tables with dimensions and minimum distances of these codes; others can be found via our home page.     

Duality between compactness and discreteness beyond pontryagin duality

One of the most striking results of Pontryagin’s duality theory is the duality between compact and discrete locally compact abelian groups. This duality also persists in part for objects associated with noncommutative topological groups. In particular, it is well known that the dual space of a compact topological group is discrete, while the dual space of a discrete group is quasicompact (i.e., it satisfies the finite covering theorem but is not necessarily Hausdorff). The converse of the former assertion is also true, whereas the converse of the latter is not (there are simple examples of nondiscrete locally compact solvable groups of height 2 whose dual spaces are quasicompact and non-Hausdorff (they are T ₁ spaces)). However, in the class of locally compact groups all of whose irreducible unitary representations are finite-dimensional, a group is discrete if and only if its dual space is quasicompact (and is automatically a T ₁ space). The proof is based on the structural theorem for locally compact groups all of whose irreducible unitary representations are finite-dimensional. Certain duality between compactness and discreteness can also be revealed in groups that are not necessarily locally compact but are unitarily, or at least reflexively, representable, provided that (in the simplest case) the irreducible representations of a group form a sufficiently large family and have jointly bounded dimensions. The corresponding analogs of compactness and discreteness cannot always be easily identified, but they are still duals of each other to some extent.     

A groupoid approach to discrete inverse semigroup algebras

Let K be a commutative ring with unit and S an inverse semigroup. We show that the semigroup algebra KS can be described as a convolution algebra of functions on the universal étale groupoid associated to S by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal C^∗-algebra. It provides a convenient topological framework for understanding the structure of KS, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality.Using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup S that can be induced from associated groups as precisely those satisfying a certain “finiteness condition.” This “finiteness condition” is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent.     

Generalized Shalika models of p-adic SO4n and functoriality

Let F denote a p-adic local field of characteristic zero. In this paper, we investigate the structures of irreducible admissible representations of SO_4n (F) having nonzero generalized Shalika models and find relations between the generalized Shalika models and the local Arthur parameters, which support our conjectures on the local Arthur parametrization and the local Langlands functoriality in terms of the dual group associated to the spherical variety, which is attached to the generalized Shalika models.     

对称群S5的表示环

本文利用有限群特征标理论计算了对称群S₅的所有不可约复表示的幂公式.根据求解幂公式过程中得到的S₅任意两个不可约表示张量积的分解情况,作者刻画了S₅上表示环r(S₅)及其若干结构性质,如极小生成元关系式表达、单位群、本原幂等元、行列式与Casimir数.     

Low discrepancy sequences failing Poissonian pair correlations

Let G be a locally compact totally disconnected topological group. Under a necessary mild assumption, we show that the irreducible unitary representations of G are uniformly admissible if and only if the irreducible smooth representations of G are uniformly admissible. An analogous result for *-algebras is also established. We further show that the property of having uniformly admissible irreducible smooth representations is inherited by finite-index subgroups and overgroups of G.     

Brauer algebras of simply laced type

The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A _{n − 1} on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph Q, called the Brauer algebra of type Q, and study its structure in the cases where Q is a Coxeter graph of simply laced spherical type (so its connected components are of type A _{n − 1}, D _n , E₆, E₇, E₈). We find its irreducible representations and its dimension, and show that the algebra is cellular. The algebra is generically semisimple and contains the group algebra of the Coxeter group of type Q as a subalgebra. It is a ring homomorphic image of the Birman-Murakami-Wenzl algebra of type Q; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.     

Construct irreducible representations of quantum groups U q (ƒ m (K))

In this paper, we construct families of irreducible representations for a class of quantum groups U _q (ƒ _m (K)). First, we give a natural construction of irreducible weight representations for U _q (ƒ _m (K)) using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of U _q (ƒ _m (K)). As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.      

Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory

We establish an isomorphism between the vertex and spinor representations of affine Lie algebras for types D_l⁽¹⁾and D_{l + 1}⁽²⁾. We also study decomposition of spinor representations using the infinite family of Casimir operators and prove that they are either irreducible or have two irreducible components. We show that the vertex and spinor constructions of the representations can be reformulated in the language of two-dimensional quantum field theory. In this physical context, the two constructions yield the generalized sine-Gordon and Thirring models, respectively, already in renormalized form. The isomorphism of representations implies an equivalence of these two models which is known in quantum field theory as the boson-fermion correspondence