Finite semisimple group algebra of a normally monomial group

The wedge-direct sum of a sequence of semisimple algebras (over an arbitrary field) was introduced in [Xu, B. (2019). Wedge-direct sums of table algebras and applications to association schemes, I. Commun. Algebra 47(8):3309–3328]. The group algebras (over the complex field) of finite groups are a special class of semisimple algebras. In [Xu, B. (2020). Wedge-direct sums of table algebras and applications to association schemes, II. J. Pure Appl. Algebra 224(11):106402], the wedge-direct sum of a sequence of group algebras of finite p-groups is used to obtain the characterization and classification of a class of p-table algebras. In this article, we continue the research in above mentioned articles and characterize, in terms of table algebras, the wedge-direct sum of a sequence of group algebras of arbitrary finite groups. Applications to association schemes will also be discussed.

Semisimple finite group algebra of a generalized strongly monomial group

Tensorial sets adapted to sequences of finite subgroups are applied to the crystal field problem, and a general method for generating sequence‐adapted molecular tensors using finite group algebra is formulated. All subgroup sequences of the abstract finite group G(24), isomorphic to the octahedral, O, tetrahedral, Td, and symmetric, S(4), groups are tabulated with explicit isomorphisms provided. The sequences fall into eight equivalence classes. A catalog of irreducible representations of G(24) adapted to a member of each of the eight sequence classes is given together with the transformations which generate representations adapted to all other sequences. With this data it is possible to systematically generate tensorial sets adapted to any sequence of a realization of G(24). Unitary transformations which adapt conventional forms of first‐ and second‐rank irreducible tensorial sets of the rotation group to the eight sequences of the octahedral group are provided. Forms suitable for use with magnetic fields are included. The problem of a d1 ion in a trigonal crystal field is treated with sequence‐adapted molecular tensors, and the utility of different sequences for descent in symmetry is discussed.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Residually finite quantum group algebras

Kitaev’s quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras.

A deformation-theoretic version of Maschke's theorem for modular group algebras: The commutative case

I construct an algebraic model for a typical fiber on a 1 + 1 dimensional spacetime. The vector space comprising the fiber is composed of elements x ⊗ x formed from the direct product of two copies of an element x in the D 2 = C 2 ⊗ C 2 finite group algebra over the real numbers. The fiber contains subspaces whose elements can be associated with the tangent and momentum vectors of trajectories in the manifold. The fiber also contains a subspace whose elements are associated with the local flow of action of each trajectory. The condition of minimum action translates into a constraint on the original vector x in the direct product structure.KeywordsTangent spacemanifoldfiber differential geometry1 + 1 spacetime

The varieties for some Specht modules

Important algebraic calculations for n-variables polynomial codes

A basis for the unitary subgroup of the group of units in a finite commutative group algebra

Nontriviality of the first Hochschild cohomology of some block algebras of finite groups

We use the Perron-Frobenius Theorem to define, study and, in some sense, classify special simple modules over arbitrary finite dimensional positively based algebras. For group algebras of finite Weyl groups with respect to the Kazhdan-Lusztig basis, this agrees with Lusztig's notion of a special module introduced in [Lu1].

Supports of irreducible spherical representations of rational Cherednik algebras of finite Coxeter groups

A table algebra was defined in [1] in order to consider in a uniform way the common properties of conjugacy classes and irreducible characters. Non-commutative table algebras were introduced in [5]. They generalize properties of such well-known objects as coherent and Hecke algebras. Here we extend the main definition of a non-commutative table algebra by letting the ground field be an integral domain. We call these algebrasgeneralized table algebras (GT-algebras, in brief). It is worth mentioning that this class of algebras includes generic Hecke-Iwahori algebras of finite Coxeter groups. We develop the general theory for this type of algebras which includes their representation theory and theory of closed subsets. We also study the properties of primitive integral table algebras.