Representations of 2-transitive topological groups

Simple transitive 2-representations of 2-categories associated to self-injective cores

Given a finite dimensional algebra $A$, we consider certain sets of idempotents of $A$, called self-injective cores, to which we associate 2-subcategories of the 2-category of projective bimodules over $A$. We classify the simple transitive 2-representations of such 2-subcategories, vastly generalizing the main result of arXiv:1805.05724.

Transitive 2-representations of finitary 2-categories

In this article, we define and study the class of simple transitive $2$-representations of finitary $2$-categories. We prove a weak version of the classical Jordan-Hölder Theorem where the weak composition subquotients are given by simple transitive $2$-representations. For a large class of finitary $2$-categories we prove that simple transitive $2$-representations are exhausted by cell $2$-representations. Finally, we show that this large class contains finitary quotients of $2$-Kac-Moody algebras.

Simple transitive $2$-representations and Drinfeld center for some finitary $2$-categories

We classify all simple transitive $2$-representations for two classes of finitary $2$-categories associated with tree path algebras and also for one class of fiat $2$-categories associated with truncated polynomial rings. Additionally, we compute the Drinfeld centers for all these $2$-categories.

Simple transitive 2-representations for some 2-subcategories of Soergel bimodules

We classify simple transitive $2$-representations of certain $2$-sub\-ca\-te\-go\-ri\-es of the $2$-category of Soergel bimodules over the coinvariant algebra in Coxeter types $B_2$ and $I_2(5)$. In the $I_2(5)$ case it turns out that simple transitive $2$-representations are exhausted by cell $2$-representations. In the $B_2$ case we show that, apart from cell $2$-representations, there is a unique, up to equivalence, additional simple transitive $2$-representation and we give an explicit construction of this $2$-representation.

Simple transitive 2-representations of small quotients of Soergel bimodules

In all finite Coxeter types but $I_2(12)$, $I_2(18)$ and $I_2(30)$, we classify simple transitive $2$-rep\-re\-sen\-ta\-ti\-ons for the quotient of the $2$-category of Soergel bimodules over the coinvariant algebra which is associated to the two-sided cell that is the closest one to the two-sided cell containing the identity element. It turns out that, in most of the cases, simple transitive $2$-representations are exhausted by cell $2$-representations. However, in Coxeter types $I_2(2k)$, where $k\geq 3$, there exist simple transitive $2$-representations which are not equivalent to cell $2$-r…

Pathological and Omega-transitive Representations of Free Groups

Given a linear order $Ω$ its automorphism group $\Aut(Ω)$ forms a lattice-ordered group via pointwise order. Assuming the continuum to be a regular cardinal, we show that \emph{pathological} and \emph{$ω$-transitive} (i.e. highly transitive) representations of free groups abound within \emph{large} permutation groups of linear orders. Consequently, under the Generalized Continuum Hypothesis it is then true that given any linear order $Ω$ for which $|Ω| = $ cof$(Ω) = \aleph_i$ ($i \in \N$) then any permutation group that is large in $\Aut(Ω)$ contains an $ω$-transitive representation of $G_{\al…

Two-color Soergel calculus and simple transitive 2-representations

In this paper we complete the $\mathrm{ADE}$-like classification of simple transitive $2$-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In particular, we use bipartite graphs and zigzag algebras of $\mathrm{ADE}$ type to give an explicit construction of a graded (non-strict) version of all these $2$-representations. Moreover, we give simple combinatorial criteria for when two such $2$-representations are equivalent and for when their Grothendieck groups give rise to isomorphic representations. Finally, our construction also gives a large cl…

Simple transitive 2-representations via (co)algebra 1-morphisms

For any fiat 2-category C, we show how its simple transitive 2-representations can be constructed using coalgebra 1-morphisms in the injective abelianization of C. Dually, we show that these can also be constructed using algebra 1-morphisms in the projective abelianization of C. We also extend Morita-Takeuchi theory to our setup and work out several examples explicitly.

有限 Coxeter 类型的 Soergel 双模的简单传递 $2$-表示

Simple transitive $2$-representations of Soergel bimodules for finite Coxeter types

该研究考察有限 Coxeter 类型 Soergel 双模的简单传递 $2$-表示，证明其等价类有限，并在除 $H_3$、$H_4$ 外的所有类型中完成分类。

On Transitive Systems of Subspaces in a Hilbert Space

Methods of *-representations in Hilbert space are applied to study of systems of $n$ subspaces in a linear space. It is proved that the problem of description of $n$-transitive subspaces in a finite-dimensional linear space is *-wild for $n \geq 5$.

Analogues of centralizer subalgebras for fiat 2-categories and their 2-representations

The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive $2$-representations with a fixed apex $\mathcal{J}$ of a fiat $2$-category $\cC$ and the set of equivalence classes of faithful simple transitive $2$-representations of the fiat $2$-subquotient of $\cC$ associated with a diagonal $\mathcal{H}$-cell in $\mathcal{J}$. As an application, we classify simple transitive $2$-representations of various categories of Soergel bimodules, in particular, completing the classification in types $B_3$ and $B_4$.

2-representations of small quotients of Soergel bimodules in infinite types

We determine for which Coxeter types the associated small quotient of the $2$-category of Soergel bimodules is finitary and, for such a small quotient, classify the simple transitive $2$-representations (sometimes under the additional assumption of gradability). We also describe the underlying categories of the simple transitive $2$-representations. For the small quotients of general Coxeter types, we give a description for the cell $2$-representations.

2-Representations and Associated Coalgebra 1-Morphisms for Locally Wide Finitary 2-Categories

We define locally wide finitary 2-categories by relaxing the definition of finitary 2-categories to allow infinitely many objects and isomorphism classes of 1-morphisms and infinite dimensional hom-spaces of 2-morphisms. After defining related concepts including transitive 2-representations in this setting, we provide a new method of constructing coalgebra 1-morphisms associated to transitive 2-representations of locally wide weakly fiat 2-categories, and demonstrate that any such transitive 2-representation is equivalent to a certain subcategory of the category of comodule 1-morphisms over th…

Permutation Representations of the Orbits of the Automorphism Group of a Finite Module over Discrete Valuation Ring

Consider a discrete valuation ring $R$ whose residue field is finite of cardinality at least $3$. For a finite torsion module, we consider transitive subsets $O$ under the action of the automorphism group of the module. We prove that the associated permutation representation on the complex vector space $C[O]$ is multiplicity free. This is achieved by obtaining a complete description of the transitive subsets of $O\times O$ under the diagonal action of the automorphism group.