Related papers: On infinite groups admitting a faithful irreducibl…
We give a characterization of connected solvable groups in terms of the existence of representations with certain geometric properties. The existence of such representations for the group of upper triangular matrices played an important…
We determine the minimal degree of a faithful permutation representation for each group of order $p^6$ where $p$ is an odd prime. We also record how to obtain such a representation.
In this paper, we study the spectrality of infinite convolutions generated by infinitely many admissible pairs which may not be compactly supported, where the spectrality means the corresponding square integrable function space admits a…
Let $n$ be a positive integer, and let $R$ be a (possibly infinite dimensional) finitely presented algebra over a computable field of characteristic zero. We describe an algorithm for deciding (in principle) whether $R$ has at most finitely…
By [R. Bautista, P. Gabriel, A.V Roiter., L. Salmeron, Representation-finite algebras and multiplicative basis. Invent. Math. 81 (1985) 217-285.], a finite-dimensional algebra having finitely many isoclasses of indecomposable…
We introduce the concept of quantifying the extent to which a finitely generated group is residually finite. The quantification is carried out for some examples including free groups, the first Grigorchuk group, finitely generated nilpotent…
In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are…
Non-deductive reasoning systems are often {\em representation dependent}: representing the same situation in two different ways may cause such a system to return two different answers. Some have viewed this as a significant problem. For…
We show that finite quasisimple groups of Lie type in characteristic $p$ with an irreducible representation of prime degree $r$ over a finite field of characteristic $p$ have orders bounded above by a function of $r$, independent of $p$. We…
We give a simple formula for the essential dimension of a finite pseudo-reflection group at a prime p and determine the absolute essential dimension for most irreducible pseudo-reflection groups. We also study the "poor man's essential…
We survey recent work ranging around the question in how far a group, or a property of a group, is determined by the set of finite quotient groups. Our focus lies on $S$-arithmetic groups, branch groups, and their relatives.
The following theorem is proved: Let $G$ be a finite group and $\pi_e(G)$ be the set of element orders in $G$. If $\pi_e(G) \cap \{2\}=\emptyset$; or $\pi_e(G) \cap \{3, 4\}=\emptyset$; or $\pi_e(G) \cap \{3,5\}=\emptyset$, then $G$ is…
We call a group $G$ {\it algorithmically finite} if no algorithm can produce an infinite set of pairwise distinct elements of $G$. We construct examples of recursively presented infinite algorithmically finite groups and study their…
We find upper and lower bounds of the multiplicities of irreducible admissible representations $\pi$ of a semisimple Lie group $G$ occurring in the induced representations $Ind_H^G\tau$ from irreducible representations $\tau$ of a closed…
Each irreducible representation of the affine group of a finite field has a unique maximal inductive algebra, and it is self adjoint.
We study finite dimensional representations of the projective modular group. Various explicit dimension formulas are given.
The point of view of these notes on the topic is to bring out the flavour that Representation Theory is an extension of the first course on Group Theory. We also emphasize the importance of the base field. These notes cover completely the…
This is a survey article on the theory of finite complex reflection groups. No proofs are given but numerous references are included.
We introduce the class of projective reflection groups which includes all complex reflection groups. We show that several aspects involving the combinatorics and the representation theory of all non exceptional irreducible complex…
We prove that every finite dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.