501 research outputs found
Representation Theory of Finite Semigroups, Semigroup Radicals and Formal Language Theory
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 given to obtain many new results, as well as easier proofs of several
results in the literature, involving: triangularizability of finite semigroups;
which semigroups have (split) basic semigroup algebras, two-sided semidirect
product decompositions of finite monoids; unambiguous products of rational
languages; products of rational languages with counter; and \v{C}ern\'y's
conjecture for an important class of automata
Strongly solvable spherical subgroups and their combinatorial invariants
A subgroup H of an algebraic group G is said to be strongly solvable if H is
contained in a Borel subgroup of G. This paper is devoted to establishing
relationships between the following three combinatorial classifications of
strongly solvable spherical subgroups in reductive complex algebraic groups:
Luna's general classification of arbitrary spherical subgroups restricted to
the strongly solvable case, Luna's 1993 classification of strongly solvable
wonderful subgroups, and the author's 2011 classification of strongly solvable
spherical subgroups. We give a detailed presentation of all the three
classifications and exhibit interrelations between the corresponding
combinatorial invariants, which enables one to pass from one of these
classifications to any other.Comment: v3: 58 pages, revised according to the referee's suggestions; v4:
numbering of sections changed to agree with the published versio
Quivers of monoids with basic algebras
We compute the quiver of any monoid that has a basic algebra over an
algebraically closed field of characteristic zero. More generally, we reduce
the computation of the quiver over a splitting field of a class of monoids that
we term rectangular monoids (in the semigroup theory literature the class is
known as ) to representation theoretic computations for group
algebras of maximal subgroups. Hence in good characteristic for the maximal
subgroups, this gives an essentially complete computation. Since groups are
examples of rectangular monoids, we cannot hope to do better than this.
For the subclass of -trivial monoids, we also provide a semigroup
theoretic description of the projective indecomposables and compute the Cartan
matrix.Comment: Minor corrections and improvements to exposition were made. Some
theorem statements were simplified. Also we made a language change. Several
of our results are more naturally expressed using the language of Karoubi
envelopes and irreducible morphisms. There are no substantial changes in
actual result
The algebra of adjacency patterns: Rees matrix semigroups with reversion
We establish a surprisingly close relationship between universal Horn classes
of directed graphs and varieties generated by so-called adjacency semigroups
which are Rees matrix semigroups over the trivial group with the unary
operation of reversion. In particular, the lattice of subvarieties of the
variety generated by adjacency semigroups that are regular unary semigroups is
essentially the same as the lattice of universal Horn classes of reflexive
directed graphs. A number of examples follow, including a limit variety of
regular unary semigroups and finite unary semigroups with NP-hard variety
membership problems.Comment: 30 pages, 9 figure
Equivariant compactifications of reductive groups
We study equivariant projective compactifications of reductive groups
obtained by closing the image of a group in the space of operators of a
projective representation. We describe the structure and the mutual position of
their orbits under the action of the doubled group by left/right
multiplications, the local structure in a neighborhood of a closed orbit, and
obtain some conditions of normality and smoothness of a compactification. Our
methods of research use the theory of equivariant embeddings of spherical
homogeneous spaces and of reductive algebraic semigroups.Comment: 30 pages, AmSLaTeX. Bibliography: 36 item
- …