Related papers: Automata and cells in affine Weyl groups
Both cellular automata (CA) and lattice-gas automata (LG) provide finite algorithmic presentations for certain classes of infinite dynamical systems studied by symbolic dynamics; it is customary to use the term `cellular automaton' or…
A reduced word of a permutation $w$ is a minimal length expression of $w$ as a product of simple transpositions. We examine the computational complexity, formulas and (randomized) algorithms for their enumeration. In particular, we prove…
Regular languages -- the languages accepted by deterministic finite automata -- are known to be precisely the languages recognized by finite monoids. This characterization is the origin of algebraic language theory. In this paper, we…
Regular expressions in an Automata Theory and Formal Languages course are mostly treated as a theoretical topic. That is, to some degree their mathematical properties and their role to describe languages is discussed. This approach fails to…
FC is a first-order logic that reasons over all factors of a finite word using concatenation, and can define non-regular languages like that of all squares (ww). In this paper, we establish that there are regular languages that are not…
Symbolic automata are finite state automata that support potentially infinite alphabets, such as the set of rational numbers, generally applied to regular expressions/languages over finite words. In symbolic automata (or automata modulo…
In this work we study a non-linear generalization based on affine transformations of probabilistic and quantum automata proposed recently by D\'iaz-Caro and Yakary{\i}lmaz \cite{DCY16A} referred as affine automata. First, we present…
To each category C of modules of finite length over a complex simple Lie algebra g, closed under tensoring with finite dimensional modules, we associate and study a category Aff(C)_\kappa of smooth modules (in the sense of Kazhdan and…
In this paper we show that the lowest two-sided ideal of an affine Hecke algebra is affine cellular for all choices of parameters. We explicitely describe the cellular basis and we show that the basis elements have a nice decomposition when…
We study sources of isomorphisms of additive cellular automata on finite groups (called index-group). It is shown that many isomorphisms (called regular) of automata are reducible to the isomorphisms of underlying algebraic structures (such…
Most Formal Languages and Automata Theory courses explore the duality between computation models to recognize words in a language and computation models to generate words in a language. For students unaccustomed to formal statements, these…
This work considers weak deterministic B\"uchi automata reading encodings of non-negative reals in a fixed base. A Real Number Automaton is an automaton which recognizes all encoding of elements of a set of reals. It is explained how to…
The states of a deterministic finite automaton A can be identified with collections of words in Pf(L(A)) -- the set of prefixes of words belonging to the regular language accepted by A. But words can be ordered and among the many possible…
Given an affine Coxeter group $W$, the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements that was introduced by Shi to study Kazhdan-Lusztig cells for $W$. In particular, Shi showed that each…
The decomposition of representations of compact classical Lie groups into representations of finite subgroups is discussed. A Mathematica package is presented that can be used to compute these branching rules using the Weyl character…
We classify the elements of $W(\tilde{A}_n)$ by giving a canonical reduced expression for each, using basic tools among which affine length. We give some direct consequences for such a canonical form: a description of left multiplication by…
Extended affine Weyl groups are the Weyl groups of extended affine root systems. Finite presentations for extended affine Weyl groups are known only for nullities $\leq 2$, where for nullity 2 there is only one known such presentation. We…
Let v and w be nontrivial words in two free groups. We prove that, for all sufficiently large finite non-abelian simple groups G, there exist subsets C of v(G) and D of w(G) of size such that every element of G can be realized in at least…
Given a regular language L over an ordered alphabet $\Sigma$, the set of lexicographically smallest (resp., largest) words of each length is itself regular. Moreover, there exists an unambiguous finite-state transducer that, on a given word…
We study the representation theory of finite W-algebras. After introducing parabolic subalgebras to describe the structure of W-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the…