2,333 research outputs found
An automaton-theoretic approach to the representation theory of quantum algebras
We develop a new approach to the representation theory of quantum algebras
supporting a torus action via methods from the theory of finite-state automata
and algebraic combinatorics. We show that for a fixed number , the
torus-invariant primitive ideals in quantum matrices can be seen as
a regular language in a natural way. Using this description and a semigroup
approach to the set of Cauchon diagrams, a combinatorial object that
paramaterizes the primes that are torus-invariant, we show that for fixed,
the number of torus-invariant primitive ideals in quantum matrices
satisfies a linear recurrence in over the rational numbers. In the case we give a concrete description of the torus-invariant primitive ideals
and use this description to give an explicit formula for the number P(3,n).Comment: 31 page
Factorizations of languages and commutativity conditions
Representations of languages as a product (catenation) of languages are investigated, where the factor languages are "prime", that is, cannot be decomposed further in a nontrivial manner. In general, such prime decompositions do not necessarily exist. If they exist, they are not necessarily unique - the number of factors can vary even exponentially. The paper investigates prime decompositions, as well as the commuting of the factors, especially for the case of finite languages. In particular, a technique about commuting is developed in Section 4, where the factorization of languages L1 and L2 is discussed under the assumption L1L2 = L2L1
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
LIPIcs
A regular language L of finite words is composite if there are regular languages Lâ,Lâ,âŚ,L_t such that L = â_{i = 1}^t L_i and the index (number of states in a minimal DFA) of every language L_i is strictly smaller than the index of L. Otherwise, L is prime. Primality of regular languages was introduced and studied in [O. Kupferman and J. Mosheiff, 2015], where the complexity of deciding the primality of the language of a given DFA was left open, with a doubly-exponential gap between the upper and lower bounds. We study primality for unary regular languages, namely regular languages with a singleton alphabet. A unary language corresponds to a subset of â, making the study of unary prime languages closer to that of primality in number theory. We show that the setting of languages is richer. In particular, while every composite number is the product of two smaller numbers, the number t of languages necessary to decompose a composite unary language induces a strict hierarchy. In addition, a primality witness for a unary language L, namely a word that is not in L but is in all products of languages that contain L and have an index smaller than Lâs, may be of exponential length. Still, we are able to characterize compositionality by structural properties of a DFA for L, leading to a LogSpace algorithm for primality checking of unary DFAs
The matroid secretary problem for minor-closed classes and random matroids
We prove that for every proper minor-closed class of matroids
representable over a prime field, there exists a constant-competitive matroid
secretary algorithm for the matroids in . This result relies on the
extremely powerful matroid minor structure theory being developed by Geelen,
Gerards and Whittle.
We also note that for asymptotically almost all matroids, the matroid
secretary algorithm that selects a random basis, ignoring weights, is
-competitive. In fact, assuming the conjecture that almost all
matroids are paving, there is a -competitive algorithm for almost all
matroids.Comment: 15 pages, 0 figure
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