298 research outputs found
Generalized Thue-Morse words and palindromic richness
We prove that the generalized Thue-Morse word defined for
and as , where denotes the sum of digits in the base-
representation of the integer , has its language closed under all elements
of a group isomorphic to the dihedral group of order consisting of
morphisms and antimorphisms. Considering simultaneously antimorphisms , we show that is saturated by -palindromes
up to the highest possible level. Using the terminology generalizing the notion
of palindromic richness for more antimorphisms recently introduced by the
author and E. Pelantov\'a, we show that is -rich. We
also calculate the factor complexity of .Comment: 11 page
Group-Like algebras and Hadamard matrices
We give a description in terms of square matrices of the family of group-like
algebras with . In the case that and is
not 2 and does not divide the dimension of the algebra, this translation take
us to Hadamard matrices and, particularly, to examples of biFrobenius algebras
satisfying and that are not Hopf algebras. Finally, we
generalize some known results on separability and coseparability valid for
finite dimensional Hopf algebras to this special class of biFrobenius algebras
with , presenting a version of Maschke's theorem for this
family
Languages invariant under more symmetries: overlapping factors versus palindromic richness
Factor complexity and palindromic complexity of
infinite words with language closed under reversal are known to be related by
the inequality for any \,. Word for which
the equality is attained for any is usually called rich in palindromes. In
this article we study words whose languages are invariant under a finite group
of symmetries. For such words we prove a stronger version of the above
inequality. We introduce notion of -palindromic richness and give several
examples of -rich words, including the Thue-Morse sequence as well.Comment: 22 pages, 1 figur
- …