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The algebraic and geometric classification of nilpotent anticommutative algebras
Publication . Kaygorodov, Ivan; Khrypchenko, Mykola; Lopes, Samuel A.; CMA - Centro de Matemática e Aplicações; Elsevier Science B.V., Amsterdam.
We give algebraic and geometric classifications of 6-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are 14 one-parameter families of 6-dimensional nilpotent anticommutative algebras, complemented by 130 additional isomorphism classes. The corresponding geometric variety is irreducible and determined by the Zariski closure of a one-parameter family of algebras. In particular, there are no rigid 6-dimensional complex nilpotent anticommutative algebras.
Associative spectra of graph algebras II
Publication . Lehtonen, Erkko; Waldhauser, Tamás; CMA - Centro de Matemática e Aplicações; Springer Verlag
A necessary and sufficient condition is presented for a graph algebra to satisfy a bracketing identity. The associative spectrum of an arbitrary graph algebra is shown to be either constant or exponentially growing.
Algebraic Theory of Quasi-crystals: A Generalization of the Hypoplactic Monoid and a Littelmann Path Model
Publication . Guilherme, Ricardo Jorge Pratas; Malheiro, António; Cain, Alan
The plactic monoid was first introduced by Lascoux and Schützenberger based on
work by Schensted and Knuth, resulting in a description of the plactic monoid by
a presentation and by Young tableaux and an insertion algorithm. Then, due to
work by Kashiwara on crystal bases, the plactic monoid was obtained by identifying
vertices in the crystal graph associated to the general linear Lie algebra. This
process proved to be fruitful, as it enriched the structure of the already acclaimed
plactic monoid and allowed its generalization based on other crystal graphs.
Analogously, the hypoplactic monoid was introduced by Krob and Thibon via
a presentation and via quasi-ribbon tableaux and an insertion algorithm. Recent
work by Cain and Malheiro showed a construction of the hypoplactic monoid
by identifying vertices in a quasi-crystal graph derived from the crystal graph
associated to the general linear Lie algebra. Although this construction is based
on Kashiwara’s work, it cannot be extended to other crystal graphs, and it lacks
connections to representation theory such as a Littelmann path model.
In this thesis we present an algebraic theory of quasi-crystals. We associate
a quasi-crystal graph to each quasi-crystal and describe a one-to-one correspondence
between seminormal quasi-crystals and quasi-crystal graphs. We show
that a hypoplactic congruence can be defined over any seminormal quasi-crystal,
from which a general notion of hypoplactic monoid emerges. We prove that the
construction of the hypoplactic monoid proposed by Cain and Malheiro can be
placed in context as the hypoplactic monoid associated with the general linear Lie
algebra. Based on this framework, we make a study of the hypoplactic monoid
associated to the symplectic Lie algebra. Finally, We present a Littelmann path
model for quasi-crystals.
HS-stability and complex products in involution semigroups
Publication . Bodor, Bertalan; Lehtonen, Erkko; Quinn-Gregson, Thomas; Verhulst, Nikolaas; CMA - Centro de Matemática e Aplicações; Springer
When does the complex product of a given number of subsets of a group generate the same subgroup as their union? We answer this question in a more general form by introducing HS-stability and characterising the HS-stable involution subsemigroup generated by a subset of a given involution semigroup. We study HS-stability for the special cases of regular ∗-semigroups and commutative involution semigroups.
Permutation reconstruction from a few large patterns
Publication . Gouveia, Maria João; Lehtonen, Erkko; CMA - Centro de Matemática e Aplicações; American Mathematical Society
Every permutation of rank n ≥ 5 is reconstructible from any ⌈n/2⌉ + 2 of its (n − 1)-patterns.
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Entidade financiadora
Fundação para a Ciência e a Tecnologia
Programa de financiamento
3599-PPCDT
Número da atribuição
PTDC/MAT-PUR/31174/2017