71 research outputs found
Generalized adjoint actions
The aim of this paper is to generalize the classical formula
by replacing
with any formal power series . We
also obtain combinatorial applications to -exponentials, -binomials, and
Hall-Littlewood polynomials.Comment: 5 pages, LaTeX, typos corrected, to appear in Journal of Lie Theor
Algebras of Quasi-Pl\"ucker Coordinates are Koszul
Motivated by the theory of quasi-determinants, we study non-commutative
algebras of quasi-Pl\"ucker coordinates. We prove that these algebras provide
new examples of non-homogeneous quadratic Koszul algebras by showing that their
quadratic duals have quadratic Gr\"obner bases.Comment: 15 pages, AMS tex, tikz figures. V2: small corrections based on
referee report. V3: Further minor changes. To appear in J. Pure Appl. Algebr
Noncommutative Catalan numbers
The goal of this paper is to introduce and study noncommutative Catalan
numbers which belong to the free Laurent polynomial algebra in
generators. Our noncommutative numbers admit interesting (commutative and
noncommutative) specializations, one of them related to Garsia-Haiman
-versions, another -- to solving noncommutative quadratic equations. We
also establish total positivity of the corresponding (noncommutative) Hankel
matrices and introduce accompanying noncommutative binomial coefficients.Comment: 12 pages AM LaTex, a picture and proof of Lemma 3.6 are added,
misprints correcte
Algebras associated to acyclic directed graphs
We construct and study a class of algebras associated to generalized layered
graphs, i.e. directed graphs with a ranking function on their vertices. Each
finite directed acyclic graph admits countably many structures of a generalized
layered graph. We construct linear bases in such algebras and compute their
Hilbert series. Our interest to generalized layered graphs and algebras
associated to those graphs is motivated by their relations to factorizations of
polynomials over noncommutative rings.Comment: 20 pages, Latex; an expanded and corrected version; to appear in
"Advances of Applied Mathematics
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