97 research outputs found
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
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
Mystic Reflection Groups
This paper aims to systematically study mystic reflection groups that emerged
independently in the paper [Selecta Math. (N.S.) 14 (2009), 325-372,
arXiv:0806.0867] by the authors and in the paper [Algebr. Represent. Theory 13
(2010), 127-158, arXiv:0806.3210] by Kirkman, Kuzmanovich and Zhang. A detailed
analysis of this class of groups reveals that they are in a nontrivial
correspondence with the complex reflection groups . We also prove
that the group algebras of corresponding groups are isomorphic and classify all
such groups up to isomorphism
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