Preprint
A permutation based approach to the $q$-deformation of the Dynkin Operator
13 Apr 2025
Abstract
Introduced by Solomon, the descent algebra is a significant subalgebra of the
group algebra of the symmetric group $\mathbf{k}S_n$ related to many important
algebraic and combinatorial topics. It contains all the classical Lie
idempotents of $\mathbf{k}S_n$, in particular the Dynkin operator, a
fundamental tool for studying the free Lie algebra. We look at a
$q$-deformation of the Dynkin operator and study its action over the descent
algebra with classical combinatorial tools like Solomon's Mackey formula. This
leads to elementary proofs that the operator is indeed an idempotent for $q=1$
as well as to interesting formulas and algebraic structures especially when $q$
is a root of unity.
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Details
- Title
- A permutation based approach to the $q$-deformation of the Dynkin Operator
- Creators
- Darij Grinberg - Drexel UniversityEkaterina A Vassilieva - Institut Polytechnique de Paris
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022048368004721