A permutation based approach to the $q$-deformation of the Dynkin Operator
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915240660172800 |
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| author | Grinberg, Darij Vassilieva, Ekaterina A. |
| author_facet | Grinberg, Darij Vassilieva, Ekaterina A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09709 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A permutation based approach to the $q$-deformation of the Dynkin Operator Grinberg, Darij Vassilieva, Ekaterina A. Combinatorics 20C30, 05A05, 05A19 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. |
| title | A permutation based approach to the $q$-deformation of the Dynkin Operator |
| topic | Combinatorics 20C30, 05A05, 05A19 |
| url | https://arxiv.org/abs/2504.09709 |