Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2406.13728 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912188666478592 |
|---|---|
| author | Hicks, Angela McCloskey, Robert |
| author_facet | Hicks, Angela McCloskey, Robert |
| contents | The noncommutative symmetric functions $\textbf{NSym}$ were first defined abstractly by Gelfand et al. in 1995 as the free associative algebra generated by noncommuting indeterminants $\{\boldsymbol{e}_n\}_{n\in \mathbb{N}}$ that were taken as a noncommutative analogue of the elementary symmetric functions. The resulting space was thus a variation on the traditional symmetric functions $Λ$. Giving noncommutative analogues of generating function relations for other bases of $Λ$ allowed Gelfand et al. to define additional bases of $\textbf{NSym}$ and then determine change-of-basis formulas using quasideterminants. In this paper, we aim for a self-contained exposition that expresses these bases concretely as functions in infinitely many noncommuting variables and avoids quasideterminants. Additionally, we look at the noncommutative analogues of two different interpretations of change-of-basis in $Λ$: both as a product of a minimal number of matrices, mimicking Macdonald's exposition of $Λ$ in Symmetric Functions and Hall Polynomials, and as statistics on brick tabloids, as in work by Eğecioğlu and Remmel, 1990. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13728 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Combinatorial Perspective on the Noncommutative Symmetric Functions Hicks, Angela McCloskey, Robert Combinatorics 05E05 The noncommutative symmetric functions $\textbf{NSym}$ were first defined abstractly by Gelfand et al. in 1995 as the free associative algebra generated by noncommuting indeterminants $\{\boldsymbol{e}_n\}_{n\in \mathbb{N}}$ that were taken as a noncommutative analogue of the elementary symmetric functions. The resulting space was thus a variation on the traditional symmetric functions $Λ$. Giving noncommutative analogues of generating function relations for other bases of $Λ$ allowed Gelfand et al. to define additional bases of $\textbf{NSym}$ and then determine change-of-basis formulas using quasideterminants. In this paper, we aim for a self-contained exposition that expresses these bases concretely as functions in infinitely many noncommuting variables and avoids quasideterminants. Additionally, we look at the noncommutative analogues of two different interpretations of change-of-basis in $Λ$: both as a product of a minimal number of matrices, mimicking Macdonald's exposition of $Λ$ in Symmetric Functions and Hall Polynomials, and as statistics on brick tabloids, as in work by Eğecioğlu and Remmel, 1990. |
| title | A Combinatorial Perspective on the Noncommutative Symmetric Functions |
| topic | Combinatorics 05E05 |
| url | https://arxiv.org/abs/2406.13728 |