Matrix symmetric and quasi-symmetric functions and noncommutative representation theory
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909541433606144 |
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| author | Foissy, Loïc Malvenuto, Claudia Patras, Frédéric |
| author_facet | Foissy, Loïc Malvenuto, Claudia Patras, Frédéric |
| contents | A fundamental result by L. Solomon in algebraic combinatorics and representation theory states that Mackey formulas for products of characters of a symmetric group, or equivalently the computation of tensor products of representations thereof, can be lifted to the corresponding Solomon's descent algebra, a subalgebra of the group algebra with a very rich structure. Motivated by the structure of the product formula in these algebras and by other results and ideas in the field, we introduce and investigate in the present article a two dimensional analogue of descent algebras based on packed integer matrices that inherits most of their fundamental properties. One of the various bialgebra structures we introduce on packed integer matrices identifies with a bialgebra recently introduced by J. Diehl and L. Schmitz to define a two dimensional generalisation of Chen's iterated integrals signatures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14417 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Matrix symmetric and quasi-symmetric functions and noncommutative representation theory Foissy, Loïc Malvenuto, Claudia Patras, Frédéric Combinatorics Rings and Algebras 05E05, 16T30 A fundamental result by L. Solomon in algebraic combinatorics and representation theory states that Mackey formulas for products of characters of a symmetric group, or equivalently the computation of tensor products of representations thereof, can be lifted to the corresponding Solomon's descent algebra, a subalgebra of the group algebra with a very rich structure. Motivated by the structure of the product formula in these algebras and by other results and ideas in the field, we introduce and investigate in the present article a two dimensional analogue of descent algebras based on packed integer matrices that inherits most of their fundamental properties. One of the various bialgebra structures we introduce on packed integer matrices identifies with a bialgebra recently introduced by J. Diehl and L. Schmitz to define a two dimensional generalisation of Chen's iterated integrals signatures. |
| title | Matrix symmetric and quasi-symmetric functions and noncommutative representation theory |
| topic | Combinatorics Rings and Algebras 05E05, 16T30 |
| url | https://arxiv.org/abs/2503.14417 |