Topology and monoid representations II: left regular bands of groups and Hsiao's monoid of ordered $G$-partitions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909160911667200 |
|---|---|
| author | Steinberg, Benjamin |
| author_facet | Steinberg, Benjamin |
| contents | The goal of this paper is to use topological methods to compute $\mathrm{Ext}$ between irreducible representations of von Neumann regular monoids in which Green's $\mathscr L$- and $\mathscr J$-relations coincide (e.g., left regular bands). Our results subsume those of S.~Margolis, F.~Saliola, and B.~Steinberg, \emph{Combinatorial topology and the global dimension of algebras arising in combinatorics}, J. Eur. Math. Soc. (JEMS), \textbf{17}, 3037--3080 (2015).
Applications include computing $\mathrm{Ext}$ between arbitrary simple modules and computing a quiver presentation for the algebra of Hsiao's monoid of ordered $G$-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product $G\wr S_n$). We show that this algebra is Koszul, compute its Koszul dual and compute minimal projective resolutions of all the simple modules using topology. More generally, these results work for CW left regular bands of abelian groups. These results generalize the results of S.~Margolis, F.~V. Saliola, and B.~Steinberg. \emph{Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry}, Mem. Amer. Math. Soc., \textbf{274}, 1--135, (2021). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topology and monoid representations II: left regular bands of groups and Hsiao's monoid of ordered $G$-partitions Steinberg, Benjamin Representation Theory Group Theory Rings and Algebras 20M25, 20M30, 20M50, 16S37, 16G99, 05E10, 05E45 The goal of this paper is to use topological methods to compute $\mathrm{Ext}$ between irreducible representations of von Neumann regular monoids in which Green's $\mathscr L$- and $\mathscr J$-relations coincide (e.g., left regular bands). Our results subsume those of S.~Margolis, F.~Saliola, and B.~Steinberg, \emph{Combinatorial topology and the global dimension of algebras arising in combinatorics}, J. Eur. Math. Soc. (JEMS), \textbf{17}, 3037--3080 (2015). Applications include computing $\mathrm{Ext}$ between arbitrary simple modules and computing a quiver presentation for the algebra of Hsiao's monoid of ordered $G$-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product $G\wr S_n$). We show that this algebra is Koszul, compute its Koszul dual and compute minimal projective resolutions of all the simple modules using topology. More generally, these results work for CW left regular bands of abelian groups. These results generalize the results of S.~Margolis, F.~V. Saliola, and B.~Steinberg. \emph{Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry}, Mem. Amer. Math. Soc., \textbf{274}, 1--135, (2021). |
| title | Topology and monoid representations II: left regular bands of groups and Hsiao's monoid of ordered $G$-partitions |
| topic | Representation Theory Group Theory Rings and Algebras 20M25, 20M30, 20M50, 16S37, 16G99, 05E10, 05E45 |
| url | https://arxiv.org/abs/2404.03475 |