The module structure of the Solomon-Tits algebra of the symmetric group
| dc.creator | Schocker, Manfred | |
| dc.date | 2005-05-09 | |
| dc.date | 2005-05-12 | |
| dc.date.accessioned | 2026-07-07T05:19:42Z | |
| dc.date.available | 2026-07-07T05:19:42Z | |
| dc.description | Let $(W,S)$ be a finite Coxeter system. Tits defined an associative product on the set $Σ$ of simplices of the associated Coxeter complex. The corresponding semigroup algebra is the Solomon-Tits algebra of $W$. It contains the Solomon algebra of $W$ as the algebra of invariants with respect to the natural action of $W$ on $Σ$. For the symmetric group $S_n$, there is a 1-1 correspondence between $Σ$ and the set of all set compositions (or ordered set partitions) of $\{1,...,n\}$. The product on $Σ$ has a simple combinatorial description in terms of set compositions. We study in detail the representation theory of the Solomon-Tits algebra of $S_n$ over an arbitrary field, and show how our results relate to the corresponding results on the Solomon algebra of $S_n$. This includes the construction of irreducible and principal indecomposable modules, a description of the Cartan invariants, of the Ext-quiver, and of the descending Loewy series. Our approach builds on a (twisted) Hopf algebra structure on the direct sum of all Solomon-Tits algebras. | |
| dc.description | 50 pages, several minor changes/additions, most notably in Remark 6.5 (2) and Section 9 | |
| dc.identifier | https://arxiv.org/abs/math/0505137 | |
| dc.identifier | http://arxiv.org/abs/math/0505137 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75114 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Representation Theory | |
| dc.subject | 20M30 (Primary) 05E15, 16W30, 17A30, 20F55 (Secondary) | |
| dc.title | The module structure of the Solomon-Tits algebra of the symmetric group | |
| dc.type | text |