The $M$-triangle of generalised non-crossing partitions for the types $E_7$ and $E_8$
| dc.creator | Krattenthaler, Christian | |
| dc.date | 2006-01-27 | |
| dc.date | 2006-10-25 | |
| dc.date.accessioned | 2026-07-07T06:59:22Z | |
| dc.date.available | 2026-07-07T06:59:22Z | |
| dc.description | The $M$-triangle of a ranked locally finite poset $P$ is the generating function $\sum_{u,w\in P} ^{}μ(u,w) x^{\rk u}y^{\rk w}$, where $μ(.,.)$ is the Möbius function of $P$. We compute the $M$-triangle of Armstrong's poset of $m$-divisible non-crossing partitions for the root systems of type $E_7$ and $E_8$. For the other types except $D_n$ this had been accomplished in the earlier paper "The $F$-triangle of the generalised cluster complex." Altogether, this almost settles Armstrong's $F=M$ Conjecture predicting a surprising relation between the $M$-triangle of the $m$-divisible partitions poset and the $F$-triangle (a certain refined face count) of the generalised cluster complex of Fomin and Reading, the only gap remaining in type $D_n$. Moreover, we prove a reciprocity result for this $M$-triangle, again with the possible exception of type $D_n$. Our results are based on the calculation of certain decomposition numbers for the reflection groups of types $E_7$ and $E_8$, which carry in fact finer information than does the $M$-triangle. The decomposition numbers for the other exceptional reflection groups had been computed in the earlier paper. We present a conjectured formula for the type $A_n$ decomposition numbers. | |
| dc.description | AmS-TeX; 34 pages; journal version. Proofs of Lemma 5 and Proposition 6 simplified. Note at the end expanded | |
| dc.identifier | https://arxiv.org/abs/math/0601676 | |
| dc.identifier | http://arxiv.org/abs/math/0601676 | |
| dc.identifier | Séminaire Lotharingien Combin. 54 (2006), Article B54l, 34 pages | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/107726 | |
| dc.subject | Combinatorics | |
| dc.subject | Group Theory | |
| dc.subject | Primary 05E15; Secondary 05A05 05A15 05A19 06A07 20F55 | |
| dc.title | The $M$-triangle of generalised non-crossing partitions for the types $E_7$ and $E_8$ | |
| dc.type | text |