Boolean intervals in the weak Bruhat order of a finite Coxeter group
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914711479517184 |
|---|---|
| author | Adenbaum, Ben Elder, Jennifer Harris, Pamela E. Mori, J. Carlos Martínez |
| author_facet | Adenbaum, Ben Elder, Jennifer Harris, Pamela E. Mori, J. Carlos Martínez |
| contents | Given a Coxeter group $W$ with Coxeter system $(W,S)$, where $S$ is finite. We provide a complete characterization of Boolean intervals in the weak order of $W$ uniformly for all Coxeter groups in terms of independent sets of the Coxeter graph. Moreover, we establish that the number of Boolean intervals of rank $k$ in the weak order of $W$ is ${i_k(Γ_W)\cdot|W|}\,/\,2^{k}$, where $Γ_W$ is the Coxeter graph of $W$ and $i_k(Γ_W)$ is the number of independent sets of size $k$ of $Γ_W$ when $W$ is finite. Specializing to $A_n$, we recover the characterizations and enumerations of Boolean intervals in the weak order of $A_n$ given in arXiv:2306.14734. We provide the analogous results for types $C_n$ and $D_n$, including the related generating functions and additional connections to well-known integer sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07989 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boolean intervals in the weak Bruhat order of a finite Coxeter group Adenbaum, Ben Elder, Jennifer Harris, Pamela E. Mori, J. Carlos Martínez Combinatorics 05A05, 05A15, 05A19, 06A07, 20F55 Given a Coxeter group $W$ with Coxeter system $(W,S)$, where $S$ is finite. We provide a complete characterization of Boolean intervals in the weak order of $W$ uniformly for all Coxeter groups in terms of independent sets of the Coxeter graph. Moreover, we establish that the number of Boolean intervals of rank $k$ in the weak order of $W$ is ${i_k(Γ_W)\cdot|W|}\,/\,2^{k}$, where $Γ_W$ is the Coxeter graph of $W$ and $i_k(Γ_W)$ is the number of independent sets of size $k$ of $Γ_W$ when $W$ is finite. Specializing to $A_n$, we recover the characterizations and enumerations of Boolean intervals in the weak order of $A_n$ given in arXiv:2306.14734. We provide the analogous results for types $C_n$ and $D_n$, including the related generating functions and additional connections to well-known integer sequences. |
| title | Boolean intervals in the weak Bruhat order of a finite Coxeter group |
| topic | Combinatorics 05A05, 05A15, 05A19, 06A07, 20F55 |
| url | https://arxiv.org/abs/2403.07989 |