A two-sided analogue of the Coxeter complex
The electronic journal of combinatorics, Tome 25 (2018) no. 4
For any Coxeter system $(W,S)$ of rank $n$, we study an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. For finite $W$, this complex is first described in work of Hultman. Faces are indexed by triples $(I,w,J)$, where $I$ and $J$ are subsets of the set $S$ of simple generators, and $w$ is a minimal length representative for the parabolic double coset $W_I w W_J$. There is exactly one maximal face for each element of the group $W$. The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the $h$-polynomial is given by the "two-sided" $W$-Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in $W$.
DOI :
10.37236/8015
Classification :
05E16, 20F55, 51F15
Mots-clés : Coxeter group, Coxeter complex, Eulerian polynomial, contingency table
Mots-clés : Coxeter group, Coxeter complex, Eulerian polynomial, contingency table
Affiliations des auteurs :
T. Kyle Petersen 1
@article{10_37236_8015,
author = {T. Kyle Petersen},
title = {A two-sided analogue of the {Coxeter} complex},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {4},
doi = {10.37236/8015},
zbl = {1441.05243},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8015/}
}
T. Kyle Petersen. A two-sided analogue of the Coxeter complex. The electronic journal of combinatorics, Tome 25 (2018) no. 4. doi: 10.37236/8015
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