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Register a new userOn the Cyclically Fully Commutative Elements of Coxeter Groups
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Let W be an arbitrary Coxeter group. If two elements have expressions that are cyclic shifts of each other (as words), then they are conjugate (as group elements) in W. We say that w is cyclically fully commutative (CFC) if every cyclic shift of any reduced expression for w is fully commutative (i.e., avoids long braid relations). These generalize Coxeter elements in that their reduced expressions can be described combinatorially by acyclic directed graphs, and cyclically shifting corresponds to source-to-sink
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Coxeter groups are a special class of groups generated by involutions. They play important roles in the various areas of mathematics. This survey particularly focuses on how one use Coxeter groups to construct interesting examples of discrete subgroups of Lie group.
Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,...,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=si_1si_2 cdotssi_d$ of a Coxeter element $c$ of $W$, such that $si_i$ is a Coxeter element in a subgroup of type $T_i$ in $W$, $i=1,2,...,d$, and such that the factorisation is minimal in the sense that the sum of the ranks of the $T_i$s, $i=1,2,...,d$, equals the rank of $W$. For the exceptional types, these decomposition numbers have been computed by the first author. The type $A_n$ decomposition numbers have been computed by Goulden and Jackson, albeit using a somewhat different language. We explain how to extract the type $B_n$ decomposition numbers from results of Bona, Bousquet, Labelle and Leroux on map enumeration. Our formula for the type $D_n$ decomposition numbers is new. These results are then used to determine, for a fixed positive integer $l$ and fixed integers $r_1le r_2le ...le r_l$, the number of multi-chains $pi_1le pi_2le ...le pi_l$ in Armstrongs generalised non-crossing partitions poset, where the poset rank of $pi_i$ equals $r_i$, and where the block structure of $pi_1$ is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non-crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type $D_n$ generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrongs $F=M$ Conjecture in type $D_n$.
In the 1990s, J.H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the topograph, Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pells equation. It appears that the crux of his method is the coincidence between the arithmetic group $PGL_2({mathbb Z})$ and the Coxeter group of type $(3,infty)$. There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conways topograph, and generalizations to other arithmetic Coxeter groups. This includes a study of arithmetic flags and variants of binary quadratic forms.
We prove that certain families of Coxeter groups and inclusions $W_1hookrightarrow W_2hookrightarrow...$ satisfy homological stability, meaning that in each degree the homology $H_ast(BW_n)$ is eventually independent of $n$. This gives a uniform treatment of homological stability for the families of Coxeter groups of type $A_n$, $B_n$ and $D_n$, recovering existing results in the first two cases, and giving a new result in the third. The key step in our proof is to show that a certain simplicial complex with $W_n$-action is highly connected. To do this we show that the barycentric subdivision is an instance of the basic construction, and then use Daviss description of the basic construction as an increasing union of chambers to deduce the required connectivity.
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