Commuting involution graphs have been studied for finite Coxeter groups and for affine groups of ... more Commuting involution graphs have been studied for finite Coxeter groups and for affine groups of classical type. The purpose of this short note is to establish some general results for commuting involution graphs in affine Coxeter groups, and to deal with types F̃_4 and G̃_2. Types Ẽ_6 Ẽ_7 and Ẽ_8 are more substantial and we will address these in a forthcoming paper.
Let G be a group and S a non-empty subset of G. If ab / ∈ S for any a,b ∈ S, then S is called sum... more Let G be a group and S a non-empty subset of G. If ab / ∈ S for any a,b ∈ S, then S is called sum-free. We show that if S is maximal by inclusion and no proper subset generates 〈S 〉 then |S | ≤ 2. We determine all groups with a maximal (by inclusion) sum-free set of size at most 2 and all of size 3 where there exists a ∈ S such that a / ∈ 〈S \ {a}〉. 1
In this paper we prove that for W a finite Coxeter group and C a conjugacy class of W, there is a... more In this paper we prove that for W a finite Coxeter group and C a conjugacy class of W, there is always an element of C of maximal length in C which has excess zero. An element {w\in W} has excess zero if there exist elements {\sigma,\tau\in W} such that {\sigma^{2}=\tau^{2}=1,w=\sigma\tau} and {\ell(w)=\ell(\sigma)+\ell(\tau)} , {\ell} being the length function on W.
The commuting graph ${\cal C}(G,X)$, where $G$ is a group and $X$ a subset of $G$, has $X$ as its... more The commuting graph ${\cal C}(G,X)$, where $G$ is a group and $X$ a subset of $G$, has $X$ as its vertex set with two distinct elements of $X$ joined by an edge when they commute in $G$. Here the diameter and disc structure of ${\cal C}(G,X)$ is investigated when $G$ is the symmetric group and $X$ a conjugacy class of $G$.
The excess of an element $w$ of a finite Coxeter group $W$ is the minimal value of $l(x) + l(y) -... more The excess of an element $w$ of a finite Coxeter group $W$ is the minimal value of $l(x) + l(y) - l(w)$, where $x$, $y$ are elements of $W$ such that $x^2 = y^2 = 1$ and $w = xy$. Every element of a finite Coxeter group is either an involution or the product of two involutions, so the concept of excess is well defined. It can be extended to strongly real classes of infinite Coxeter groups. Earlier work by the authors showed that every conjugacy class of a finite Coxeter group contains an element of minimal length and excess zero. The current paper shows that each conjugacy class also contains an element of maximal length and excess zero.
Let $S$ be a non-empty subset of a group $G$. We say $S$ is product-free if $S\cap SS=\varnothing... more Let $S$ be a non-empty subset of a group $G$. We say $S$ is product-free if $S\cap SS=\varnothing$, and $S$ is locally maximal if whenever $T$ is product-free and $S\subseteq T$, then $S=T$. Finally $S$ fills $G$ if $G^*\subseteq S \sqcup SS$ (where $G^*$ is the set of all non-identity elements of $G$), and $G$ is a filled group if every locally maximal product-free set in $G$ fills $G$. Street and Whitehead (in `Group Ramsey Theory', J. Comb. Theory Series A, 17 (1974) 219-226) investigated filled groups and gave a classification of filled abelian groups. In this paper, we obtain some results about filled groups in the non-abelian case, including a classification of filled groups of odd order. Street and Whitehead conjectured that the finite dihedral group of order $2n$ is not filled when $n=6k+1$ ($k\geq 1$). We disprove this conjecture on dihedral groups, and in doing so obtain a classification of locally maximal product-free sets of sizes 3 and 4 in dihedral groups.
There are given some properties of Coxeter groups and subgroups using the length of elements and ... more There are given some properties of Coxeter groups and subgroups using the length of elements and subsets. (The length of an element is the number of factors in its minimal presentation.) In such a way there are considered conjugacy classes, subgroups, cosets, X-posets. There are also given some open questions.
For a Coxeter group W and a subset X of W, let N(X)={α∈Φ + ∣w·α∈Φ - forsomew∈X}, where Φ + and Φ ... more For a Coxeter group W and a subset X of W, let N(X)={α∈Φ + ∣w·α∈Φ - forsomew∈X}, where Φ + and Φ - are, respectively, the positive and negative roots of the root system Φ of W. The Coxeter length of X, l(X), is defined to be the cardinality of N(X). Suppose that X is a subgroup of W. For right cosets Xg and Xh, write Xg∼Xh whenever Xgt=Xh for some reflection t∈W and l(Xg)=l(Xh). Then ∼ generates an equivalence relation on the right cosets of X in W. Let 𝔛 be the set of its equivalence classes. Let x,x ' ∈𝔛. Write x⇝x ' if there is a right coset Xg∈x and a right coset Xh∈x ' such that Xgt=Xh for some reflection t in W and l(Xg)≤l(Xh). The partial order ⪯ on 𝔛 is defined by x⪯x ' if and only if there exist x 1 ,⋯,x m ∈𝔛 such that x⇝x 1 ⇝⋯⇝x m ⇝x ' . Then 𝔛 is called the X-poset of W. The authors obtain a number of results on X-posets, e.g., the X-posets represent a generalization of the Bruhat order on right cosets of standard parabolic subgroups, and also that cer...
Let X be a subgroup of a Coxeter group W. In [Turk. J. Math. 36, No. 1, 77-93 (2012; Zbl 1253.200... more Let X be a subgroup of a Coxeter group W. In [Turk. J. Math. 36, No. 1, 77-93 (2012; Zbl 1253.20041)], the authors developed the notion of X-posets, which are defined on certain equivalence classes of the (right) cosets of X in W. These posets can be thought of as a generalization of the well-known Bruhat order of W. This article provides a catalogue of all the X-posets for various small Coxeter groups.
Let $W$ be a finite Coxeter group and $X$ a subset of $W$. The length polynomial $L_{W,X}(t)$ is ... more Let $W$ be a finite Coxeter group and $X$ a subset of $W$. The length polynomial $L_{W,X}(t)$ is defined by $L_{W,X}(t) = \sum_{x \in X} t^{\ell(x)}$, where $\ell$ is the length function on $W$. In this article we derive expressions for the length polynomial where $X$ is any conjugacy class of involutions, or the set of all involutions, in any finite Coxeter group $W$. In particular, these results correct errors in the paper "Permutation statistics on involutions", W.M.B. Dukes., European J. Combin. 28 (2007), 186--198. for the involution length polynomials of Coxeter groups of type $B_n$ and $D_n$. Moreover, we give a counterexample to a unimodality conjecture of Dukes.
The commuting graph C(G; X) , where G is a group and X a subset of G, has X as its vertex set wit... more The commuting graph C(G; X) , where G is a group and X a subset of G, has X as its vertex set with two distinct elements of X joined by an edge when they commute in G. Here the diameter and disc structure of C(G; X) is investigated when G is the symmetric group and X a conjugacy class
Commuting involution graphs have been studied for finite Coxeter groups and for affine groups of ... more Commuting involution graphs have been studied for finite Coxeter groups and for affine groups of classical type. The purpose of this short note is to establish some general results for commuting involution graphs in affine Coxeter groups, and to deal with types F̃_4 and G̃_2. Types Ẽ_6 Ẽ_7 and Ẽ_8 are more substantial and we will address these in a forthcoming paper.
Let G be a group and S a non-empty subset of G. If ab / ∈ S for any a,b ∈ S, then S is called sum... more Let G be a group and S a non-empty subset of G. If ab / ∈ S for any a,b ∈ S, then S is called sum-free. We show that if S is maximal by inclusion and no proper subset generates 〈S 〉 then |S | ≤ 2. We determine all groups with a maximal (by inclusion) sum-free set of size at most 2 and all of size 3 where there exists a ∈ S such that a / ∈ 〈S \ {a}〉. 1
In this paper we prove that for W a finite Coxeter group and C a conjugacy class of W, there is a... more In this paper we prove that for W a finite Coxeter group and C a conjugacy class of W, there is always an element of C of maximal length in C which has excess zero. An element {w\in W} has excess zero if there exist elements {\sigma,\tau\in W} such that {\sigma^{2}=\tau^{2}=1,w=\sigma\tau} and {\ell(w)=\ell(\sigma)+\ell(\tau)} , {\ell} being the length function on W.
The commuting graph ${\cal C}(G,X)$, where $G$ is a group and $X$ a subset of $G$, has $X$ as its... more The commuting graph ${\cal C}(G,X)$, where $G$ is a group and $X$ a subset of $G$, has $X$ as its vertex set with two distinct elements of $X$ joined by an edge when they commute in $G$. Here the diameter and disc structure of ${\cal C}(G,X)$ is investigated when $G$ is the symmetric group and $X$ a conjugacy class of $G$.
The excess of an element $w$ of a finite Coxeter group $W$ is the minimal value of $l(x) + l(y) -... more The excess of an element $w$ of a finite Coxeter group $W$ is the minimal value of $l(x) + l(y) - l(w)$, where $x$, $y$ are elements of $W$ such that $x^2 = y^2 = 1$ and $w = xy$. Every element of a finite Coxeter group is either an involution or the product of two involutions, so the concept of excess is well defined. It can be extended to strongly real classes of infinite Coxeter groups. Earlier work by the authors showed that every conjugacy class of a finite Coxeter group contains an element of minimal length and excess zero. The current paper shows that each conjugacy class also contains an element of maximal length and excess zero.
Let $S$ be a non-empty subset of a group $G$. We say $S$ is product-free if $S\cap SS=\varnothing... more Let $S$ be a non-empty subset of a group $G$. We say $S$ is product-free if $S\cap SS=\varnothing$, and $S$ is locally maximal if whenever $T$ is product-free and $S\subseteq T$, then $S=T$. Finally $S$ fills $G$ if $G^*\subseteq S \sqcup SS$ (where $G^*$ is the set of all non-identity elements of $G$), and $G$ is a filled group if every locally maximal product-free set in $G$ fills $G$. Street and Whitehead (in `Group Ramsey Theory', J. Comb. Theory Series A, 17 (1974) 219-226) investigated filled groups and gave a classification of filled abelian groups. In this paper, we obtain some results about filled groups in the non-abelian case, including a classification of filled groups of odd order. Street and Whitehead conjectured that the finite dihedral group of order $2n$ is not filled when $n=6k+1$ ($k\geq 1$). We disprove this conjecture on dihedral groups, and in doing so obtain a classification of locally maximal product-free sets of sizes 3 and 4 in dihedral groups.
There are given some properties of Coxeter groups and subgroups using the length of elements and ... more There are given some properties of Coxeter groups and subgroups using the length of elements and subsets. (The length of an element is the number of factors in its minimal presentation.) In such a way there are considered conjugacy classes, subgroups, cosets, X-posets. There are also given some open questions.
For a Coxeter group W and a subset X of W, let N(X)={α∈Φ + ∣w·α∈Φ - forsomew∈X}, where Φ + and Φ ... more For a Coxeter group W and a subset X of W, let N(X)={α∈Φ + ∣w·α∈Φ - forsomew∈X}, where Φ + and Φ - are, respectively, the positive and negative roots of the root system Φ of W. The Coxeter length of X, l(X), is defined to be the cardinality of N(X). Suppose that X is a subgroup of W. For right cosets Xg and Xh, write Xg∼Xh whenever Xgt=Xh for some reflection t∈W and l(Xg)=l(Xh). Then ∼ generates an equivalence relation on the right cosets of X in W. Let 𝔛 be the set of its equivalence classes. Let x,x ' ∈𝔛. Write x⇝x ' if there is a right coset Xg∈x and a right coset Xh∈x ' such that Xgt=Xh for some reflection t in W and l(Xg)≤l(Xh). The partial order ⪯ on 𝔛 is defined by x⪯x ' if and only if there exist x 1 ,⋯,x m ∈𝔛 such that x⇝x 1 ⇝⋯⇝x m ⇝x ' . Then 𝔛 is called the X-poset of W. The authors obtain a number of results on X-posets, e.g., the X-posets represent a generalization of the Bruhat order on right cosets of standard parabolic subgroups, and also that cer...
Let X be a subgroup of a Coxeter group W. In [Turk. J. Math. 36, No. 1, 77-93 (2012; Zbl 1253.200... more Let X be a subgroup of a Coxeter group W. In [Turk. J. Math. 36, No. 1, 77-93 (2012; Zbl 1253.20041)], the authors developed the notion of X-posets, which are defined on certain equivalence classes of the (right) cosets of X in W. These posets can be thought of as a generalization of the well-known Bruhat order of W. This article provides a catalogue of all the X-posets for various small Coxeter groups.
Let $W$ be a finite Coxeter group and $X$ a subset of $W$. The length polynomial $L_{W,X}(t)$ is ... more Let $W$ be a finite Coxeter group and $X$ a subset of $W$. The length polynomial $L_{W,X}(t)$ is defined by $L_{W,X}(t) = \sum_{x \in X} t^{\ell(x)}$, where $\ell$ is the length function on $W$. In this article we derive expressions for the length polynomial where $X$ is any conjugacy class of involutions, or the set of all involutions, in any finite Coxeter group $W$. In particular, these results correct errors in the paper "Permutation statistics on involutions", W.M.B. Dukes., European J. Combin. 28 (2007), 186--198. for the involution length polynomials of Coxeter groups of type $B_n$ and $D_n$. Moreover, we give a counterexample to a unimodality conjecture of Dukes.
The commuting graph C(G; X) , where G is a group and X a subset of G, has X as its vertex set wit... more The commuting graph C(G; X) , where G is a group and X a subset of G, has X as its vertex set with two distinct elements of X joined by an edge when they commute in G. Here the diameter and disc structure of C(G; X) is investigated when G is the symmetric group and X a conjugacy class
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