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Let B be a real hyperplane arrangement which is stable under the action of a Coxeter group W. Then B acts naturally on the set of chambers of B. We assume that B is disjoint from the Coxeter arrangement A=A(W) of W. In this paper, we show that the W-orbits of the set of chambers of B are in one-to-one correspondence with the chambers of C=Acup B which are contained in an arbitrarily fixed chamber of A. From this fact, we find that the number of W-orbits of the set of chambers of B is given by the number of chambers of C divided by the order of W. We will also study the set of chambers of C which are contained in a chamber b of B. We prove that the cardinality of this set is equal to the order of the isotropy subgroup W_b of b. We illustrate these results with some examples, and solve an open problem in Kamiya, Takemura and Terao [Ranking patterns of unfolding models of codimension one, Adv. in Appl. Math. (2010)] by using our results.
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Let $A$ be an irreducible Coxeter arrangement and $W$ be its Coxeter group. Then $W$ naturally acts on $A$. A multiplicity $bfm : Arightarrow Z$ is said to be equivariant when $bfm$ is constant on each $W$-orbit of $A$. In this article, we prove that the multi-derivation module $D(A, bfm)$ is a free module whenever $bfm$ is equivariant by explicitly constructing a basis, which generalizes the main theorem of cite{T02}. The main tool is a primitive derivation and its covariant derivative. Moreover, we show that the $W$-invariant part $D(A, bfm)^{W}$ for any multiplicity $bfm$ is a free module over the $W$-invariant subring.
Let $W$ be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As a consequence, we extend the Hodge filtration, indexed by nonnegative integers, into a filtration indexed by all integers. This filtration coincides with the filtration by the order of poles. The results are translated into the derivation case.
Let $A$ be an irreducible Coxeter arrangement and $bfk$ be a multiplicity of $A$. We study the derivation module $D(A, bfk)$. Any two-dimensional irreducible Coxeter arrangement with even number of lines is decomposed into two orbits under the action of the Coxeter group. In this paper, we will {explicitly} construct a basis for $D(A, bfk)$ assuming $bfk$ is constant on each orbit. Consequently we will determine the exponents of $(A, bfk)$ under this assumption. For this purpose we develop a theory of universal derivations and introduce a map to deal with our exceptional cases.
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
As a visualization of Cartier and Foatas partially commutative monoid theory, G.X. Viennot introduced heaps of pieces in 1986. These are essentially labeled posets satisfying a few additional properties. They naturally arise as models of reduced words in Coxeter groups. In this paper, we introduce a cyclic version, motivated by the idea of taking a heap and wrapping it into a cylinder. We call this object a toric heap, as we formalize it as a labeled toric poset, which is a cyclic version of an ordinary poset. To define the concept of a toric extension, we develop a morphism in the category of toric heaps. We study toric heaps in Coxeter theory, in view of the fact that a cyclic shift of a reduced word is simply a conjugate by an initial or terminal generator. This allows us to formalize and study a framework of cyclic reducibility in Coxeter theory, and apply it to model conjugacy. We introduce the notion of torically reduced, which is stronger than being cyclically reduced for group elements. This gives rise to a new class of elements called torically fully commutative (TFC), which are those that have a unique cyclic commutativity class, and comprise a strictly bigger class than the cyclically fully commutative (CFC) elements. We prove several cyclic analogues of results on fully commutative (FC) elements due to Stembridge. We conclude with how this framework fits into recent work in Coxeter groups, and we correct a minor flaw in a few recently published theorems.
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