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Register a new userEquivariant multiplicities of Coxeter arrangements and invariant bases
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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.
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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.
When $W$ is a finite Coxeter group acting by its reflection representation on $E$, we describe the category ${mathsf{Perv}}_W(E_{mathbb C}, {mathcal{H}}_{mathbb C})$ of $W$-equivariant perverse sheaves on $E_{mathbb C}$, smooth with respect to the stratification by reflection hyperplanes. By using Kapranov and Schechtmans recent analysis of perverse sheaves on hyperplane arrangements, we find an equivalence of categories from ${mathsf{Perv}}_W(E_{mathbb C}, {mathcal{H}}_{mathbb C})$ to a category of finite-dimensional modules over an algebra given by explicit generators and relations. We also define categories of equivariant perverse sheaves on affine buildings, e.g., $G$-equivariant perverse sheaves on the Bruhat--Tits building of a $p$-adic group $G$. In this setting, we find that a construction of Schneider and Stuhler gives equivariant perverse sheaves associated to depth zero representations.
We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant multiplicity is combinatorially computable.
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.
Let $q$ be a positive integer. In our recent paper, we proved that the cardinality of the complement of an integral arrangement, after the modulo $q$ reduction, is a quasi-polynomial of $q$, which we call the characteristic quasi-polynomial. In this paper, we study general properties of the characteristic quasi-polynomial as well as discuss two important examples: the arrangements of reflecting hyperplanes arising from irreducible root systems and the mid-hyperplane arrangements. In the root system case, we present a beautiful formula for the generating function of the characteristic quasi-polynomial which has been essentially obtained by Ch. Athanasiadis and by A. Blass and B. Sagan. On the other hand, it is hard to find the generating function of the characteristic quasi-polynomial in the mid-hyperplane arrangement case. We determine them when the dimension is less than six.
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