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Register a new userBases for the derivation modules of two-dimensional multi-Coxeter arrangements and universal derivations
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Authors
Atsushi Wakamiko
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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.
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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 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.
In his affirmative answer to the Edelman-Reiner conjecture, Yoshinaga proved that the logarithmic derivation modules of the cones of the extended Shi arrangements are free modules. However, all we know about the bases is their existence and degrees. In this article, we introduce two distinguished bases for the modules. More specifically, we will define and study the simple-root basis plus (SRB+) and the simple-root basis minus (SRB-) when a primitive derivation is fixed. They have remarkable properties relevant to the simple roots and those properties characterize the bases.
Let $mathcal{L}$ be the derivation Lie algebra of ${mathbb C}[t_1^{pm 1},t_2^{pm 1}]$. Given a triangle decomposition $mathcal{L} =mathcal{L}^{+}oplusmathfrak{h}oplusmathcal{L}^{-}$, we define a nonsingular Lie algebra homomorphism $psi:mathcal{L}^{+}rightarrowmathbb{C}$ and the universal Whittaker $mathcal{L}$-module $W_{psi}$ of type $psi$. We obtain all Whittaker vectors and submodules of $W_{psi}$, and all simple Whittaker $mathcal{L}$-modules of type $psi$.
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