共查询到20条相似文献,搜索用时 15 毫秒
1.
We consider hyperplane arrangements generated by generic points and study their intersection lattices. These arrangements are known to be equivalent to discriminantal arrangements. We show a fundamental structure of the intersection lattices by decomposing the poset ideals as direct products of smaller lattices corresponding to smaller dimensions. Based on this decomposition we compute the M?bius functions of the lattices and the characteristic polynomials of the arrangements up to dimension six. 相似文献
2.
Let V be Euclidean space. Let be a finite irreducible reflection group. Let be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For choose such that . The arrangement is known to be free: the derivation module is a free S-module with generators of degrees equal to the exponents of W. In this paper we prove an analogous theorem for the submodule of defined by . The degrees of the basis elements are all equal to the Coxeter number. The module may be considered a deformation of the derivation module for the Shi arrangement, which is conjectured to be free. The proof
is by explicit construction using a derivation introduced by K. Saito in his theory of flat generators.
Received: March 13, 1997 相似文献
3.
We calculate the skew-symmetric cohomology of the complement of a discriminantal hyperplane arrangement with coefficients in local systems arising in the context of the representation theory of the Lie algebra
. For a discriminantal arrangement in k, the skew-symmetric cohomology is nontrivial in dimension k–1 precisely when the 'master function' which defines the local system on the complement has nonisolated criticalpoints. In symmetric coordinates, the critical set is a union of lines. Generically, the dimension of this nontrivial skew-symmetric cohomology group is equal to the number of critical lines. 相似文献
4.
We construct lattices with quadratic structure over the integers in quadratic number fields having the property that the rank
of the quadratic structure is constant and equal to the rank of the lattice in all reductions modulo maximal ideals. We characterize
the case in which such lattices are free. The construction gives a representative of the genus of such lattices as an orthogonal
sum of “standard” pieces of ranks 1–4 and covers the case of the discriminant of the real quadratic number field congruent
to 1 modulo 8 for which a general construction was not known.
相似文献
5.
Let Z be a centrally symmetric polygon with integer side lengths. We answer the following two questions:
- When is the associated discriminantal hyperplane arrangementfree in the sense of Saito and Terao?
- When areall of the tilings of Z by unit rhombicoherent in the sense of Billera and Sturmfels?
6.
Hyperplane arrangements of rank 3 admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note we prove that arrangements which admit a locally heavy flag satisfy Terao's conjecture which is a generalization of the statement above to arbitrary dimension. To this end we extend results characterizing the freeness of multiarrangements with a heavy hyperplane to those satisfying the weaker notion of a locally heavy hyperplane. As a corollary we give a new proof that irreducible arrangements with a generic hyperplane are totally nonfree. In another application we show that an irreducible multiarrangement of rank 3 with at least two locally heavy hyperplanes is not free. 相似文献
7.
Daniel C. Cohen Peter Orlik 《Transactions of the American Mathematical Society》2005,357(8):3031-3050
We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a complex rank one local system. We define formal Gauss-Manin connection matrices in the Aomoto complex and prove that, for all arrangements and all local systems, these formal connection matrices specialize to Gauss-Manin connection matrices.
8.
The characteristic polynomial of a multiarrangement 总被引:1,自引:0,他引:1
Takuro Abe 《Advances in Mathematics》2007,215(2):825-838
Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates ‘global’ data to ‘local’ data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free. 相似文献
9.
Marc-Antoine Leclerc 《Algebras and Representation Theory》2016,19(5):1043-1057
In the present paper we extend the construction of the formal (affine) Demazure algebra due to Hoffnung, Malagón-López, Savage and Zainoulline in two directions. First, we introduce and study the notion of a formal Demazure lattice in the Kac-Moody setting and show that all the definitions and properties of the formal (affine) Demazure operators and algebras hold for such lattices. Second, we show that for the hyperbolic formal group law the formal Demazure algebra is isomorphic (after extending the coefficients) to the Hecke algebra. 相似文献
10.
Discriminantal arrangement, 3 × 3 minors of Plücker matrix and hypersurfaces in Grassmannian Gr(3,n)
We show that points in specific degree-2 hypersurfaces in the Grassmannian correspond to generic arrangements of n hyperplanes in with associated discriminantal arrangement having intersections of multiplicity three in codimension two. 相似文献
11.
P. H. Schmitt 《Algebra Universalis》1983,17(1):135-142
We prove that the class of existentially complete lattices is not an elementary class; thus the theory of lattices does not have a model-companion. Finally we observe that there is a locally finite finitely generic lattice. 相似文献
12.
We construct a formal connection on the Aomoto complex of an arrangement of hyperplanes, and use it to study the Gauss–Manin connection for the moduli space of the arrangement in the cohomology of a complex rank one local system. We prove that the eigenvalues of the Gauss–Manin connection are integral linear combinations of the weights which define the local system. 相似文献
13.
Jonathan Wiens Sergey Yuzvinsky 《Transactions of the American Mathematical Society》1997,349(4):1653-1662
The paper is devoted to computation of the cohomology of the complex of logarithmic differential forms with coefficients in rational functions whose poles are located on the union of several hyperplanes of a linear space over a field of characteristic zero. The main result asserts that for a vast class of hyperplane arrangements, including all free and generic arrangements, the cohomology algebra coincides with the Orlik-Solomon algebra. Over the field of complex numbers, this means that the cohomologies coincide with the cohomologies of the complement of the union of the hyperplanes. We also prove that the cohomologies do not change if poles of arbitrary multiplicity are allowed on some of the hyperplanes. In particular, this gives an analogue of the algebraic de Rham theorem for an arbitrary arrangement over an arbitrary field of zero characteristic.
14.
Hidehiko Kamiya Akimichi Takemura Hiroaki Terao 《Journal of Algebraic Combinatorics》2008,27(3):317-330
We study central hyperplane arrangements with integral coefficients modulo positive integers q. We prove that the cardinality of the complement of the hyperplanes is a quasi-polynomial in two ways, first via the theory
of elementary divisors and then via the theory of the Ehrhart quasi-polynomials. This result is useful for determining the
characteristic polynomial of the corresponding real arrangement. With the former approach, we also prove that intersection
lattices modulo q are periodic except for a finite number of q’s.
This work was supported by the MEXT and the JSPS. 相似文献
15.
强正则剩余格值逻辑系统L~N及其完备性 总被引:7,自引:0,他引:7
正则剩余格是一类重要的模糊逻辑代数系统,而常见的模糊逻辑形式系统大多数带有非联接词,并且相应的Lindenbaum代数都是正则剩余格.本文以强正则剩余格为语义,建立了一个一般的命题演算形式系统LN,并且证明了这个系统的完备性.几种常见的带有非联接词的模糊逻辑形式系统都是系统LN的扩张. 相似文献
16.
17.
For any arrangement of hyperplanes in ℂℙ3, we introduce the soul of this arrangement. The soul, which is a pseudo-complex, is determined by the combinatorics of the
arrangement of hyperplanes. In this paper, we give a sufficient combinatoric condition for two arrangements of hyperplanes
to be diffeomorphic to each other. In particular we have found sufficient conditions on combinatorics for the arrangement
of hyperplanes whose moduli space is connected. This generalizes our previous result on hyperplane point arrangements in ℂℙ3.
This work was partially supported by NSA grant and NSF grant 相似文献
18.
正则剩余格是一类重要的模糊逻辑代数系统,而常见的模糊逻辑形式系统大多数带有非联接词,并且相应的Lindenbaum代数都是正则剩余格.本文以强正则剩余格为语义,建立了一个一般的命题演算形式系统LN,并且证明了这个系统的完备性.几种常见的带有非联接词的模糊逻辑形式系统都是系统LN的扩张. 相似文献
19.
《Discrete Mathematics》2019,342(1):233-249
A Weyl arrangement is the hyperplane arrangement defined by a root system. Saito proved that every Weyl arrangement is free. The Weyl subarrangements of type are represented by simple graphs. Stanley gave a characterization of freeness for this type of arrangements in terms of their graph. In addition, the Weyl subarrangements of type can be represented by signed graphs. A characterization of freeness for them is not known. However, characterizations of freeness for a few restricted classes are known. For instance, Edelman and Reiner characterized the freeness of the arrangements between type and type . In this paper, we give a characterization of the freeness and supersolvability of the Weyl subarrangements of type under certain assumption. 相似文献
20.
M. S. Baouendi P. Ebenfelt Linda Preiss Rothschild 《Journal of the American Mathematical Society》2000,13(4):697-723
It is shown that a formal mapping between two real-analytic hypersurfaces in complex space is convergent provided that neither hypersurface contains a nontrivial holomorphic variety. For higher codimensional generic submanifolds, convergence is proved e.g. under the assumption that the source is of finite type, the target does not contain a nontrivial holomorphic variety, and the mapping is finite. Finite determination (by jets of a predetermined order) of formal mappings between smooth generic submanifolds is also established.