On the Milnor Monodromy of the Irreducible Complex Reflection Arrangements

A central question in arrangement theory is to determine whether the characteristic polynomial Δq of the algebraic monodromy acting on the homology group Hqfalse(F(A),double-struckCfalse) of the Milnor fiber of a complex hyperplane arrangement scriptA is determined by the intersection lattice L(A). Under simple combinatorial conditions, we show that the multiplicities of the factors of Δ1 corresponding to certain eigenvalues of order a power of a prime p are equal to the Aomoto–Betti numbers βpfalse(scriptAfalse), which in turn are extracted from L(A). When scriptA defines an arrangement of projective lines with only double and triple points, this leads to a combinatorial formula for the algebraic monodromy. To obtain these results, we relate nets on the underlying matroid of scriptA to resonance varieties in positive characteristic. Using modular invariants of nets, we find a new realizability obstruction (over double-struckC) for matroids, and estimate the number of essential components in the first complex resonance variety of scriptA. Our approach also reveals a rather unexpected connection of modular resonance with the geometry of prefixSL2false(double-struckCfalse)‐representation varieties, which are governed by the Maurer–Cartan equation.

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“…Recent results from [17,23,40] establish the validity of this conjecture, in the strong form (8), for all complex reflection arrangements.…”

Section: Resultsmentioning

confidence: 79%

“…In fact, it can be checked that Conjecture 1.9 holds in the strong form (8), for all full monomial arrangements. For a complete proof of Conjecture 1.9(8) for all complex reflection arrangements, we refer to [17,23,40].…”

Section: More Examplesmentioning

confidence: 99%

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

Papadima

Suciu

2017

Proc. London Math. Soc.

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“…for m ≥ 3, see Corollary 4.5. This result was previously established in [6], using completely different techniques, namely residues of rational differential forms with poles along the line arrangement A. In the final section we apply the same approach to the exceptional reflection arrangement A(G 31 ), consisting of 60 planes in P 3 .…”

Section: Introductionmentioning

confidence: 66%

“…The following result was established in [6] using completely different techniques, namely residues of rational differential forms with poles along the line arrangement A. A(m, m, 3) contains a point of multiplicity m and m triple points.…”

Section: A Vanishing Resultsmentioning

confidence: 99%

A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements

Bailet¹,

Dimca²,

Yoshinaga³

2018

manuscripta math.

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“…Proof One can check that the elements listed above that lie in

S \otimes \land^{m} V^{*} \otimes V

have degrees adding to

prefixΔ false(\land^{m} V^{*} \otimes V false) = 60 (3 ((), \binom{3}{m - 1}) + ((), \binom{3}{m})), for 0 ⩽ m ⩽ 4 .

Thus, the theorem follows from Corollary after one checks that each matrix of coefficients for

0 ⩽ m ⩽ 4

is nonsingular. We did this in Mathematica, using explicit choices of basic invariant polynomials

f_{1}, f_{2}, f_{3}, f_{4}

of degrees 8,12,20,24 and basic derivations

θ_{1}, θ_{2}, θ_{3}, θ_{4}

of degrees 1,13,17,29, constructed as prescribed by Orlik and Terao (using Maschke's invariants

F_{8}, F_{12}, F_{20}

, with

F_{24} = det Hessian false(F_{8} false)

; see also Dimca and Sticlaru and also [, p. 285]).

□

…”

Section: The Reflection Group G31mentioning

confidence: 99%

Invariant derivations and differential forms for reflection groups

Reiner

Shepler

2019

Proc. London Math. Soc.

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Classical invariant theory of a complex reflection group W beautifully describes the W‐invariant polynomials, the W‐invariant differential forms, and the relative invariants of any W‐representation. When W is a duality (or well‐generated) group, we give an explicit description of the isotypic component within the differential forms of the irreducible reflection representation. This resolves a conjecture of Armstrong, Rhoades, and the first author, and relates to Lie‐theoretic conjectures and results of Bazlov, Broer, Joseph, Reeder, and Stembridge, and also Deconcini, Papi, and Procesi. We establish this result by examining the space of W‐invariant differential derivations; these are derivations whose coefficients are not just polynomials, but differential forms with polynomial coefficients. For every complex reflection group W, we show that the space of invariant differential derivations is finitely generated as a module over the invariant differential forms by the basic derivations together with their exterior derivatives. When W is a duality group, we show that the space of invariant differential derivations is free as a module over the exterior subalgebra of W‐invariant forms generated by all but the top‐degree exterior generator. (The basic invariant of highest degree is omitted.) Our arguments for duality groups do not rely on any reflection group classification.

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On the Milnor Monodromy of the Irreducible Complex Reflection Arrangements

Cited by 7 publications

References 46 publications

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements

Invariant derivations and differential forms for reflection groups

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