Internal zonotopal algebras and the monomial reflection groups G(m,1,n)

Berget, Andrew

doi:10.1016/j.jcta.2018.05.001

Cited by 7 publications

(13 citation statements)

References 24 publications

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“…[n] , define To prove Theorem 6.21, we will first prove that G n,1 is isomorphic to a different induced representation, and then appeal to Lemma 6.22. Both the proof of Theorem 6.21 and Lemma 6.22 will use techniques developed by Berget [8] and in the case of Lemma 6.22, by Douglass-Tomlin [20]. When n is odd, we use similar methods to Berget in [8,Corollary 9.2].…”

Section: Let λ ±mentioning

confidence: 99%

“…Both the proof of Theorem 6.21 and Lemma 6.22 will use techniques developed by Berget [8] and in the case of Lemma 6.22, by Douglass-Tomlin [20]. When n is odd, we use similar methods to Berget in [8,Corollary 9.2]. In this case, −η is a Coxeter element with eigenvalue −e 2πi/n , implying that η, −1 = Z (∅,(n)) .…”

Section: Let λ ±mentioning

confidence: 99%

“…In addition to drawing upon and generalizing techniques of Berget [8], Moseley [37], and Moseley-Proudfoot-Young [38], the primary novelties in our methodology will be to (1) study the "lifted" spaces Y d n+1 to deduce information about the spaces Z d n and (2) to utilize various filtrations in the signed Heaviside and signed cyclic Heaviside functions. 1.3.…”

Section: Introductionmentioning

confidence: 99%

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A Type B analog of the Whitehouse representation

Brauner¹

2022

Preprint

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We give a Type B analog of Whitehouse's lifts of the Eulerian representations from Sn to Sn+1 by introducing a family of Bn-representations that lift to Bn+1. As in Type A, we interpret these representations combinatorially via a family of orthogonal idempotents in the Mantaci-Reutenauer algebra, and topologically as the graded pieces of the cohomology of a certain Z2-orbit configuration space of R 3 . We show that the lifted Bn+1-representations also have a configuration space interpretation, and further parallel the Type A story by giving analogs of many of its notable properties, such as connections to equivariant cohomology and the Varchenko-Gelfand ring. Contents 1. Introduction 1.1. Type A Motivation 1.2. Type B analog 1.3. Outline of the paper 2. Background 2.1. Type A revisited 2.2. Orbit configuration spaces and hidden actions 2.3. The hyperoctahedral group 3. The d = 1 case in Type B 3.1. Signed cyclic Heaviside functions and Y 1 n+1 3.2. A presentation for H * Z 1 n 4. The d = 3 case in Type B 4.1. Tools from Topology 4.2. Tools from representation theory 4.3. Presentation for H * Z 3 n 5. T -equivariant cohomology 5.1. Background 5.2. Applications to Z 3 n and Y 3 n+1 5.3. Presentation of equivariant cohomology 5.4. Lifting to Y 3 n+1 6. Connection to the Mantaci-Reutenauer algebra 6.1. The Vazirani idempotents 6.2. Induced representations 6.3. Consequences 6.4. Further

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Section: Let λ ±mentioning

confidence: 99%

Section: Let λ ±mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Type B analog of the Whitehouse representation

Brauner¹

2022

Preprint

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“…These ideals were independently introduced in two different ways. The first definition was used by F. Ardila and A. Postnikov [2]; it originates from the algebras generated by the curvature forms of tautological Hermitian linear bundles [4,29], see also papers [5,6,15,16,17,23,24,27,28,30], where the quotient algebras by these ideals were discussed by details. At the same time a similar definition and the term were established by O. Holtz and A. Ron [13]; Their definitions originates from Box-Splines and from Dahmen-Micchelli's space [1,9,11], see also the papers [10,12,14,19,20,21,22,31].…”

Section: Introductionmentioning

confidence: 99%

Classification of External Zonotopal Algebras

Nenashev

2019

Electron. J. Combin.

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In this paper we work with power algebras associated to hyperplane arrangements. There are three main types of these algebras, namely, external, central, and internal zonotopal algebras. We classify all external algebras up to isomorphism in terms of zonotopes. Also, we prove that unimodular external zonotopal algebras are in one to one correspondence with regular matroids. For the case of central algebras we formulate a conjecture.

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“…There are three types of algebras. We will work with the definition from F. Ardila and A. Postnikov [2]; it comes from algebras generated by the curvature forms of tautological Hermitian linear bundles [3,28], see also papers [4,5,14,15,16,22,23,26,27,29], where people work with quotients algebras by these ideals. At the same time the definition and the name was established by O. Holtz and A. Ron [12]; it comes from Box-Splines and from Dahmen-Micchelli space [1,8,10], see also the papers [9,11,13,18,19,20,21,30].…”

Section: Introductionmentioning

confidence: 99%

Classification of external Zonotopal algebras

Nenashev¹

2018

Preprint

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In this paper we work with power algebras associated to hyperplane arrangements. There are three main types of these algebras, namely, external, central, and internal zonotopal algebras. We classify all external algebras up to isomorphism in terms of zonotopes. Also, we prove that unimodular external zonotopal algebras are in one to one correspondence with regular matroids.For the case of central algebras we formulate a conjecture.

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Internal zonotopal algebras and the monomial reflection groups G(m,1,n)

Cited by 7 publications

References 24 publications

A Type B analog of the Whitehouse representation

A Type B analog of the Whitehouse representation

Classification of External Zonotopal Algebras

Classification of external Zonotopal algebras

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