HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH

AbstractWe settle a conjecture of Goresky, Kottwitz and MacPherson related to Koszul duality, i.e., to the correspondence between differential graded modules over the exterior algebra and those over the symmetric algebra.

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On Hilbert functions of graded modules

Mathematische Nachrichten ◽

10.1002/mana.19800960123 ◽

1980 ◽

Vol 96 (1) ◽

pp. 301-309 ◽

Cited By ~ 1

Author(s):

P. K. Shukla

Keyword(s):

Hilbert Functions ◽

Graded Modules

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On the Structure of a Class of Graded Modules Related to Symmetric Pairs

Algebra Colloquium ◽

10.1142/s1005386706000289 ◽

2006 ◽

Vol 13 (02) ◽

pp. 315-328 ◽

Cited By ~ 1

Author(s):

Gang Han

Keyword(s):

Lie Algebra ◽

Adjoint Representation ◽

Exterior Algebra ◽

Symmetric Algebra ◽

Module Structure ◽

Cartan Decomposition ◽

Isotropy Representation ◽

Semisimple Lie Algebra ◽

Graded Modules ◽

Special Case

Let [Formula: see text] be the Cartan decomposition of a real semisimple Lie algebra and [Formula: see text] be its complexification. Let [Formula: see text] be the corresponding isotropy representation, and the exterior algebra [Formula: see text] becomes a graded [Formula: see text]-module by extending ν. We study a graded [Formula: see text]-submodule C of [Formula: see text] and get two important decompositions of the [Formula: see text]-module [Formula: see text]. Let [Formula: see text] be the symmetric algebra over [Formula: see text]. Then [Formula: see text] also has an [Formula: see text]-module structure, which is [Formula: see text]-equivariant, and C is a space of generators for this module. Our results generalize Kostant's results in the special case that ν is the adjoint representation of a semisimple Lie algebra.

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Cohomological degrees and Hilbert functions of graded modules

American Journal of Mathematics ◽

10.1353/ajm.1998.0019 ◽

1998 ◽

Vol 120 (3) ◽

pp. 493-504 ◽

Cited By ~ 18

Author(s):

Wolmer V. Vasconcelos ◽

Luisa Rodrigues Doering ◽

Tor Gunston

Keyword(s):

Hilbert Functions ◽

Graded Modules

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A generalization of Kruskal–Katona’s theorem

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2020-0018 ◽

2020 ◽

Vol 28 (2) ◽

pp. 35-51

Author(s):

Luca Amata ◽

Marilena Crupi

Keyword(s):

Vector Space ◽

Crucial Role ◽

Hilbert Function ◽

Exterior Algebra ◽

Hilbert Functions ◽

Finitely Generated ◽

Finite Dimensional ◽

Homogeneous Basis

AbstractLet K be a field, E the exterior algebra of a finite dimensional K-vector space, and F a finitely generated graded free E-module with homogeneous basis g1, . . ., gr such that deg g1 ≤ deg g2 ≤ · · · ≤ deg gr. We characterize the Hilbert functions of graded E–modules of the type F/M, with M graded submodule of F. The existence of a unique lexicographic submodule of F with the same Hilbert function as M plays a crucial role.

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