Exterior powers of the standard E6-module: An elementary approach

For the standard [Formula: see text]-dimensional representation [Formula: see text] of the exceptional group [Formula: see text] of type [Formula: see text] we prove that [Formula: see text] is a Donkin pair if and only if the characteristic of a ground field is greater than [Formula: see text]. We also develop an elementary approach to describe submodule structure of any exterior power of [Formula: see text].

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Polynomial ring representations of endomorphisms of exterior powers

Collectanea mathematica ◽

10.1007/s13348-020-00310-5 ◽

2021 ◽

Author(s):

Ommolbanin Behzad ◽

André Contiero ◽

Letterio Gatto ◽

Renato Vidal Martins

Keyword(s):

Lie Algebra ◽

Polynomial Ring ◽

Vertex Operators ◽

Exterior Power ◽

Infinite Dimensional ◽

Symmetric Structure ◽

Rational Polynomials ◽

Exterior Powers ◽

Vertex Representation ◽

Countably Infinite

AbstractAn explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

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Degree of reductivity of a modular representation

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500231 ◽

2017 ◽

Vol 19 (03) ◽

pp. 1650023 ◽

Cited By ~ 1

Author(s):

Martin Kohls ◽

Müfi̇t Sezer

Keyword(s):

Fixed Point ◽

Lower Bound ◽

Cyclic Subgroup ◽

Representation Formula ◽

Modular Representation ◽

Dimensional Representation ◽

Finite Dimensional Representation ◽

Finite Dimensional ◽

Modular Groups ◽

Sharp Lower Bound

For a finite-dimensional representation [Formula: see text] of a group [Formula: see text] over a field [Formula: see text], the degree of reductivity [Formula: see text] is the smallest degree [Formula: see text] such that every nonzero fixed point [Formula: see text] can be separated from zero by a homogeneous invariant of degree at most [Formula: see text]. We compute [Formula: see text] explicitly for several classes of modular groups and representations. We also demonstrate that the maximal size of a cyclic subgroup is a sharp lower bound for this number in the case of modular abelian [Formula: see text]-groups.

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Reconstruction of Objects from their Projections Using Orthogonal Functions

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100071788 ◽

1973 ◽

Vol 31 ◽

pp. 268-269

Author(s):

Elrnar Zeitler

Keyword(s):

Unit Area ◽

Three Dimensional ◽

One Dimension ◽

Spatial Density ◽

Dimensional Representation ◽

Orthogonal Functions ◽

Object A ◽

Plane Normal ◽

Dimensional Object ◽

Spatial Density Distribution

Considering any finite three-dimensional object, a “projection” is here defined as a two-dimensional representation of the object's mass per unit area on a plane normal to a given projection axis, here taken as they-axis. Since the object can be seen as being built from parallel, thin slices, the relation between object structure and its projection can be reduced by one dimension. It is assumed that an electron microscope equipped with a tilting stage records the projectionWhere the object has a spatial density distribution p(r,ϕ) within a limiting radius taken to be unity, and the stage is tilted by an angle 9 with respect to the x-axis of the recording plane.

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Revealing gross three-dimensional structure in the SEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100127712 ◽

1987 ◽

Vol 45 ◽

pp. 656-657

Author(s):

Sterling P. Newberry

Keyword(s):

Freeze Drying ◽

Three Dimensional ◽

Dimensional Structure ◽

Small Object ◽

Final Solution ◽

Dimensional Representation ◽

X Ray ◽

Three Dimensional Representation ◽

Freeze Dried ◽

Critical Point Drying

The beautiful three dimensional representation of small object surfaces by the SEM leads one to search for ways to open up the sample and look inside. Could this be the answer to a better microscopy for gross biological 3-D structure? We know from X-Ray microscope images that Freeze Drying and Critical Point Drying give promise of adequately preserving gross structure. Can we slice such preparations open for SEM inspection? In general these preparations crush more readily than they slice. Russell and Dagihlian got around the problem by “deembedding” a section before imaging. This some what defeats the advantages of direct dry preparation, thus we are reluctant to accept it as the final solution to our problem. Alternatively, consider fig 1 wherein a freeze dried onion root has a window cut in its surface by a micromanipulator during observation in the SEM.

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