Counting Fine Gradings on Matrix Algebras and on Classical Simple Lie Algebras

Kochetov, Mikhail; Parsons, Nicholas R.; Sadov, Sergey

doi:10.1142/s0218196713500434

Cited by 6 publications

(8 citation statements)

References 14 publications

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“…• For p = 2, [18] is a very lucid review of the cases where the solution of Problem 1.2 has been found; for further details, see [1,19]; for examples of applications of certain G-gradings, see [20].…”

Section: Arxiv:171100638v3 [Mathrt] 10 Dec 2018mentioning

confidence: 99%

On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2

Krutov¹,

Lebedev²

2018

SIGMA

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The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finitedimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of Lie algebras and their simple relatives: of special linear series, its projectivizations, and projectivizations of the derived Lie algebras of two inequivalent orthogonal series (except for o Π (8)). The classification of gradings is new, but all of the corresponding superizations are known. For the simple derived Zassenhaus algebras of height n > 1, there is an (n − 2)-parametric family of modulo 2 gradings; all but one of the corresponding simple Lie superalgebras are new. Our classification also proves non-triviality of a deformation of a simple 3|2-dimensional Lie superalgebra (new result).

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Section: Arxiv:171100638v3 [Mathrt] 10 Dec 2018mentioning

confidence: 99%

On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2

Krutov¹,

Lebedev²

2018

SIGMA

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“…We are not aware of any publications presenting this result. A predesigned application of Theorem 6 is the improvement of the estimate derived in [16], which has already been mentioned above.…”

Section: Introductionmentioning

confidence: 99%

“…Kochetov who invited me to participate in this project. The problem of deriving nontrivial estimates for the average size of the stabilizer arose during our discussion of the possible improvement of the upper estimate in Theorem 6.2 of the work [16] (which is to be published separately).…”

Section: Introductionmentioning

confidence: 99%

Asymptotically free action of permutation groups on subsets and multisets

Sadov

2015

Discrete Mathematics and Applications

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Let be a permutation group acting on a nite set of cardinality . The number of orbits of the induced action of on the set of all -element subsets of obeys the trivial estimates | |/| | ≤ | / | ≤ | |. In this paper the upper estimate is improved in terms of the minimal degree of the group or the minimal degree of its subset with small complement. In particular, using the universal estimates obtained by Bochert for the minimal degree of a group and by Babai-Pyber for the order of a group, in terms of only we demonstrate that if is a 2-transitive group other than the full symmetric or the alternating groups, and are large enough, and the ratio / is bounded away from 0 and 1, then | / | ≈ | |/| |. Similar results hold true for the induced action of on the set ( ) of all -element multisets with elements drawn from , provided that the ratio /( + ) is uniformly bounded away from 0 and 1.Keywords: permutation group, regular orbits, average size of the stabilizer, minimal degree of a group, asymptotics of the number of orbits, enumeration of a ne con gurations, enumeration of graphs, asymptotically free action.

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“…More recent work has resulted in a classification of gradings for many classical simple Lie algebras (see [1], [8], [12] and [13]). A method to induce good gradings on Lie superalgebras, Lie algebras, and matrix algebras using directed graphs was developed in [5], [10], [11], and [20].…”

mentioning

confidence: 99%

“…With an increased understanding of how to induce gradings, we can count the number of gradings of a particular type. For full matrix algebras, fine gradings are counted in [13] and elementary gradings are counted in [5].…”

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confidence: 99%

Induced Good Gradings of Structural Matrix Rings

DeWitt¹,

Price²

2017

Preprint

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Our approach to structural matrix rings defines them over preordered directed graphs. A grading of a structural matrix ring is called a good grading if its standard unit matrices are homogeneous. For a group G, a G-grading set is a set of arrows with the property that any assignment of these arrows to elements of G uniquely determines an induced good grading. One of our main results is that a G-grading set exists for any transitive directed graph if G is a group of prime order. This extends a result of Kelarev. However, an example of Molli Jones shows there are directed graphs which do not have G-grading sets for any cyclic group G of even order greater than 2. Finally, we count the number of nonequivalent elementary gradings by a finite group of a full matrix ring over an arbitrary field.

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Counting Fine Gradings on Matrix Algebras and on Classical Simple Lie Algebras

Cited by 6 publications

References 14 publications

On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2

On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2

Asymptotically free action of permutation groups on subsets and multisets

Induced Good Gradings of Structural Matrix Rings

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