Geodesic
Commutative nilpotent subalgebras with nilpotency index $n-1$ in the algebra of matrices of order $n$
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIX, Tome 453 (2016), pp. 219-242 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper establishes the existence of an element with nilpotency index $n-1$ in the algebra of upper niltriangular matrices $N_n(\mathbb F)$ over a field $\mathbb F$ with at least $n$ elements for all $n\ge5$ and, as a corollary, also in the full matrix algebra $M_n(\mathbb F)$. This result implies an improvement with respect to the basic field of known classification theorems due to D. A. Suprunenko, R. I. Tyschkevich, and I. A. Pavlov for algebras of the class considered. 
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     author = {O. V. Markova},
     title = {Commutative nilpotent subalgebras with nilpotency index $n-1$ in the algebra of matrices of order~$n$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {219--242},
     year = {2016},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_453_a14/}
}
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O. V. Markova. Commutative nilpotent subalgebras with nilpotency index $n-1$ in the algebra of matrices of order $n$. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIX, Tome 453 (2016), pp. 219-242. http://geodesic.mathdoc.fr/item/ZNSL_2016_453_a14/

[1] R. C. Courter, “The dimension of maximal commutative subalgebras of $K_n$”, Duke Math. J., 32 (1965), 225–232 | DOI | MR | Zbl

[2] M. Gerstenhaber, “On dominance and varieties of commuting matrices”, Annals Math., 73:2 (1961), 324–348 | DOI | MR | Zbl

[3] A. E. Guterman, O. V. Markova, “Commutative matrix subalgebras and length function”, Linear Algebra Appl., 430 (2009), 1790–1805 | DOI | MR | Zbl

[4] N. Jacobson, “Schur's theorems on commutative matrices”, Bull. Amer. Math. Soc., 50 (1944), 431–436 | DOI | MR | Zbl

[5] T. J. Laffey, “The minimal dimension of maximal commutative subalgebras of full matrix algebras”, Linear Algebra Appl., 71 (1985), 199–212 | DOI | MR | Zbl

[6] T. J. Laffey, S. Lazarus, “Two-generated commutative matrix subalgebras”, Linear Algebra Appl., 147 (1991), 249–273 | DOI | MR | Zbl

[7] O. V. Markova, “Kharakterizatsiya kommutativnykh matrichnykh podalgebr maksimalnoi dliny nad proizvolnym polem”, Vestn. Mosk. un-ta. Ser. 1. Matematika. Mekhanika, 2009, no. 5, 53–55 | Zbl

[8] I. A. Pavlov, “O kommutativnykh nilpotentnykh algebrakh matrits”, Dokl. Akad. nauk BSSR, 11:10 (1967), 870–872 | MR | Zbl

[9] A. Paz, “An application of the Cayley–Hamilton theorem to matrix polynomials in several variables”, Linear Multilinear Algebra, 15 (1984), 161–170 | DOI | MR | Zbl

[10] I. Schur, “Zur Theorie der Vertauschbären Matrizen”, J. reine angew. Math., 130 (1905), 66–76 | MR | Zbl

[11] Youngkwon Song, “A construction of maximal commutative subalgebra of matrix algebras”, J. Korean Math. Soc., 40:2 (2003), 241–250 | DOI | MR | Zbl

[12] D. A. Suprunenko, R. I. Tyshkevich, Perestanovochnye matritsy, 2-e izd., URSS, M., 2003