Classification of commutative subalgebras of length $n-2$ in the algebra of $n\times n$ matrices over algebraically closed fields

O. V. Markova

Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 1, pp. 133-159.

Résumé

In this paper, commutative subalgebras of length $n-2$ in the algebra of matrices of order $n$ over algebraically closed fields are classified up to similarity. In other terms, commutative algebras having length one less than the maximum value are described. It is shown that for an arbitrary fixed order of matrices, the number of pairwise non-conjugate algebras of the indicated type is finite. Using the number of partitions of natural numbers, a formula is obtained for the number of different algebras as a function of the order of the matrices.

Bibliographie

[1] Gantmakher F. R., Teoriya matrits, Nauka, M., 1988 | MR

[2] Lambek I., Koltsa i moduli, Mir, M., 1971 | MR

[3] Maltsev A. I., Osnovy lineinoi algebry, Nauka, M., 1975 | MR

[4] Markova O. V., “Verkhnyaya otsenka dliny kommutativnykh algebr”, Matem. sb., 200:12 (2009), 41–62 | DOI | Zbl

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

[6] Markova O. V., “O nekotorykh svoistvakh funktsii dliny”, Matem. zametki, 87:1 (2010), 83–91 | DOI | MR | Zbl

[7] Markova O. V., “Klassifikatsiya matrichnykh podalgebr dliny $1$”, Fundament. i prikl. matem., 17:1 (2012), 169–188 | Zbl

[8] Markova O. V., “Funktsiya dliny i matrichnye algebry”, Fundament. i prikl. matem., 17:6 (2012), 65–173

[9] Markova O. V., “Kommutativnye nilpotentnye podalgebry indeksa nilpotentnosti $n-1$ v algebre matrits poryadka $n$”, Zap. nauchn. sem. POMI, 453, 2016, 219–242

[10] Markova O. V., “Funktsiya dliny i odnovremennaya triangulizuemost par matrits”, Zap. nauchn. sem. POMI, 514, 2022, 126–137

[11] Markova O. V., “Kommutativnye matrichnye algebry, porozhdennye tsiklicheskimi matritsami”, Zap. nauchn. sem. POMI, 524, 2023, 112–124

[12] Markova O. V., Novochadov D. Yu., “Sistemy porozhdayuschikh polnoi matrichnoi algebry, soderzhaschie tsiklicheskie matritsy”, Zap. nauchn. sem. POMI, 504, 2021, 157–171

[13] On-Line Encyclopedia of Integer Sequences, OEIS

[14] Pavlov I. A., “O kommutativnykh nilpotentnykh algebrakh matrits”, DAN BSSR, 11:10 (1967), 870–872 | Zbl

[15] Pirs R., Assotsiativnye algebry, Mir, M., 1986

[16] Suprunenko D. A., Tyshkevich R. I., Perestanovochnye matritsy, URSS, M., 2003

[17] Endryus G., Teoriya razbienii, Nauka, M., 1982

[18] Al'pin Yu. A., Ikramov Kh. D., “Reducibility theorems for pairs of matrices as rational criteria”, Linear Algebra Appl., 313 (2000), 155–161 | DOI | MR | Zbl

[19] Brown W. C., Call F. W., “Maximal commutative subalgebras of $n\times n$ matrices”, Commun. Algebra, 21:12 (1993), 4439–4460 | DOI | MR | Zbl

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

[21] Dolinar G., Guterman A., Kuzma B., Oblak P., “Extremal matrix centralizers”, Linear Algebra Appl., 438:7 (2013), 2904–2910 | DOI | MR | Zbl

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

[23] Gustafson W. H., “On maximal commutative algebras of linear transformations”, J. Algebra, 42:2 (1976), 557–563 | DOI | MR | Zbl

[24] Guterman A. E., Laffey T., Markova O. V., Šmigoc H., “A resolution of Paz's conjecture in the presence of a nonderogatory matrix”, Linear Algebra Appl., 543 (2018), 234–250 | DOI | MR | Zbl

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

[26] Guterman A. E., Markova O. V., Mehrmann V., “Lengths of quasi-commutative pairs of matrices”, Linear Algebra Appl., 498 (2016), 450–470 | DOI | MR | Zbl

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

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

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

[30] Longstaff W. E., “Burnside's theorem: irreducible pairs of transformations”, Linear Algebra Appl., 382 (2004), 247–269 | DOI | MR | Zbl

[31] Longstaff W. E., “On minimal sets of $(0,1)$-matrices whose pairwise products form a basis for $\mathrm{M}a_n(\mathbb{F})$”, Bull. Austral. Math. Soc., 98:3 (2018), 402–413 | DOI | MR | Zbl

[32] Longstaff W. E., “Irreducible families of complex matrices containing a rank-one matrix”, Bull. Austral. Math. Soc., 102:2 (2020), 226–236 | DOI | MR | Zbl

[33] Pappacena C. J., “An upper bound for the length of a finite-dimensional algebra”, J. Algebra, 197 (1997), 535–545 | DOI | MR | Zbl

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

[35] Schur I., “Zur Theorie der vertauschbaren Matrizen”, J. Reine Angew. Math., 130 (1905), 66–76 | DOI | MR

[36] Shitov Ya., “An improved bound for the lengths of matrix algebras”, Algebra Number Theory, 13:6 (2019), 1501–1507 | DOI | MR | Zbl

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

[38] Spencer A. J. M., Rivlin R. S., “The theory of matrix polynomials and its applications to the mechanics of isotropic continua”, Arch. Ration. Mech. Anal., 2 (1959), 309–336 | DOI | MR | Zbl

[39] Spencer A. J. M., Rivlin R. S., “Further results in the theory of matrix polynomials”, Arch. Ration. Mech. Anal., 4 (1960), 214–230 | DOI | MR | Zbl

[40] Wadsworth A., “The algebra generated by two commuting matrices”, Linear Multilinear Algebra, 27 (1990), 159–162 | DOI | MR | Zbl