Commutative matrix subalgebras generated by nonderogatory matrices

O. V. Markova

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXVI, Tome 524 (2023), pp. 112-124

Résumé

The paper is devoted to studying the question on the number of subalgebras in the matrix algebra that generated by nonderogatory matrices and are distinct up to similarity. A criterion for similarity of matrix algebras of the type indicated is established in terms of isomorphisms of quotient algebras of the algebra of polynomials. The existence of infinite fields for which the number of distinct algebras is infinite for all values of the matrix order is established. For algebraically closed fields, for the field of real numbers, and for finite fields of sufficiently large cardinality, the number of distinct algebras generated by nonderogatory matrices is determined as a function of the matrix order.

Bibliographie

[1] E. B. Vinberg, Kurs algebry, 3-e izd., Faktorial Press, M., 2002

[2] F. R. Gantmakher, Teoriya Matrits, 4-e izd., Nauka, M., 1988 | MR

[3] R. Lidl, G. Niderraiter, Konechnye polya, V 2-kh tomakh, Mir, M., 1988

[4] O. V. Markova, “Kharakterizatsiya kommutativnykh matrichnykh podalgebr maksimalnoi dliny nad proizvolnym polem”, Vestn. Mosk. un-ta. Ser.1. Matematika. Mekhanika, 5 (2009), 53–55 | Zbl

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

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

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

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

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

[10] N. A. Brigham, “A general asymptotic formula for partition functions”, Proc. Amer. Math. Soc., 1 (1950), 182–191 | DOI | MR | Zbl

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

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

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

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

[15] R. Horn, C. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, 2013 | MR | Zbl

[16] V. Kotesovec, Asymptotics of Sequence A002513, 2019

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

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

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

[20] On-Line Encyclopedia of Integer Sequences (OEIS)

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

[22] 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

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