The paper studies two numerical characteristics of matrix incidence algebras over finite fields associated with generating sets of such algebras: the minimal cardinality of a generating set and the length of an algebra. Generating sets are understood in the usual sense, the identity of the algebra being considered a word of length 0 in generators, and also in the strict sense, where this assumption is not used. A criterion for a subset to generate an incidence algebra in the strict sense is obtained. For all matrix incidence algebras, the minimum cardinality of a generating set and a generating set in the strict sense are determined as functions of the field cardinality and the order of the matrices. Some new results on the lengths of such algebras are obtained. In particular, the length of the algebra of “almost” diagonal matrices is determined, and a new upper bound for the length of an arbitrary matrix incidence algebra is obtained.
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Yu. A. Al’pin and Kh. D. Ikramov, “Reducibility theorems for pairs of matrices as rational criteria,” Linear Algebra Appl., 313, 155–161 (2000).
A. Guterman, O. Markova, and V. Mehrmann, “Lengths of quasi-commutative pairs of matrices,” Linear Algebra Appl., 498, 450–470 (2016).
A. Guterman, O. Markova, and V. Mehrmann, “Length realizability for pairs of quasi-commuting matrices,” Linear Algebra Appl., 568, 135–154 (2019).
A. Guterman, T. Laffey, O. Markova, and H. Šmigoc, “A resolution of Paz’s conjecture in the presence of a nonderogatory matrix,” Linear Algebra Appl., 543, 234–250 (2018).
L. Halbeisen, M. Hamilton, and P. Ružička, “Minimal generating sets of groups, rings, and fields,” Quaest. Math., 30, 355–363 (2007).
M. C. Iovanov and G. D. Koffi, Incidence Algebras and Their Representation Theory, arXiv:1702.03356 (2017).
T. Laffey, O. Markova, and H. Šmigoc, “The effect of assuming the identity as a generator on the length of the matrix algebra,” Linear Algebra Appl., 498, 378–393 (2016).
V. Lomonosov and P. Rosenthal, “The simplest proof of Burnside’s theorem on matrix algebras,” Linear Algebra Appl., 383, 45–47 (2004).
W. E. Longstaff and P. Rosenthal, “Generators of matrix incidence algebras,” Austral. J. Combin., 22, 117–121 (2000).
W. E. Longstaff and P. Rosenthal, “On the lengths of irreducible pairs of complex matrices,” Proc. Amer. Math. Soc., 139, 3769–3777 (2011).
V. E. Marenich, “Conjugation properties in incidence algebras,” Fundam. Prikl. Mat., 9, 111–123 (2003).
O. V. Markova, “Length computation of matrix subalgebras of a special type,” Fundam. Prikl. Mat., 13, 165–197 (2007).
O. V. Markova, “An upper bound for the length of commutative algebras,” Mat. Sb., 200, 41–62 (2009).
O. V. Markova, “On the relationship between the length of an algebra and the index of nilpotency of its Jacobson radical,” Mat. Zametki, 94, 682–688 (2013).
O. V. Markova, “Commutative nilpotent subalgebras with nilpotency index n − 1 in the algebra of matrices of order n,” Zap. Nauchn. Semin. POMI, 453, 219–242 (2016).
N. A. Načev, “Polynomial identities in incidence algebras,” Usp. Mat. Nauk, 32, 233–234 (1977).
A. Paz, “An application of the Cayley–Hamilton theorem to matrix polynomials in several variables,” Linear Multilinear Algebra, 15, 161–170 (1984).
R. S. Piers, Associative Algebras, Springer-Verlag (1982).
G.-C. Rota, “On the foundations of combinatorial theory, I. Theory of Möbius functions,” Z. Wahrscheinlichkeitsrechnung, 2, 340-368 (1964).
D. A. Smith, “Incidence functions as generalized arithmetic functions. I,” Duke Math. J., 34, 617-633 (1967).
E. Spiegel and C. J. O’Donnel, Incidence Algebras, Marcel Dekker (1997).
D. T. Tapkin, “Generalized matrix rings and generalization of incidence algebras,” Chebyshev. Sb., 16, 422–449 (2015).
M. Radjabalipour, P. Rosenthal, and B. R. Yahaghi, “Burnside’s theorem for matrix rings over division rings,” Linear Algebra Appl., 383, 29–44 (2004).
D. P. Zhelobenko, Compact Lie Groups and Their Representations [in Russian], Nauka, Moscow (1970).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 472, 2018, pp. 120–144.
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Kolegov, N.A., Markova, O.V. Systems of Generators of Matrix Incidence Algebras over Finite Fields. J Math Sci 240, 783–798 (2019). https://doi.org/10.1007/s10958-019-04396-6
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DOI: https://doi.org/10.1007/s10958-019-04396-6