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Length computation of matrix subalgebras of special type. (English. Russian original) Zbl 1163.16017
J. Math. Sci., New York 155, No. 6, 908-931 (2008); translation from Fundam. Prikl. Mat. 13, No. 4, 165-197 (2007).
Let \(A\) be a finite-dimensional algebra over a field \(F\). For each set \(S\subseteq A\) which generates \(A\) as an \(F\)-algebra, define \(l(S)\) to be the least integer \(h\geq 0\) such that there is an \(F\)-basis of \(A\) consisting of words of length \(\leq h\) in \(S\). The ‘length’ \(l(A)\) of \(A\) is the maximum of \(l(S)\) over all finite generating sets \(S\).
In general the length of a subalgebra of \(A\) can be larger than the length of \(A\). If \(A\) is the full matrix algebra \(M_n(F)\), then it is known that \(l(M_n(F))=O(n^{3/2})\) and it is conjectured that \(l(M_n(F))=2n-2\) [see C. J. Pappacena, J. Algebra 197, No. 2, 535-545 (1997; Zbl 0888.16008) and A. Paz, Linear Multilinear Algebra 15, 161-170 (1984; Zbl 0536.15007)].
In the present paper the author gives short proofs that the conjecture holds for \(n=2\) and \(3\), and computes the lengths of various classes of subalgebras of \(M_n(F)\).

MSC:
16S50 Endomorphism rings; matrix rings
15A30 Algebraic systems of matrices
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
16P10 Finite rings and finite-dimensional associative algebras
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References:
[1] W. C. Brown and F. W. Call, ”Maximal commutative subalgebras of n {\(\times\)} n matrices,” Commun. Algebra, 21, No. 12, 4439–4460 (1993). · Zbl 0796.16021
[2] R. C. Courter, ”The dimension of maximal commutative subalgebras of K n ,” Duke Math. J., 32, 225–232 (1965). · Zbl 0141.03101
[3] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge (1991). · Zbl 0729.15001
[4] T. J. Laffey, ”Simultaneous reduction of sets of matrices under similarity,” Linear Algebra Appl., 84, 123–138 (1986). · Zbl 0609.15004
[5] W. E. Longstaff, ”Burnside’s theorem: Irreducible pairs of transformations,” Linear Algebra Appl., 382, 247–269 (2004). · Zbl 1059.15006
[6] W. E. Longstaff and P. Rosenthal, ”Generators of matrix incidence algebras,” Aust. J. Combin., 22, 117–121 (2000). · Zbl 0967.15011
[7] O. V. Markova, ”On the length of upper-triangular matrices,” Russ. Math. Surv., 60, No. 5, 984–985 (2005). · Zbl 1140.16306
[8] C. J. Pappacena, ”An upper bound for the length of a finite-dimensional algebra,” J. Algebra, 197, 535–545 (1997). · Zbl 0888.16008
[9] A. Paz, ”An application of the Cayley-Hamilton theorem to matrix polynomials in several variables,” Linear Multilinear Algebra, 15, 161–170 (1984). · Zbl 0536.15007
[10] I. Schur, ”Zur Theorie der vertauschbaren Matrizen,” J. Reine Angew. Math., 130, 66–76 (1905). · JFM 36.0140.01
[11] A. J. M. Spencer and R. S. Rivlin, ”The theory of matrix polynomials and its applications to the mechanics of isotropic continua,” Arch. Ration. Mech. Anal., 2, 309–336 (1959). · Zbl 0095.25101
[12] A. J. M. Spencer and R. S. Rivlin, ”Further results in the theory of matrix polynomials,” Arch. Ration. Mech. Anal., 4, 214–230 (1960). · Zbl 0095.25103
[13] A. Wadsworth, ”The algebra generated by two commuting matrices,” Linear Multilinear Algebra, 27, 159–162 (1990). · Zbl 0703.15016
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