Geodesic
The lengths of matrix incidence algebras over small finite fields
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIV, Tome 504 (2021), pp. 102-135 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The paper considers the problem of computing the lengths of matrix incidence algebras over a field whose cardinality is strictly less than the matrix size $n$. For $n=3,4$, the lengths of all such algebras are determined over the field of two elements. In the case where the ground field and the number $n$ are arbitrary but the Jacobson radical of the algebra has nilpotency index $2$, an upper bound for the length is provided. In addition, the incidence algebras isomorphic to a direct sum of triangular matrix algebras of order $2$ and an algebra of diagonal matrices are considered. It is shown that the lengths of these algebras over the field of two elements can equal only two different numbers, which can be determined explicitly. Moreover, the diagonal number of a matrix incidence algebra is introduced and bounded above. 
@article{ZNSL_2021_504_a6,
     author = {N. A. Kolegov and O. V. Markova},
     title = {The lengths of matrix incidence algebras over small finite fields},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {102--135},
     year = {2021},
     volume = {504},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_504_a6/}
}
TY  - JOUR
AU  - N. A. Kolegov
AU  - O. V. Markova
TI  - The lengths of matrix incidence algebras over small finite fields
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2021
SP  - 102
EP  - 135
VL  - 504
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2021_504_a6/
LA  - ru
ID  - ZNSL_2021_504_a6
ER  - 
%0 Journal Article
%A N. A. Kolegov
%A O. V. Markova
%T The lengths of matrix incidence algebras over small finite fields
%J Zapiski Nauchnykh Seminarov POMI
%D 2021
%P 102-135
%V 504
%U http://geodesic.mathdoc.fr/item/ZNSL_2021_504_a6/
%G ru
%F ZNSL_2021_504_a6
N. A. Kolegov; O. V. Markova. The lengths of matrix incidence algebras over small finite fields. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIV, Tome 504 (2021), pp. 102-135. http://geodesic.mathdoc.fr/item/ZNSL_2021_504_a6/

[1] R. Brusamarello, E. Z. Fornaroli, E. A. Santulo Jr., “Classification of involutions on finitary incidence algebras”, Int. J. Algebra Comput., 24:8 (2014), 1085–1098 | DOI | MR | Zbl

[2] 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

[3] A. E. Guterman, T. J. 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

[4] I. Kaygorodov, M. Khrypchenko, F. Wei, “Higher derivations of finitary incidence algebras”, Algebr. Represent. Theory, 22 (2019), 1331–1341 | DOI | MR | Zbl

[5] N. A. Kolegov, “On real algebras generated by positive and nonnegative matrices”, Linear Algebra Appl., 611 (2021), 46–65 | DOI | MR | Zbl

[6] N. A. Kolegov, On the lengths of matrix incidence algebras with radicals of square zero, Preprint

[7] N. A. Kolegov, O. V. Markova, “Kommutativnost matrits s tochnostyu do matrichnogo mnozhitelya”, Zap. nauchn. semin. POMI, 482 (2019), 151–168

[8] N. A. Kolegov, O. V. Markova, “Sistemy porozhdayuschikh matrichnykh algebr intsidentnosti nad konechnymi polyami”, Zap. nauchn. semin. POMI, 472 (2018), 120–144

[9] V. Lomonosov, P. Rosenthal, “The simplest proof of Burnside's theorem on matrix algebras”, Linear Algebra Appl., 383 (2004), 45–47 | DOI | MR | Zbl

[10] 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 | Zbl

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

[12] W. E. Longstaff, P. Rosenthal, “Generators of matrix incidence algebras”, Austral. J. Combin., 22 (2000), 117–121 | MR | Zbl

[13] W. E. Longstaff, P. Rosenthal, “On the lengths of irreducible pairs of complex matrices”, Proc. Amer. Math. Soc., 139:11 (2011), 3769–3777 | DOI | MR | Zbl

[14] O. V. Markova, “O dline algebry verkhnetreugolnykh matrits”, UMN, 60:5 (2005), 177–178 | MR | Zbl

[15] O. V. Markova, “Vychislenie dlin matrichnykh podalgebr spetsialnogo vida”, Fund. prikl. matem., 13:4 (2007), 165–197

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

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

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

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

[20] 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

[21] R. Pirs, Assotsiativnye algebry, Mir, 1986

[22] G.-C. Rota, “On the foundations of combinatorial theory, I. Theory of Möbius functions”, Z. Wahrscheinlichkeitsrechnung, 2 (1964), 340–368 | DOI | Zbl

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

[24] E. Spiegel, C. J. O'Donnel, Incidence Algebras, Marcel Dekker, 1997 | Zbl