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Counting fine gradings on matrix algebras and on classical simple Lie algebras. (English) Zbl 1286.17026

A grading of an algebra \({\mathcal A}\) by an abelian group \(G\) is a vector space decomposition \(\Gamma:{\mathcal A}=\bigoplus_{g\in G}{\mathcal A}_g\), such that \({\mathcal A}_g{\mathcal A}_h\subseteq {\mathcal A}_{gh}\) for all \(g,h\in G\). Given two gradings \(\Gamma:{\mathcal A}=\bigoplus_{g\in G}{\mathcal A}_g\) and \(\Gamma':{\mathcal A}=\bigoplus_{h\in H}{\mathcal A}'_h\), \(\Gamma'\) is said to be a refinement of \(\Gamma\) if any nonzero homogeneous component \({\mathcal A}_g\) of \(\Gamma\) is a direct sum of some homogeneous components of \(\Gamma'\). Fine gradings are those gradings which do not admit proper refinements. The fine gradings reveal important properties of the structure of an algebra and of its automorphism group scheme.
The recent classification results on gradings on simple Lie algebras and on (associative) matrix algebras over algebraically closed fields (see the monograph by the reviewer and the first author [Gradings on simple Lie algebras. Mathematical Surveys and Monographs 189. Providence, RI: AMS (2013; Zbl 1281.17001)]) allow the computation of the number of fine gradings (up to equivalence) on these algebras. This computation reduces to counting orbits of actions of certain finite groups. For matrix algebras, the number of fine gradings is expressed in terms of the partition function and the multiplicities of the prime factors of the degree of the algebra.
For \(X\in\{A,B,C,D\}\), the exact numbers of fine gradings \(N_X(r)\) on the simple Lie algebras of type \(X_r\) for \(r\leq 100\) are displayed carefully in several tables, which show the irregular behavior of these numbers (with the exception of type \(B\), where the situation is quite simple: \(N_B(r)=r+1\)). For instance, \(N_D(98)\simeq 5\times 10^7\), \(N_D(99)\simeq 1.5\times 10^4\) and \(N_D(100)\simeq 4.3\times 10^{10}\).
Also, the asymptotic behavior of the average \(\hat N_X(n)={1\over n}\sum_{j=1}^n N_X(j)\) is studied thoroughly. For matrix algebras this is derived from the known asymptotics of the number of abelian groups of order \(\leq n\). The average turns out to be of the form \(a\,\log(n)+O(1)\) for a constant \(a\) depending on the characteristic of the ground field. For types \(A\), \(C\) or \(D\), \(\hat N_X\) exhibits a growth faster than any polynomial but slower than any exponential. This is shown by relating it to the asymptotics of certain binomial coefficients.

MSC:

17B70 Graded Lie (super)algebras
16W50 Graded rings and modules (associative rings and algebras)
65A05 Tables in numerical analysis

Citations:

Zbl 1281.17001

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References:

[1] DOI: 10.1006/jabr.2000.8643 · Zbl 0988.16033 · doi:10.1006/jabr.2000.8643
[2] Y. Bahturin and M. Zaicev, Polynomial Identities and Combinatorial Methods, Lecture Notes in Pure and Applied Mathematics 235 (Dekker, New York, 2003) pp. 101–139. · Zbl 1053.16032
[3] DOI: 10.1007/978-1-4612-0731-3 · doi:10.1007/978-1-4612-0731-3
[4] DOI: 10.1016/j.laa.2006.01.017 · Zbl 1146.17027 · doi:10.1016/j.laa.2006.01.017
[5] Draper C., Rev. Mat. Iberoamericana 25 pp 841– (2009)
[6] DOI: 10.1016/j.jalgebra.2010.09.018 · Zbl 1213.17030 · doi:10.1016/j.jalgebra.2010.09.018
[7] DOI: 10.4171/RMI/691 · Zbl 1303.17019 · doi:10.4171/RMI/691
[8] DOI: 10.1016/j.jalgebra.2012.05.008 · Zbl 1267.16037 · doi:10.1016/j.jalgebra.2012.05.008
[9] Elduque A., Serdica Math. J. 38 pp 7– (2012)
[10] DOI: 10.1090/surv/189 · doi:10.1090/surv/189
[11] Erdös P., Acta Sci. Math. (Szeged) 7 pp 95– (1935)
[12] DOI: 10.1016/S0024-3795(97)10039-8 · Zbl 0939.17020 · doi:10.1016/S0024-3795(97)10039-8
[13] Heath-Brown D. R., Astérisque 200 pp 153– (1991)
[14] DOI: 10.1007/s10440-008-9386-0 · Zbl 1230.17025 · doi:10.1007/s10440-008-9386-0
[15] Liu H.-Q., Acta Arith. 59 pp 261– (1991)
[16] DOI: 10.1063/1.1383788 · Zbl 1032.17050 · doi:10.1063/1.1383788
[17] DOI: 10.1063/1.1508434 · Zbl 1060.17016 · doi:10.1063/1.1508434
[18] Svobodová M., SIGMA 4 pp 13– (2008)
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