Matrix identities with forms. (English) Zbl 1294.16016
Let \(R\) be the algebra generated by the generic \(n\times n\) matrices \(X_k\), \(k\geq 1\), over an infinite field. Denote by \(\sigma_t(A)\) the coefficients of the characteristic polynomial of \(A\), it is well known that the algebra of matrix \(GL_n\)-invariants \(R^{GL_n}\) is generated by the \(\sigma_t(A)\) where \(A\) are monomials in the \(X_k\). The algebra of concomitants (also called the algebra of \(n\times n\) generic matrices with forms), is generated by the \(X_k\) and \(fE\) where \(E\) is the identity matrix and \(f\in R^{GL_n}\). The identities with forms generalize the well known and widely studied trace identities. The paper under review studies the ideal \(T_n\) of identities with forms for the \(n\times n\) matrices. The author proves that \(T_n\) is finitely generated (as a \(T\)-ideal). Then this is transferred to the ideal of identities with forms for the algebra generated by generic matrices together with their transposes. As consequences the author deduces that the algebras of matrix \(GL_n\)- and \(O_n\)-invariants are generated by the known free relations and partial linearizations of the coefficients \(\sigma_t\) of the characteristic polynomial.
Reviewer: Plamen Koshlukov (Campinas)
MSC:
| 16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |
| 16R30 | Trace rings and invariant theory (associative rings and algebras) |
| 13A50 | Actions of groups on commutative rings; invariant theory |
Keywords:
algebras of matrices; identities with forms; generic matrices; characteristic polynomials; finitely based T-ideals; algebras of matrix invariants; trace identitiesReferences:
| [1] | Amitsur, S. A., On the characteristic polynomial of a sum of matrices, Linear Mult. Algebra, 8, 177-182 (1980) · Zbl 0429.15006 |
| [2] | Donkin, S., Invariants of several matrices, Invent. Math., 110, 389-401 (1992) · Zbl 0826.20036 |
| [3] | Donkin, S., Invariant functions on matrices, Math. Proc. Cambridge Philos. Soc., 113, 23-43 (1993) · Zbl 0826.20015 |
| [4] | Domokos, M.; Kuzmin, S. G.; Zubkov, A. N., Rings of matrix invariants in positive characteristic, J. Pure Appl. Algebra, 176, 61-80 (2002) · Zbl 1024.16014 |
| [5] | Domokos, M., Vector invariants of a class pseudo-reflection groups and multisymmetric syzygies, J. Lie Theory, 19, 507-525 (2009) · Zbl 1221.13007 |
| [6] | Koshlukov, P., Basis of the identities of the matrix algebra of order two over a field of characteristic \(p \neq 2\), J. Algebra, 241, 410-434 (2001) · Zbl 0988.16015 |
| [7] | Colombo, J.; Koshlukov, P., Central polynomials in the matrix algebra of order two, Linear Algebra Appl., 377, 53-67 (2004) · Zbl 1044.16016 |
| [8] | Colombo, J.; Koshlukov, P., Identities with involution for the matrix algebra of order two in characteristic \(p\), Israel J. Math., 146, 337-355 (2005) · Zbl 1076.16022 |
| [9] | Lopatin, A. A., Free relations for matrix invariants in modular case, J. Pure Appl. Algebra, 216, 427-437 (2012) · Zbl 1235.13004 |
| [10] | Lopatin, A. A., Relations between \(O ( n )\)-invariants of several matrices, Algebra Repr. Theory, 15, 855-882 (2012) · Zbl 1272.16024 |
| [11] | Procesi, C., The invariant theory of \(n \times n\) matrices, Adv. Math., 19, 306-381 (1976) · Zbl 0331.15021 |
| [12] | Razmyslov, Yu. P., The existence of a finite basis for the identities of the matrix algebra of order two over a field of characteristic zero, Algebra Logika, 12, 1, 83-113 (1973), (in Russian); Algebra Logic 12 (1973), 47-63 (Engl. transl.) · Zbl 0282.17003 |
| [13] | Razmyslov, Yu. P., Trace identities of full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR Ser. Mat., 38, 4, 723-756 (1974), (in Russian) · Zbl 0311.16016 |
| [14] | Samoilov, L. M., On the nilindex of the radical of a relatively free associative algebra, Mat. Zametki, 82, 4, 583-592 (2007), (in Russian); Math. Notes 82 (4) (2007), 522-530 (Engl. transl.) · Zbl 1153.16021 |
| [15] | Samoilov, L. M., On the radical of a relatively free associative algebra over fields of positive characteristic, Mat. Sb., 199, 5, 81-126 (2008), (in Russian); Sb. Math. 199 (5) (2008), 707-753 (Engl. transl.) · Zbl 1168.16010 |
| [16] | Sibirskii, K. S., Algebraic invariants of a system of matrices, Sibirsk. Mat. Zh., 9, 1, 152-164 (1968), (in Russian) · Zbl 0273.15024 |
| [17] | Zubkov, A. N., On a generalization of the Razmyslov-Procesi theorem, Algebra Logic, 35, 4, 241-254 (1996) · Zbl 0941.16012 |
| [18] | Zubkov, A. N., Invariants of an adjoint action of classical groups, Algebra Logic, 38, 5, 299-318 (1999) · Zbl 0940.14033 |
| [19] | Zubkov, A. N., Invariants of mixed representations of quivers II: defining relations and applications, J. Algebra Appl., 4, 3, 287-312 (2005) · Zbl 1082.16023 |
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