Centralizer matrix algebras and symmetric polynomials of partitions. (English) Zbl 1509.16041
The main object under study, in this paper, is the centralizer algebra
\[
S_n(C, k)=\{a\in \operatorname{M}_n(k)\colon ac=ca \ \text{ for all } c\in C \}
\]
of a non-empty set \(C\) of matrices in \(\operatorname{M}_n(k)\) over a field \(k\).
Given an algebraically closed field \(K\), the authors show, in [C. Xi and J. Zhang, Linear Algebra Appl. 622, 215–249 (2021; Zbl 1473.16024)], that \(S_n(\{c\}, K)\) is a cellular algebra (in the sense of [J. J. Graham and G. I. Lehrer, Invent. Math. 123, No. 1, 1–34 (1996; Zbl 0853.20029)]) and the extension \(S_n({c}, K)\subset M_n(K)\) is a Frobenius extension.
In this paper, the authors prove that a centralizer algebra \(S_n(\{c\}, k)\) of a matrix \(c\in M_n(k)\), over a field \(k\), is Frobenius-finite (in the sense of [W. Hu and C. Xi, Rev. Mat. Iberoam. 34, No. 1, 59–110 (2018; Zbl 1431.16012)]), \(1\)-minimal Auslander-Gorenstein (in the sense of [O. Iyama and Ø. Solberg, Adv. Math. 326, 200–240 (2018; Zbl 1432.16012)]), and gendo-symmetric (in the sense of [M. Fang and S. Koenig, Trans. Am. Math. Soc. 368, No. 7, 5037–5055 (2016; Zbl 1409.16006)]). Further, they prove that \(S_n(c, k)\subset M_n(k)\) is a separable Frobenius extension.
Also, the authors give a characterisation for a centralizer algebra to be semi-simple over a field of positive characteristic in terms of the cycle type of the permutation. Over algebraically closed fields, the authors characterise when two semi-simple centraliser algebras of permutation matrices can be isomorphic and Morita equivalent in terms of combinatoric data of cycle types of the underlying permutations. The latter is reduced to the computation of the number of the distinct eigenvalues of the involved permutation matrices while the former is reduced to the study of an equivalence relation in the set of partitions of a natural number \(n\).
Given an algebraically closed field \(K\), the authors show, in [C. Xi and J. Zhang, Linear Algebra Appl. 622, 215–249 (2021; Zbl 1473.16024)], that \(S_n(\{c\}, K)\) is a cellular algebra (in the sense of [J. J. Graham and G. I. Lehrer, Invent. Math. 123, No. 1, 1–34 (1996; Zbl 0853.20029)]) and the extension \(S_n({c}, K)\subset M_n(K)\) is a Frobenius extension.
In this paper, the authors prove that a centralizer algebra \(S_n(\{c\}, k)\) of a matrix \(c\in M_n(k)\), over a field \(k\), is Frobenius-finite (in the sense of [W. Hu and C. Xi, Rev. Mat. Iberoam. 34, No. 1, 59–110 (2018; Zbl 1431.16012)]), \(1\)-minimal Auslander-Gorenstein (in the sense of [O. Iyama and Ø. Solberg, Adv. Math. 326, 200–240 (2018; Zbl 1432.16012)]), and gendo-symmetric (in the sense of [M. Fang and S. Koenig, Trans. Am. Math. Soc. 368, No. 7, 5037–5055 (2016; Zbl 1409.16006)]). Further, they prove that \(S_n(c, k)\subset M_n(k)\) is a separable Frobenius extension.
Also, the authors give a characterisation for a centralizer algebra to be semi-simple over a field of positive characteristic in terms of the cycle type of the permutation. Over algebraically closed fields, the authors characterise when two semi-simple centraliser algebras of permutation matrices can be isomorphic and Morita equivalent in terms of combinatoric data of cycle types of the underlying permutations. The latter is reduced to the computation of the number of the distinct eigenvalues of the involved permutation matrices while the former is reduced to the study of an equivalence relation in the set of partitions of a natural number \(n\).
Reviewer: Tiago Cruz (Bonn)
MSC:
| 16W22 | Actions of groups and semigroups; invariant theory (associative rings and algebras) |
| 16S50 | Endomorphism rings; matrix rings |
| 05A17 | Combinatorial aspects of partitions of integers |
| 15A30 | Algebraic systems of matrices |
| 11C08 | Polynomials in number theory |
| 16K99 | Division rings and semisimple Artin rings |
Keywords:
centralizer algebras; Frobenius extension; Gorenstein algebra; greatest common divisor; invariant matrix algebra; partition; symmetric polynomialReferences:
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