Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices

Houzhi He; Huabo Xu

doi:10.3842/umzh.v76i12.7976

Houzhi He School of Science, Beijing University of Civil Engineering and Architecture, China
Huabo Xu School of Science, Beijing University of Civil Engineering and Architecture, China

DOI: https://doi.org/10.3842/umzh.v76i12.7976

Keywords: Invariant subring; $\sigma$-$\varphi$ permutation matrix; Cellular algebra

Abstract

UDC 512.5

Let $n$ be a positive integer, let $R$ be a (unitary associative) ring, and let $M_n(R)$ be the ring of all $n$ by $n$ matrices over $R.$ For a permutation $\sigma$ in the symmetry group $\Sigma_n$ and a ring automorphism $\varphi$ of $R,$ we introduce the definition of $\sigma$-$\varphi$ permutation matrices. The set $B_n(\sigma, \varphi, R)$ of all $\sigma$-$\varphi$ permutation matrices is proved to be a subring of $M_n(R).$ We show that the extension $B_n(\sigma, \varphi, R) \subseteq M_n(R)$ is a separable Frobenius extension. Moreover, if $R$ is a commutative cellular algebra over the invariant subring $R^\varphi$ of $R,$ then $B_n(\sigma, \varphi, R)$ is also a cellular algebra over $R^\varphi.$

References

A. Cantoni, P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra and Appl., 13, № 3, 275–288 (1976). DOI: https://doi.org/10.1016/0024-3795(76)90101-4

B. Gbrtzen, Finitistic dimensions of ring extensions, Comm. Algebra, 10, № 9, 993–1001 (1982). DOI: https://doi.org/10.1080/00927878208822761

I. J. Good, The inverse of a centrosymmetric matrix, Technometrics, 12, № 4, 925–928 (1970). DOI: https://doi.org/10.1080/00401706.1970.10488743

J. J. Graham, G. I. Lehrer, Cellular algebras, Invent. Math., 123, 1–34 (1996). DOI: https://doi.org/10.1007/BF01232365

J. Joseph, Advanced modern algebra, Prentice Hall, New Jersey (2003).

L. Kadison, New examples of Frobenius extensions, Univ. Lecture Ser., 14, Amer. Math. Soc., Providence, RI (1999). DOI: https://doi.org/10.1090/ulect/014

S. Köenig, C. C. Xi, Affine cellular algebras, Adv. Math., 229, № 1, 139–182 (2012). DOI: https://doi.org/10.1016/j.aim.2011.08.010

X. G. Li, C. C. Xi, Derived and stable equivalences of centralizer matrix algebras; arXiv (2023): 2312.08794.

J. R. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors, Amer. Math. Monthly, 92, № 10, 711–717 (1985). DOI: https://doi.org/10.1080/00029890.1985.11971719

C. C. Xi, S. J. Yin, Cellularity of centrosymmetric matrix algebras and Frobenius extensions, Linear Algebra and Appl., 590, 317–329 (2020). DOI: https://doi.org/10.1016/j.laa.2020.01.002

C. C. Xi, J. B. Zhang, Structure of centralizer matrix algebras, Linear Algebra and Appl., 622, 215–249 (2021). DOI: https://doi.org/10.1016/j.laa.2021.03.034

C. C. Xi, J. B. Zhang, Centralizer matrix algebras and symmetric polynomials of partitions, J. Algebra, 609, 688–717 (2022). DOI: https://doi.org/10.1016/j.jalgebra.2022.06.037

H. B. Xu, Derived equivalences and higher $K$-groups of a class of KLR algebras, Comm. Algebra, 48, № 12, 5360–5371 (2020). DOI: https://doi.org/10.1080/00927872.2020.1788047