The canonical representation of the Klein group $K_4=\mathbb Z_2\oplus\mathbb Z_2$ on the space $\mathbb C^*=\mathbb C\setminus\{0\}$ induces a representation of this group on the ring $\mathscr L= C[z,z^{-1}]$, $z\in\mathbb C^*$, of Laurent polynomials and, as a consequence, a representation of the group $K_4$ on the automorphism group of the group $G=GL(4,\mathscr L)$ by means of the elementwise action. The semidirect product $\widehat G= G\ltimes K_4$ is considered together with a realization of the group $\widehat G$ as a group of semilinear automorphisms of the free $4$-dimensional $\mathscr L$-module $\mathscr M^4$. A three-parameter family of representations $\mathfrak R$ of $K_4$ in the group $\widehat G$ and a three-parameter family of elements $\mathfrak X\in\mathscr M^4$ with polynomial coordinates of degrees $2(\ell-1)$, $2\ell$, $2(\ell-1)$, and $2\ell$, where $\ell$ is an arbitrary positive integer (one of the three parameters), are constructed. It is shown that, for any given family of parameters, the vector $\mathfrak X$ is a fixed point of the corresponding representation $\mathfrak R$. An algorithm for calculating the polynomials that are the components of $\mathfrak X$ was obtained in a previous paper of the authors, in which it was proved that these polynomials give explicit formulas for automorphisms of the solution space of the doubly confluent Heun equation.
@article{MZM_2018_103_3_a2,
author = {V. M. Buchstaber and S. I. Tertychnyi},
title = {Representations of the {Klein} {Group}},
journal = {Matemati\v{c}eskie zametki},
pages = {346--363},
year = {2018},
volume = {103},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_2018_103_3_a2/}
}
TY - JOUR
AU - V. M. Buchstaber
AU - S. I. Tertychnyi
TI - Representations of the Klein Group
JO - Matematičeskie zametki
PY - 2018
SP - 346
EP - 363
VL - 103
IS - 3
UR - http://geodesic.mathdoc.fr/item/MZM_2018_103_3_a2/
LA - ru
ID - MZM_2018_103_3_a2
ER -
%0 Journal Article
%A V. M. Buchstaber
%A S. I. Tertychnyi
%T Representations of the Klein Group
%J Matematičeskie zametki
%D 2018
%P 346-363
%V 103
%N 3
%U http://geodesic.mathdoc.fr/item/MZM_2018_103_3_a2/
%G ru
%F MZM_2018_103_3_a2
V. M. Buchstaber; S. I. Tertychnyi. Representations of the Klein Group. Matematičeskie zametki, Tome 103 (2018) no. 3, pp. 346-363. http://geodesic.mathdoc.fr/item/MZM_2018_103_3_a2/
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