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This article is cited in 4 scientific papers (total in 4 papers)
Representations of the Klein Group
Determined by Quadruples of Polynomials
Associated with the Double Confluent Heun Equation
V. M. Buchstabera, S. I. Tertychnyib a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b All-Russian Scientific Research Institute of Physical-Technical and Radiotechnical Measurements, Mendeleevo, Moscow region
Abstract:
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.
Keywords:
semilinear mappings, ring of Laurent polynomials, representations of the Klein group, doubly confluent Heun equation.
Received: 18.08.2017 Revised: 08.09.2017
Citation:
V. M. Buchstaber, S. I. Tertychnyi, “Representations of the Klein Group
Determined by Quadruples of Polynomials
Associated with the Double Confluent Heun Equation”, Mat. Zametki, 103:3 (2018), 346–363; Math. Notes, 103:3 (2018), 357–371
Linking options:
https://www.mathnet.ru/eng/mzm11682https://doi.org/10.4213/mzm11682 https://www.mathnet.ru/eng/mzm/v103/i3/p346
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