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Mat. Zametki, 2018, Volume 103, Issue 3, Pages 346–363 (Mi mz11682)  

This article is cited in 2 scientific papers (total in 2 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.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-01-00192
This work was supported in part by the Russian Foundation for Basic Research under grant 17-01-00192.


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English version:
Mathematical Notes, 2018, 103:3, 357–371

Bibliographic databases:

UDC: 512.715+512.643+517.926.4
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

Citation in format AMSBIB
\by V.~M.~Buchstaber, S.~I.~Tertychnyi
\paper Representations of the Klein Group
Determined by Quadruples of Polynomials
Associated with the Double Confluent Heun Equation
\jour Mat. Zametki
\yr 2018
\vol 103
\issue 3
\pages 346--363
\jour Math. Notes
\yr 2018
\vol 103
\issue 3
\pages 357--371

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    This publication is cited in the following articles:
    1. Glutsyuk A.A., “On Constrictions of Phase-Lock Areas in Model of Overdamped Josephson Effect and Transition Matrix of the Double-Confluent Heun Equation”, J. Dyn. Control Syst., 25:3 (2019), 323–349  crossref  isi
    2. S. I. Tertychnyi, “Solution space monodromy of a special double confluent Heun equation and its applications”, Theoret. and Math. Phys., 201:1 (2019), 1426–1441  mathnet  crossref  crossref  adsnasa  isi  elib
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