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

This article is cited in 3 scientific papers (total in 3 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. A. A. Glutsyuk, “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  mathscinet  adsnasa  isi  elib
    3. S. I. Tertychniy, “Symmetries of the space of solutions to special double confluent Heun equations of integer order”, J. Math. Phys., 60:10 (2019), 103501  crossref  mathscinet  zmath  isi
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