RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

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

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.

DOI: https://doi.org/10.4213/mzm11682

Full text: PDF file (654 kB)
First page: PDF file
References: PDF file   HTML file

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

Bibliographic databases:

Document Type: Article
UDC: 512.715+512.643+517.926.4
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
\Bibitem{BucTer18} \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 \mathnet{http://mi.mathnet.ru/mz11682} \crossref{https://doi.org/10.4213/mzm11682} \elib{http://elibrary.ru/item.asp?id=32641318} \transl \jour Math. Notes \yr 2018 \vol 103 \issue 3 \pages 357--371 \crossref{https://doi.org/10.1134/S0001434618030033} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000430553100003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85046353382}