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 Zap. Nauchn. Sem. POMI, 2018, Volume 469, Pages 64–95 (Mi znsl6606)

The unimodularity of the induced toric tilings

V. G. Zhuravlevab

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: Induced tilings $\mathcal T=\mathcal T|_\mathrm{Kr}$ of the $d$-dimensional torus $\mathbb T^d$, generated by the embedded karyon $\mathrm{Kr}$, are considered. The operations of differentiation are defined $\sigma\colon\mathcal T\to\mathcal T^\sigma$, as a result we get again induced partitions $\mathcal T^\sigma=\mathcal T|_{\mathrm{Kr}^\sigma}$ of the same torus $\mathbb T^d$, generated by the derived karyon $\mathrm{Kr}^\sigma$. In the language of the karyons $\mathrm {Kr}$ the derivations of $\sigma$ reduce to a combination of geometric transformations of the space $\mathbb R^d$. It is proved that if the karyon $\mathrm{Kr}$ is unimodular, then it generates an induced tiling $\mathcal T=\mathcal T|_\mathrm{Kr}$ and the derivative karyon $\mathrm{Kr}^\sigma$ is unimodular again. So there exists the corresponding derivative tiling $\mathcal T^\sigma=\mathcal T|_{\mathrm {Kr}^\sigma}$. Using unimodular karyons one can build an infinite family of induced tilings $\mathcal T=\mathcal T(\alpha,\mathrm{Kr}_*)$ depending on a shift vector $\alpha$ of the torus $\mathbb T^d$ and the initial karyon $\mathrm{Kr}_*$. Two algorithms are presented for constructing such unimodular karyons of $\mathrm{Kr}_*$.

Key words and phrases: shift vector, induced tilings, induced toric tilings, oblique shift, derived karyon, exchange transformation of a torus, derived tilings, contraction along a straight line.

 Funding Agency Grant Number Russian Science Foundation 14-11-00433

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English version:
Journal of Mathematical Sciences (New York), 2019, 242:4, 509–530

Bibliographic databases:

UDC: 511.3

Citation: V. G. Zhuravlev, “The unimodularity of the induced toric tilings”, Algebra and number theory. Part 1, Zap. Nauchn. Sem. POMI, 469, POMI, St. Petersburg, 2018, 64–95; J. Math. Sci. (N. Y.), 242:4 (2019), 509–530

Citation in format AMSBIB
\Bibitem{Zhu18} \by V.~G.~Zhuravlev \paper The unimodularity of the induced toric tilings \inbook Algebra and number theory. Part~1 \serial Zap. Nauchn. Sem. POMI \yr 2018 \vol 469 \pages 64--95 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6606} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3885096} \transl \jour J. Math. Sci. (N. Y.) \yr 2019 \vol 242 \issue 4 \pages 509--530 \crossref{https://doi.org/10.1007/s10958-019-04493-6} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85072101016} 

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