Calculating entries of unitary $SL_3$-friezes

Publikation: Arbeitspapier/Preprint › Preprint

Autorschaft

Organisationseinheiten

Forschungs-netzwerk anzeigen

Details

OriginalspracheEnglisch
PublikationsstatusElektronisch veröffentlicht (E-Pub) - 15 Apr. 2024

Abstract

In this article we consider tame $ SL_3 $-friezes that arise by specializing a cluster of Pl\"ucker variables in the coordinate ring of the Grassmannian $ \mathscr{G}(3,n) $ to $ 1 $. We show how to calculate arbitrary entries of such friezes from the cluster in question. Let $ \mathscr{F} $ be such a cluster. We study the set $ \mathscr{F}_x $ of cluster variables in $ \mathscr{F} $ that share a given index $ x $ and derive a structure Theorem for $ \mathscr{F}_x $. These sets prove central to calculating the first and last non-trivial rows of the frieze. After that, simple recursive formulas can be used to calculate all remaining entries.

Fachgebiet (basierend auf ÖFOS 2012)

Zitieren

Calculating entries of unitary $SL_3$-friezes. / Surmann, Lucas.
2024.

Publikation: Arbeitspapier/Preprint › Preprint

Surmann, L. (2024). Calculating entries of unitary $SL_3$-friezes. Vorabveröffentlichung online.
Surmann L. Calculating entries of unitary $SL_3$-friezes. 2024 Apr 15. Epub 2024 Apr 15.
Download
@techreport{b536c07fda224e379f3d5e45aa6401e1,
title = "Calculating entries of unitary $SL_3$-friezes",
abstract = " In this article we consider tame $ SL_3 $-friezes that arise by specializing a cluster of Pl\{"}ucker variables in the coordinate ring of the Grassmannian $ \mathscr{G}(3,n) $ to $ 1 $. We show how to calculate arbitrary entries of such friezes from the cluster in question. Let $ \mathscr{F} $ be such a cluster. We study the set $ \mathscr{F}_x $ of cluster variables in $ \mathscr{F} $ that share a given index $ x $ and derive a structure Theorem for $ \mathscr{F}_x $. These sets prove central to calculating the first and last non-trivial rows of the frieze. After that, simple recursive formulas can be used to calculate all remaining entries. ",
keywords = "math.CO, 13F60, 14M15, 05C99, 05E99, 51M20",
author = "Lucas Surmann",
year = "2024",
month = apr,
day = "15",
language = "English",
type = "WorkingPaper",

}

Download

TY - UNPB

T1 - Calculating entries of unitary $SL_3$-friezes

AU - Surmann, Lucas

PY - 2024/4/15

Y1 - 2024/4/15

N2 - In this article we consider tame $ SL_3 $-friezes that arise by specializing a cluster of Pl\"ucker variables in the coordinate ring of the Grassmannian $ \mathscr{G}(3,n) $ to $ 1 $. We show how to calculate arbitrary entries of such friezes from the cluster in question. Let $ \mathscr{F} $ be such a cluster. We study the set $ \mathscr{F}_x $ of cluster variables in $ \mathscr{F} $ that share a given index $ x $ and derive a structure Theorem for $ \mathscr{F}_x $. These sets prove central to calculating the first and last non-trivial rows of the frieze. After that, simple recursive formulas can be used to calculate all remaining entries.

AB - In this article we consider tame $ SL_3 $-friezes that arise by specializing a cluster of Pl\"ucker variables in the coordinate ring of the Grassmannian $ \mathscr{G}(3,n) $ to $ 1 $. We show how to calculate arbitrary entries of such friezes from the cluster in question. Let $ \mathscr{F} $ be such a cluster. We study the set $ \mathscr{F}_x $ of cluster variables in $ \mathscr{F} $ that share a given index $ x $ and derive a structure Theorem for $ \mathscr{F}_x $. These sets prove central to calculating the first and last non-trivial rows of the frieze. After that, simple recursive formulas can be used to calculate all remaining entries.

KW - math.CO

KW - 13F60, 14M15, 05C99, 05E99, 51M20

M3 - Preprint

BT - Calculating entries of unitary $SL_3$-friezes

ER -