Universität zu Köln

Genz, Volker, Koshevoy, Gleb and Schumann, Bea (2021). Combinatorics of canonical bases revisited: type A. Sel. Math.-New Ser., 27 (4). CHAM: SPRINGER INTERNATIONAL PUBLISHING AG. ISSN 1420-9020

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Abstract

We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type A. Using geometric objects called rhombic tilings we derive a Crossing Formula for the action of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein-Fomin-Zelevinsky we prove an enhanced version of the Anderson-Mirkovic conjecture for the crystal structure on MV polytopes. We establish a duality between Kashiwara's string and Lusztig's parametrization, revealing that each of them is controlled by the crystal structure of the other. We identify the potential functions of the unipotent radical of a Borel subgroup of SLn defined by Berenstein-Kazhdan and Gross-Hacking-Keel-Kontsevich, respectively, with a function arising from the crystal structure on Lusztig data.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Genz, Volker UNSPECIFIED UNSPECIFIED UNSPECIFIED Koshevoy, Gleb UNSPECIFIED UNSPECIFIED UNSPECIFIED Schumann, Bea UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-585736
DOI:	10.1007/s00029-021-00658-x
Journal or Publication Title:	Sel. Math.-New Ser.
Volume:	27
Number:	4
Date:	2021
Publisher:	SPRINGER INTERNATIONAL PUBLISHING AG
Place of Publication:	CHAM
ISSN:	1420-9020
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language MIRKOVIC-VILONEN POLYTOPES; CRYSTAL-STRUCTURE; STRING CONE; PARAMETRIZATIONS; SET Multiple languages Mathematics, Applied; Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/58573