Illinois Journal of Mathematics

SLk-tilings of the plane

François Bergeron and Christophe Reutenauer

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Abstract

We study properties of (bi-infinite) arrays having all adjacent k×k adjacent minors equal to one. If we further add the condition that all adjacent (k−1)×(k−1) minors be nonzero, then these arrays are necessarily of rank k. It follows that we can explicit construct all of them. Several nice properties are made apparent. In particular, we revisit, with this perspective, the notion of frieze patterns of Coxeter. This shed new light on their properties. A connexion is also established with the notion of T-systems of Statistical Physics.

Article information

Source
Illinois J. Math. Volume 54, Number 1 (2010), 263-300.

Dates
First available in Project Euclid: 9 March 2011

Permanent link to this document
http://projecteuclid.org/euclid.ijm/1299679749

Mathematical Reviews number (MathSciNet)
MR2776996

Subjects
Primary: 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14]
Secondary: 11C20: Matrices, determinants [See also 15B36] 05A05: Permutations, words, matrices 05E10: Combinatorial aspects of representation theory [See also 20C30]

Citation

Bergeron, François; Reutenauer, Christophe. SL k -tilings of the plane. Illinois J. Math. 54 (2010), no. 1, 263--300. http://projecteuclid.org/euclid.ijm/1299679749.


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