Illinois Journal of Mathematics

$SL_k$-tilings of the plane

François Bergeron and Christophe Reutenauer

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We study properties of (bi-infinite) arrays having all adjacent $k\times k$ adjacent minors equal to one. If we further add the condition that all adjacent $(k-1)\times(k-1)$ 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

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

First available in Project Euclid: 9 March 2011

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Zentralblatt MATH identifier

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]


Bergeron, François; Reutenauer, Christophe. $SL_k$-tilings of the plane. Illinois J. Math. 54 (2010), no. 1, 263--300. doi:10.1215/ijm/1299679749.

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