##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Christophe Reutenauer

#### Universite du Quebec `a Montreal

## $Sl_2$-Tilings

##### Abstract:

Call SL2-tiling a filling of the discrete plane by elements of a ring (the coefficients) in such a way that each connected 2 by 2 submatrix has determinant 1. Similar objects have been studied by Coxeter and Conway; they call them frieze-patterns. Given a bi-infinite word on {x, y}, interpreted as a path in the discrete plane, called the frontier, put 1s at its vertices. Then one may uniquely complete this picture to an SL2-tiling; it turns out that the coefficients of the tiling are all positive integers; we prove this by giving explicit matrix product formulas for these coefficients. Our constructions are motivated by the so-called "frises", associated to acyclic digraphs. In a joint work with I. Assem and D. Smith, we showed that the sequences of the frise all satisfy a linear recursion if and only if the digraph is a Dynkin diagram, or an affine diagram, with an acyclic orientation.

Host: Adriano Garsia

### February 23, 2010

### 3:00 PM

### AP&M 7321

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