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$\mathrm{SL}_2$-tilings, infinite triangulations, and continuous cluster categories

Peter Jørgensen (Newcastle University)
Onsdag, 7 november, 2018, at 15:15-16:15, in Aud. D4 (1531-219)
An $\mathrm{SL}_2$-tiling is an infinite grid of positive integers such that each adjacent $2x2$-submatrix has determinant $1$.  These tilings were introduced by Assem, Reutenauer, and Smith for combinatorial purposes.

We will show how each $\mathrm{SL}_2$-tiling can be obtained, by a procedure called Conway-Coxeter counting, from an infinite triangulation of the disc with four accumulation points.  We will see how properties of the tilings are reflected in the triangulations.  For instance, the entry $1$ of a tiling always gives an arc of the corresponding triangulation, and $1$ can occur infinitely often in a tiling.  On the other hand, if a tiling has no entry equal to $1$, then the minimal entry of the tiling is unique, and the minimal entry can be seen as a more complex pattern in the triangulation.

The infinite triangulations also give rise to cluster tilting subcategories of a certain cluster category with infinite clusters related to the continuous cluster categories of Igusa and Todorov.  We will show how $\mathrm{SL}_2$-tilings can be viewed as the corresponding cluster characters.

This is joint work with Christine Bessenrodt and Thorsten Holm.
Organiseret af: math