Combinatorical and Algebraic Structures Seminar

Session details

Date: 30.3.2007
Speaker: Petr Ambrož, FJFI, České vysoké učení technické
Title: Matrices of morphisms preserving 3iet words
Abstract: (spoluautoři: Z. Masáková, E. Pelantová)
We study matrices of morphisms preserving the family of words coding 3-interval exchange transformations. It is well known (Berstel, Mignosi and S\'e\'ebold) that matrices of morphisms preserving sturmian words (i.e.\ words coding 2-interval exchange transforations with the maximal possible subword complexity) form the monoid \[ \{\boldsymbol{M}\in\mathbb{N}^{2\times 2}\;|\;\det\boldsymbol{M}=\pm1\} = \{ \boldsymbol{M}\in\mathbb{N}^{2\times 2}\;|\; \boldsymbol{M}\boldsymbol{E}\boldsymbol{M}^T = \pm\boldsymbol{E}\}\,, \] where $\boldsymbol{E} = (\!\begin{smallmatrix}0&1\\-1&0\end{smallmatrix})$. We prove that in the case of 3-interval exchange transformations the matrices preserving words coding these transformations and having the maximal possible subword complexity belong to the monoid \[ \{\boldsymbol{M}\in\mathbb{N}^{3\times 3}\;|\; \boldsymbol{M}\boldsymbol{E}\boldsymbol{M}^T = \pm\boldsymbol{E}\: \text{ and } \det\boldsymbol{M}=\pm 1\}\,, \] where $\boldsymbol{E} = \Big(\!\begin{smallmatrix}0&1&1\\-1&0&1\\-1&-1&0\end{smallmatrix}\Big)$.

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