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Varga Kristóf: One-dimensional strong affine representations of the polycyclic monoids.

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Wednesday, 7. June 2023, 10:40 - 11:10
Let D = (d_0, d_1, . . . , d_(n-1)) be a complete system of residues modulo n and let us consider the functions f_i(x) = nx + d_i on the integers. These functions give rise to a so-called one-dimensional strong affine representation of the polycyclic monoid P_n. This representation can be visualized by an edge-labeled directed graph: we draw an arrow with label i from x to f_i(x) Every connected component of such a graph contains exactly one cycle that can be uniquely identified by the word obtained by recording the labels along the edges of the cycle. It can be shown that the set of words corresponding to the cycles determine the edge-labeled graph up to isomorphism as well as the representation up to equivalence.

Our main problem is describing the set of words corresponding to a one-dimensional strong affine representation of P_n. We give a complete characterization for the sets of words induced by arithmetic sequences starting with zero, i.e., if D = (0, h, 2h, . . . ,(n-1)h) for some positive integer h relatively prime to n. In the case of the bicyclic monoid P_2, this covers all one-dimensional strong affine representations.


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