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Samu Potka: Cyclic sieving

Time: Fri 2019-11-08 13.15 - 14.00

Location: Room 3418, Lindstedtsvägen 25, 4th floor, Department of Mathematics, KTH

Participating: Samu Potka

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Abstract: The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in 2004. The ingredients are a finite set X, a cyclic group C acting on X, and a polynomial f(q) with integer coefficients and satisfying f(1) = |X|. The triple (X, C, f(q)) is said to exhibit the cyclic sieving phenomenon if f(q) evaluated at certain roots of unity gives the number of elements of X fixed by powers of a generator of C. We will discuss this curious phenomenon and an example instance on circular Dyck paths (joint work with Per Alexandersson and Svante Linusson) which also provides an example on bijective/enumerative combinatorics.

In this talk we will study HS(M) by defining an action of the automorphism group aut(G) of G. This action lets us study the symmetries of G in terms of homology, which is susceptible to computations. By replacing G with any finite abstract simplicial complex we deepen the study of graph symmetries. Abstract simplicial complexes are generalisations of graphs, and there are natural such complexes associated to graphs which unveil more complicated symmetries.