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The cyclic sieving phenomenon: a survey
The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a
2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and
f(q) be a polynomial in q with nonnegative integer coefficients. Then the
triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we
have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g,
and w is a root of unity chosen to have the same order as g. It might seem
improbable that substituting a root of unity into a polynomial with integer
coefficients would have an enumerative meaning. But many instances of the
cyclic sieving phenomenon have now been found. Furthermore, the proofs that
this phenomenon hold often involve interesting and sometimes deep results from
representation theory. We will survey the current literature on cyclic sieving,
providing the necessary background about representations, Coxeter groups, and
other algebraic aspects as needed.Comment: 48 pages, 3 figures, the sedcond version contains numerous changes
suggested by colleagues and the referee. To appear in the London Mathematical
Society Lecture Note Series. The third version has a few smaller change
Invariant tensors and the cyclic sieving phenomenon
We construct a large class of examples of the cyclic sieving phenomenon by
expoiting the representation theory of semi-simple Lie algebras. Let be a
finite dimensional representation of a semi-simple Lie algebra and let be
the associated Kashiwara crystal. For , the triple which
exhibits the cyclic sieving phenomenon is constructed as follows: the set
is the set of isolated vertices in the crystal ; the map is a generalisation of promotion acting on standard tableaux of
rectangular shape and the polynomial is the fake degree of the Frobenius
character of a representation of related to the natural action
of on the subspace of invariant tensors in .
Taking to be the defining representation of gives the
cyclic sieving phenomenon for rectangular tableaux
Dihedral Sieving Phenomena
Cyclic sieving is a well-known phenomenon where certain interesting
polynomials, especially -analogues, have useful interpretations related to
actions and representations of the cyclic group. We propose a definition of
sieving for an arbitrary group and study it for the dihedral group
of order . This requires understanding the generators of the representation
ring of the dihedral group. For odd, we exhibit several instances of
dihedral sieving which involve the generalized Fibonomial coefficients,
recently studied by Amdeberhan, Chen, Moll, and Sagan. We also exhibit an
instance of dihedral sieving involving Garsia and Haiman's -Catalan
numbers.Comment: 10 page
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