229 research outputs found
Refined Catalan and Narayana cyclic sieving
We prove several new instances of the cyclic sieving phenomenon (CSP) on
Catalan objects of type A and type B. Moreover, we refine many of the known
instances of the CSP on Catalan objects. For example, we consider
triangulations refined by the number of "ears", non-crossing matchings with a
fixed number of short edges, and non-crossing configurations with a fixed
number of loops and edges.Comment: Updated version, minor change
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
The toggle group, homomesy, and the Razumov-Stroganov correspondence
The Razumov-Stroganov correspondence, an important link between statistical
physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello,
relates the ground state eigenvector of the O(1) dense loop model on a
semi-infinite cylinder to a refined enumeration of fully-packed loops, which
are in bijection with alternating sign matrices. This paper reformulates a key
component of this proof in terms of posets, the toggle group, and homomesy, and
proves two new homomesy results on general posets which we hope will have
broader implications.Comment: 14 pages, 10 figures, final versio
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