A formula is obtained for the dimension a(n; m) of the space of degree m invariants of the regular representation of the n-th order cyclic group. This formula implies in particular that a(n; m) = a(m; n). Moreover, the automorphism group of the multiplicative semigroup generated by invariant monomials of the cyclic group in that representation is determined.

We completely characterize possible indices of quasi-cyclic subcodes in a cyclic code for a very broad class of cyclic codes. We present enumeration results for quasi-cyclic subcodes of a fixed index and show that the problem of enumeration is equivalent to enumeration of certain vector subspaces in finite fields. In particular, we present enumeration results for quasi-cyclic subcodes of the si...

We construct a large class of examples of the cyclic sieving phenomenon by exploiting the representation theory of semi-simple Lie algebras. Let M be a finite dimensional representation of a semi-simple Lie algebra and let B be the associated Kashiwara crystal. For r > 0, the triple (X, c, P ) which exhibits the cyclic sieving phenomenon is constructed as follows: the set X is the set of isolat...

We show that any discrete opinion pooling procedure with positive weights can be asymptotically approximated by DeGroot’s procedure whose communication digraph is a Hamiltonian cycle with loops. In this cycle, the weight of each arc (which is not a loop) is inversely proportional to the influence of the agent the arc leads to.

A representation of information on cyclic events has been proposed which is advantageous for computing environments where a distributed set of Receivers reacts to cyclic events generated by distributed sources. In such scenario no immanent central information repository exist on event timing or volume. Receivers are able to learn the event cycles without communicating with each other, merely on...

We prove a collection of conjectures of D. White [37], as well as some related conjectures of Abuzzahab-Korson-Li-Meyer [1] and of Reiner and White [21], [37], regarding the cyclic sieving phenomenon of Reiner, Stanton, and White [22] as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of ...