Math Thesis Archive

ABSTRACT OF THE DISSERTATION

The Combinatorics of the Permutation Enumeration of Wreath Products between Cyclic and Symmetric Groups

Jennifer D. Wagner

Brenti introduced a homomorphism $\xi:\Lambda \rightarrow Q(x)$ defined on the the elementary symmetric functions by

$\xi(e_k(X)) = \frac{(1-x)^k}{k!}$

where $\Lambda$ is the space of homogeneous polynomials in an infinite number of variables $X = (x_1,x_2, \ldots )$ which are constant under all permutations of these variables. He proved that the homomorphism $\xi$ has the remarkable property that when it is applied to a homogeneous symmetric function $h_k(X)$, the result is the well-known Eulerian polynomial, which is also the generating function for the number of descents of a permutation. In addition, if $\xi$ is applied to a power symmetric function, the result is a generating function for another permutation statistic. Beck and Remmel used combinatorial interpretations of the transition matrices between bases of $\Lambda$ to give combinatorial proofs of these and other related identities, including $q$-analogs. In addition, they used these combinatorial methods to develop an analog of Brenti's permutation enumeration for $B_n$, the hyperoctahedral group consisting of signed permutations. In the dissertation we extend Brenti, Beck and Remmel's results to wreath products $C_kxS_n$ between cyclic and symmetric groups, which can be considered as groups of permutations signed with $k$^{th}$ roots of unity. The key steps in our extension to $C_kxS_n$ include the following.

We develop the representation theory of $C_kxS_n$ in an appropriate way, including the definition of a characteristic map from the class functions on $C_kxS_n$ to a space of symmetric functions, and an extension of lambda-ring notation to take into account the complex signs.

We determine combinatorial interpretations of the transition matrices between bases of the appropriate space.

We define appropriate statistics on the elements or $C_kxS_n$. Since there are a number of ways to define such statistics, we are forced to choose among several possible definitions.

We use combinatorial methods to define an analog of $\xi$, which when applied to certain basis elements, gives the desired generating functions on elements of $C_kxS_n$.

We give combinatorial proofs of the desired identities. The proofs include interpretation of sums in terms of combinatorial objects, and the performance of involutions on the objects.