Computing Isotypic Projections with the Lanczos Iteration

When the isotypic subspaces of a representation are viewed as the eigenspaces of a symmetric linear transformation, isotypic projections may be achieved as eigenspace projections and computed using the Lanczos iteration. In this paper, we show how this approach gives rise to an efficient isotypic projection method for permutation representations of distance transitive graphs and the symmetric group

Generalized Spectral Analysis for Large Sets of Approval Voting Data

Generalized Spectral analysis of approval voting data uses representation theory and the symmetry of the data to project the approval voting data into orthogonal and interpretable subspaces. Unfortunately, as the number of voters grows, the data space becomes prohibitively large to compute the decomposition of the data vector. To attack these large data sets we develop a method to partition the data set into equivalence classes, in order to drastically reduce the size of the space while retaining the necessary characteristics of the data set. We also make progress on the needed statistical tools to explain the results of the spectral analysis. The standard spectral analysis will be demonstrated, and our partitioning technique is applied to U.S. Senate roll call data

Separating Sets for the Alternating and Dihedral Groups

This thesis presents the results of an investigation into the representation theory of the alternating and dihedral groups and explores how their irreducible representations can be distinguished with the use of class sums

Spectral Analysis of the Supreme Court

The focus of this paper is the linear algebraic framework in which the spectral analysis of voting data like that above is carried out. As we will show, this framework can be used to pinpoint voting coalitions in small voting bodies like the United States Supreme Court. Our goal is to show how simple ideas from linear algebra can come together to say something interesting about voting. And what could be more simple than where our story begins— with counting

Markov Bases for Noncommutative Harmonic Analysis of Partially Ranked Data

Given the result $v_0$ of a survey and a nested collection of summary statistics that could be used to describe that result, it is natural to ask which of these summary statistics best describe $v_0$. In 1998 Diaconis and Sturmfels presented an approach for determining the conditional significance of a higher order statistic, after sampling a space conditioned on the value of a lower order statistic. Their approach involves the computation of a Markov basis, followed by the use of a Markov process with stationary hypergeometric distribution to generate a sample.This technique for data analysis has become an accepted tool of algebraic statistics, particularly for the study of fully ranked data. In this thesis, we explore the extension of this technique for data analysis to the study of partially ranked data, focusing on data from surveys in which participants are asked to identify their top $k$ choices of $n$ items. Before we move on to our own data analysis, though, we present a thorough discussion of the Diaconis–Sturmfels algorithm and its use in data analysis. In this discussion, we attempt to collect together all of the background on Markov bases, Markov proceses, Gröbner bases, implicitization theory, and elimination theory, that is necessary for a full understanding of this approach to data analysis

Simplified preferences, voting, and the power of combination.

In this paper we interpreted the decision to vote for a particular party as a process of delegation to decision makers having a simplified system of preferences. Each person in a population votes for the political party that place priority on one or more issues that they consider important. Moreover, on the basis of a survey on preferences of population, we have simulated a delegation procedure which chart the selection process of a particular party. Finally, making use of noncommutative harmonic analysis, we decomposed the delegation function, and isolated the effect of a particular affinity, or a combination of either the pair of items that characterize a party. We used noncommutative harmonic analysis as an application of some results obtained by Michael E. Orrison and Brian L. %%@ Lawson in relation to spectral analysis applied in voting in political committees.

Decimation-in-Frequency Fast Fourier Transforms for the Symmetric Group

In this thesis, we present a new class of algorithms that determine fast Fourier transforms for a given finite group G. These algorithms use eigenspace projections determined by a chain of subgroups of G, and rely on a path algebraic approach to the representation theory of finite groups developed by Ram (26). Applying this framework to the symmetric group, Sn, yields a class of fast Fourier transforms that we conjecture to run in O(n2n!) time. We also discuss several future directions for this research

Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II

We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms %\cite{sovi}, we make explicit use of the path algebra connection to the construction of Gel'fand-Tsetlin bases and work in the setting of quivers. We relate this framework to the construction of a {\em configuration space} derived from a Bratteli diagram. In this setting the complexity of an algorithm for computing a Fourier transform reduces to the calculation of the dimension of the associated configuration space. Our methods give improved upper bounds for computing the Fourier transform for the general linear groups over finite fields, the classical Weyl groups, and homogeneous spaces of finite groups, while also recovering the best known algorithms for the symmetric group and compact Lie groups.Comment: 53 pages, 5 appendices, 34 figure