Michael E. Orrison

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.

Voting, the Symmetric Group, and Representation Theory

We show how voting may be viewed naturally from an algebraic perspective by viewing voting profiles as elements of certain well-studied

Quantum enhancements and biquandle brackets

\mathbb{Q}S_n

Linear rank tests of uniformity: understanding inconsistent outcomes and the construction of new tests

-modules. By using only a handful of simple combinatorial objects (e.g., tabloids) and some basic ideas from representation theory (e.g., Schurs Lemma), this allows us to recast and extend some well-known results in the field of voting theory.

Generalized Condorcet winners

We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY-PT polynomial as special cases, providing an explicit unification of these apparently unrelated types of invariants. We provide examples demonstrating that the new invariants are not determined by the biquandle counting invariant, the knot quandle, the knot group or the traditional skein invariants.

A Unifying Framework for Teaching Nonparametric Statistical Tests

Several nonparametric tests exist to test for differences among alternatives when using ranked data. Testing for differences among alternatives amounts to testing for uniformity over the set of possible permutations of the alternatives. Well-known tests of uniformity, such as the Friedman test or the Anderson test, are based on the impact of the usual limiting theorems (e.g. central limit theorem) and the results of different summary statistics (e.g. mean ranks, marginals, and pairwise ranks). Inconsistencies can occur among statistical tests’ outcomes – different statistical tests can yield different outcomes when applied to the same ranked data. In this paper, we describe a conceptual framework that naturally decomposes the underlying ranked data space. Using the framework, we explain why test results can differ and how their differences are related. In practice, one may choose a test based on the power or the structure of the ranked data. We discuss the implications of these choices and illustrate that for data meeting certain conditions, no existing test is effective in detecting nonuniformity. Finally, using a real data example, we illustrate how to construct new linear rank tests of uniformity.

Reflections Acting Efficiently on a Building

In an election, a Condorcet winner is a candidate who would win every two-candidate subelection against any of the other candidates. In this paper, we extend the idea of a Condorcet winner to subelections consisting of three or more candidates. We then examine some of the relationships between the resulting generalized Condorect winners.

Dead Heat: The 2006 Public Choice Society Election

Abstract Increased importance is being placed on statistics at both the K-12 and undergraduate level. Research divulging effective methods to teach specific statistical concepts is still widely sought after. In this paper, we focus on best practices for teaching topics in nonparametric statistics at the undergraduate level. To motivate the work, we consider the problem of n rankings: m alternatives are fully ranked by a sample of n judges. Through this problem, we addresses how to teach nonparametric methods under a unifying framework that connects nonparametric methods to their parametric counterparts, utilizes basic techniques from linear algebra, and can empower students to make their own statistical tests.

An eigenspace approach to decomposing representations of finite groups

Abstract We show how Radon transforms may be used to apply efficiently the class sum of reflections in the finite general linear group GLn (𝔽 q ) to vectors in permutation modules arising from the action of GLn(𝔽 q ) on the building of type An −1 (𝔽 q ).