Overview


My primary research is in finite group theory and the representation theory of finite groups.  Group theory has applications to many other areas of mathematics, as well as to other science and engineering fields, including chemistry, physics, cryptography, and coding theory. The study of group theory was motivated by the desire to understand the symmetry of an object, whether it be in nature, art, communication networks, or any other place that symmetry might play a role. Representation theory is a tool used to better undersantd the structure of groups and symmetries it represents. Roughly speaking, representations give us a way to view, in some sense, an abstract group as a group of matrices whose structure is often easier to understand. In particular, we are interested in the irreducible representations, which are in a sense the building blocks of all representations.

My doctoral thesis research involves the irreducible representations of finite groups of Lie type, sometimes called finite reductive groups.  These groups are analogues of Lie groups over finite fields and form the largest collection of finite simple groups. This class of groups can be further separated into the finite classical groups and the so-called exceptional groups. The classical groups can be realized as certain groups of matrices over finite fields. In my thesis, I look at various problems concerning the cross-characteristic representations of finite classical groups, with a particular emphasis on the symplectic group Sp(6,q), with q even. My advisor is Dr. Pham Huu Tiep. 

For my undergraduate honors thesis, I began research involving strong reality in finite Coxeter groups (a problem proposed by Dr. Ryan Vinroot, now at College of William and Mary) and was advised by Dr. Klaus Lux.  We later finished the problem, and the results recently appeared in Communications in Algebra.

The summer of 2008, I participated in an REU at Central Michigan University.  My group's project was on investigating the properties of various forms of graph coloring, under the advisement of Dr. Lon Mitchell.  The results were published in the Electronic Journal of Combinatorics.

I am also interested in exploring the applications of algebra, such as to cryptography and coding theory.

Publications

Schaeffer Fry, A.Sp(6, 2^a) is `Good` for the McKay, Alperin Weight, and Related Local-Global Conjectures . Submitted 2012. Available on arXiv.

Schaeffer Fry, A.Cross-Characteristic Representations of Sp(6, 2^a) and Their Restrictions to Proper Subgroups. Journal of Pure and Applied Algebra (2012), doi: 10.1016/j.jpaa.2012.11.011 (in press). Available on arXiv.

Lux, K., Schaeffer Fry, A., Vinroot, W.R. Strong Reality Properties of Normalizers of Parabolic
Subgroups in Finite Coxeter Groups. Comm. Algebra. 40 (2012), no. 8, 3056-3070. Available here.

Haynes, G., Mitchell, L., Park, C., Schaeffer, A., Webster, J. Orthogonal Vector Coloring. Electronic
Journal of Combinatorics. v 17. 2010.

Computations


Here is GAP code to check that all elements of the normalizer of a parabolic subgroup of an irreducible Coxeter group are strongy real.  Here is GAP code to veriy that all Frobenius-Schur indicators are 1.  These were used in my paper with Dr. Ryan Vinroot and Dr. Klaus Lux.

Upcoming Talks

Schaeffer Fry, A. Title TBD. Special Session on p-Local Group Theory, Fusion Systems, and Representation Theory - AMS Fall Central Sectional Meeting 2013. St. Louis, MO. October 19-20, 2013 (Invited).

Schaeffer Fry, A. Title TBD Southwest Group Theory Day. Tucson, AZ. November 2, 2013 (Invited).

Recent Talks

Schaeffer Fry, A. Irreducible Representations of Finite Groups of Lie Type. Final Dissertation Defense, UA Dept. of Mathematics. Tucson, AZ. April 3, 2013.

Schaeffer Fry, A. Irreducible Representations of Finite Groups of Lie Type - A Practice for My Defense. UA Dept. of Mathematics Graduate Student Colloquium. Tucson, AZ. March 27, 2013.

Schaeffer Fry, A. Sp(6, 2^a) is `Good` for the McKay, Alperin Weight, and Related Local-Global Conjectures. Special Session on Groups, Representations, and Applications - Joint Mathematics Meetings 2013. San Diego, CA. January 11, 2013 (Invited). Abstract: Avalaible here.

Schaeffer Fry, A. Cross-characteristic Representations of Sp(6, 2^a) and their Restrictions to Maximal Subgroups. AWM Poster Session - Joint Mathematics Meetings 2013. San Diego, CA. January 11, 2013. Abstract: Available here.

Schaeffer Fry, A. Local-Global Conjectures in Representation Theory and the Goodness of Sp(6, 2^a). Algebra and Number Theory Seminar, University of Arizona. Tucson, AZ. December 4, 2012. Abstract: Available here.

Schaeffer Fry, A. On the Largest Irreducible Representations of the Finite Unitary Groups. Special Session on Representations of Groups and Algebras - AMS Fall 2012 Western Sectional Meeting. Tucson, AZ. October 28, 2012. (Invited) Abstract: Avalaible here.

Schaeffer Fry, A. Cross-characteristic Representations of Sp(6, 2^a) and their Restrictions to Maximal Subgroups. Poster Session - University of Washington Summer School and Conference on Cohomology and Support in Representation Theory. Seattle, WA. August 1, 2012.

Schaeffer Fry, A. Cross-characteristic Representations of Sp(6, 2^a) and their Restrictions to Maximal Subgroups. Special Session on Linear and Permuation Representations - AMS Spring 2012 Western Sectional Meeting. Honolulu, HI. March 3, 2012 (Invited)

Schaeffer Fry, A. Cross-characteristic Representations of Sp(6, 2^a) and their Restrictions to Maximal Subgroups. Algebra and Number Theory Seminar, University of Arizona. Tucson, AZ. February 21, 2012.

Schaeffer Fry, A. The Irreducible Restriction Problem: Motivation, Overview, and Results. Univer- sity of Arizona Graduate Student Colloquium. Tucson, Arizona. November 16, 2011.

Schaeffer Fry, A. On the Largest Irreducible Representations of the Finite Unitary Groups. Poster Session, 2011 Alliance Field of Dreams Conference. Arizona State University. Tempe, Arizona. Ocober 14, 2011.

Schaeffer Fry, A. Maximal Subgroups of the Finite Classical Groups and Representations. University
of Arizona Graduate Student Colloquium. Tucson, Arizona. April 20, 2011.

Schaeffer Fry, A. Nonsense With the General Linear Group: the "Field" with One Element and
Other Such Absurdities. Univ. of Arizona Dept. of Mathematics Recruitment Workshop. Tucson,
Arizona. March 7, 2011.

Schaeffer Fry, A. On the Largest Irreducible Representations of the Finite Simple Groups of Lie
Type. University of Arizona Algebra and Number Theory Seminar. Tucson, Arizona. March 1,
2011.

Schaeffer, A. Complex Representations of the Finite Unitary Groups and Their Restrictions to
Certain Subgroups. Research Tutorial Group Mini-Conference. Tucson, Arizona. December 10,
2009.

Schaeffer, A. Strong Reality in Coxeter Groups. Nebraska Conference for Undergraduate Women in
Mathematics (NCUWM). Lincoln, Nebraska. January 31, 2009.

Schaeffer, A. Vector Coloring. Mathematical Association of America (MAA) MathFest. Madison,
Wisconsin. July 31, 2008.