research

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 advisor was Dr. Pham Huu Tiep. My thesis and postdoctoral 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. I have been particularly interested in problems concerning local-global conjectures in representation theory and problems involving the structure of finite classical groups and their representations.  

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 , J. Algebra (2014), http://dx.doi.org/10.1016/j.jalgebra.2013.12.007. (in press) 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 and Workshops

Mathematics Colloquium - College of William and Mary. Williamsburg, VA. February 21, 2014. Title: TBD (Invited)

Mathematics Colloquium - Kent State University. Kent, OH. February 27, 2014. Title: TBD (Invited)

BIRS Workshop on Global/Local Conjectures in Representation Theory of Finite Groups. Banff, Alberta, Canada. March, 2014. (Invited)

Recent Talks

Self-Normalizing Sylow 2-Subgroups and Odd Characters Fixed by Galois Automorphisms. Southwest Group Theory Day. Tucson, AZ. November 2, 2013 (Invited). Abstract: Available here.

Odd Characters and Self-Normalizing Sylow 2-Subgroups. 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). Abstract: Available here.

Local-Global Conjectures in Representation Theory, Character Tables, and Self-Normalizing Sylow Subgroups. Group Theory Seminar, Michigan State University. East Lansing, MI. September 24 and October 1, 2013. (Two Parts)

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

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.

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.

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.

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.

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.

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.

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)

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.

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

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.

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

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.

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.

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

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

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