Representations of finite groups

Mostly I study the representations of solvable groups. In particular I am interested in the relationship between representations in characteristic zero and representations in characteristic p. I also am interested in the representations of the symmetric groups and their related combinatorics. Here are some published papers, some preprints, and some notes:

PUBLISHED PAPERS:

• Cossey, J.P. "A construction of two distinct canonical sets of lifts of Brauer characters in a p-solvable group", Arch. Math 87 (2006), 385-389. This paper shows that the $B_{\pi}$ characters are not the same as the $N_{\pi}$ characters in general, though the sets are the same if $|G|$ is odd.
  
• Cossey, J.P. "Bounds on the number of lifts of a Brauer character in a p-solvable group", Journal of Algebra 312 (2007), 699-708. This paper gives lower bounds on the number of lifts of a Brauer character for an arbitrary p-solvable group, and gives upper bounds in the case that G has odd order.
  
• Cossey, J.P. and Lewis, Mark , "Fong characters and chains of normal subgroups", Arch. Math 89 (2007), 193-201. This paper studies the behavior of Fong characters with respect to normal subgroups. Hopefully we'll be able to solve the more general problem soon.
  
• Cossey, J.P., " Vertices of pi-irreducible characters of groups of odd order" , Communications in Algebra 36 (2008), 3972-3979. This paper shows that if G has odd order, then the vertices of lifts of Brauer characters of G have the same uniqueness property that the vertices of the Brauer characters have.
  
• Cossey, J.P. "Vertices and normal subgroups of solvable groups", Journal of Algebra 321 (2009), 2962-2969. This paper develops some properties of the vertices of Brauer characters in groups of odd order with respect to normal subgroups, and in particular generalizes the Alperin weight conjecture if G has odd order.
  
• Cossey, J.P. "Vertex subgroups and vertex pairs in solvable groups" in the "Proceedings of the Conference on Character Theory of Finite Groups in honor of I.M. Isaacs," Contemporary Math 524 (2010) 17-32. This is an expository paper on some of the stuff Mark Lewis and I have been working on for a while. Though a lot had happened since this paper was submitted.
  
• Cossey, J.P., Lewis, M., and Navarro, G. The number of lifts of a Brauer character with a normal vertex, Journal of Algebra 328 (2010) 384-387. This paper bounds the number of lifts of a Brauer character in a solvable group in the case that the vertex subgroup is normal or abelian.
  
• Cossey, J.P. and Lewis, M., Lifts of partial characters with cyclic defect groups, Journal of the Australian Mathematical Society 89 (2010), 145-163. A paper with Mark Lewis about lifts of partial characters with cyclic defect groups (preprint).
  
• Cossey, J.P., "A bijection for the Alperin weight conjecture in S_n", Algebras and Representation Theory 14 (2011), 391-402. This paper gives a specific bijection between a certain set of partitions and the weights of the symmetric group, which extends the work of Alperin and Fong that proved the Alperin weight conjecture for the symmetric groups.
  
• Cossey, J.P., "Navarro vertices and normal subgroups in groups of odd order", Rocky Mountain Journal of Mathematics 42 (2012), 59-70. Some results about the behavior of vertex characters of lifts of Brauer characters in groups of odd order, including some results which prove when the constituents of the restriction of a lift of a Brauer character to a normal subgroup are lifts.
  
• Cossey, J.P. and Lewis, M. , "Inductive pairs and lifts in solvable groups" Journal of Group Theory 14 (2011), 201-212. (Here's a preprint.)
  
• Cossey, J.P. and Lewis, M. , "Lifts and vertex pairs in solvable groups", Proceedings of the Edinburgh Mathematical Society 55 (2012), 143-153.
  
• Cossey, J.P., Isaacs, I.M., and Lewis, M. , Induction and restriction of characters and Hall subgroups, Journal of Algebra 383 (2013), 129-143.
  
• Cossey, J.P. and Nguyen, H.N., Controlling composition factors of a finite group by its character degree ratio, Journal of Algebra (403), 185-200.

PREPRINTS, arXiv, and NOTES:

• Some results about nilpotent blocks of solvable groups. There are still plenty of open questions here.
  
• A paper (with Mark Lewis) about diameters of Brauer graphs of blocks of solvable groups. This paper has been submitted.
  
• Some more results about the behavior of lifts of Brauer characters with respect to induction and restriction from/to normal subgroups in groups of odd order.
  
• A result on Fong characters and normal subgroups (This gets many of the same results as the paper I cowrote with Mark Lewis, above, but via a different approach.)
  
• Here is a result about the uniqueness of lifts of irreducible Specht modules of the symmetric group. Apparently this is a new proof of an old result, but I'm working on extending some of the new ideas in this paper.
  
• Mark Lewis and I got some results about bounding the number of lifts of a Brauer character in the case that the vertex subgroup is abelian. Later, Gabriel Navarro helped us find a shorter argument, so we won't be publishing this one probably. But there are some interesting ideas in there I think.

Work with students
I've supervised a number of student research projects. Here are the final reports.

• A beginning of an attempt to determine a generating function for Fayers partitions and their corresponding representations of the symmetric group. This is work I did with Brendan Fry at the University of Arizona as part of an undergraduate research project.
  
• I supervised an undergraduate research project on the representations of the braid group with Jacob White in Spring of 2006 at the University of Arizona, and here is the final report that Jacob turned in. I think it's pretty nice, and it asks some interesting questions.
  

Miscellaneous
Here are some other topics I have been interested in.

• For a class my final year of grad school I wrote a little expository paper about the connection between representations of finite groups and wireless multiple antenna designs.

Slides from some talks I've given

• A talk on the McKay conjecture for the ANTS seminar.
  
• A talk on vertices of lifts of Brauer characters in groups of odd order I gave at the Zassenhaus Conference in St. Louis in spring of 2007.
  
• A talk on representations of the symmetric group I gave at Central Michigan University
  
• A talk on representations of solvable groups I gave at the University of Akron.
  
• Extending a character correspondence in groups of odd order for the May 2009 Zassenhaus group theory conference.
  
• Normal subgroups and a character correspondence in groups of odd order for the Conference on Character Theory in honor of I.M. Isaacs in Valencia, Spain in June 2009.