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 psolvable group", Arch. Math 87 (2006), 385389.
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 psolvable
group", Journal of Algebra 312 (2007), 699708. This paper gives
lower bounds on the number of lifts of a Brauer character for an
arbitrary psolvable 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), 193201. 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 piirreducible characters of groups of odd order"
, Communications in Algebra 36 (2008), 39723979. 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), 29622969. 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) 1732. 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) 384387. 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), 145163. 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), 391402. 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), 5970. 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), 201212. (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), 143153.
 Cossey, J.P., Isaacs, I.M., and Lewis, M. , Induction and restriction of characters and Hall subgroups, Journal of Algebra 383 (2013), 129143.
 Cossey, J.P. and Nguyen, H.N., Controlling composition factors of a finite group by its character degree ratio, Journal of Algebra (403), 185200.
PREPRINTS, arXiv, and NOTES:
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