Research

In brief, I am interested in noncommutative structures in algebra. My research thus far has focused on quasideterminants and Hopf algebras, but, frankly, if your problem has the word "noncommutative" somewhere in its description, you've piqued my interest.

Recent Communications

My C.V. is often more up-to-date than the lists below. If there is no link, then a digital copy is unavailable.

Presentations

New Hopf structures on planar binary trees, FPSAC'09, Austria, July 2009. [poster]
Nichols algebras in positive characteristic, CMS Summer Meeting, St. John's, Newfoundland, June 2009. [slides]
Hopf structures on binary trees (variations on a theme), AMS Section Meeting, Raleigh, April 2009. [slides]
Rational and irrational series over the free group, AMS Joint Meetings, Washington, D.C., January 2009. [slides]
New Hopf structures on planar binary trees, CMS Winter Meetings, Ottawa, December 2008. [slides]
Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables, 20th International Conference on Formal Power Series and Algebraic Combinatorics, Valparaiso, Chile, June, 2008. [slides]
On matrix inversion using mixed information, AMS Sectional Meeting, Baton Rouge, LA, March, 2008. [slides]
The Markoff condition and central words, AMS/MAA Joint Meetings, San Diego, CA, January 2008. [poster][YMN Poster Session]
Yangian flags via quasideterminants, Advanced Course on Quasideterminants and Universal Localization, CRM, Barcelona, Spain February, 2007. [slides]
PO-set paths and q-commuting minors, Sèminaire de combinatoire et d'informatique mathématique, LaCIM-UQAM, Montréal, April 2006. [slides]
Noncommutative flag varieties and Yangians, AMS Sectional Meeting, Eugene, OR, November, 2005. [slides]
A quasideterminantal approach to noncommutative flag varieties, Dissertation Defense, Rutgers University, April, 2005. [slides]

Papers, proceedings, etc

(with S. Forcey & F. Sottile) Hopf structures on the multiplihedra. [preprint]
(with T. Lam & F. Sottile) Skew Littlewood-Richardson rules from Hopf algebra [preprint]
(with S. Forcey & F. Sottile) New Hopf structures on binary trees. [preprint][to appear, DMTCS Proc. (FPSAC 2009)]
(with C. Cibils & S. Witherspoon) Hopf quivers and Nichols algebras in positive characteristic. [preprint][Proc. AMS, to appear]
(with J. Berstel, C. Reutenauer, F.V. Saliola) Combinatorics on words: Christoffel words and repetitions in words. [preprint][CRM Monograph Series, v.27, AMS, 2008]
(with A. Glen & F.V. Saliola) A note on the Markoff condition and central words. [preprint][Inform. Process. Lett. `08]
(with F. Bergeron) Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables. [preprint][DMTCS Proc. (FPSAC 2008)]
Quasideterminants and q-commuting minors. [preprint]
Flag varieties for the Yangian Y(gl_n). [preprint]
(with E.J. Taft) A class of left quantum groups modeled after $SL$ _q(n). [preprint][JPAA `07]
Quantum- and quasi-Plücker coordinates. [preprint][J.Algebra `06]
(with D.F. Anderson, A. Frazier, P.S. Livingston) The zero-divisor graph of a commutative ring, II. [preprint][Ideal theoretic methods in commutative algebra, pp.61--72, Lecture Notes in Pure and Appl. Math., v.220, Dekker, 2001]

Research Interests, Briefly

Quasideterminants

While following courses as a graduate student, one question I often asked myself was, "but what can we say when things don't commute?" Obviously, I was elated to discover the work of Gelfand & Retakh in this direction. In 1991, they introduced to the world what Cayley (1845), and others had been searching for… a proper determinant-like tool for the noncommutative setting. Their goal has been to provide explict formulas and objects with which to work---bringing the al-jabr back into the world of noncommutative algebra. In this they have been extremely successful. Since 1991, the quasideterminant has appeared as part of the story---if not THE story---in numerous seemingly diverse areas: Casimir operators in Lie theory, quantum determinants for quantum groups, the theory of noncommutative symmetric functions and the factorization of noncommutative polynomials. What's more, there's even a Cramer's rule with which to do noncommutative linear algebra!

In my dissertation, I introduce the notion of "amenable determinant" and use it, together with quasideterminants, to define (flag) varieties for a great many type A (for $GL$ _n) noncommutative settings. There is some promise that quasideterminants can also provide flag varieties for other types, and for Schubert subvarieties of these. An open question is whether quasideterminantal constructs alone can "completely describe" these noncommutative varieties or if specific results in each setting are needed to provide the proper flag analogs there.

I. Gel'fand, S. Gel'fand, V. Retakh, and R. Wilson, Quasideterminants, Advances in Mathematics, Volume 193, Issue 1, 1 May 2005, Pages 56-141. [preprint]

I. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, and J.-Y. Thibon, Noncommutative Symmetric Functions, Advances in Math 112 (1995), no. 2, 218--348. [preprint]

Hopf algebras

The Hopf algebra phenomenon was first explored in algebraic topology (Heinz Hopf introduced them in his study of

Sⁿ). The second wave of Hopf algebras were launched from Lie- and Algebraic-group theory. These are very nice, and continue to be a source of interesting mathematics (e.g., quantum groups & pointed Hopf algebras). The latest wave of Hopf algebras were introduced by G.-C. Rota and his contemporaries in the service of combinatorics. These are my favorites! Singling out one paper on this subject that I admire would be wrong, but I'll do it anyway (see below). One timeless example, so so important in representation theory, is the ring of symmetric functions. A newer, equally beautiful example: the ring NSym of noncommutative symmetric functions (see Gelfand, Krob, et.al., above). It would be interesting to find more uses for the quasideterminant in the study of NSym and the like.

N. Andruskiewitsch, H-J Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. of Math, to appear. [preprint]

M. Aguiar, N. Bergeron, F. Sottile, Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math. 142 (2006), no. 1, 1--30. [preprint]

Representation theory

The interplay between combinatorics and Lie theory at work in the representation theory of irreducible representations of Lie groups and Lie algebras is a source of endless amazement for me. Again, it would be an injustice for me to single out two papers on this subject which I admire, but I will do it anyway.

S. Sahi, A new formula for weight multiplicities and characters, Duke Mathematical Journal 101 (2000), no. 1, 77--84. [preprint]

C. Lenart, A. Postnikov, Affine Weyl groups in K-theory and representation theory, Int. Math. Res. Not. IMRN 2007, no. 12, Art. ID rnm038, 65 pp. [preprint]