In brief, I am interested in noncommutative
structures in algebra and combinatorics. My research has seldom strayed
far from quasideterminants, Hopf algebras, or representation theory, but, frankly, if your
problem has the word "noncommutative" somewhere in its description,
you've piqued my interest.
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 noncommutative settings in type A ()
. 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, Adv. Math. 193 (2005), no. 1, 56--141. [preprint]
I. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, and J.-Y. Thibon, Noncommutative Symmetric Functions, Adv. Math. 112 (1995), no. 2, 218--348. [preprint]
M. Konvalinka, I. Pak, Noncommutative extensions of the MacMahon Master Theorem, Adv. Math. 216 (2007), no. 1, 29--61. [preprint]
The Hopf algebra
phenomenon was first explored in algebraic topology, where Heinz Hopf (1941) used them
in his study of n-dimensional spheres. 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). (See [Andruskiewitsch–Santos]
for the beginnings of the theory.) The latest wave of Hopf algebras were
introduced by G.-C. Rota and his contemporaries in the service of
combinatorics. These are my favorites. One timeless example, so, so important
in geometry and representation theory, is the ring of
A newer, equally beautiful example is the ring NSym of
noncommutative symmetric functions (introduced by Malvenuto & Reutenauer; see Gelfand, Krob, et.al., above).
The adoption of Hopf algebraic techniques by the algebraic combinatorics
community is proceeding at a snail's pace. This despite the fact that these
techniques can greatly organize and simplify combinatorial arguments. Were I to
speculate, I'd say the pace is a reflection of taste… “fewer
lines of paper” is not “simpler” if we can prove it by traditional means and one needs several chapters of [Montgomery] to prove it with your techniques. In recent work with Lam and Sottile, we give an instance where Hopf algebra techniques are the only way (not just the simpler way). See the bibliography entries below for further applications. Here's hoping more will appear in the near future.
M. Aguiar, N. Bergeron, F. Sottile,
Combinatorial Hopf algebras and generalized Dehn-Sommerville relations,
Compos. Math. 142 (2006), no. 1, 1--30.
N. Andruskiewitsch, H-J Schneider,
On the classification of finite-dimensional pointed Hopf algebras,
Ann. of Math. (2) 171 (2010), no. 1, 375--417.
Structures in Feynman graphs: Hopf algebras and symmetries,
in Graphs and patterns in mathematics and theoretical physics, 43--78, Proc. Sympos. Pure Math., 73, Amer. Math. Soc., 2005. [preprint]
The fact that the inner workings of complex (semisimple) Lie groups can be boiled down to
discrete, often finite, bits of combinatorial data would come as no surprise
to Doron Zeilberger—who
believes that the world is finite—but it’s an endless source of
fascination for me. The interplay between combinatorics and representation theory has
a long history, stretching back to Killing and Lie (1880s), passing through Cartan and Dynkin (1940s), and still kicking today.
A relatively modern spin on the interplay is the notion of supercharacters. There are finite groups G which are known to have “wild” representation theories (characters). As developed by C. André and N. Yan (and organized by Diaconis and Isaacs), supercharacter theory aims to make the study of their representations more manageable. Perhaps it is not surprising (see Zelevinsky's book below), but combinatorial Hopf algebras enter the picture here. At least in the case when G is the group of upper-triangular matrices over a finite field. Stay tuned, more is surely to come!
S. Sahi, A new formula for weight multiplicities and
characters, Duke Math. J. 101 (2000), no. 1, 77--84.
C. Lenart, A. Postnikov, Affine Weyl groups in K-theory and
Int. Math. Res. Not. IMRN 2007, no. 12, Art. ID rnm038, 65 pp.
A. Zelevinsky, Zelevinsky, Andrey V. Representations of finite classical groups. A Hopf algebra approach. Lecture Notes in Mathematics, 869. Springer-Verlag, Berlin-New York, 1981. [book]