Aaron Lauve’s

RESEARCH INTERESTS

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.

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 noncommutative settings in type A ($GL$n) . 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]

Hopf Algebras

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 symmetric functions. 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. [preprint]

N. Andruskiewitsch, H-J Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. of Math. (2) 171 (2010), no. 1, 375--417. [preprint]

D. Kreimer, 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]

Representation Theory

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!