My work has always tried to unite the true with the beautiful and when I had to choose
one or the other,
I usually chose the beautiful. -- Hermann Weyl.
My area of research is algebraic
combinatorics and representation theory. My focus is in applying
combinatorics to answer algebraic questions related to the Coxeter
groups and their related algebras,
e.g. Hecke algebras, Brauer algebras, partition algebras. I am
particularly interested in the interplay
between Markov traces, centralizer algebras, quantum groups, Hopf
algebras, symmetric functions,
and knot theory. I have a special fondness towards Young diagrams and
the symmetric group.
PUBLICATIONS AND PREPRINTS:
The Hecke algebra of type B at roots of unity,
Markov traces and subfactors.
Ph.D. Thesis, UCSD 1999.
Weights
of Markov traces on Hecke algebras, J.
Reine Angew Math. 508, (1999)
157-178.
Markov traces and Hecke algebras at roots of unity, Proceedings of the Formal Power
Series and Algebraic Combinatorics, 1999. pp. 405-416.
Hecke algebras of type B and type D and Subfactors, Pacific Journal of Math. Vol
199, No. 1 (2001) 137-161.
(with H. Wenzl) q-Centralizer algebras for spin
groups, Journal of Algebra,
253 (2002)
237-275.
(with M. Orrison and D. Rockmore) Rooted trees and
the representation theory of
iterated wreath products of
abelian groups, Advances in Applied
Math. 33, Issue 3,
(2004), 531-547.
On the algebraic decomposition of centralizer
algebras of the hyperoctahedral group,
Contemporary Math. 376, (2005), 345-357.
(with
C. Ballantine) On the Kronecker product $s_{(n-p,p)}\ast s_{\lambda}$,
Elec. Journal of Combinatorics. Vol. 12, (2005), #R28, 1-26.
(with M. Aguiar and K. Nyman) New Results on the
Peak algebra,
J. of
Algebraic Combinatorics. Vol. 23, No.2 (2006), pg. 149-188
(with A. Ram) Affine Braids, Markov traces
and the category O, Algebraic groups
and homogeneous spaces, 423-473, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res.,
Mumbai, 2007.
http://lanl.arxiv.org/abs/math.RT/0401317
(with C. Ballantine) A combinatorial interpretation for the
coefficients in the Kronecker
product $s_{(n,n-p)}\ast s_{\lambda}$.
S\'em. Lothar. Combin. 54A (2005/06), Art. B54Af, 29 pp.
On the partition algebras of complex reflection groups, Journal of Algebra,
313 (2007) 590-616.
(with M. Aguiar) The Hopf Algebra of Uniform block Permutations.,
J. Algebraic Combin. 28
(2008), no. 1, 115--138.
An extended abstract of 13 pages appeared in:
Proceedings of the Formal Power Series and Algebraic Combinatorics 2005
Available at:
http://lanl.arxiv.org/abs/math.RA/0505199.
(with A. Mathas)
Cyclotomic Solomon Algebras
Adv. Math. 219 (2008), no. 2, 450--487.
(with E. Briand and M. Rosas)
Reduced Kronecker coefficients and counter-examples
to Mulmuley's saturation conjecture SH, with an appendix by Ketan Mulmuley.
To appear in
computational complexity
(with E. Briand and M. Rosas)
Quasipolynomial formulas for the
Kronecker coefficients
indexed by two two-row shapes FPSAC 2009, Hagenberg, Austria, Discrete Mathematics and
Theoretical Computer Science proc. AK, 2009, 241-252
(with E. Briand and M. Rosas)
The stability of the Kronecker product of Schur functions
submitted for publication
(with M. Zabrocki) The Hilbert series for the ring
of Hook Schur functions.
Preprint available at:
http://lanl.arxiv.org/abs/math.CO/0008152