My
research interests are on the combinatorial aspects of the algebra and
geometry. I am interested in the algebraic
monoids (and algebraic groups), the theory of the Macdonald
polynomials (hence the Hilbert schemes), and the combinatorics
related to the moduli spaces.
Articles:
® A Proof of the
$q,t$-Square Conjecture (with N. Loehr),
Journal of Comb. Theory, Series A, 113 (2006), no. 7,
1419--1434}.
We prove a combinatorial formula conjectured by Loehr and
Warrington
for the coefficient of the sign character in $\nabla(p_n)$ .
Here $\nabla$ denoted the Bergeron-Garsia nable
operator, and $p_n$ is a power-sum symmetric function. The
combinatorial
formula enumerates lattice paths in an
$n\times n$ square according to two suitable statistics.
® Combinatorics of the Shatz
polygons (with A. Dhillon), submitted.
For every lattice point in the right half plane there is an
associated poset of Harder-Narasimhan polygons. In this article, we
study the
combinatorial properties of these posets. We show that on each vertical
direction there are only finitely many non-isomorphic posets of
Harder-Narasimhan polygons. Furthermore, we determine the exact number
of isomorphism classes of the augmented posets of Harder-Narasimhan
polygons on each vertical direction.
® Nested Hilbert schemes
and the nested $q,t$-Catalan series.
http://arxiv.org/abs/0711.0763
In this paper we study the tangent spaces of the smooth nested
Hilbert
scheme
$ Hil{n,n-1}$ of points in the plane, and give a general
formula for computing
the Euler characteristic of a $\TT^2$-equivariant locally free sheaf on
$\Hil{n,n-1}$. Applying our result to a
particular sheaf, we conjecture that
the result is a polynomial in the variables $q$ and $t$ with
non-negative
integer coefficients . We call
this conjecturally positive polynomial as
\textsl{the nested $q,t$-Cat alan series}, for it has many conjectural
properties similar to that of
the $q,t $-Catalan series.
® Asymptotic computation
of the probability for two subspaces over a
finite field to have the same dimension (with M. Bona, R.
Pemantle and H.Wilf).
In this paper, we calculate the statement that is in the title, and we
express the
result in terms of the ''theta function.''
® H-polynomials and rook
polynomials (with L. Renner), submitted.
® The Bruhat-Chevalley
ordering on the rook monoid (with L. Renner), submitted.
® The rook monoid is
lexicographically shellable, to be submitted
-available upon request.
® Some plethystic
identities and Kostka-Foulkes polynomials, to be submitted -available
upon request.
® Equivariant $K$-theory
for the rook monoids (with L. Renner), in
preparation.
Here is the Hasse diagram of the rook
monoid for n=3.
(Thanks to John Stembridge's SF package.)
