SOME PREPRINTS
- Bruhat-Chevalley order on the rook monoid (with L. Renner).
http://arxiv.org/abs/0803.0491
The rook monoid $R_n$ is the finite monoid whose elements are the 0-1 matrices with at most
one nonzero entry in each row and column. The group of invertible elements of $R_n$ is
isomorphic to the symmetric group $S_n$. The natural extension to $R_n$ of the Bruhat-Chevalley
ordering on the symmetric group is defined in \cite{Renner86}. In this paper, we find an efficient,
combinatorial description of the Bruhat-Chevalley ordering on $R_n$. We also give a useful, combinatorial
formula for the length function on $R_n$.
- The rook monoid is lexicographically shellable. (The link to the preprint will be provided, soon.)
The rook monoid $R_n$ is the finite monoid whose elements are the $0-1$ matrices with at most one
1 in each row and column. It is known that $R_n$ is the parametrizing object for the $B\times B$ orbits
on $n\times n$ matrices, where $B$ is the group of invertible upper triangular matrices.
The Bruhat-Renner ordering on $R_n$ is the partial ordering which determines the containment relations
of these orbits. In this article we investigate the properties of this partial ordering. In particular,
we show that $R_n$ is an $EL$-shellable poset.
- H-polynomials and rook polynomials (with L. Renner). Internat. J. Algebra Comput. 18 (2008), no. 5, 935--949.
The purpose of this paper is twofold. First, we describe a useful procedure for computing the $H$-polynomials
of reductive monoids. Second, we use this procedure to compute the $H$-polynomial of the monoid of $n\times n$
matrices in terms of the $q$-analogues of the rook polynomials of Garsia and Remmel.
- R-polynomials of finite monoids of Lie type (with K. Aker and M. Taskin).
http://arxiv.org/abs/0811.4382
This paper concerns the combinatorics of the orbit Hecke algebra associated with the orbit of a two sided Weyl
group action on the Renner monoid of a finite monoid of Lie type, $M$. It is shown by Putcha in \cite{Putcha97}
that the Kazhdan-Lusztig involution (\cite{KL79}) can be extended to the orbit Hecke algebra which enables one
to define the $R$-polynomials of the intervals contained in a given orbit. Using the $R$-polynomials, we calculate
the M\"obius function of the Bruhat-Chevalley ordering on the orbits. Furthermore, we provide a necessary condition
for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group.
- Supersolvable cross section lattices. A slightly updated version will replace it, soon.
The purpose of this article is to investigate the combinatorial
properties of the cross section lattice of a $J$-irreducible monoid
associated with a semisimple algebraic group of one of the types $A_n$, $B_n$,
or $C_n$.
Our main tool is a theorem of Putcha and Renner which identifies the
cross section lattice in the Boolean lattice of subsets of the nodes of a
Dynkin diagram.
We determine the join irreducibles of the cross section lattice.
Exploiting this we find characterizations of the relatively complemented intervals.
By a result of Putcha, this determines the M\"{o}bius function for
$\Lambda$. We show that an interval of the cross section lattice is Boolean
if and only if it is relatively complemented if and only if it is atomic.
We characterize distributive cross section lattices, showing that
they are products of chains. We determine which cross section lattices are supersolvable,
and furthermore, we compute the characteristic polynomials of these
supersolvable cross section lattices.
- The nested Hilbert scheme of points in the plane and nested Catalan numbers.
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.
- A proof of the $q,t$-Square conjecture (with N. Loehr). J. Combin. Theory Ser. 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 $\nabla$ denotes the BergeronÐGarsia nabla operator, and $p_n$
is a power-sum symmetric function. The combinatorial formula enumerates lattice paths in an $n?n$ square
according to two suitable statistics.
- Some plethystic identities and Koska-Foulkes polynomials.
- From parking functions to Gelfand pairs A slightly updated version will replace the link, soon.
NEW:
- Enumeration in the rook monoid (with L. Renner). (The link will be provided, soon.)
- The Farahat-Higman ring of some finite symmetric spaces. (with K. Aker). (The link will be provided, soon.)
NOTES:
- Some notes on Demazure character formula. (I will keep adding to it.)
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