78 research outputs found
On the Fully Commutative Elements of Coxeter Groups
Let W be a Coxeter group. We define an element w ∈ W to be fully commutative if any reduced expression for w can be obtained from any other by means of braid relations that only involve commuting generators. We give several combinatorial characterizations of this property, classify the Coxeter groups with finitely many fully commutative elements, and classify the parabolic quotients whose members are all fully commutative. As applications of the latter, we classify all parabolic quotients with the property that (1) the Bruhat ordering is a lattice, (2) the Bruhat ordering is a distributive lattice, (3) the weak ordering is a distributive lattice, and (4) the weak ordering and Bruhat ordering coincide.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46269/1/10801_2005_Article_415276.pd
Liouville integrability of a class of integrable spin Calogero-Moser systems and exponents of simple Lie algebras
In previous work, we introduced a class of integrable spin Calogero-Moser
systems associated with the classical dynamical r-matrices with spectral
parameter, as classified by Etingof and Varchenko for simple Lie algebras. Here
the main purpose is to establish the Liouville integrability of these systems
by a uniform method
Schubert Polynomials for the affine Grassmannian of the symplectic group
We study the Schubert calculus of the affine Grassmannian Gr of the
symplectic group. The integral homology and cohomology rings of Gr are
identified with dual Hopf algebras of symmetric functions, defined in terms of
Schur's P and Q-functions. An explicit combinatorial description is obtained
for the Schubert basis of the cohomology of Gr, and this is extended to a
definition of the affine type C Stanley symmetric functions. A homology Pieri
rule is also given for the product of a special Schubert class with an
arbitrary one.Comment: 45 page
Combinatorics of -orbits and Bruhat--Chevalley order on involutions
Let be the group of invertible upper-triangular complex
matrices, the space of upper-triangular complex matrices with
zeroes on the diagonal and its dual space. The group acts
on by , , ,
.
To each involution in , the symmetric group on letters, one
can assign the -orbit . We present a
combinatorial description of the partial order on the set of involutions
induced by the orbit closures. The answer is given in terms of rook placements
and is dual to A. Melnikov's results on -orbits on .
Using results of F. Incitti, we also prove that this partial order coincides
with the restriction of the Bruhat--Chevalley order to the set of involutions.Comment: 27 page
Weak splittings of quotients of Drinfeld and Heisenberg doubles
We investigate the fine structure of the simplectic foliations of Poisson
homogeneous spaces. Two general results are proved for weak splittings of
surjective Poisson submersions from Heisenberg and Drinfeld doubles. The
implications of these results are that the torus orbits of symplectic leaves of
the quotients can be explicitly realized as Poisson-Dirac submanifolds of the
torus orbits of the doubles. The results have a wide range of applications to
many families of real and complex Poisson structures on flag varieties. Their
torus orbits of leaves recover important families of varieties such as the open
Richardson varieties.Comment: 20 pages, AMS Late
Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams
We consider the problem of finding the number of matrices over a finite field
with a certain rank and with support that avoids a subset of the entries. These
matrices are a q-analogue of permutations with restricted positions (i.e., rook
placements). For general sets of entries these numbers of matrices are not
polynomials in q (Stembridge 98); however, when the set of entries is a Young
diagram, the numbers, up to a power of q-1, are polynomials with nonnegative
coefficients (Haglund 98).
In this paper, we give a number of conditions under which these numbers are
polynomials in q, or even polynomials with nonnegative integer coefficients. We
extend Haglund's result to complements of skew Young diagrams, and we apply
this result to the case when the set of entries is the Rothe diagram of a
permutation. In particular, we give a necessary and sufficient condition on the
permutation for its Rothe diagram to be the complement of a skew Young diagram
up to rearrangement of rows and columns. We end by giving conjectures
connecting invertible matrices whose support avoids a Rothe diagram and
Poincar\'e polynomials of the strong Bruhat order.Comment: 24 pages, 9 figures, 1 tabl
Zero-one Schubert polynomials
We prove that if σ∈Sm is a pattern of w∈Sn, then we can express the Schubert polynomial w as a monomial times σ (in reindexed variables) plus a polynomial with nonnegative coefficients. This implies that the set of permutations whose Schubert polynomials have all their coefficients equal to either 0 or 1 is closed under pattern containment. Using Magyar's orthodontia, we characterize this class by a list of twelve avoided patterns. We also give other equivalent conditions on w being zero-one. In this case, the Schubert polynomial w is equal to the integer point transform of a generalized permutahedron
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