10 research outputs found
Generalization of Schensted insertion algorithm to the cases of hooks and semi-shuffles
Given an rc-graph of permutation and an rc-graph of permutation
, we provide an insertion algorithm, which defines an rc-graph in the case when is a shuffle with the descent at and has no
descents greater than or in the case when is a shuffle, whose shape is
a hook. This algorithm gives a combinatorial rule for computing the generalized
Littlewood-Richardson coefficients in the two cases mentioned
above.Comment: 22 page
Flag arrangements and triangulations of products of simplices
We investigate the line arrangement that results from intersecting d complete
flags in C^n. We give a combinatorial description of the matroid T_{n,d} that
keeps track of the linear dependence relations among these lines. We prove that
the bases of the matroid T_{n,3} characterize the triangles with holes which
can be tiled with unit rhombi. More generally, we provide evidence for a
conjectural connection between the matroid T_{n,d}, the triangulations of the
product of simplices Delta_{n-1} x \Delta_{d-1}, and the arrangements of d
tropical hyperplanes in tropical (n-1)-space. Our work provides a simple and
effective criterion to ensure the vanishing of many Schubert structure
constants in the flag manifold, and a new perspective on Billey and Vakil's
method for computing the non-vanishing ones.Comment: 39 pages, 12 figures, best viewed in colo
Growth Diagrams for the Schubert Multiplication
We present a partial generalization to Schubert calculus on flag varieties of
the classical Littlewood-Richardson rule, in its version based on
Schuetzenberger's jeu de taquin. More precisely, we describe certain structure
constants expressing the product of a Schubert and a Schur polynomial. We use a
generalization of Fomin's growth diagrams (for chains in Young's lattice of
partitions) to chains of permutations in the so-called k-Bruhat order. Our work
is based on the recent thesis of Beligan, in which he generalizes the classical
plactic structure on words to chains in certain intervals in k-Bruhat order.
Potential applications of our work include the generalization of the
S_3-symmetric Littlewood-Richardson rule due to Thomas and Yong, which is based
on Fomin's growth diagrams
Littlewood-Richardson coefficients for reflection groups
In this paper we explicitly compute all Littlewood-Richardson coefficients
for semisimple or Kac-Moody groups G, that is, the structure coefficients of
the cohomology algebra H^*(G/P), where P is a parabolic subgroup of G. These
coefficients are of importance in enumerative geometry, algebraic combinatorics
and representation theory. Our formula for the Littlewood-Richardson
coefficients is given in terms of the Cartan matrix and the Weyl group of G.
However, if some off-diagonal entries of the Cartan matrix are 0 or -1, the
formula may contain negative summands. On the other hand, if the Cartan matrix
satisfies for all , then each summand in our formula
is nonnegative that implies nonnegativity of all Littlewood-Richardson
coefficients. We extend this and other results to the structure coefficients of
the T-equivariant cohomology of flag varieties G/P and Bott-Samelson varieties
Gamma_\ii(G).Comment: 51 pages, AMSLaTeX, typos correcte