88 research outputs found
A symmetry theorem on a modified jeu de taquin
For their bijective proof of the hook-length formula for the number of
standard tableaux of a fixed shape Novelli, Pak and Stoyanovskii define a
modified jeu de taquin which transforms an arbitrary filling of the Ferrers
diagram with (tabloid) into a standard tableau. Their definition
relies on a total order of the cells in the Ferrers diagram induced by a
special standard tableau, however, this definition also makes sense for the
total order induced by any other standard tableau. Given two standard tableaux
of the same shape we show that the number of tabloids which result in
if we perform modified jeu de taquin with respect to the total order induced by
is equal to the number of tabloids which result in if we perform
modified jeu de taquin with respect to . This symmetry theorem extends to
skew shapes and shifted skew shapes.Comment: 8 page
Monodromy and K-theory of Schubert curves via generalized jeu de taquin
We establish a combinatorial connection between the real geometry and the
-theory of complex Schubert curves , which are
one-dimensional Schubert problems defined with respect to flags osculating the
rational normal curve. In a previous paper, the second author showed that the
real geometry of these curves is described by the orbits of a map on
skew tableaux, defined as the commutator of jeu de taquin rectification and
promotion. In particular, the real locus of the Schubert curve is naturally a
covering space of , with as the monodromy operator.
We provide a local algorithm for computing without rectifying the
skew tableau, and show that certain steps in our algorithm are in bijective
correspondence with Pechenik and Yong's genomic tableaux, which enumerate the
-theoretic Littlewood-Richardson coefficient associated to the Schubert
curve. We then give purely combinatorial proofs of several numerical results
involving the -theory and real geometry of .Comment: 33 pages, 12 figures including 2 color figures; to appear in the
Journal of Algebraic Combinatoric
Reductions of Young tableau bijections
We introduce notions of linear reduction and linear equivalence of bijections
for the purposes of study bijections between Young tableaux. Originating in
Theoretical Computer Science, these notions allow us to give a unified view of
a number of classical bijections, and establish formal connections between
them.Comment: 42 pages, 15 figure
The octahedron recurrence and gl(n) crystals
We study the hive model of gl(n) tensor products, following Knutson, Tao, and
Woodward. We define a coboundary category where the tensor product is given by
hives and where the associator and commutor are defined using a modified
octahedron recurrence. We then prove that this category is equivalent to the
category of crystals for the Lie algebra gl(n). The proof of this equivalence
uses a new connection between the octahedron recurrence and the Jeu de Taquin
and Schutzenberger involution procedures on Young tableaux.Comment: 25 pages, 19 figures, counterexample to Yang-Baxter equation adde
Increasing and Decreasing Sequences in Fillings of Moon Polyominoes
We present an adaptation of jeu de taquin and promotion for arbitrary
fillings of moon polyominoes. Using this construction we show various symmetry
properties of such fillings taking into account the lengths of longest
increasing and decreasing chains. In particular, we prove a conjecture of Jakob
Jonsson. We also relate our construction to the one recently employed by
Christian Krattenthaler, thus generalising his results.Comment: fixed typo
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