227 research outputs found
On an index two subgroup of puzzle and Littlewood-Richardson tableau Z2 x S3-symmetries
We consider an action of the dihedral group Z2 × S3 on Littlewood-
Richardson tableaux which carries a linear time action of a subgroup of index two.
This index two subgroup action on Knutson-Tao-Woodward puzzles is the group
generated by the puzzle mirror reflections with label swapping. One shows that, as
happens in puzzles, half of the twelve symmetries of Littlewood-Richardson coefficients
may also be exhibited on Littlewood-Richardson tableaux by surprisingly easy
maps. The other hidden half symmetries are given by a remaining generator which
enables to reduce those symmetries to the Sch¨utzenberger involution. Purbhoo
mosaics are used to map the action of the subgroup of index two on Littlewood-
Richardson tableaux into the group generated by the puzzle mirror reflections with
label swapping. After Pak and Vallejo one knows that Berenstein-Zelevinsky triangles,
Knutson-Tao hives and Littlewood-Richardson tableaux may be put in correspondence
by linear algebraic maps. We conclude that, regarding the symmetries,
the behaviour of the various combinatorial models for Littlewood-Richardson coefficients
is similar, and the bijections exhibiting them are in a certain sense unique
A bijection between Littlewood-Richardson tableaux and rigged configurations
A bijection is defined from Littlewood-Richardson tableaux to rigged
configurations. It is shown that this map preserves the appropriate statistics,
thereby proving a quasi-particle expression for the generalized Kostka
polynomials, which are q-analogues of multiplicities in tensor products of
irreducible general linear group modules indexed by rectangular partitions.Comment: 66 pages, AMS-LaTeX, requires xy.sty and related file
Promotion operator on rigged configurations of type A
Recently, the analogue of the promotion operator on crystals of type A under
a generalization of the bijection of Kerov, Kirillov and Reshetikhin between
crystals (or Littlewood--Richardson tableaux) and rigged configurations was
proposed. In this paper, we give a proof of this conjecture. This shows in
particular that the bijection between tensor products of type A_n^{(1)}
crystals and (unrestricted) rigged configurations is an affine crystal
isomorphism.Comment: 37 page
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