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 Robinson-Schensted Correspondence and A2A_2-web Bases

We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to $[n,n,n]$: the {\em reduced web} basis associated to Kuperberg's combinatorial description of the spider category; and the {\em left cell basis} for the left cell construction of Kazhdan and Lusztig. In the case of $[n,n]$, the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the images of these bases under classical maps: the {\em Robinson-Schensted algorithm} between permutations and Young tableaux and {\em Khovanov-Kuperberg's bijection} between Young tableaux and reduced webs. One main result uses Vogan's generalized $\tau$-invariant to uncover a close structural relationship between the web basis and the left cell basis. Intuitively, generalized $\tau$-invariants refine the data of the inversion set of a permutation. We define generalized $\tau$-invariants intrinsically for Kazhdan-Lusztig left cell basis elements and for webs. We then show that the generalized $\tau$-invariant is preserved by these classical maps. Thus, our result allows one to interpret Khovanov-Kuperberg's bijection as an analogue of the Robinson-Schensted correspondence. Despite all of this, our second main result proves that the reduced web and left cell bases are inequivalent; that is, these bijections are not $S_{3n}$-equivariant maps.Comment: 34 pages, 23 figures, minor corrections and revisions in version

Linear time equivalence of Littlewood--Richardson coefficient symmetry maps

Benkart, Sottile, and Stroomer have completely characterized by Knuth and dual Knuth equivalence a bijective proof of the conjugation symmetry of the Littlewood-Richardson coefficients. Tableau-switching provides an algorithm to produce such a bijective proof. Fulton has shown that the White and the Hanlon-Sundaram maps are versions of that bijection. In this paper one exhibits explicitly the Yamanouchi word produced by that conjugation symmetry map which on its turn leads to a new and very natural version of the same map already considered independently. A consequence of this latter construction is that using notions of Relative Computational Complexity we are allowed to show that this conjugation symmetry map is linear time reducible to the Schutzenberger involution and reciprocally. Thus the Benkart-Sottile-Stroomer conjugation symmetry map with the two mentioned versions, the three versions of the commutative symmetry map, and Schutzenberger involution, are linear time reducible to each other. This answers a question posed by Pak and Vallejo.Comment: accepted in Discrete Mathematics and Theoretical Computer Science Proceeding

Monodromy and K-theory of Schubert curves via generalized jeu de taquin

We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S(\lambda_\bullet)$, 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 $\omega$ 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 $\mathbb{RP}^1$, with $\omega$ as the monodromy operator. We provide a local algorithm for computing $\omega$ 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 $K$-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the $K$-theory and real geometry of $S(\lambda_\bullet)$.Comment: 33 pages, 12 figures including 2 color figures; to appear in the Journal of Algebraic Combinatoric

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

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