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

International audience We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves Spλ‚q, 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 ω 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 RP1, with ω as the monodromy operator.We provide a fast, local algorithm for computing ω 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. Using this bijection, we give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of Spλ‚q.

Jeu de taquin, uniqueness of rectification and ultradiscrete KP

In this article, we study tropical-theoretic aspects of the ‘rectification algorithm’ on skew Young tableaux. It is shown that the algorithm is interpreted as a time evolution of some tropical integrable system. By using this fact, we construct a new combinatorial map that is essentially equivalent to the rectification algorithm. Some of properties of the rectification can be seen more clearly via this map. For example, the uniqueness of a rectification boils down to an easy combinatorial problem. Our method is mainly based on the two previous researches: the theory of geometric tableaux by Noumi–Yamada, and the study on the relationship between jeu de taquin slides and the ultradiscrete KP equation by Mikami and Katayama–Kakei.

Unique Rectification in $d$-Complete Posets: Towards the $K$-Theory of Kac-Moody Flag Varieties

The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of 'unique rectification targets' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to $K$-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's '$d$-complete posets' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of $d$-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain $K$-theoretic Schubert structure constants in the Kac-Moody setting.

Promotion of Increasing Tableaux: Frames and Homomesies

A key fact about M.-P. Schützenberger's (1972) promotion operator on rectangular standard Young tableaux is that iterating promotion once per entry recovers the original tableau. For tableaux with strictly increasing rows and columns, H. Thomas and A. Yong (2009) introduced a theory of $K$-jeu de taquin with applications to $K$-theoretic Schubert calculus. The author (2014) studied a $K$-promotion operator $\mathcal{P}$ derived from this theory, but observed that this key fact does not generally extend to $K$-promotion of such increasing tableaux. Here, we show that the key fact holds for labels on the boundary of the rectangle. That is, for $T$ a rectangular increasing tableau with entries bounded by $q$, we have $\mathsf{Frame}(\mathcal{P}^q(T)) = \mathsf{Frame}(T)$, where $\mathsf{Frame}(U)$ denotes the restriction of $U$ to its first and last row and column. Using this fact, we obtain a family of homomesy results on the average value of certain statistics over $K$-promotion orbits, extending a $2$-row theorem of J. Bloom, D. Saracino, and the author (2016) to arbitrary rectangular shapes.

Minimal Orbits of Promotion

We give a bijection between the symmetric group $S_n$, and the set of standard Young tableaux of rectangular shape $m^n$, $m \geq n$, that have order $n$ under jeu de taquin promotion.

One-dimensional Schubert Problems with Respect to Osculating Flags

We consider Schubert problems with respect to flags osculating the rational normal curve. These problems are of special interest when the osculation points are all real. In this case, for zerodimensional Schubert problems, the solutions are “ as real as possible”. Recent work by Speyer has extended the theory to the moduli space allowing the points to collide. This gives rise to smooth covers (ℝ), with structure and monodromy described by Young tableaux and jeu de taquin.In this paper, we give analogous results on one-dimensional Schubert problems over .Their(real) geometry turns out to be described by orbits of Schützenberger promotion and a related operation involving tableau evacuation. Over M 0,r, our results show that the real points of the solution curves are smooth.We also find a new identity involving “first-order” K-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof.

EQUIVARIANT -THEORY OF GRASSMANNIANS

We address a unification of the Schubert calculus problems solved by Buch [A Littlewood–Richardson rule for the $K$-theory of Grassmannians, Acta Math. 189 (2002), 37–78] and Knutson and Tao [Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J.119(2) (2003), 221–260]. That is, we prove a combinatorial rule for the structure coefficients in the torus-equivariant $K$-theory of Grassmannians with respect to the basis of Schubert structure sheaves. This rule is positive in the sense of Anderson et al. [Positivity and Kleiman transversality in equivariant $K$-theory of homogeneous spaces, J. Eur. Math. Soc.13 (2011), 57–84] and in a stronger form. Our work is based on the combinatorics of genomic tableaux and a generalization of Schützenberger’s [Combinatoire et représentation du groupe symétrique, in Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976, Lecture Notes in Mathematics, 579 (Springer, Berlin, 1977), 59–113] jeu de taquin. Using our rule, we deduce the two other combinatorial rules for these coefficients. The first is a conjecture of Thomas and Yong [Equivariant Schubert calculus and jeu de taquin, Ann. Inst. Fourier (Grenoble) (2013), to appear]. The second (found in a sequel to this paper) is a puzzle rule, resolving a conjecture of Knutson and Vakil from 2005.