A-ordered tableaux

In resolution proof procedures refinements based on A-orderings of literals have a long tradition and are well investigated. In tableau proof procedures such refinements were only recently introduced by the authors of the present paper. In this paper we prove the following results: we give a completeness proof of A-ordered ground clause tableaux which is a lot easier to follow than the previous one. The technique used in the proof is extended to the non-clausal case as well as to the non-ground case and we introduce an ordered version of Hintikka sets that shares the model existence property of standard Hintikks sets. We show that A-ordered tableaux are a proof confluent refinement of tableaux and that A-ordered tableaux together with the connection refinement yield an incomplete proof procedure. We introduce A-ordered first-order NNF tableaux, prove their completeness, and we briefly discuss implementation issues

Positroid Stratification of Orthogonal Grassmannian and ABJM Amplitudes

A novel understanding of scattering amplitudes in terms of on-shell diagrams and positive Grassmannian has been recently established for four dimensional Yang-Mills theories and three dimensional Chern-Simons theories of ABJM type. We give a detailed construction of the positroid stratification of orthogonal Grassmannian relevant for ABJM amplitudes. On-shell diagrams are classified by pairing of external particles. We introduce a combinatorial aid called `OG tableaux' and map each equivalence class of on-shell diagrams to a unique tableau. The on-shell diagrams related to each other through BCFW bridging are naturally grouped by the OG tableaux. Introducing suitably ordered BCFW bridges and positive coordinates, we construct the complete coordinate charts to cover the entire positive orthogonal Grassmannian for arbitrary number of external particles. The graded counting of OG tableaux suggests that the positive orthogonal Grassmannian constitutes a combinatorial polytope.Comment: 32 pages, 23 figures; v2. minor corrections; v3. several clarifications and minor improvement

A-ordered tableaux

In resolution proof procedures refinements based on A-orderings of literals have a long tradition and are well investigated. In tableau proof procedures such refinements were only recently introduced by the authors of the present paper. In this paper we prove the following results: we give a completeness proof of A-ordered ground clause tableaux which is a lot easier to follow than the previous one. The technique used in the proof is extended to the non-clausal case as well as to the non-ground case and we introduce an ordered version of Hintikka sets that shares the model existence property of standard Hintikks sets. We show that A-ordered tableaux are a proof confluent refinement of tableaux and that A-ordered tableaux together with the connection refinement yield an incomplete proof procedure. We introduce A-ordered first-order NNF tableaux, prove their completeness, and we briefly discuss implementation issues

Inhomogeneous lattice paths, generalized Kostka polynomials and An−1_{n-1} supernomials

Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works and by Kashiwara's theory of crystal bases we define a statistic on paths yielding two novel classes of polynomials. One of these provides a generalization of the Kostka polynomials while the other, which we name the A$_{n-1}$ supernomial, is a $q$-deformation of the expansion coefficients of products of Schur polynomials. Many well-known results for Kostka polynomials are extended leading to representations of our polynomials in terms of a charge statistic on Littlewood-Richardson tableaux and in terms of fermionic configuration sums. Several identities for the generalized Kostka polynomials and the A$_{n-1}$ supernomials are proven or conjectured. Finally, a connection between the supernomials and Bailey's lemma is made.Comment: 44 pages, figures, AMS-latex; improved version to appear in Commun. Math. Phys., references added, some statements clarified, relation to other work specifie

Combinatorics of Oscillating Tableaux

In this paper, we first introduce the RSK algorithm, which gives a correspondence between integer sequences and standard tableaux. Then we introduce Schensted’s theorem and Greene’s theorem that describe how the shape of the standard tableau is determined by the sequence. We list four different bijections constructed by using the RSK insertion. The first one is a bijection between vacillating tableaux and pairs (P, T), where P is a set of ordered pairs and T is a standard tableau. The second one is a bijection between set partitions of [n] and vacillating tableaux. The third one is about partial matchings and up-down tableaux and the final one is from sequences to pairs (T, P), where T is still a standard tableau and P is a special oscillating tableau. We analyze some combinatorial statistics via these bijections

Accessible Proof of Standard Monomial Basis for Coordinatization of Schubert Sets of Flags

The main results of this paper are accessible with only basic linear algebra. Given an increasing sequence of dimensions, a flag in a vector space is an increasing sequence of subspaces with those dimensions. The set of all such flags (the flag manifold) can be projectively coordinatized using products of minors of a matrix. These products are indexed by tableaux on a Young diagram. A basis of "standard monomials" for the vector space generated by such projective coordinates over the entire flag manifold has long been known. A Schubert variety is a subset of flags specified by a permutation. Lakshmibai, Musili, and Seshadri gave a standard monomial basis for the smaller vector space generated by the projective coordinates restricted to a Schubert variety. Reiner and Shimozono made this theory more explicit by giving a straightening algorithm for the products of the minors in terms of the right key of a Young tableau. Since then, Willis introduced scanning tableaux as a more direct way to obtain right keys. This paper uses scanning tableaux to give more-direct proofs of the spanning and the linear independence of the standard monomials. In the appendix it is noted that this basis is a weight basis for the dual of a Demazure module for a Borel subgroup of GL(n). This paper contains a complete proof that the characters of these modules (the key polynomials) can be expressed as the sums of the weights for the tableaux used to index the standard monomial bases.Comment: 18 page