Semistandard Tableaux Research Articles

Lenart's Bijection via Bumpless Pipe Dreams

Pipe dreams and bumpless pipe dreams for vexillary permutations are each known to be in bijection with certain semistandard tableaux via maps due to Lenart and Weigandt, respectively. Recently, Gao and Huang have defined a bijection between the former two sets. In this note we show for vexillary permutations that the Gao-Huang bijection preserves the associated tableaux, giving a new proof of Lenart's result. Our methods extend to give a recording tableau for any bumpless pipe dream.

Free Action and Cyclic Sieving on Skew Semi-standard Young Tableaux

In this note, we provide a short proof of Theorem 4.3 in the paper titled Crystals, semistandard tableaux and cyclic sieving phenomenon, by Y.-T. Oh and E. Park, which concerns a cyclic sieving phenomenon on semi-standard Young tableaux. We also extend their result to skew shapes.

Candidate for the crystal B(−∞) for the queer Lie superalgebra

It is shown that the direct limit of the semistandard decomposition tableau model for polynomial representations of the queer Lie superalgebra exists, which is believed to be the crystal for the upper half of the corresponding quantum group. An extension of this model to describe the direct limit combinatorially is given. Furthermore, it is shown that the polynomial representations may be recovered from the limit in most cases.

Determinantal and Pfaffian identities for ninth variation skew Schur functions and Q-functions

Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald’s ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel’s Pfaffian outside decomposition identity. As corollaries to this we derive Pfaffian identities generalising those of Józefiak–Pragacz, Nimmo, and most recently Okada. As a preamble to this we present a parallel development based on (unshifted) semistandard tableaux that leads to a ninth variation version of the outside decomposition determinantal identity of Hamel and Goulden. In this case the corollaries we offer include determinantal identities generalising the Schur and skew Schur function identities of Jacobi–Trudi, Giambelli, Lascoux–Pragacz, Stembridge, and Okada.

The Berenstein–Kirillov group and cactus groups

Berenstein and Kirillov have studied the action of Bender–Knuth moves on semistandard tableaux. Losev has studied a cactus group action in Kazhdan–Lusztig theory; in type A this action can also be identified in the work of Henriques and Kamnitzer. We establish the relationship between the two actions. We show that the Berenstein–Kirillov group is a quotient of the cactus group. We use this to derive previously unknown relations in the Berenstein–Kirillov group.We also determine precise implications between subsets of relations in the two groups, which yields a presentation for cactus groups in terms of Bender–Knuth generators.

Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux

We study Gröbner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by matching field tableaux in the sense of Sturmfels and Zelevinsky in [31]. We prove that these ideals are all quadratically generated and they yield a SAGBI basis of the Plücker algebra. This leads to a new family of toric degenerations of Grassmannians. Moreover, we apply our results to construct a family of Gröbner degenerations of Schubert varieties inside Grassmannians. We provide a complete characterization of toric ideals among these degenerations in terms of the combinatorics of matching fields, permutations and semi-standard tableaux.

Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions

The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters such that the generalization still defines symmetric functions. We outline two self-contained proofs of this fact, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2.

A ratio of alternants formula for loop Schur functions

Lam and Pylyavskyy introduced loop symmetric functions as a generalization of symmetric functions. They defined loop Schur functions as generating functions over semistandard tableaux with respect to a `colored weight,' and they proved a Jacobi--Trudi-style determinantal formula for these generating functions. We prove that loop Schur functions can be expressed as a ratio of `loop alternants,' extending the analogy with Schur functions. As an application, we give a new proof of the loop version of the Murnaghan--Nakayama rule.

Abacus Proofs of Cauchy Product Identities for Schur Polynomials

The well-known Cauchy Product Identities state that $$\prod _{i,j} (1-x_iy_j)^{-1}= \sum _{\lambda } s_{\lambda }(X) s_{\lambda }(Y)$$ and $$\prod _{i,j} (1+x_iy_j)=\sum _{\lambda } s_{\lambda }(X)s_{\lambda '}(Y)$$ , where $$s_{\lambda }$$ denotes a Schur polynomial. The classical bijective proofs of these identities use the Robinson–Schensted–Knuth (RSK) algorithm to map matrices with entries in $${\mathbb {Z}}_{\ge 0}$$ or $$\{0,1\}$$ to pairs of semistandard tableaux. This paper gives new involution-based proofs of the Cauchy Product Identities using the abacus model for antisymmetrized Schur polynomials. These proofs provide a novel combinatorial perspective on these formulas in which carefully engineered sign cancellations gradually impose more and more structure on collections of matrices.

A Notion of Inversion Number Associated to Certain Quiver Flag Varieties

We define an algebraic variety $X(d,A)$ consisting of matrices whose rows and columns are partial flags. This is a smooth, projective variety, and we describe it as an iterated bundle of Grassmannian varieties. Moreover, we show that $X(d,A)$ has a cell decomposition, in which the cells are parametrized by certain matrices of sets and their dimensions are given by a notion of inversion number. On the other hand, we consider the Spaltenstein variety of partial flags which are stabilized by a given nilpotent endomorphism. We partition this variety into locally closed subvarieties which are affine bundles over certain varieties called $Y_T$, parametrized by semistandard tableaux $T$. We show that the varieties $Y_T$ are in fact isomorphic to varieties of the form $X(d,A)$. We deduce that each variety $Y_T$ has a cell decomposition, in which the cells are parametrized by certain row-increasing tableaux obtained by permuting the entries in the columns of $T$ and their dimensions are given by the inversion number recently defined by P. Drube for such row-increasing tableaux.

Sliding Presentation of the Jeux de Taquin for Classical Lie Groups

The simple \(GL(n,\mathbb {C})\)-modules are described by using semistandard Young tableaux. Any semistandard skew tableau can be transformed into a well defined semistandard tableau by a combinatorial operation, the Schutzenberger jeu de taquin. Associated to the classical Lie groups \(SP(2n,\mathbb {C})\), \(SO(2n+1,\mathbb {C})\), there are other notions of semistandard Young tableaux and jeux de taquin. In this paper, we study these various jeux de taquin, proving that each of them has a simple and explicit formulation as a step-by-step sliding. Any of these jeux de taquin is the restriction of the orthogonal one, associated to \(SO(2n+1,\mathbb {C})\).

A theory of pictures for quasi-posets

The theory of pictures between posets is known to encode much of the combinatorics of symmetric group representations and related topics such as Young diagrams and tableaux. Many reasons, combinatorial (e.g. since semi-standard tableaux can be viewed as double quasi-posets) and topological (quasi-posets identify with finite topologies) lead to extend the theory to quasi-posets. This is the object of the present article.

A Bijective Proof of Vershik’s Relations for the Kostka Numbers

We turn the tableau insertion algorithm for semistandard tableaux to a sequence called a buming route. We also define a reverse bumping route by the reverse insertion algorithm. These sequences are used to give a bijective proof of Vershik's relations for the Kostka numbers.

Pieri Rules for Classical Groups and Equinumeration between Generalized Oscillating Tableaux and Semistandard Tableaux

We present several equinumerous results between generalized oscillating tableaux and semistandard tableaux and give a representation-theoretic proof to them. As one of the key ingredients of the proof, we provide Pieri rules for the symplectic and orthogonal groups.

$q$-randomized Robinson–Schensted–Knuth correspondences and random polymers

We introduce and study q -randomized Robinson–Schensted–Knuth (RSK) correspondences which interpolate between the classical ( q=0 ) and geometric q \nearrow1 ) RSK correspondences (the latter ones are sometimes also called tropical). For 0<q<1 our correspondences are randomized, i.e., the result of an insertion is a certain probability distribution on semistandard Young tableaux. Because of this randomness, we use the language of discrete time Markov dynamics on two-dimensional interlacing particle arrays (these arrays are in a natural bijection with semistandard tableaux). Our dynamics act nicely on a certain class of probability measures on arrays, namely, on q -Whittaker processes (which are t=0 versions of Macdonald processes of Borodin–Corwin [8]). We present four Markov dynamics which for q=0 reduce to the classical row or column RSK correspondences applied to a random input matrix with independent geometric or Bernoulli entries. Our new two-dimensional discrete time dynamics generalize and extend several known constructions. (1) The discrete time q -TASEPs studied by Borodin–Corwin [7] arise as one-dimensional marginals of our „column" dynamics. In a similar way, our“row" dynamics lead to discrete time q -PushTASEPs – new integrable particle systems in the Kardar–Parisi–Zhang universality class. We employ these new one-dimensional discrete time systems to establish a Fredholm determinantal formula for the two-sided continuous time q -PushASEP conjectured by Corwin–Petrov [23]. (2) In a certain Poisson-type limit (from discrete to continuous time), our two-dimensional dynamics reduce to the q -randomized column and row Robinson–Schensted correspondences introduced by O’Connell–Pei [59] and Borodin–Petrov [15], respectively. (3) In a scaling limit as q\nearrow1 , two of our four dynamics on interlacing arrays turn into the geometric RSK correspondences associated with log-Gamma (introduced by Seppäläinen [70] or strict-weak (introduced independently by O’Connell–Ortmann [58] and Corwin–Seppäläinen–Shen [25] directed random lattice polymers.

Bijections between oscillating tableaux and (semi)standard tableaux via growth diagrams

We prove that the number of oscillating tableaux of length n with at most k columns, starting at ∅ and ending at the one-column shape (1m), is equal to the number of standard Young tableaux of size n with m columns of odd length, all columns of length at most 2k. This refines a conjecture of Burrill, which it thereby establishes. We prove as well a “Knuth-type” extension stating a similar equi-enumeration result between generalised oscillating tableaux and semistandard tableaux.

Refined Dual Stable Grothendieck Polynomials and Generalized Bender-Knuth Involutions

The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the $K$-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries $1$ and $2$.

Relating the right key to the type A filling map and minimal defining chains

There are several combinatorial methods that can be used to produce type A Demazure characters (key polynomials). The alcove path model of Lenart and Postnikov provides a procedure that inputs a semistandard tableau T and outputs a saturated chain in the Bruhat order. The final permutation in this Bruhat chain determines a family of Demazure characters for which T contributes its weight. Separately, the right key of T introduced by Lascoux and Schützenberger also determines a family of Demazure characters for which T contributes its weight. In this paper we show that the final permutation in this Bruhat chain corresponds bijectively to the right key of the tableau. We will also show that the minimal defining sequence for T, as introduced by Lakshmibai, Musili, and Seshadri, is a sub-chain of this Bruhat chain. Most notably their final elements coincide. From this it follows that the generating sets for the Demazure characters produced by these three methods are equivalent.

JACOBI-TRUDI TYPE FORMULA FOR PARABOLICALLY SEMISTANDARD TABLEAUX

The notion of a parabolically semistandard tableau is a generalisation of Young tableau, which explains combinatorial aspect of various Howe dualities of type A. We prove a Jacobi-Trudi type formula for the character of parabolically semistandard tableaux of a given generalised partition shape by using non-intersecting lattice paths.

Extremal Loop Weight Modules for U q ( sl ̂ ∞ ) $\mathcal {U}_{q}(\hat {sl}_{\infty })$

We construct by fusion product new families of irreducible representations of the quantum affinization $\mathcal {U}_{q}(\hat {sl}_{\infty })$ . The action is defined via the Drinfeld coproduct and is related to the crystal structure of semi-standard tableaux of type A ∞ . We call these representations extremal loop weight modules. The main motivations are applications to quantum toroidal algebras $\mathcal {U}_{q}(sl_{n+1}^{tor})$ : we prove the conjectural link between $\mathcal {U}_{q}(\hat {sl}_{\infty })$ and $\mathcal {U}_{q}(sl_{n+1}^{tor})$ stated in Hernandez (J. Algebra 329, 147–162, 2011) for these families of representations. We recover in this way the extremal loop weight modules obtained in Mansuy (arXiv: 1305.3481 ).