Semistandard Tableaux Research Articles

The Combinatorics of $\mathsf{A_2}$-Webs

The nonelliptic $\mathsf{A_2}$-webs with $k$ "$+$''s on the top boundary and $3n-2k$ "$-$''s on the bottom boundary combinatorially model the space $\mathsf{Hom}_{\mathfrak{sl}_3}(\mathsf{V}^{\otimes (3n-2k)}, \mathsf{V}^{\otimes k})$ of $\mathfrak{sl}_3$-module maps on tensor powers of the natural 3-dimensional $\mathfrak{sl}_3$-module $\mathsf{V}$, and they have connections with the combinatorics ofSpringer varieties. Petersen, Pylyavskyy, and Rhodes showed that the set of such $\mathsf{A_2}$-webs and the set of semistandard tableaux of shape $(3^n)$ and type $\{1^2,\dots,k^2,k+1,\dots, 3n-k\}$ have the same cardinalities. In this work, we use the $\sf{m}$-diagrams introduced by Tymoczko and the Robinson-Schensted correspondence to construct an explicit bijection, different from the one given by Russell, between these two sets. In establishing our result, we show that the pair of standard tableaux constructed using the notion of path depth is the same as the pair constructed from applying the Robinson-Schensted correspondence to a $3\,2\,1$-avoiding permutation. We also obtain a bijection between such pairs of standard tableaux and Westbury's $\mathsf{A_2}$ flow diagrams.

Cyclic sieving of increasing tableaux and small Schröder paths

An increasing tableau is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a bijection between rectangular 2-row increasing tableaux and small Schröder paths. We demonstrate relations between the jeu de taquin for increasing tableaux developed by H. Thomas and A. Yong and the combinatorics of tropical frieze patterns. We then use this jeu de taquin to present new instances of the cyclic sieving phenomenon of V. Reiner, D. Stanton, and D. White, generalizing results of D. White and of J. Stembridge.

Super tableaux and a branching rule for the general linear Lie superalgebra

In this note, we formulate and prove a branching rule for simple polynomial modules of the Lie superalgebra . Our branching rules depend on the conjugacy class of a Borel subalgebra. A Gelfand–Tsetlin basis of a polynomial module associated to each Borel subalgebra is obtained in terms of generalized semistandard tableaux.

An explicit bijection between semistandard tableaux and non-elliptic sl 3 webs

The sl 3 spider is a diagrammatic category used to study the representation theory of the quantum group U q (sl 3). The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov–Kuperberg observe that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection via a recursive growth algorithm. Recently, Tymoczko gave a simple version of this bijection in the case that the tableaux are standard and used it to study rotation and join of webs. This article builds on Tymoczko’s bijection to give a simple and explicit algorithm for constructing all non-elliptic sl 3 webs. As an application, we generalize results of Petersen–Pylyavskyy–Rhoades and Tymoczko proving that, for all non-elliptic sl 3 webs, rotation corresponds to jeu de taquin promotion and join corresponds to shuffling.

A $q$-weighted version of the Robinson-Schensted algorithm

We introduce a q-weighted version of the Robinson-Schensted (column insertion) algorithm which is closely connected to q-Whittaker functions (or Macdonald polynomials with t=0) and reduces to the usual Robinson-Schensted algorithm when q=0. The q-insertion algorithm is `randomised', or `quantum', in the sense that when inserting a positive integer into a tableau, the output is a distribution of weights on a particular set of tableaux which includes the output which would have been obtained via the usual column insertion algorithm. There is also a notion of recording tableau in this setting. We show that the distribution of weights of the pair of tableaux obtained when one applies the q-insertion algorithm to a random word or permutation takes a particularly simple form and is closely related to q-Whittaker functions. In the case $0\le q<1$, the q-insertion algorithm applied to a random word also provides a new framework for solving the q-TASEP interacting particle system introduced (in the language of q-bosons) by Sasamoto and Wadati (1998) and yields formulas which are equivalent to some of those recently obtained by Borodin and Corwin (2011) via a stochastic evolution on discrete Gelfand-Tsetlin patterns (or semistandard tableaux) which is coupled to the q-TASEP process. We show that the sequence of P-tableaux obtained when one applies the q-insertion algorithm to a random word defines another, quite different, evolution on semistandard tableaux which is also coupled to the q-TASEP process.

On Homomorphisms Indexed by Semistandard Tableaux

We describe techniques which may be used to compute the homomorphism space between Specht modules for the Hecke algebras of type A. We prove a q-analogue of a result of Fayers and Martin and show how it may be applied to construct homomorphisms between Specht modules. In particular, we show that in certain cases the dimension of the homomorphism space is given by the corank of a matrix whose entries we write down explicitly.

A note on bideterminants for Schur superalgebras

Let S ( m | n , r ) Z be a Z -form of a Schur superalgebra S ( m | n , r ) generated by elements ξ i , j . We solve a problem of Muir and describe a Z -form of a simple S ( m | n , r ) -module D λ , Q over the field Q of rational numbers, under the action of S ( m | n , r ) Z . This Z -form is the Z -span of modified bideterminants [ T ℓ : T i ] defined in this work. We also prove that each [ T ℓ : T i ] is a Z -linear combination of modified bideterminants corresponding to ( m | n ) -semistandard tableaux T i .

Semistandard tableaux associated with generalized labellings of posets

We describe a correspondence between a family of labelled partially ordered sets and semistandard Young tableaux. Moreover, we define some operations among labelled posets which naturally correspond to operations among the associated semistandard Young tableaux.

Robinson–Schensted–Knuth Correspondence and Weak Polynomial Identities of M1,1(E)

In this paper, it is proved that the ideal Iw of the weak polynomial identities of the superalgebra M1,1(E) is generated by the proper polynomials [x1, x2, x3] and [x2, x1] [x3, x1] [x4, x1]. This is proved for any infinite field F of characteristic different from 2. Precisely, if B is the subalgebra of the proper polynomials of F&lt;X&gt;, we determine a basis and the dimension of any multihomogeneous component of the quotient algebra B / (B ∩ Iw). We also compute the Hilbert series of this algebra. One of the main tools of this paper is a variant we found of the Robinson–Schensted–Knuth correspondence defined for single semistandard tableaux of double shape.

Linear extension of the Robinson–Schensted algorithm

AbstractThe Robinson–Schensted (RS) algorithm demonstrates a bijection between the set of magnetic configurations f and the set of pairs of tableaux: a semistandard Weyl tableau P(f) accompanied by a standard Young tableau Q(f). We show that it is not only a bijection between sets, but it can be extended to a linear unitary transformation within the space of all quantum states of the magnet. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Proof of two conjectures of Zuber on fully packed loop configurations

Two conjectures of Zuber (On the counting of fully packed loops configurations. Some new conjectures, Electronic J. Combin 11 (2004)) on the enumeration of configurations in the fully packed loop model on the square grid with periodic boundary conditions, which have a prescribed linkage pattern, are proved. Following an idea of de Gier (Loops, matchings and alternating-sign matrices, Discrete Math., to appear), the proofs are based on bijections between such fully packed loop configurations and rhombus tilings, and the hook-content formula for semistandard tableaux.

Kinematics of the Heisenberg chain and irreducible bases of the Weyl duality

Quantum kinematics of the linear Heisenberg ring, consisting of N crystal nodes, each with spin s, is presented in terms of the Weyl duality between actions of the symmetric and unitary groups in the space of quantum states of the magnet. This space is spanned on magnetic configurations which gives rise to an application of the combinatorial Robinson–Schensted–Knuth algorithm for a unique classification of irreducible basis of the duality of Weyl in terms of a pair of tableaux: a standard Young tableau in the alphabet of nodes, accompanied by a semistandard Weyl tableau in the alphabet of spins. Similarities and distinctions between various group-theoretic and combinatorial objects are discussed within the context of the magnetic interpretation. In particular, the role of the spectrum of Jucys–Murphy operators in the clasification and construction of magnetic eigenstates corresponding to Young tableaux is illustrated.

Distribution of the shape of Markovian random words

The distribution of the shape λ of the semi-standard tableau of a random word in k letters is asymptotically given by the distribution of the spectrum of a random traceless k×k Gaussian Unitary Ensemble (GUE) matrix provided that these letters are independent with uniform distribution. Kuperberg (2002) conjectured that this result by Johansson (2001) remains valid if the letters of the word are generated by an irreducible Markov chain on the alphabet with cyclic transition matrix. In this paper we give a proof of this conjecture for an alphabet with k=2 letters.

Branching formula for q-Littlewood–Richardson coefficients

A generalized inversion statistic is introduced on k-tuples of semistandard tableaux. It is shown that the cospin of a semistandard k-ribbon tableau is equal to the generalized inversion number of its k-quotient. This leads to a branching formula for the q-analogue of Littlewood–Richardson coefficients defined by Lascoux, Leclerc, and Thibon. This branching formula generalizes a recurrence of Garsia and Procesi involving Kostka–Foulkes polynomials.

Random words, quantum statistics, central limits, random matrices

Recently Tracy and Widom conjectured [math.CO/9904042] and Johansson proved [math.CO/9906120] that the expected shape \lambda of the semi-standard tableau produced by a random word in k letters is asymptotically the spectrum of a random traceless k by k GUE matrix. In this article we give two arguments for this fact. In the first argument, we realize the random matrix itself as a quantum random variable on the space of random words, if this space is viewed as a quantum state space. In the second argument, we show that the distribution of \lambda is asymptotically given by the usual local limit theorem, but the resulting Gaussian is disguised by an extra polynomial weight and by reflecting walls. Both arguments more generally apply to an arbitrary finite-dimensional representation V of an arbitrary simple Lie algebra g. In the original question, V is the defining representation of g = su(k).

A Combinatorial Proof of a Recursion for the q-Kostka Polynomials

The Kostka numbers Kλμ play an important role in symmetric function theory, representation theory, combinatorics and invariant theory. The q-Kostka polynomials Kλμ(q) are the q-analogues of the Kostka numbers and generalize and extend the mathematical meaning of the Kostka numbers. Lascoux and Schützenberger proved one can attach a non-negative integer statistic called charge to a semistandard tableau of shape λ and content μ such that the Kostka polynomial Kλμ(q) is the generating function for charge on those semistandard tableaux. We will give two new descriptions of charge and prove several new properties of this statistic. These new descriptions of charge will be used to give a combinatorial proof of a content reducing recursion for the q-Kostka polynomials originally proved by A. M. Garsia and C. Procesi (1992, Adv. in Math.94, 82–138).

A New Plethysm Formula for Symmetric Functions

This paper gives a new formula for the plethysm of power-sum symmetric functions and Schur symmetric functions with one part. The form of the main result is that for μ v b,p_\mu(\underline x) \circ s_a(\underline x) = \sum_{T} \underline \omega^{{\rm maj}_\mu(T)} s_{{\rm sh} (T)}(\underline x) where the sum is over semistandard tableaux T of weight ab, {\underline\omega} is a root of unity, and majμ(T) is a major index like statistic on semistandard tableaux. An Sb-representation, denoted Sλ,b, is defined. In the special case when λ v b, Sλ,b is the Specht module corresponding to λ. It is shown that the character of Sλ,b on elements of cycle type μ is \sum_T {\underline\omega}^{{\rm maj}_\mu (T)} where the sum is over semistandard tableaux T of shape λ and weight ab. Moreover, the eigenvalues of the action of an element of cycle type μ acting on Sλ,b are l{\underline \omega}^{{\rm maj}_\mu(T)} : Tr. This generalizes J. Stembridge‘s result [11] on the eigenvalues of elements of the symmetric group acting on the Specht modules.