Young Tableaux Research Articles

Ground States of theSU(N)Heisenberg Model

The SU(N) Heisenberg model with various single-row representations is investigated by quantum Monte Carlo simulations. While the zero-temperature phase boundary agrees qualitatively with the theoretical predictions based on the 1/N expansion, some unexpected features are also observed. For N> or =5 with the fundamental representation, for example, it is suggested that the ground states possess exact or approximate U(1) degeneracy. In addition, for the representation of Young tableau with more than one column, the ground state shows no valence-bond-solid order even at N greater than the threshold value.

Markov Measures on Young Tableaux and Induced Representations of the Infinite Symmetric Group

We show that the class of so‐called Markov representations of the infinite symmetric group ${\mathfrak G}_{\bf N}$, associated with Markov measures on the space of infinite Young tableaux, coincides with the class of simple representations, i.e., inductive limits of representations with simple spectrum. The spectral measure of an arbitrary representation of ${\mathfrak G}_{\bf N}$ with simple spectrum is equivalent to a multi‐Markov measure on the space of Young tableaux. We also show that the representations of ${\mathfrak G}_{\bf N}$ induced from the identity representations of two‐block Young subgroups are Markov and find explicit formulas for the transition probabilities of the corresponding Markov measures. The induced representations are studied with the help of the tensor model of two‐row representations of the symmetric groups; in particular, we deduce explicit formulas for the Gelfand–Tsetlin basis in the tensor models.

A new Young tableau method of obtaining equivalent-electron Young basis

In this paper, the projection function of regular Young tableau of equivalent-electrons is proved to be a Slauer function, and the operation rules of Young's permutation operators operating on the Slater function are given. The Young operators can be distinguished as elimination operator and effective permutation operator by the normalization of Young basis, then the constructive rules of permutation operators of the orthogonal normalization of Young basis is given. By this method, the number of Young's permutation operators is reduced and the new Young tableau method of obtaining equivalent-electron Young basis is represented.

Limit shapes for random square Young tableaux

Our main result is a limit shape theorem for the two-dimensional surface defined by a uniform random n × n square Young tableau. The analysis leads to a calculus of variations minimization problem that resembles the minimization problems studied by Logan–Shepp, Vershik–Kerov, and Cohn–Larsen–Propp. We solve this problem by developing a general technique for solving variational problems of this kind. An extension to rectangular Young tableaux is also given. We also apply the main result to show that the location of a particular entry in the tableau is in the limit governed by a semicircle distribution, and to the study of extremal Erdös–Szekeres permutations, namely permutations of the numbers 1 , 2 , … , n 2 whose longest monotone subsequence is of length n.

On the sign-imbalance of skew partition shapes

Let the sign of a skew standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. We examine how the sign property is transferred by the skew Robinson–Schensted correspondence invented by Sagan and Stanley. The result is a remarkably simple generalization of the ordinary non-skew formula. The sum of the signs of all standard tableaux on a given skew shape is the sign-imbalance of that shape. We generalize previous results on the sign-imbalance of ordinary partition shapes to skew ones.

On the combinatorics of crystal graphs, I. Lusztig's involution

In this paper, we continue the development of a new combinatorial model for the irreducible characters of a complex semisimple Lie group. This model, which will be referred to as the alcove path model, can be viewed as a discrete counterpart to the Littelmann path model. It leads to an extensive generalization of the combinatorics of irreducible characters from Lie type A (where the combinatorics is based on Young tableaux, for instance) to arbitrary type; our approach is type-independent. The main results of this paper are: (1) a combinatorial description of the crystal graphs corresponding to the irreducible representations (this result includes a transparent proof, based on the Yang–Baxter equation, of the fact that the mentioned description does not depend on the choice involved in our model); (2) a combinatorial realization (which is the first direct generalization of Schützenberger's involution on tableaux) of Lusztig's involution on the canonical basis exhibiting the crystals as self-dual posets; (3) an analog for arbitrary root systems, based on the Yang–Baxter equation, of Schützenberger's sliding algorithm, which is also known as jeu de taquin (this algorithm has many applications to the representation theory of the Lie algebra of type A).

Holographic Wilson loops

We show that all half-BPS Wilson loop operators in N=4 SYM -- which are labeled by Young tableaus -- have a gravitational dual description in terms of D5-branes or alternatively in terms of D3-branes in AdS_5xS^5. We prove that the insertion of a half-BPS Wilson loop operator in the cal N=4 SYM path integral is achieved by integrating out the degrees of freedom on a configuration of bulk D5-branes or alternatively on a configuration of bulk D3-branes. The bulk D5-brane and D3-brane descriptions are related by bosonization.

Realizations of crystal [formula omitted] using Young tableaux and Young walls

Young tableaux and Young walls are combinatorial schemes realizing the irreducible highest weight crystal B ( λ ) . We modify the notions of Young tableau and Young wall to give a realization of the crystal B ( ∞ ) . For the case A n ( 1 ) , the limit B ∞ of a coherent family of perfect crystals B l is realized as the classical crystal of the set of equivalence classes of slices and the crystal B ( ∞ ) is realized as the affine crystal consisting of reduced Young walls, which, in turn, are concatenations of slices. We also give a new realization of the same crystal over special linear Lie algebra, in the language of Young tableau.

Permutations with short monotone subsequences

We consider permutations of 1 , 2 , … , n 2 whose longest monotone subsequence is of length n and are therefore extremal for the Erdős–Szekeres theorem. Such permutations correspond via the Robinson–Schensted correspondence to pairs of square n × n Young tableaux. We show that all the bumping sequences are constant and therefore these permutations have a simple description in terms of the pair of square tableaux. We deduce a limit shape result for the plot of values of the typical such permutation, and other properties of these extremal permutations.

Asymptotics for the Distributions of Subtableaux in Young and Up-Down Tableaux

Let $\mu$ be a partition of $k$, and $T$ a standard Young tableau of shape $\mu$. McKay, Morse, and Wilf show that the probability a randomly chosen Young tableau of $N$ cells contains $T$ as a subtableau is asymptotic to $f^\mu/k!$ as $N$ goes to infinity, where $f^\mu$ is the number of all tableaux of shape $\mu$. We use a random-walk argument to show that the analogous asymptotic probability for randomly chosen Young tableaux with at most $n$ rows is proportional to $\prod_{1\le i < j\le n}\bigl((\mu_i-i)-(\mu_j-j)\bigr)$; as $n$ goes to infinity, the probabilities approach $f^\mu/k!$ as expected. We have a similar formula for up-down tableaux; the probability approaches $f^\mu/k!$ if $\mu$ has $k$ cells and thus the up-down tableau is actually a standard tableau, and approaches 0 if $\mu$ has fewer than $k$ cells.

Stationary or static space-times and Young tableaux

Algebraic curvature tensors possess generators which can be formed from symmetric or alternating tensors S, A or tensors θ with an irreducible (2 1)-symmetry. In differential geometry examples of curvature formulas are known which contain generators on the basis of S or A realized by differentiable tensor fields in a natural way. We show that certain curvature formulas for stationary or static space-times contain such differentiable realizations of generators based on θ. The tensor θ is connected with the timelike Killing vector field of the space-time. θ lies in a special symmetry class from the in.nite family of irreducible (2 1)- symmetry classes. We determine characteristics of this class. In particular, this class allows a maximal reduction of the length of the curvature formulas. We use a projection formalism by Vladimirov, Young symmetrizers and Littlewood-Richardson products. Computer calculations were carried out by means of the packages Ricci and PERMS.

On the Classification Theorems of Almost-Hermitian or Homogeneous Kähler Structures

A proof by Young tableaux and symmetrizers is given of the classification theorems by Gray and Hervella of almost-Hermitian structures and

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.

Queues, stores, and tableaux

Consider the single-server queue with an infinite buffer and a first-in–first-out discipline, either of type M/M/1 or Geom/Geom/1. Denote by 𝒜 the arrival process and by s the services. Assume the stability condition to be satisfied. Denote by 𝒟 the departure process in equilibrium and by r the time spent by the customers at the very back of the queue. We prove that (𝒟, r) has the same law as (𝒜, s), which is an extension of the classical Burke theorem. In fact, r can be viewed as the sequence of departures from a dual storage model. This duality between the two models also appears when studying the transient behaviour of a tandem by means of the Robinson–Schensted–Knuth algorithm: the first and last rows of the resulting semistandard Young tableau are respectively the last instant of departure from the queue and the total number of departures from the store.

Queues, stores, and tableaux

Consider the single-server queue with an infinite buffer and a first-in–first-out discipline, either of type M/M/1 or Geom/Geom/1. Denote by 𝒜 the arrival process and by s the services. Assume the stability condition to be satisfied. Denote by 𝒟 the departure process in equilibrium and by r the time spent by the customers at the very back of the queue. We prove that (𝒟, r) has the same law as (𝒜, s), which is an extension of the classical Burke theorem. In fact, r can be viewed as the sequence of departures from a dual storage model. This duality between the two models also appears when studying the transient behaviour of a tandem by means of the Robinson–Schensted–Knuth algorithm: the first and last rows of the resulting semistandard Young tableau are respectively the last instant of departure from the queue and the total number of departures from the store.

The octahedron recurrence and [formula omitted] crystals

We study the hive model of g l 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 g l n . The proof of this equivalence uses a new connection between the octahedron recurrence and the Jeu de Taquin and Schützenberger involution procedures on Young tableaux.

Chess Tableaux

A chess tableau is a standard Young tableau in which, for all $i$ and $j$, the parity of the entry in cell $(i,j)$ equals the parity of $i+j+1$. Chess tableaux were first defined by Jonas Sjöstrand in his study of the sign-imbalance of certain posets, and were independently rediscovered by the authors less than a year later in the completely different context of composing chess problems with interesting enumerative properties. We prove that the number of $3\times n$ chess tableaux equals the number of Baxter permutations of $n-1$, as a corollary of a more general correspondence between certain three-rowed chess tableaux and certain three-rowed Dulucq-Guibert nonconsecutive tableaux. The correspondence itself is proved by means of an explicit bijection. We also outline how lattice paths, or rat races, can be used to obtain generating functions for chess tableaux. We conclude by explaining the connection to chess problems, and raising some unanswered questions, e.g., there are striking numerical coincidences between chess tableaux and the Charney-Davis statistic; is there a combinatorial explanation?

The Combinatorics of Orbital Varieties Closures of Nilpotent Order 2 in sl${}_n$

We consider two partial orders on the set of standard Young tableaux. The first one is induced to this set from the weak right order on symmetric group by Robinson-Schensted algorithm. The second one is induced to it from the dominance order on Young diagrams by considering a Young tableau as a chain of Young diagrams. We prove that these two orders of completely different nature coincide on the subset of Young tableaux with 2 columns or with 2 rows. This fact has very interesting geometric implications for orbital varieties of nilpotent order 2 in special linear algebra $sl_n.$

Discrete strip-concave functions, Gelfand–Tsetlin patterns, and related polyhedra

Discrete strip-concave functions considered in this paper are, in fact, equivalent to an extension of Gelfand–Tsetlin patterns to the case when the pattern has a not necessarily triangular but convex configuration. They arise by releasing one of the three types of rhombus inequalities for discrete concave functions (or “hives”) on a “convex part” of a triangular grid. The paper is devoted to a combinatorial study of certain polyhedra related to such functions or patterns, and results on faces, integer points and volumes of these polyhedra are presented. Also some relationships and applications are discussed. In particular, we characterize, in terms of valid inequalities, the polyhedral cone formed by the boundary values of discrete strip-concave functions on a grid having trapezoidal configuration. As a consequence of this result, necessary and sufficient conditions on a pair of vectors to be the shape and content of a semi-standard skew Young tableau are obtained.

THE COINVARIANT ALGEBRA OF THE SYMMETRIC GROUP AS A DIRECT SUM OF INDUCED MODULES

AbstractLet be the coinvariant algebra of the symmetric group . The al gebra hasa natural gradation. For a xed ( 1 ), let ( ; ) (0 1 ) be the directsum of all the homogeneous components of whose degrees are co ngruent tomodulo . In this article, we will show that for each there exis ts a subgroupof and a representation ( ; ) of such that each ( ; ) is induced by ( ; ). 1. IntroductionThroughout this article, we follow [5] for fundamental term inology on partitions,Young tableaux and symmetric functions.A partition of a positive integer is a weakly decreasing sequence = ( 12 ) of nonnegative integers with 1 + 2 + + = . We also denote the parti-tion by (1 1 2 2 ), where is the multiplicity of in for 1 . If isa partition of , we simply write . The Young diagram of a partition is a setof points= ( ) Z 2 1in which we regard the coordinates increase from left to righ t, and from top to bottom.Let [ ] denote the set of integers 1 2 . A standard tableau of shape isa bijection : [ ] with the condition that the assigned numbers strictly increasealong both the rows and the columns in . We illustrate the Youn g diagram anda standard tableau for = (3 2 2) 7 in the following:= ,1 3 4= 2 56 7We denote by STab( ) the set of all the standard tableaux of sha pe .For a standard tableau of shape , dene the descent set Des( ) byDes( ) := [ 1] +1 is located in a lower row than in