The numbers $f_λ$ of standard tableaux of shape $λ\vdash n$ satisfy 2 fundamental recursions: $f_λ= \sum f_{λ^-}$ and $(n + 1)f_λ=\sum f_{λ^+}$, where $λ^-$ and $λ^+$ run over all shapes obtained from $λ$ by adding or removing a square respectively. The first of these recursions is trivial; the second can be proven algebraically from the first. These recursions together imply algebraically the dimension formula $n! =\sum f_λ^2$ for the irreducible representations of $S_n$. We show that a combinatorial analysis of this classical algebraic argument produces an infinite family of algorithms, among which are the classical Robinson-Schensted row and column insertion algorithms. Each of our algorithms yields a bijective proof of the dimension formula.
We show how the sign of a permutation can be deduced from the tableaux induced by the permutation under the Robinson-Schensted-Knuth correspondence. The result yields a simple proof of a conjecture on the squares of imbalances raised by Stanley.
We describe a relationship between Robinson-Schensted algorithms defined for standard domino tableaux of unequal rank. The principal idea is the moving-through operation defined on standard domino tableaux by D. Garfinkle. When restricted to involutions, this answers a question posed by M.A.A. van Leeuwen.
We investigate the evolution in time of the position of a fixed number in the insertion tableau when the Robinson-Schensted-Knuth algorithm is applied to a sequence of random numbers. When the length of the sequence tends to infinity, a typical trajectory after scaling converges uniformly in probability to some deterministic curve.
We exhibit a weight-preserving bijection between semi-standard Young tableaux and semi-skyline augmented fillings to provide a combinatorial proof that the Schur functions decompose into nonsymmetric functions indexed by compositions. The insertion procedure involved in the proof leads to an analogue of the Robinson-Schensted-Knuth Algorithm for semi-skyline augmented fillings. This procedure commutes with the RSK algorithm, and therefore retains many of its properties.
We generalize a formula obtained independently by A. Reifegerste and J. Sjöstrand for the sign of a permutation under the classical Robinson-Schensted map to a family of domino Robinson-Schensted algorithms.
The Stanley-Stembridge conjecture associates a symmetric function to each natural unit interval order $\mathcal P$. In this paper, we define relations à la Knuth on the symmetric group for each $\mathcal P$ and conjecture that the associated $\mathcal P$-Knuth equivalence classes are Schur-positive, refining theorems of Gasharov, Brosnan-Chow, and Guay-Paquet. The resulting equivalence graphs fit into the framework of D graphs studied by Assaf. Furthermore, we conjecture that the Schur expansion is given by column-readings of $\mathcal P$-tableaux that occur in the equivalence class. We prove …
In this paper we generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence.
In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line.
We consider the Robinson-Schensted-Knuth algorithm applied to a random input and investigate the shape of the bumping route (in the vicinity of the $y$-axis) when a specified number is inserted into a large Plancherel-distributed tableau. We show that after a projective change of the coordinate system the bumping route converges in distribution to the Poisson process.
We present an algorithm which computes the annihilator in ${\rm U}(\frak{sl}(\infty))$ of any simple highest weight $\frak{sl}(\infty)$-module $L_\frak b(λ)$. This algorithm is based on an infinite version of the Robinson-Schensted algorithm.
We prove a limit shape theorem describing the asymptotic shape of bumping routes when the Robinson-Schensted algorithm is applied to a finite sequence of independent, identically distributed random variables with the uniform distribution $U[0,1]$ on the unit interval, followed by an insertion of a deterministic number $α$. The bumping route converges after scaling, in the limit as the length of the sequence tends to infinity, to an explicit, deterministic curve depending only on $α$. This extends our previous result on the asymptotic determinism of Robinson-Schensted insertion, and answers a q…
We present a probabilistic generalization of the Robinson--Schensted correspondence in which a permutation maps to several different pairs of standard Young tableaux with nonzero probability. The probabilities depend on two parameters $q$ and $t$, and the correspondence gives a new proof of the squarefree part of the Cauchy identity for Macdonald polynomials. By specializing $q$ and $t$ in various ways, one recovers both the row and column insertion versions of the Robinson--Schensted correspondence, as well as several $q$- and $t$-deformations of row and column insertion which have been intro…
We study the Steinberg variety associated to matrix Schubert varieties, and develop a Robinson-Schensted type correspondence, $τ\leftrightarrow(Λ,\mathsf Q,\mathsf P)$. Here $τ$ is a partial permutation of size $p\times q$, $Λ$ an admissible signed Young diagram of size $p+q$, and $\mathsf P$ (resp. $\mathsf Q$) a standard Young tableau of size $p$ (resp. $q$) whose shape is determined by $Λ$. By embedding the matrix Schubert variety into a Schubert variety, we find a close relationship between the combinatorics of the classical Robinson-Schensted-Knuth correspondence and our bijection. We als…
In [F. Caselli, Involutory reflection groups and their models, J. Algebra 24 (2010), 370--393] there is constructed a uniform Gelfand model for all non-exceptional irreducible complex reflection groups which are involutory. Such model can be naturally decomposed into the direct sum of submodules indexed by $S_n$-conjugacy classes, and we present here a general result that relates the irreducible decomposition of these submodules with the projective Robinson-Schensted correspondence. This description also reflects in a very explicit way the existence of split representations for these groups.
Although the Robinson-Schensted-Knuth correspondence is a classical subject, its study is still active because of new development in last two decades. In this field, fundamental results are sometimes proved by using machineries which may be bypassed. Hence we still feel a lack of good source for these elementary materials. This paper is intended to fill in this lack about the RSK correspondence and the left cells of the symmetric group. It is mostly survey-like, but shows the variety of fields the RSK correspondence is related to.
We construct new "standard modules" for the representations of general linear groups over a local non-archimedean field. The construction uses a modified Robinson-Schensted-Knuth correspondence for Zelevinsky's multisegments. Typically, the new class categorifies the basis of Doubilet, Rota, and Stein for matrix polynomial rings, indexed by bitableaux. Hence, our main result provides a link between the dual canonical basis (coming from quantum groups) and the DRS basis.
In his study of Kazhdan-Lusztig cells in affine type $A$, Shi has introduced an affine analog of Robinson-Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combinatorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Lusztig and Xi, we extend the algorithm to a bijection between extended affine symmetric group and triples $(P, Q, ρ)$ where $P$ and $Q$ are tabloids and $ρ$ is a dominant weight. The weights $ρ$ get …
The purpose of this note is to give an insertion scheme proof of the formula, $$p_μ= \sum_{λ\vdash k} χ^λ(μ)s_λ,\formula$$ where $p_μ$ is the power sum symmetric function, $s_λ$ is the Schur function and $χ^λ(μ)$ is the irreducible character of the symmetric group $S_k$ indexed by the partition $λ$ and evaluated at a permutation of cycle type $μ=(μ_1,\ldots,μ_\ell)$. The proof of this formula is by direct application of the Robinson-Schensted-Knuth insertion scheme and a recent formula of Roichman.
We consider Robinson-Schensted-Knuth algorithm applied to a random input and study the growth of the bottom rows of the corresponding Young diagrams. We prove multidimensional Poisson limit theorem for the resulting Plancherel growth process. In this way we extend the result of Aldous and Diaconis to more than just one row. This result can be interpreted as convergence of the multi-line Hammersley process to its stationary distribution which is given by a collection of independent Poisson point processes.
In O'Connell-Pei(2013) a q-weighted version of the Robinson-Schensted algorithm was introduced. In this paper we show that this algorithm has a symmetry property analogous to the well known symmetry property of the normal Robinson-Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin(1979,1986,1994,1995). This approach, which uses "growth graphs", can also be applied to a wider class of insertion algorithms which have a branching structure, including some of the other q-weighted versions of the Robinson-Schensted algorithm which have recently b…
By using an elementary matrix approach, based on the technique of discrete Toda equation, we construct subtraction-free rational and piecewise linear transformations associated with various combinatorial algorithms, including the RSK correspondence. We also investigate birational Weyl group actions related to those algorithms.
In symmetric groups, a two-sided cell is the set of all permutations which are mapped by the Robinson-Schensted correspondence on a pair of tableaux of the same shape. In this article, we show that the set of permutations in a two-sided cell which have a minimal number of inversions is the set of permutations which have a maximal number of inversions in conjugated Young subgroups. We also give an interpretation of these sets with particular tableaux, called reading tableaux. As corollary, we give the set of elements in a two-sided cell which have a maximal number of inversions.
The box-ball system is studied from the viewpoint of combinatorics of words and tableaux. Each state of the box-ball system can be transformed into a pair of tableaux $(P,Q)$ by the Robinson-Schensted-Knuth correspondence. In the language of tableaux, the $P$-symbol gives rise to a conserved quantity of the box-ball system, and the $Q$-symbol evolves independently of the $P$-symbol. The time evolution of the $Q$-symbol is described explicitly in terms of the box-labels.
We present a probabilistic generalization of the Robinson--Schensted correspondence in which a permutation maps to several different pairs of standard Young tableaux with nonzero probability. The probabilities depend on two parameters $q$ and $t$, and the correspondence gives a new proof of the squarefree part of the Cauchy identity for Macdonald polynomials (i.e., the equality of the coefficients of $x_1 \cdots x_n y_1 \cdots y_n$ on either side, which are related to permutations and standard Young tableaux). By specializing $q$ and $t$ in various ways, one recovers the row and column inserti…
We investigate Robinson-Schensted-Knuth algorithm (RSK) and Schützenberger's jeu de taquin in the infinite setup. We show that the recording tableau in RSK defines an isomorphism of the following two dynamical systems: (i) a sequence of i.i.d. random letters equipped with Bernoulli shift, and (ii) a random infinite Young tableau (with the distribution given by Vershik-Kerov measure, corresponding to some Thoma character of the infinite symmetric group) equipped with jeu de taquin transformation. As a special case we recover the results on non-colliding random walks and multidimensional Pitman …
Taking transposes of Standard Young Tableaux defines a natural involution on the set $I(n)$ of involutions of length $n$ via the the Robinson-Schensted correspondence. In some cases, this involution can be defined without resorting to the Robinson-Schensted correspondence. As a byproduct, we get an interesting generalization of layered permutations.
We give an explicit Grobner basis for the ideal of the tangent cone at any T-fixed point of a Richardson variety in the Grassmannian, thus generalizing a result of Kodiyalam-Raghavan and Kreiman-Lakshmibai. Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the bounded RSK (BRSK). We use the Grobner basis result to deduce a formula which computes the multiplicity of the Richardson variety at any T-fixed point by counting families of nonintersecting lattice paths, thus generalizing a result first proved by Krattehthaler.
Kashiwara's crystal graphs have a natural monoid structure that arises by identifying words labelling vertices that appear in the same position of isomorphic components. The celebrated plactic monoid (the monoid of Young tableaux), arises in this way from the crystal graph for the $q$-analogue of the general linear Lie algebra $\mathfrak{gl}_{n}$, and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence. The authors previously constructed an analogous `quasi-crystal' structure for the related hypo…
Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. In the particular case of the crystal graph for the $q$-analogue of the special linear Lie algebra $\mathfrak{sl}_{n}$, this monoid is the celebrated plactic monoid, whose elements can be identified with Young tableaux. The crystal graph and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence and so prov…
Using growth diagrams, we define a skew domino Schensted correspondence which is a domino analogue of the skew Robinson-Schensted correspondence due to Sagan and Stanley. The color-to-spin property of Shimozono and White is extended. As an application, we give a simple generating function for the weighted sum of skew domino tableaux, which is a generalization of Stanley's sign-imbalance formula. The generating function gives a method to calculate the generalized sign-imbalance formula. We also extend Sj{ö}strand's theorems on sign-imbalance of skew shapes.
We suggest an hierarchy of all the results known so far about the connection of the asymptotics of combinatorial or representation theoretic problems with ``beta=2 ensembles'' arising in the random matrix theory. We show that all such results are, essentially, degenerations of one general situation arising from so-called generalized regular representations of the infinite symmetric group.
Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutating relation of generalized Schur operators implies Pieri's f…
We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to $[n,n,n]$: the {\em reduced web} basis associated to Kuperberg's combinatorial description of the spider category; and the {\em left cell basis} for the left cell construction of Kazhdan and Lusztig. In the case of $[n,n]$, the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the images of these bases under classical maps: the {\em Robinson-Schensted algorithm} between perm…
We prove a restriction of an analogue of the Robinson--Schensted--Knuth correspondence for semi-skyline augmented fillings, due to Mason, to multisets of cells of a staircase possibly truncated by a smaller staircase at the upper left end corner, or at the bottom right end corner. The restriction to be imposed on the pairs of semi-skyline augmented fillings is that the pair of shapes, rearrangements of each other, satisfies an inequality in the Bruhat order, w.r.t. the symmetric group, where one shape is bounded by the reverse of the other. For semi-standard Young tableaux the inequality means…
In [O'Connell and Yor (2002)] a path-transformation G was introduced with the property that, for X belonging to a certain class of random walks on the integer lattice, the transformed walk G(X) has the same law as that of the original walk conditioned never to exit a type-A Weyl chamber. In this paper, we show that G is closely related to the Robinson-Schensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of X and G(X). The corresponding…
The set of orbits of $GL(V)$ in $Fl(V)\times Fl(V)\times V$ is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in t…
In [Matveev-Petrov 2016](arXiv:1504.00666) a $q$-deformed Robinson-Schensted-Knuth algorithm ($q$RSK) was introduced. In this article we give reformulations of this algorithm in terms of the Noumi-Yamada description, growth diagrams and local moves. We show that the algorithm is symmetric, namely the output tableaux pairs are swapped in a sense of distribution when the input matrix is transposed. We also formulate a $q$-polymer model based on the $q$RSK, prove the corresponding Burke property, which we use to show a strong law of large numbers for the partition function given stationary bounda…
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…
We introduce and study q-randomized Robinson-Schensted-Knuth (RSK) correspondences which interpolate between the classical (q=0) and geometric (q->1) 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 …
We give the first two-dimensional pictorial presentation of Berele's correspondence \cite{Berele}, an analogue of the Robinson-Schensted (R-S) correspondence \cite{Robinson, Schensted} for the symplectic group $Sp(2n, \Cpx )$. From the standpoint of representation theory, the R-S correspondence combinatorially describes the irreducible decomposition of the tensor powers of the natural representation of $GL(n,\Cpx)$. Berele's insertion algorithm gives the bijection that describes the irreducible decomposition of the tensor powers of the natural representation of $Sp(2n, \Cpx)$. Two-dimensional …
In this paper, we introduce the ghost-box-ball system, which is an extended version of the classical soliton cellular automaton. It is initially motivated as a mechanism for making precise a connection between the Schensted insertion (of the Robinson-Schensted-Knuth correspondence) and the dynamical process of the box-ball system. In addition to this motivation, we explore generalisations of classical notions of the box-ball system, including the solitonic phenomenon, the asymptotic sorting property, and the invariant shape construction. We analyse the ghost-box-ball system beyond its initial …
The conjugacy classes of nilpotent $n\times n$ matrices can be parametrised by partitions $λ$ of $n$, and for a nilpotent $η$ in the class parametrised by $λ$, the variety $F_η$ of $η$-stable flags has its irreducible components parametrised by the standard Young tableaux of shape $λ$. We indicate how several algorithmic constructions defined for Young tableaux have significance in this context, thus extending Steinberg's result that the relative position of flags generically chosen in the irreducible components of $F_η$ parametrised by tableaux $P$ and $Q$, is the permutation associated to $(…
Permutations avoiding all patterns of a given shape (in the sense of Robinson-Schensted-Knuth) are considered. We show that the shapes of all such permutations are contained in a suitable thick hook, and deduce an exponential growth rate for their number.
We generalize the Robinson-Schensted-Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length $k$ with entries in $[n]$ and pairs of tableaux of the same shape with one being a standard Young tableau of size $n$ and the other being a standard multiset…
The Little map and the Edelman-Greene insertion algorithm, a generalization of the Robinson-Schensted correspondence, are both used for enumerating the reduced decompositions of an element of the symmetric group. We show the Little map factors through Edelman-Greene insertion and establish new results about each map as a consequence. In particular, we resolve some conjectures of Lam and Little.
Patience Sorting is a combinatorial algorithm that can be viewed as an iterated, non-recursive form of the Schensted Insertion Algorithm. In recent work the authors extended Patience Sorting to a full bijection between the symmetric group and certain pairs of combinatorial objects (called pile configurations) that are most naturally defined in terms of generalized permutation pattern and barred pattern avoidance. This Extended Patience Sorting Algorithm is very similar to the Robinson-Schensted-Knuth (or RSK) Correspondence, which is itself built from repeated application of the Schensted Inse…
