Robinson-Schensted Correspondence Research Articles

A Bump Statistic on Permutations Resulting from the Robinson–Schensted Correspondence

AbstractIn this paper, we investigate a permutation statistic that was independently introduced by Romik (Funct Anal Appl 39(2):52–155, 2005). This statistic counts the number of bumps that occur during the execution of the Robinson–Schensted procedure when applied to a given permutation. We provide several interpretations of this bump statistic that include the tableaux shape and also as an extremal problem concerning permutations and increasing subsequences. Several aspects of this bump statistic are investigated from both structural and enumerative viewpoints.

Multiparameter colored partition category and the product of the reduced Kronecker coefficients

We introduce and study a multiparameter colored partition category CPar(x) by extending the construction of the partition category, over an algebraically closed field k of characteristic zero and for a multiparameter x∈kr. The morphism spaces in CPar(x) have bases in terms of partition diagrams whose parts are colored by elements of the multiplicative cyclic group Cr. We show that the endomorphism spaces of CPar(x) and additive Karoubi envelope of CPar(x) are generically semisimple. The category CPar(x) is rigid symmetric strict monoidal and we give a presentation of CPar(x) as a monoidal category. The path algebra of CPar(x) admits a triangular decomposition with Cartan subalgebra being equal to the direct sum of the group algebras of complex reflection groups G(r,n). We compute the structure constants for the classes of simple modules in the split Grothendieck ring of the category of modules over the path algebra of the downward partition subcategory of CPar(x) in two ways. Among other things, this gives a formula for the product of the reduced Kronecker coefficients in terms of the Littlewood–Richardson coefficients for G(r,n) and certain Kronecker coefficients for the wreath product (Cr×Cr)≀Sn. For r=1, this formula reduces to a formula for the reduced Kronecker coefficients given by Littlewood. We also give two analogues of the Robinson–Schensted correspondence for colored partition diagrams and, as an application, we classify the equivalence classes of Green's left, right and two-sided relations for the colored partition monoid in terms of these correspondences.

Monotone subsequences in locally uniform random permutations

A locally uniform random permutation is generated by sampling n points independently from some absolutely continuous distribution ρ on the plane and interpreting them as a permutation by the rule that i maps to j if the ith point from the left is the jth point from below. As n tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences give rise to a surface. We show that, under the correct scaling, for any r≥0, the largest union of ⌊rn⌋ decreasing subsequences approaches a limit surface as n tends to infinity, and the limit surface is a solution to a specific variational problem. As a corollary, we prove the existence of a limit shape for the Young diagram associated to the random permutation under the Robinson–Schensted correspondence. In the special case where ρ is the uniform distribution on the diamond |x|+|y|<1, we conjecture that the limit shape is triangular, and assuming the conjecture is true, we find an explicit formula for the limit surfaces of a uniformly random permutation and recover the famous limit shape of Vershik, Kerov and Logan, Shepp.

Asymptotic Hecke Algebras and Lusztig–Vogan Bijection via Affine Matrix-Ball Construction

Abstract The affine matrix-ball construction (abbreviated AMBC) was developed by Chmutov, Lewis, Pylyavskyy, and Yudovina as an affine generalization of the Robinson–Schensted correspondence. We show that AMBC gives a simple way to compute a distinguished involution in each Kazhdan–Lusztig cell of an affine symmetric group. We then use AMBC to give the 1st known canonical presentation for the asymptotic Hecke algebras of extended affine symmetric groups. As an application, we show that AMBC gives a conceptual way to compute the Lusztig–Vogan bijection. For the latter, we build upon prior works of Achar and Rush.

Weight Multiplicities and Young Tableaux Through Affine Crystals

The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac–Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the ( ( spin ) ) rigid tableaux, and prove that they are equinumerous to the weight multiplicities of the highest weight modules under our consideration. These new classes of Young tableaux arise from crystal basis elements for dominant maximal weights of the integrable highest weight modules over affine Kac–Moody algebras. By applying combinatorics of tableaux such as the Robinson–Schensted algorithm and new insertion schemes, and using integrals over orthogonal groups, we reveal hidden structures in the sets of weight multiplicities and obtain explicit closed formulas for the weight multiplicities. In particular we show that some special families of weight multiplicities form the Pascal, Catalan, Motzkin, Riordan and Bessel triangles.

On Five Types of Crucial Permutations with Respect to Monotone Patterns

A crucial permutation is a permutation that avoids a given set of prohibitions, but any of its extensions, in an allowable way, results in a prohibition being introduced. &#x0D; In this paper, we introduce five natural types of crucial permutations with respect to monotone patterns, notably quadracrucial permutations that are linked most closely to Erdős-Szekeres extremal permutations. The way we define right-crucial and bicrucial permutations is consistent with the definition of respective permutations studied in the literature in the contexts of other prohibitions. For each of the five types, we provide its characterization in terms of Young tableaux via the Robinson-Schensted correspondence. Moreover, we use the characterizations to prove that the number of such permutations of length $n$ is growing when $n\to\infty$, and to enumerate minimal crucial permutations in all but one case. We also provide other enumerative results.

Sign insertion and Kazhdan–Lusztig cells of affine symmetric groups

Combinatorics of Kazhdan–Lusztig cells in affine type A was originally developed by Lusztig, Shi, and Xi. Building on their work, Chmutov, Pylyavskyy, and Yudovina introduced the affine matrix-ball construction (abbreviated AMBC) which gives an analog of Robinson–Schensted correspondence for affine symmetric groups. An alternative approach to Kazhdan–Lusztig theory in affine type A was developed by Blasiak in his work on catabolism. He introduced a sign insertion algorithm and conjectured that if one fixes the two-sided cell, the recording tableau of the sign insertion process determines uniquely and is determined uniquely by the left cell. In this paper we unite these two approaches by proving Blasiak’s conjecture. In the process, we show that certain new operations we introduce called partial rotations connect the elements in the intersection of a left cell and a right cell. Lastly, we investigate the connection between Blasiak’s sign insertion and the standardization map acting on the set of semi-standard Young tableaux defined by Lascoux and Schützenberger.

Poisson limit theorems for the Robinson–Schensted correspondence and for the multi-line Hammersley process

We consider the Robinson–Schensted–Knuth algorithm applied to a random input and study the growth of the bottom rows of the corresponding Young diagrams. We prove a 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.

Monodromy in Kazhdan–Lusztig cells in affine type A

We use the affine Robinson-Schensted correspondence to describe the structure of bidirected edges in the Kazhdan-Lusztig cells in affine type A. Equivalently, we give a comprehensive description of the Knuth equivalence classes of affine permutations.

A Fast Fourier Transform for the Johnson Graph

The set X of k-subsets of an n-set has a natural graph structure where two k-subsets are neighbors if and only if the size of their intersection is \(k-1\). This is known as the Johnson graph. The symmetric group \(S_n\) acts on the space of complex functions on X and this space has a multiplicity-free decomposition as sum of irreducible representations of \(S_n\), so it has a well-defined Gelfand–Tsetlin basis up to scalars. The Fourier transform on the Johnson graph is defined as the change of basis matrix from the delta function basis to the Gelfand–Tsetlin basis. The direct application of this matrix to a generic vector requires \(\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\) arithmetic operations. We show that, in analogy with the classical Fast Fourier Transform on the discrete circle, this matrix can be factorized as a product of \(n-1\) orthogonal matrices, each one with at most two nonzero elements in each column. The factorization is based on the construction of \(n-1\) intermediate bases which are parametrized via the Robinson–Schensted insertion algorithm. This factorization shows that the number of arithmetic operations required to apply this matrix to a generic vector is bounded above by \(2(n-1) \left( {\begin{array}{c}n\\ k\end{array}}\right) \). We give an algorithm that constructs all these factors using at most \(289(n-1)\left( {\begin{array}{c}n\\ k\end{array}}\right) \) arithmetic operations. The coefficients of these matrices are rational numbers and the construction does not depend on numerical methods. Instead, they are obtained by solving small linear systems with integer coefficients derived from the Jucys–Murphy operators. In particular we avoid the use of square roots. As a consequence, we show that the problem of computing all the weights of the isotypic components of a given function can be solved in \(O(n \left( {\begin{array}{c}n\\ k\end{array}}\right) )\) operations, improving the previous bound \(O(k^2 \left( {\begin{array}{c}n\\ k\end{array}}\right) )\) when k asymptotically dominates \(\sqrt{n}\). The same improvement is achieved for the problem of computing the isotypic projection onto a single component.

Intersecting principal Bruhat ideals and grades of simple modules

We prove that the grades of simple modules indexed by boolean permutations, over the incidence algebra of the symmetric group with respect to the Bruhat order, are given by Lusztig's $\mathbf{a}$-function. Our arguments are combinatorial, and include a description of the intersection of two principal order ideals when at least one permutation is boolean. An important object in our work is a reduced word written as minimally many runs of consecutive integers, and one step of our argument shows that this minimal quantity is equal to the length of the second row in the permutation's shape under the Robinson-Schensted correspondence. We also prove that a simple module over the above-mentioned incidence algebra is perfect if and only if its index is the longest element of a parabolic subgroup.Mathematics Subject Classifications: 20F55, 06A07, 05E15

The number of inversions of permutations with fixed shape

The Robinson-Schensted correspondence can be viewed as a map from permutations to partitions. In this work, we study the number of inversions of permutations corresponding to a fixed partition $\lambda$ under this map. Hohlweg characterized permutations having shape $\lambda$ with the minimum number of inversions. Here, we give the first results in this direction for higher numbers of inversions. We give explicit conjectures for both the structure and the number of permutations associated to $\lambda$ where the extra number of inversions is less than the length of the smallest column of $\lambda$. We prove the result when $\lambda$ has two columns.

Robinson–Schensted correspondence for unit interval orders

The Stanley-Stembridge conjecture associates a symmetric function to each natural unit interval order $\mathcal P$. In this paper, we define relations \`a 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 these conjectures for $\mathcal P$ avoiding two specific suborders by introducing $\mathcal P$-analog of Robinson-Schensted insertion, giving an answer to a long standing question of Chow.

Primitive elements of the Hopf algebras of tableaux

The character theory of symmetric groups, and the theory of symmetric functions, both make use of the combinatorics of Young tableaux, such as the Robinson–Schensted algorithm, Schützenberger’s “jeu de taquin”, and evacuation. In 1995 Poirier and the second author introduced some algebraic structures, different from the plactic monoid, which induce some products and coproducts of tableaux, with homomorphisms. Their starting point are the two dual Hopf algebras of permutations, introduced by the authors in 1995. In 2006 Aguiar and Sottile studied in more detail the Hopf algebra of permutations: among other things, they introduce a new basis, by Möbius inversion in the poset of weak order, that allows them to describe the primitive elements of the Hopf algebra of permutations. In the present Note, by a similar method, we determine the primitive elements of the Poirier–Reutenauer algebra of tableaux, using a partial order on tableaux defined by Taskin.

QRSt: A Probabilistic Robinson–Schensted Correspondence for Macdonald Polynomials

Abstract 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 insertion versions of the Robinson–Schensted correspondence, several $q$- and $t$-deformations of row and column insertion which have been introduced in recent years in connection with $q$-Whittaker and Hall–Littlewood processes, and the Plancherel measure on partitions. Our construction is based on Fomin’s growth diagrams and the recently introduced notion of a probabilistic bijection between weighted sets.

Irreducible components of exotic Springer fibres II : The exotic Robinson–Schensted algorithm

Kato's exotic nilpotent cone was introduced as a substitute for the ordinary nilpotent cone of type C with cleaner properties. The geometric Robinson-Schensted correspondence is obtained by parametrizing the irreducible components of the Steinberg variety (the conormal variety for the action of a semisimple group on two copies of its flag variety); in type A the bijection coincides with the classical Robinson-Schensted algorithm for the symmetric group. Here we give a combinatorial description of the bijection obtained by using the exotic nilpotent cone instead of ordinary type C nilpotent cone in the geometric Robinson-Schensted correspondence; we refer this as the "exotic Robinson-Schensted bijection". This is interesting from a combinatorial perspective, and not a naive extension of the type A Robinson-Schensted bijection.

A Generalization of Steinberg Theory and an Exotic Moment Map

Abstract For a reductive group $G$, Steinberg established a map from the Weyl group to the set of nilpotent $G$-orbits by using moment maps on double flag varieties. In particular, in the case of the general linear group, it provides a geometric interpretation of the Robinson–Schensted correspondence between permutations and pairs of standard tableaux of the same shape. We extend Steinberg’s approach to the case of a symmetric pair $(G, K)$ to obtain two different maps, namely a generalized Steinberg map and an exotic moment map. Although the framework is general, in this paper we focus on the pair $(G,K) = (\textrm{GL}_{2n}({\mathbb{C}}), \textrm{GL}_n({\mathbb{C}}) \times \textrm{GL}_n({\mathbb{C}}))$. Then the generalized Steinberg map is a map from partial permutations to the pairs of nilpotent orbits in $ \mathfrak{gl}_n({\mathbb{C}}) $. It involves a generalization of the classical Robinson–Schensted correspondence to the case of partial permutations. The other map, the exotic moment map, establishes a combinatorial map from the set of partial permutations to that of signed Young diagrams, that is, the set of nilpotent $ K$-orbits in the Cartan space $(\textrm{Lie}(G)/\textrm{Lie}(K))^* $. We explain the geometric background of the theory and combinatorial algorithms, which produce the above-mentioned maps.

Matrix-Ball Construction of affine Robinson-Schensted correspondence

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 combi- natorial 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 Honeywill, 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 a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights.

The ABC of p-cells

Parallel to the very rich theory of Kazhdan–Lusztig cells in characteristic 0, we try to build a similar theory in positive characteristic. We study cells with respect to the p-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see Jensen and Williamson in Categorification and higher representation theory, American Mathematical Society, Providence, 2017). Our main technical tools are the star-operations introduced by Kazhdan–Lusztig (Invent. Math. 53(2):65–184, 1979) which have interesting numerical consequences for the p-canonical basis. As an application, we explicitly describe p-cells in finite type A (i.e. for symmetric groups) using the Robinson–Schensted correspondence. Moreover, we show that Kazhdan–Lusztig cells in finite types B and C decompose into p-cells for $$p > 2$$ p > 2 .

Matrix-Ball Construction of affine Robinson–Schensted correspondence

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 byproduct, 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 the extended affine symmetric group and collection of triples \((P, Q, \rho )\) where P and Q are tabloids and \(\rho \) is a dominant weight. The weights \(\rho \) get a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights.