Terms Of Young Tableaux Research Articles

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. 
 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.

A Generalized RSK for Enumerating Linear Series on n-pointed Curves

We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-d morphisms from a general genus g, n-marked curve C to ℙ r , sending the marked points on C to specified general points in ℙ r , is equal to (r+1) g for sufficiently large d. This computation may be rephrased as an intersection problem on Grassmannians, which has a natural combinatorial interpretation in terms of Young tableaux by the classical Littlewood-Richardson rule. We give a bijection, generalizing the well-known RSK correspondence, between the tableaux in question and the (r+1)-ary sequences of length g, and we explore our bijection’s combinatorial properties.

Expressions of Schur multiple zeta-functions of anti-hook type by zeta-functions of root systems

We investigate relations among Schur multiple zeta functions and zeta-functions of root systems attached to semisimple Lie algebras. Schur multiple zeta functions are defined as sums over semi-standard Young tableaux. Then, assuming the Young tableaux is of anti-hook shape, we show that they can be written in terms of modified zeta-functions of root systems of type $A$. Our proof is quite computational, but we also give a pictorial interpretation of our argument in terms of Young tableaux. It is also possible to understand that one of our theorems gives an expression of Schur multiple zeta functions by an analogue of Weyl group multiple Dirichlet series in the sense of Bump et al. By combining with a result of Nakasuji, Phuksuwan and Yamasaki, our theorems yield a new method of finding functional relations among zeta-functions of root systems.

Higher spins from exotic dualisations

At the free level, a given massless field can be described by an infinite number of different potentials related to each other by dualities. In terms of Young tableaux, dualities replace any number of columns of height hi by columns of height D − 2 − hi, where D is the spacetime dimension: in particular, applying this operation to empty columns gives rise to potentials containing an arbitrary number of groups of D − 2 extra antisymmetric indices. Using the method of parent actions, action principles including these potentials, but also extra fields, can be derived from the usual ones. In this paper, we revisit this off-shell duality and clarify the counting of degrees of freedom and the role of the extra fields. Among others, we consider the examples of the double dual graviton in D = 5 and two cases, one topological and one dynamical, of exotic dualities leading to spin three fields in D = 3.

An Explicit Formula Enumerating the Finely Bounded Trees Using Young Tableaux

Most of the tree enumeration formulas are generating functions or recurrence formulas. In this article, we show the explicit formula for the number of unlabeled rooted trees with a certain condition. The formula is described in terms of Young tableaux.

Agnostic Learning in Permutation-Invariant Domains

We generalize algorithms from computational learning theory that are successful under the uniform distribution on the Boolean hypercube {0, 1} n to algorithms successful on permutation-invariant distributions, distributions that stay invariant constant on permutating the coordinates in the instances. While the tools in our generalization mimic those used for the Boolean hypercube, the fact that permutation-invariant distributions are not product distributions presents a significant obstacle. We prove analogous results for permutation-invariant distributions; more generally, we work in the domain of the symmetric group. We define noise sensitivity in this setting and show that noise sensitivity has a nice combinatorial interpretation in terms of Young tableaux. The main technical innovations involve techniques from the representation theory of the symmetric group, especially the combinatorics of Young tableaux. We show that low noise sensitivity implies concentration on “simple” components of the Fourier spectrum and that this fact will allow us to agnostically learn halfspaces under permutation-invariant distributions to constant accuracy in roughly the same time as in the uniform distribution over the Boolean hypercube case.

Generalization of the Macdonald formula for Hall–Littlewood polynomials

We study the Gaussent–Littelmann formula for Hall–Littlewood polynomials and we develop combinatorial tools to describe the formula in a purely combinatorial way for type An, Bn and Cn. This description is in terms of Young tableaux and arises from identifying one-skeleton galleries that appear in the Gaussent–Littelmann formula with Young tableaux. Furthermore, we show by using these tools that the Gaussent–Littelmann formula and the well-known Macdonald formula for Hall–Littlewood polynomials for type An are the same up to an additional factorization.

A combinatorial description of the Gindikin–Karpelevich formula in type A

A combinatorial description of the crystal B(∞) for finite-dimensional simple Lie algebras in terms of Young tableaux was developed by J. Hong and H. Lee. Using this description, we obtain a combinatorial rule for expressing the Gindikin–Karpelevich formula as a sum over B(∞) when the underlying Lie algebra is of type A. We also interpret our description in terms of MV polytopes and irreducible components of quiver varieties.

Young tableaux and crystal [formula omitted] for the exceptional Lie algebra types

We study the crystal base B ( ∞ ) associated with the negative part of the quantum group for finite simple Lie algebras of types E 6 , E 7 , E 8 , and F 4 . After describing the crystal B ( ∞ ) as a union of highest weight crystals, we give an explicit description of B ( ∞ ) in terms of Young tableaux.

Young tableaux and crystal [formula omitted] for finite simple Lie algebras

We study the crystal base of the negative part of a quantum group. An explicit description of the crystal for quantum finite Lie algebras of types A n , B n , C n , D n , and G 2 is given in terms of Young tableaux.

The Robinson-Schensted-Knuth correspondence and the bijections of commutativity and associativity

The bijections of associativity and commutativity arise from symmetries of the Littlewood-Richardson coefficients. We define these bijections in terms of arrays and show that they coincide with analogous bijections defined in terms of discretely concave functions using the octahedron recurrence as well as with bijections defined in terms of Young tableaux. The main ingredient in the proof of their coincidence is a functional version of the Robinson-Schensted-Knuth correspondence.

Current Algebra on the Torus

We derive the N-point one-loop correlation functions for the currents of an arbitrary affine Kac-Moody algebra. The one-loop amplitudes, which are elliptic functions defined on the torus Riemann surface, are specified by group invariant tensors and certain constant tau-dependent functions. We compute the elliptic functions via a generating function, and explicitly construct the invariant tensor functions recursively in terms of Young tableaux. The lowest tensors are related to the character formula of the representation of the affine algebra. These general current algebra loop amplitudes provide a building block for open twistor string theory, among other applications.

Geometric and Combinatorial Realizations of Crystal Graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For type A n (1) , we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids.

On orbital variety closures in [formula omitted]: II. Descendants of a Richardson orbital variety

For a semisimple Lie algebra g , the orbit method attempts to assign representations of g to (coadjoint) orbits in g ∗ . Orbital varieties are particular Lagrangian subvarieties of such orbits leading to highest weight representations of g . In sl n orbital varieties are described by Young tableaux. In this paper we consider so-called Richardson orbital varieties in sl n . A Richardson orbital variety is an orbital variety whose closure is a standard nilradical. We show that in sl n a Richardson orbital variety closure is a union of orbital varieties. We give a complete combinatorial description of such closures in terms of Young tableaux. This is the second paper in the series of three papers devoted to a combinatorial description of orbital variety closures in sl n in terms of Young tableaux.

A Combinatorial Interpretation Of The Wreath Product Of Schur Functions

A combinatorial interpretation of Schur functions in terms of Young tableaux is well-known. (For example, see Littlewood [1] or Thomas [4]). The purpose of this paper is to present a combinatorial interpretation of the wreath product (or plethysm) of two Schur functions.