Young Diagrams Research Articles

Representations of rational Cherednik algebras with minimal support and torus knots

In this paper we obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen–Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type An−1 for c=mn, and an expression of its quasispherical part (i.e., the isotypic part of “hooks”) in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Etingof to give explicit formulas for the characters of the irreducible equivariant D-modules on the nilpotent cone for SLm. Our third result is the construction of the Koszul–BGG complex for the rational Cherednik algebra, which generalizes the construction of the Koszul–BGG resolution from [3] and [21], and the calculation of its homology in type A. We also show in type A that the differentials in the Koszul–BGG complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the Cherednik algebra. Finally, our fourth result is the symmetry theorem, which identifies the quasispherical components in the representations with minimal support over the rational Cherednik algebras Hmn(Sn) and Hnm(Sm). In fact, we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras. This symmetry was essentially established in [8] in the spherical case, and in [24] in the case GCD(m,n)=1, and it has a natural interpretation in terms of invariants of torus knots.

Law of Large Numbers for infinite random matrices over a finite field

Asymptotic representation theory of general linear groups $$\hbox {GL}(n,F_\mathfrak {q})$$ over a finite field leads to studying probability measures $$\rho $$ on the group $$\mathbb {U}$$ of all infinite uni-uppertriangular matrices over $$F_\mathfrak {q}$$ , with the condition that $$\rho $$ is invariant under conjugations by arbitrary infinite matrices. Such probability measures form an infinite-dimensional simplex, and the description of its extreme points (in other words, ergodic measures $$\rho $$ ) was conjectured by Kerov in connection with nonnegative specializations of Hall–Littlewood symmetric functions. Vershik and Kerov also conjectured the following Law of Large Numbers. Consider an infinite random matrix drawn from an ergodic measure coming from the Kerov’s conjectural classification and its $$n \times n$$ submatrix formed by the first rows and columns. The sizes of Jordan blocks of the submatrix can be interpreted as a (random) partition of $$n$$ , or, equivalently, as a (random) Young diagram $$\lambda (n)$$ with $$n$$ boxes. Then, as $$n\rightarrow \infty $$ , the rows and columns of $$\lambda (n)$$ have almost sure limiting frequencies corresponding to parameters of this ergodic measure. Our main result is the proof of this Law of Large Numbers. We achieve it by analyzing a new randomized Robinson–Schensted–Knuth (RSK) insertion algorithm which samples random Young diagrams $$\lambda (n)$$ coming from ergodic measures. The probability weights of these Young diagrams are expressed in terms of Hall–Littlewood symmetric functions. Our insertion algorithm is a modified and extended version of a recent construction by Borodin and Petrov (2013). On the other hand, our randomized RSK insertion generalizes a version of the RSK insertion introduced by Vershik and Kerov (SIAM J. Algebr. Discret. Math. 7(1):116–124, 1986) in connection with asymptotic representation theory of symmetric groups (which is governed by nonnegative specializations of Schur symmetric functions).

Excited Young diagrams, equivariant $K$-theory, and Schubert varieties

We give combinatorial descriptions of the restrictions to T-fixed points of the classes of structure sheaves of Schubert varieties in the T-equivariant K-theory of Grassmannians and of maximal isotropic Grassmannians of orthogonal and symplectic types. We also give formulas, based on these descriptions, for the Hilbert series and Hilbert polynomials at T-fixed points of the corresponding Schubert varieties. These descriptions and formulas are given in terms of two equivalent combinatorial models: excited Young diagrams and set-valued tableaux. The restriction fomulas are positive, in that for a Schubert variety of codimension d, the formula equals (-1)^d times a sum, with nonnegative coefficients, of monomials in the expressions (e^{-\alpha}-1), as \alpha runs over the positive roots. In types A_n and C_n the restriction formulas had been proved earlier by [Kreiman 05], [Kreiman 06] by a different method. In type A_n, the formula for the Hilbert series had been proved earlier by [Li-Yong 12]. The method of this paper, which relies on a restriction formula of [Graham 02] and [Willems 06], is based on the method used by [Ikeda-Naruse 09] to obtain the analogous formulas in equivariant cohomology. The formulas we give differ from the K-theoretic restriction formulas given by [Ikeda-Naruse 11], which use different versions of excited Young diagrams and set-valued tableaux. We also give Hilbert series and Hilbert polynomial formulas which are valid for Schubert varieties in any cominuscule flag variety, in terms of the 0-Hecke algebra.

Recursive method for the Nekrasov partition function for classical Lie groups

Nekrasov partition function for the supersymmetric gauge theories with general Lie groups is not so far known in a closed form while there is a definition in terms of the integral. In this paper, as an intermediate step to derive it, we give a recursion formula among partition functions, which can be derived from the integral. We apply the method to a toy model which reflects the basic structure of partition functions for BCD type Lie groups and obtained a closed expression for the factor associated with the generalized Young diagram.

Conformal correlators of mixed-symmetry tensors

We generalize the embedding formalism for conformal field theories to the case of general operators with mixed symmetry. The index-free notation encoding symmetric tensors as polynomials in an auxiliary polarization vector is extended to mixed-symmetry tensors by introducing a new commuting or anticommuting polarization vector for each row or column in the Young diagram that describes the index symmetries of the tensor. We determine the tensor structures that are allowed in n-point conformal correlation functions and give an algorithm for counting them in terms of tensor product coefficients. A simple derivation of the unitarity bound for arbitrary mixed-symmetry tensors is obtained by considering the conservation condition in embedding space. We show, with an example, how the new formalism can be used to compute conformal blocks of arbitrary external fields for the exchange of any conformal primary and its descendants. The matching between the number of tensor structures in conformal field theory correlators of operators in d dimensions and massive scattering amplitudes in d + 1 dimensions is also seen to carry over to mixed-symmetry tensors.

The Categorification of Fermions

In this paper, we lift Fermions to functors acting on some homotopy category by the Boson-Fermion correspondence and get the categorified relations of Fermions. In this way, both the categorified Bosons and the categorified Fermions can be viewed as functors on the same category. We also give actions of these functors on the charged Young diagrams (or equivalently Maya diagrams), so that the classical theory of Boson-Fermion correspondence is very well recovered as a result of such a categorification.

SO(N) restricted Schur polynomials

We focus on the 1/4-BPS sector of free super Yang-Mills theory with an SO(N) gauge group. This theory has an AdS/CFT (an equivalence between a conformal field theory in d-1 dimensions and type II string theory defined on an AdS space in d-dimensions) dual in the form of type IIB string theory with AdS5×RP5 geometry. With the aim of studying excited giant graviton dynamics, we construct an orthogonal basis for this sector of the gauge theory in this work. First, we demonstrate that the counting of states, as given by the partition function, and the counting of restricted Schur polynomials match by restricting to a particular class of Young diagram labels. We then give an explicit construction of these gauge invariant operators and evaluate their two-point function exactly. This paves the way to studying the spectral problem of these operators and their D-brane duals.

On ramifications of Artin–Schreier extensions of surfaces over algebraically closed fields of positive characteristic II

For a smooth surface X over an algebraically closed field of positive characteristic, we consider the ramification of an Artin–Schreier extension of X. A ramification at a point of codimension 1 of X is understood by the Swan conductor. A ramification at a closed point of X is understood by the invariant rx defined by Kato (1994) [1]. The main theme of this paper is to construct the Young diagram Y(X,D,x) which is closely related to rx and to prove Kato's conjecture Kato (1994) [1] for an upper bound of rx for a good Artin–Schreier extension.

A determinantal formula for Catalan tableaux and TASEP probabilities

We present a determinantal formula for the steady state probability of each state of the TASEP (Totally Asymmetric Simple Exclusion Process) with open boundaries, a 1D particle model that has been studied extensively and displays rich combinatorial structure. These steady state probabilities are computed by the enumeration of Catalan tableaux, which are certain Young diagrams filled with α's and β's that satisfy some conditions on the rows and columns. We construct a bijection from the Catalan tableaux to weighted lattice paths on a Young diagram, and from this we enumerate the paths with a determinantal formula, building upon a formula of Narayana that counts unweighted lattice paths on a Young diagram. Finally, we provide a formula for the enumeration of Catalan tableaux that satisfy a given condition on the rows, which corresponds to the steady state probability that in the TASEP on a lattice with n sites, precisely k of the sites are occupied by particles. This formula is an α/β generalization of the Narayana numbers.

Triangular fully packed loop configurations of excess 2

Triangular fully packed loop configurations (TFPLs) came up in the study of fully packed loop configurations on a square (FPLs) corresponding to link patterns with a large number of nested arches. To a TFPL is assigned a triple $(u,v;w)$ of $01$-words encoding its boundary conditions. A necessary condition for the boundary $(u,v;w)$ of a TFPL is $\lvert \lambda(u) \rvert +\lvert \lambda(v) \rvert \leq \lvert \lambda(w) \rvert$, where $\lambda(u)$ denotes the Young diagram associated with the $01$-word $u$. Wieland gyration, on the other hand, was invented to show the rotational invariance of the numbers $A_\pi$ of FPLs corresponding to a given link pattern $\pi$. Later, Wieland drift was defined as the natural adaption of Wieland gyration to TFPLs. The main contribution of this article is a linear expression for the number of TFPLs with boundary $(u,v;w)$ where $\lvert \lambda (w) \rvert - \lvert\lambda (u) \rvert - \lvert \lambda (v)\rvert \leq 2$ in terms of numbers of stable TFPLs that is TFPLs invariant under Wieland drift. These stable TFPLs have boundary $(u^{+},v^{+};w)$ for words $u^{+}$ and $v^{+}$ such that $\lambda (u) \subseteq \lambda (u^{+})$ and $\lambda (v) \subseteq \lambda (v^{+})$. Les configurations de boucles compactes triangulaires (”triangular fully packed loop configurations”, ou TFPLs) sont apparues dans l’étude des configurations de boucles compactes dans un carré (FPLs) correspondant à des motifs de liaison avec un grand nombre d’arcs imbriqués. À chaque TPFL on associe un triplet $(u,v;w)$ de mots sur {0,1}, qui encode ses conditions aux bords. Une condition nécessaire pour le bord $(u,v;w)$ d’un TFPL est $\lvert \lambda(u) \rvert +\lvert \lambda(v) \rvert \leq \lvert \lambda(w) \rvert$, où $\lambda(u)$ désigne le diagramme de Young associé au mot $u$. D’un autre côté, la giration de Wieland a été inventée pour montrer l’invariance par rotation des nombres $A_\pi$ de FPLs correspondant à un motif de liaison donné $\pi$. Plus tard, la déviation de Wieland a été définie pour adapter de manière naturelle la giration de Wieland aux TFPLs. La contribution principale de cet article est une expression linéaire pour le nombre de TFPLs de bord $(u,v;w)$, où $\lvert \lambda (w) \rvert - \lvert\lambda (u) \rvert - \lvert \lambda (v)\rvert \leq 2$, en fonction des nombres de TFPLs stables, <i>i.e</i>., les TFPLs invariants par déviation de Wieland. Ces TFPLs stables ont pour bord $(u^{+},v^{+};w)$, avec $u^{+}$ et $v^{+}$ des mots tels que $\lambda (u) \subseteq \lambda (u^{+})$ et $\lambda (v) \subseteq \lambda (v^{+})$.

Permutation symmetry in spinor quantum gases: selection rules, conservation laws, and correlations.

Many-body systems of identical arbitrary-spin particles, with separable spin and spatial degrees of freedom, are considered. Their eigenstates can be classified by Young diagrams, corresponding to nontrivial permutation symmetries (beyond the conventional paradigm of symmetric-antisymmetric states). The present work obtains the following. (a)Selection rules for additional nonseparable (dependent on spins and coordinates) k-body interactions: the Young diagrams, associated with the initial and final states of a transition, can differ by relocation of no more than k boxes between their rows. (b)Correlation rules in which eigenstate-averaged local correlations of k particles vanish if k exceeds the number of columns (for bosons) or rows (for fermions) in the associated Young diagram. It also elucidates the physical meaning of the quantities conserved due to permutation symmetry-in 1929, Dirac identified those with characters of the symmetric group-relating them to experimentally observable correlations of several particles. The results provide a way to control the formation of entangled states belonging to multidimensional non-Abelian representations of the symmetric group. These states can find applications in quantum computation and metrology.

A Bijective Proof of Loehr-Warrington's Formulas for the Statistics $${\rm{ctot}_\frac{q}{p}}$$ ctot q p and $${\rm{mid}_\frac{q}{p}}$$ mid q p

Loehr and Warrington introduced partitional statistics \({\rm{ctot}_\frac{q}{p}}\) (D) and \({\rm{mid}_\frac{q}{p}}\) (D) and provided formulas for these statistics in terms of the boundary graph of the Young diagram D. In this paper we give a bijective proof of Loehr-Warrington's formulas by using the following simple combinatorial observation: given a Young diagram D and two numbers a and l, the number of boxes in D with the arm length a and the leg length l is one less than the number of boxes with the same properties in the complement to D. Here the complement is taken inside the positive quadrant or, equivalently, a very large rectangle.

K-theoretic boson–fermion correspondence and melting crystals

We study non-Hermitian integrable fermion and boson systems from the perspectives of Grothendieck polynomials. The models considered in this article are the five-vertex model as a fermion system and the non-Hermitian phase model as a boson system. Both models are characterized by different solutions satisfying the same Yang–Baxter relation. From our previous works on the identification between the wavefunctions of the five-vertex model and Grothendieck polynomials, we introduce skew Grothendieck polynomials and derive the addition theorem among them. Using these relations, we derive the wavefunctions of the non-Hermitian phase model as a determinant form, which can also be expressed as Grothendieck polynomials. Namely, we establish a K-theoretic boson–fermion correspondence at the level of wavefunctions. As a by-product, the partition function of the statistical mechanical model of a three-dimensional (3D) melting crystal is exactly calculated by use of the scalar products of the wavefunctions of the phase model. The resultant expression can be regarded as a K-theoretic generalization of the MacMahon function describing the generating function of the plane partitions, which interpolates the generating functions of two-dimensional (2D) and (3D) Young diagrams.

Combinatorial aspects of the conserved quantities of the tropical periodic Toda lattice

The tropical periodic Toda lattice (trop p-Toda) is a dynamical system attracting attention in the area of the interplay of integrable systems and tropical geometry. We show that the Young diagrams associated with trop p-Toda given by two very different definitions are identical. The first definition is given via a Lax representation of the discrete periodic Toda lattice, and the second one is associated with a generalization of the Kerov–Kirillov–Reshetikhin bijection in the combinatorics of the Bethe ansatz. By means of this identification, it is shown for the first time that the Young diagrams given by the latter definition are preserved under time evolution. This result is regarded as an important first step in clarifying the iso-level set structure of this dynamical system in general cases, i.e. not restricted to generic cases.

Diagonal Slices of 3D Young Diagrams in the Approach of Maya Diagrams

According to the correspondence between 2D Young diagrams and Maya diagrams and the relation between 2D and 3D Young diagrams, we construct 3D Young diagrams in the approach of Maya diagrams. Moreover, we formulate the generating function of 3D Young diagrams, which is the MacMahon function in terms of Maya diagrams.

Descent sets on 321-avoiding involutions and hook decompositions of partitions

We show that the distribution of the major index over the set of involutions in Sn that avoid the pattern 321 is given by the q-analogue of the n-th central binomial coefficient. The proof consists of a composition of three non-trivial bijections, one being the Robinson–Schensted correspondence, ultimately mapping those involutions with major index m into partitions of m whose Young diagram fits inside a ⌊n2⌋×⌈n2⌉ box. We also obtain a refinement that keeps track of the descent set, and we deduce an analogous result for the comajor index of 123-avoiding involutions.

Quasi-symmetric functions as polynomial functions on Young diagrams

We determine the most general form of a smooth function on Young diagrams, that is, a polynomial in the interlacing or multirectangular coordinates whose value depends only on the shape of the diagram. We prove that the algebra of such functions is isomorphic to quasi-symmetric functions, and give a noncommutative analog of this result.

Ordering molecular energies by moving boxes

Structures such as isoprotonic–isoelectronic molecules can be mathematically represented by Young diagrams (or integer partitions), whose columns correspond to the different nuclear charges in a molecule. We found that these Young diagrams, when ordered by majorization, produce a partially ordered set, which is equivalent to the ordering of ground-state energies of the represented molecules. This result shows that the ordering of such energies can be derived by moving boxes in Young diagrams. The correspondence between majorization order and energy ordering opens the door for obtaining new general expressions for molecular properties.

Limit shape of random convex polygonal lines: Even more universality

The paper concerns the limit shape (under some probability measure) of convex polygonal lines with vertices on Z+2, starting at the origin and with the right endpoint n=(n1,n2)→∞. In the case of the uniform measure, an explicit limit shape γ⁎:={(x1,x2)∈R+2:1−x1+x2=1} was found independently by Vershik (1994) [19], Bárány (1995) [3], and Sinaĭ (1994) [16]. Recently, Bogachev and Zarbaliev (1999) [5] proved that the limit shape γ⁎ is universal for a certain parametric family of multiplicative probability measures generalizing the uniform distribution. In the present work, the universality result is extended to a much wider class of multiplicative measures, including (but not limited to) analogs of the three meta-types of decomposable combinatorial structures — multisets, selections, and assemblies. This result is in sharp contrast with the one-dimensional case where the limit shape of Young diagrams associated with integer partitions heavily depends on the distributional type.

Ascent-Descent Young Diagrams and Pattern Avoidance in Alternating Permutations

We investigate pattern avoidance in alternating permutations and an alternating analogue of Young diagrams. In particular, using an extension of Babson and West's notion of shape-Wilf equivalence described in our recent paper (with N. Gowravaram), we generalize results of Backelin, West, and Xin and Ouchterlony to alternating permutations. Unlike Ouchterlony and Bóna's bijections, our bijections are not the restrictions of Backelin, West, and Xin's bijections to alternating permutations. This paper is the second of a two-paper series presenting the work of Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (with N. Gowravaram, arXiv:1301.6796v1). The first paper in the series is Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (with N. Gowravaram, Electronic Journal of Combinatorics 20(4):#P17, 2013).