Young Diagrams Research Articles

Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity

We continue our study of the AGT correspondence between instanton counting on C^2/Z_p and Conformal field theories with the symmetry algebra A(r,p). In the cases r=1, p=2 and r=2, p=2 this algebra specialized to: A(1,2)=H+sl(2)_1 and A(2,2)=H+sl(2)_2+NSR. As the main tool we use a new construction of the algebra A(r,2) as the limit of the toroidal gl(1) algebra for q,t tend to -1. We claim that the basis of the representation of the algebra A(r,2) (or equivalently, of the space of the local fields of the corresponding CFT) can be expressed through Macdonald polynomials with the parameters q,t go to -1. The vertex operator which naturally arises in this construction has factorized matrix elements in this basis. We also argue that the singular vectors of the $\mathcal{N}=1$ Super Virasoro algebra can be realized in terms of Macdonald polynomials for a rectangular Young diagram and parameters q,t tend to -1.

Gradings on the algebra of upper triangular matrices of size three

Let UT3(F) be the algebra of 3×3 upper triangular matrices over a field F. On UT3(F), up to isomorphism, there are at most five non-trivial elementary gradings and we study the graded polynomial identities for such gradings. In case F is of characteristic zero we give a complete description of the space of multilinear graded identities in the language of Young diagrams through the representation theory of a Young subgroup of Sn. We finally compute the multiplicities in the graded cocharacter sequence for every elementary G-grading on UT3(F).

NILPOTENT ORBITS AND CODIMENSION-2 DEFECTS OF 6d${\mathcal N} = (2, 0)$ THEORIES

We study the local properties of a class of codimension-2 defects of the 6d [Formula: see text] theories of type J = A, D, E labeled by nilpotent orbits of a Lie algebra [Formula: see text], where [Formula: see text] is determined by J and the outer-automorphism twist around the defect. This class is a natural generalization of the defects of the six-dimensional (6d) theory of type SU (N) labeled by a Young diagram with N boxes. For any of these defects, we determine its contribution to the dimension of the Higgs branch, to the Coulomb branch operators and their scaling dimensions, to the four-dimensional (4d) central charges a and c and to the flavor central charge k.

A Hurwitz theory avatar of open–closed strings

We review and explain an infinite-dimensional counterpart of the Hurwitz theory realization (Alexeevski and Natanzon, Math. Russ. Izv. 72:3–24, 2008) of algebraic open–closed-string model à la Moore and Lazaroiu, where the closed and open sectors are represented by conjugation classes of permutations and the pairs of permutations, i.e. by the algebra of Young diagrams and bipartite graphs, respectively. An intriguing feature of this Hurwitz string model is the coexistence of two different multiplications, reflecting the deep interrelation between the theory of symmetric and linear groups, S ∞ and GL(∞).

Chebyshev blossoming in Müntz spaces: Toward shaping with Young diagrams

The notion of a blossom in extended Chebyshev spaces offers adequate generalizations and extra-utilities to the tools for free-form design schemes. Unfortunately, such advantages are often overshadowed by the complexity of the resulting algorithms. In this work, we show that for the case of Müntz spaces with integer exponents, the notion of a Chebyshev blossom leads to elegant algorithms whose complexities are embedded in the combinatorics of Schur functions. We express the blossom and the pseudo-affinity property in Müntz spaces in terms of Schur functions. We derive an explicit expression for the Chebyshev–Bernstein basis via an inductive argument on nested Müntz spaces. We also reveal a simple algorithm for dimension elevation. Free-form design schemes in Müntz spaces with Young diagrams as shape parameters are discussed.

Massless hook field in AdS d+1 from the holographic perspective

We systematically consider the AdS/CFT correspondence for a simplest mixed-symmetry massless gauge field described by hook Young diagram. We introduce the radial gauge fixing and explicitly solve the Dirichlet problem for the hook field equations. Solution finding conveniently splits in two steps. We first define an incomplete solution characterized by a functional freedom and then impose the boundary conditions. The resulting complete solution is fixed unambiguously up to boundary values. Two-point correlation function of hook primary operators is found via the corresponding boundary effective action computed separately in even and odd boundary dimensions. In particular, the higher-derivative action for boundary conformal hook fields is identified with a singular part of the effective action in even dimensions. The bulk/boundary symmetry transmutation within the Dirichlet boundary problem is explicitly studied. It is shown that traces of boundary fields are Stueckelberg-like modes that can be algebraically gauged away so that boundary fields are traceless.

Root-theoretic Young Diagrams, Schubert Calculus and Adjoint Varieties

Root-theoretic Young diagrams are a conceptual framework to discuss existence of a root-system uniform and manifestly non-negative combinatorial rule for Schubert calculus. Our main results use them to obtain formulas for (co)adjoint varieties of classical Lie type. This case is the simplest after the previously solved (co)minuscule family. Yet our formulas possess both uniform and non-uniform features. Les diagrammes de Young racine-théoriques forment un cadre conceptuel qui permet de discuter l’existence de règles de calcul de Schubert explicitement non-négatives et uniformes sur les systèmes de racines. Notre principal résultat est leur utilisation pour obtenir des formules pour les variétés (co)adjointes de types classiques. C’est le cas le plus simple après celui la famille (co)minuscule, déjà résolue. Nos formules possèdent toutefois des propriétés uniformes et non-uniformes.

Wronskian solution of general nonlinear evolution equations and Young diagram prove

摘要 给出一般非线性发展方程构造Wronskian解的间接法. 根据Young图运算的性质给出了文中命题的证明, 并讨论了置换群特征标与Young图表达式系数间的关系. 关键词: 非线性发展方程 / Wronskian解 / Young图 / 特征标 Abstract In this paper, we give indirect methods constructed Wronskian solution of a general nonlinear evolution equations. Under the properties of the computing of Young diagram we have proved the proposition of this paper and discuss the relationship between the permutation group character and Young diagram expressions coefficient. Keywords: nonlinear evolution equations / Wronskian determinant solution / Young diagram / irreducible character 作者及机构信息 成建军, 张鸿庆 1. 大连理工大学数学科学学院, 大连 116024 基金项目: 国家自然科学基金(批准号: 51109031, 50921001, 50909017)、教育部基金(批准号: 20100041120037)、中央高校基本科研业务费 (批准号: DUT12LK34, DUT12LK52)和 江苏省研究生科研创新计划(批准号: CXZZ12_0815)资助的课题. Authors and contacts Cheng Jian-Jun, Zhang Hong-Qing 1. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 51109031, 50921001, 50909017), the Program ofMinistry of Education of China (Grant No. 20100041120037), the Fundamental Research Fund for the Central Universities, China (Grant Nos. DUT12LK34, DUT12LK52), and the Postgraduate Research and Innovation Project of Jiangsu Province, China (Grant No. CXZZ12_0815). 参考文献 [1] Ablowitz M J, Clarkson P A 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform (Cambridge: Cambridge University Press) [2] Rogers C, Shadwick W R 1982 Bäcklund Transformation and Their Application (New York: Academic Press) [3] Zhang H Q, Fan E G, Lin G 1998 Chin. Phys. 7 649 [4] Matveev V B, Salle M A 1991 Darboux Transformations and Solitons (Berlin: Springer) [5] Hirota R 1971 Phys. Rev. Lett. 27 1192 [6] Freeman N C, Nimmo J J C 1983 Phys. Lett. A 95 1 [7] Hirota R, Tang 1986 J. Phys. Soc. Jpn. 55 2137 [8] Zhang S Q 2008 Acta Phys. Sin. 57 1335 (in Chinese) [张善卿 2008 物理学报 57 1335] [9] OHTA Y 2003 J. Nonlinear Math. Phys. 10 143 [10] Bogoyavlenskii O I 1990 Lett. Nuovo. Cimento. Math. USSR. Izv. 34 245 [11] Toda K, Yu S J, Fukuyama F 1999 Rep. Math. Phys. 44 247 [12] Yan Z Y, Zhang H Q 2002 Comput. Math. Appl. 44 1430 [13] Tian B, Zhao K Y, Gao Y T 1997 Int. J. Engng. Sci. 35 1081 [14] Calogero F, Degasperis A 1976 Nuovo. Cimento. B 32 201 [15] Elwakil S A, El-labany S K, Zahran M A, Sabry R 2003 Z. Naturforsch. A 58 39 [16] Estvez P G, Hernaez G A 2000 J. Phys. A 33 2131 [17] Yan Z Y 2003 Czech. J. Phys. 53 89 [18] Gilson C R, Nimmo J J C 1993 Phys. Lett. A 180 337 [19] Qu C Z 1996 Theor. Phys. 25 369 施引文献

Fourier-Deligne transform and representations of the symmetric group

We calculate the Fourier-Deligne transform of the IC extension to ${\C}^{n+1}$ of the local system ${\mathcal L}_{\Lambda}$ on the cone over $\Conf_n(¶^1)$ associated to a representation $\Lambda$ of $S_n$, where the length $n-k$ of the first row of the Young diagram of $\Lambda$ is at least $\frac{|\Lambda|-1}{2}$. The answer is the IC extension to the dual vector space ${\C}^{n+1}$ of the local system ${\mathcal R}_{\lambda}$ on the cone over the $k$-th secant variety of the rational normal curve in $¶^n$, where ${\mathcal R}_{\lambda}$ corresponds to the representation $\lambda$ of $S_k$, the Young diagram of which is obtained from the Young diagram of $\Lambda$ by deleting its first row. We also prove an analogous statement for $S_n$-local systems on fibers of the Abel-Jacobi map. We use our result on the Fourier-Deligne transform to rederive a part of a result of Michel Brion on Kronecker coefficients.

On Kerov polynomials for Jack characters (extended abstract)

We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We give a partial result in this direction, showing that some quantities are polynomials in the Jack parameter $\alpha$ with prescribed degree. Our result has several interesting consequences in various directions. Firstly, we give a new proof of the fact that the coefficients of Jack polynomials expanded in the monomial or power-sum basis depend polynomially in $\alpha$. Secondly, we describe asymptotically the shape of random Young diagrams under some deformation of Plancherel measure.

Fluctuations in an evolutional model of two-dimensional Young diagrams

We discuss the non-equilibrium fluctuation problem, which corresponds to the hydrodynamic limit established by Funaki and Sasada (2010) [9], for the dynamics of two-dimensional Young diagrams associated with the uniform and restricted uniform statistics, and derive linear stochastic partial differential equations in the limit. We show that their invariant measures are identical to the Gaussian measures which appear in the fluctuation limits in the static situations.

Mixed Symmetry-Tipe (k,1) Massless Tensor Fields. Consistent Interactions Of Dual Linearized Gravity

Abstract A particular case of interactions of a single massless tensor field with the mixed symmetry corresponding to a two-column Young diagram (k,1) with k=4, dual to linearized gravity in D=7, is considered in the context of: self-couplings, cross-interactions with a Pauli-Fierz field, cross-couplings to purely matter theories, and interactions with an Abelian 1-form. The general approach relies on the deformation of the solution to the master equation from the antifield-BRST formalism by means of the local cohomology of the BRST differential.

Zeros of Wronskians of Hermite polynomials and Young diagrams

For a certain class of partitions, a simple qualitative relation is observed between the shape of the Young diagram and the pattern of zeros of the Wronskian of the corresponding Hermite polynomials. In the case of the two-term Wronskian W(Hn,Hn+k), we give an explicit formula for the asymptotic shape of the zero set as n→∞. Some empirical asymptotic formulas are given for the zero sets of three-term and four-term Wronskians.

Hook-content Formulae for Symplectic and Orthogonal Tableaux

AbstractBy considering the specialisation sλ(1, q, q2, … , qn–1) of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape λ in terms of the contents and hook lengths of the boxes in the Young diagram. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal λ-tableaux.

Off-shell Hodge dualities in linearised gravity and E 11

In a spacetime of dimension n, the dual graviton is characterised by a Young diagram with two columns, the first of length n-3 and the second of length one. In this paper we perform the off-shell dualisation relating the dual graviton to the double-dual graviton, displaying the precise off-shell field content and gauge invariances. We then show that one can further perform infinitely many off-shell dualities, reformulating linearised gravity in an infinite number of equivalent actions. The actions require supplementary mixed-symmetry fields which are contained within the generalised Kac-Moody algebra E11 and are associated with null and imaginary roots.

Homotopy transfer and self-dual Schur modules

We consider the free 2-nilpotent graded Lie algebra $$\mathfrak{g}$$ generated in degree one by a finite dimensional vector space V. We recall the beautiful result that the cohomology $$H^ \cdot \left( {\mathfrak{g},\mathbb{K}} \right)$$ of $$\mathfrak{g}$$ with trivial coefficients carries a GL(V)-representation having only the Schur modules V with self-dual Young diagrams {λ: λ = λ′} in its decomposition into GL(V)-irreducibles (each with multiplicity one). The homotopy transfer theorem due to Tornike Kadeishvili allows to equip the cohomology of the Lie algebra g with a structure of homotopy commutative algebra.

Block characters of the symmetric groups

A block character of a finite symmetric group is a positive definite function which depends only on the number of cycles in a permutation. We describe the cone of block characters by identifying its extreme rays, and find relations of the characters to descent representations and the coinvariant algebra of ${\mathfrak{S}}_{n}$ . The decomposition of extreme block characters into the sum of characters of irreducible representations gives rise to certain limit shape theorems for random Young diagrams. We also study counterparts of the block characters for the infinite symmetric group ${\mathfrak{S}}_{\infty}$ , along with their connection to the Thoma characters of the infinite linear group GL ?(q) over a Galois field.

On the Vershik–Kerov Conjecture Concerning the Shannon–McMillan–Breiman Theorem for the Plancherel Family of Measures on the Space of Young Diagrams

Vershik and Kerov conjectured in 1985 that dimensions of irreducible representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to the Plancherel family of measures on the space of Young diagrams. The statement of the Vershik–Kerov conjecture can be seen as an analogue of the Shannon–McMillan–Breiman Theorem for the non-stationary Markov process of the growth of a Young diagram. The limiting constant is then interpreted as the entropy of the Plancherel measure. The main result of the paper is the proof of the Vershik–Kerov conjecture. The argument is based on the methods of Borodin, Okounkov and Olshanski.

On the representation matrices for the symmetric group adapted to electron‐pair and electron‐group wave functions using graphical methods of spin algebras

AbstractUsing graphical methods of spin algebras, we present a direct method for the determination of the representation matrices of the symmetric group in a basis adapted to electron‐pair and electron‐group wave functions. In particular, we consider nonstandard representations adapted to $S_{n_1 } \times S_{n_2 } \times \ldots \times S_{n_\ell }$ where n1 + n2 + … + nℓ = n. The Serber coupling scheme based on S2 × S2 × …× S2 is a special case suited for electron‐pair wave functions. Our approach exploiting graphical methods of spin algebras is contrasted with the use of Young diagrams and tableaux, Yamanouchi symbols and branching diagrams. © 2012 Wiley Periodicals, Inc.

Powers of the Vandermonde determinant, Schur functions and recursive formulas

The decomposition of an even power of the Vandermonde determinant in terms of the basis of Schur functions matches the decomposition of the Laughlin wavefunction as a linear combination of Slater wavefunctions and thus contributes to the understanding of the quantum Hall effect. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, we give recursive formulas for the coefficient of the Schur function sμ in the decomposition of an even power of the Vandermonde determinant in n + 1 variables in terms of the coefficient of the Schur function sλ in the decomposition of the same even power of the Vandermonde determinant in n variables if the Young diagram of μ is obtained from the Young diagram of λ by adding a tetris type shape to the top or to the left.