A Unified Approach to Combinatorial Formulas for Schubert Polynomials

Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams for permutations known as rc-graphs as the main structure. The other structures in the literature we study include: semistandard Young tableaux, Kohnert diagrams, and balanced labelings of the diagram of a permutation. The main tools in our investigation are certain operations on rc-graphs, which correspond to the coplactic operations on tableaux, and thus define a crystal graph structure on rc-graphs; a new definition of these operations is presented. One application of these operations is a straightforward, purely combinatorial proof of a recent formula (due to Buch, Kresch, Tamvakis, and Yong), which expresses Schubert polynomials in terms of products of Schur polynomials. In spite of the fact that it refers to many objects and results related to them, the paper is mostly self-contained.     

H3势能面的键表方法计算

本文采用键表自洽场方法计算H₃体系的势能面,完成了用键表相互作用和多键表自洽场方法计算H₃体系的程序编写和计算,讨论了各种近似条件下键表方法的特点,得到一些对价键方法应用具有建设性意义的结论。     

键表的自洽场方法

讨论了键表酉群方法(BTUGA)的自洽场计算,建立了键表相互作用(BTI)和多键表自洽场(MBTSCF)方法。用该方法可使BTUGA广泛地应用于实际分子体系的量子化学计算中。     

A Littlewood-Richardson rule for factorial Schur functions

We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.

     

Vicious random walkers and a discretization of Gaussian random matrix ensembles

The vicious random walker problem on a one-dimensional lattice is considered. Many walkers take simultaneous steps on the lattice and the configurations in which two of them arrive at the same site are prohibited. It is known that the probability distribution of N walkers after M steps can be written in a determinant form. Using an integration technique borrowed from the theory of random matrices, we show that arbitrary kth order correlation functions of the walkers can be expressed as quaternion determinants whose elements are compactly expressed in terms of symmetric Hahn polynomials.     

A note on quantum products of Schubert classes in a Grassmannian

Given two Schubert classes σ_λ and σ_μ in the quantum cohomology of a Grassmannian, we construct a partition ν, depending on λ and μ, such that σ_ν appears with coefficient 1 in the lowest (or highest) degree part of the quantum product σ_λ⋆σ_μ. To do this, we show that for any two partitions λ and μ, contained in a k × (n − k) rectangle and such that the 180^∘-rotation of one does not overlap the other, there is a third partition ν, also contained in the rectangle, such that the Littlewood-Richardson number c _λμ ^ν is 1.     

Crystal bases of modified quantized enveloping algebras and a double RSK correspondence

We give a new combinatorial realization of the crystal base of the modified quantized enveloping algebras of type A+∞ or A_∞. It is obtained by describing the decomposition of the tensor product of a highest weight crystal and a lowest weight crystal into extremal weight crystals, and taking its limit using a tableaux model of extremal weight crystals. This realization induces in a purely combinatorial way a bicrystal structure of the crystal base of the modified quantized enveloping algebras and hence its Peter-Weyl type decomposition generalizing the classical RSK correspondence.     

Mixed-symmetry massless gauge fields in AdS5

Using the su(2, 2) spinor language, we describe free mixed-symmetry massless bosonic and fermionic gauge fields of arbitrary spins in the AdS₅ space. We construct manifestly covariant action functionals and derive field equations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 47–59, October, 2006.     

Pairs of noncrossing free Dyck paths and noncrossing partitions

Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length 2n and noncrossing partitions of [2n+1] with n+1 blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm.