Crystal bases for the quantum superalgebra

A crystal base theory is introduced for the quantized enveloping algebra of the general linear Lie superalgebra , and an explicit realization of the crystal base is given in terms of semistandard tableaux.     

The evaluation of matrix elements for non-canonical Weyl tableau basis states adapted toU(n 1+n 2)⊃U(n 1)×U(n 2)

Summary This is the first paper of a series of two, which enables the evaluation ofU(n) generator matrix elements in the non-canonical Weyl tableau basis adapted to subgroupU(n ₁)×U(n ₂). In this paper the explicit closed formulae for subduction coefficients are presented. These formulae will become useful through an inductive method to be presented in the second paper.     

Symplectic Shifted Tableaux and Deformations of Weyl's Denominator Formula for sp(2n)

A determinantal expansion due to Okada is used to derive both a deformation of Weyl's denominator formula for the Lie algebra sp(2n) of the symplectic group and a further generalisation involving a product of the deformed denominator with a deformation of flagged characters of sp(2n). In each case the relevant expansion is expressed in terms of certain shifted sp(2n)-standard tableaux. It is then re-expressed, first in terms of monotone patterns and then in terms of alternating sign matrices.     

A secondary semantics for Second Order Intuitionistic Propositional Logic

In this paper we propose a Kripke‐style semantics for second order intuitionistic propositional logic and we provide a semantical proof of the disjunction and the explicit definability property. Moreover, we provide a tableau calculus which is sound and complete with respect to such a semantics. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Enumeration formulas for standard Young tableaux of nearly hollow rectangular shapes

This paper considers the enumeration problem of a generalization of standard Young tableau (SYT) of truncated shape. Let $\lambda ?\mu |\left\{(i_0,j_0)\right\}$ be the SYT of shape $\lambda$ truncated by $\mu$ whose upper left cell is $(i_0,j_0)$, where $\lambda$ and $\mu$ are partitions of integers. The summation representation of the number of SYT of the truncated shape $(n+k+2,{(n+2)}^{m+1})?\left(n^m\right)|\left\{(2,2)\right\}$ is derived. Consequently, three closed formulas for SYT of hollow shapes are obtained, including the cases of (i). $m=n=1$, (ii). $k=0$, and (iii). $k=1,m=n$. Finally, an open problem is posed.     

Crossings and nestings of matchings and partitions

We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of , as well as over all matchings on . As a corollary, the number of -noncrossing partitions is equal to the number of -nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no -crossing (or with no -nesting).

     

Graphical approach to correlations in high‐spin systems

We present a graphical technique for generating and indexing spin monomials of high‐spin systems. The procedure consists of developing a graph with at most n line segments from each node in a given row to the one immediately lower, where n is the multiplicity of single‐particle spin function. The paths lead to monomials with definite M values at each node. This technique has been used to generate and diagonalize the model spin Hamiltonian. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 389–393, 1999     

Proof of the Branner-Hubbard conjecture on Cantor Julia sets

By means of a nested sequence of some critical pieces constructed by Kozlovski, Shen, and van Strien, and by using a covering lemma recently proved by Kahn and Lyubich, we prove that a component of the filled-in Julia set of any polynomial is a point if and only if its forward orbit contains no periodic critical components. It follows immediately that the Julia set of a polynomial is a Cantor set if and only if each critical component of the filled-in Julia set is aperiodic. This result was a conjecture raised by Branner and Hubbard in 1992. This work was supported by the National Natural Science Foundation of China     

Box-Ball Systems and Robinson-Schensted-Knuth Correspondence

We study a box-ball system from the viewpoint of combinatorics of words and tableaux. Each state of the box-ball system can be transformed into a pair of tableaux (P, Q) by the Robinson-Schensted-Knuth correspondence. In the language of tableaux, the P-symbol gives rise to a conserved quantity of the box-ball system, and the Q-symbol evolves independently of the P-symbol. The time evolution of the Q-symbol is described explicitly in terms of the box-labels.     

Hall-Littlewood polynomials and fixed point enumeration

We resolve affirmatively some conjectures of Reiner, Stanton, and White (2004) [12] regarding enumeration of transportation matrices which are invariant under certain cyclic row and column rotations. Our results are phrased in terms of the bicyclic sieving phenomenon introduced by Barcelo, Reiner, and Stanton (2009) [1]. The proofs of our results use various tools from symmetric function theory such as the Stanton-White rim hook correspondence (Stanton and White (1985) [18]) and results concerning the specialization of Hall-Littlewood polynomials due to Lascoux, Leclerc, and Thibon (1994, 1997) [5] and [6].