arXiv subjects
Publications and source records attributed to Tommy Wuxing Cai.
We give a condition sufficient to ensure that an amalgam of groups is generalized torsion-free. As applications, we construct a closed 3-manifold whose fundamental group is generalized torsion-free and non bi-orderable; a one-relator group which is generalized torsion-free and non bi-orderable; and a group which is generalized torsion-free and non left-orderable.
We give a new criterion which guarantees that a free group admits a bi-ordering that is invariant under a given automorphism. As an application, we show that the fundamental group of the "magic manifold" is bi-orderable, answering a question of Kin and Rolfsen.
Using the idea of the quantum inverse scattering method, we introduce the operators $\mathbf{B}(x), \mathbf{C}(x)$ and $\mathbf{\tilde{B}}(x), \mathbf{\tilde{C}}(x)$ corresponding to the off-diagonal entries of the monodromy matrix $T$ for the phase model and $i$-boson model in terms of bc fermions and neutral fermions respectively, thus giving alternative treatment of the KP and BKP hierarchies. We also introduce analogous operators $\mathbf{B}^{*}(x)$ and $\mathbf{C}^{*}(x)$ for the charged free boson system and show that they are in complete analogy to those of $bc$ fermionic fields. It is proved that the correlation function $\langle 0|\mathbf{C}(x_N)\cdots\mathbf{C}(x_1)\mathbf{B}(y_1)\cdots $ $\mathbf{B}(y_N)|0\rangle$ in the $bc$ fermionic fields is the inverse of the correlation function $\langle 0|\mathbf{C}^{*}(x_N)\cdots\mathbf{C}^{*}(x_1)\mathbf{B}^{*}(y_1)\cdots \mathbf{B}^{*}(y_N)|0\rangle$ in the charged free bosons.
We introduce a deformation of Cayley's second hyperdeterminant for even-dimensional hypermatrices. As an application, we formulate a generalization of the Jacobi-Trudi formula for Macdonald functions of rectangular shapes generalizing Matsumoto's formula for Jack functions.
We prove a conjecture of Drew Armstrong on the average maximal length of $k$-alternating subsequence of permutations. The $k=1$ case is a well-known result of Richard Stanley.
We prove a conjecture of Miller and Reiner on the Smith normal form of the operator $DU$ associated with a differential poset for the special case of Young's lattice. Equivalently, this operator can be described as $\frac{\partial}{\partial p_1}p_1$ acting on homogeneous symmetric functions of degree $n$.
Based on a generalized Newton's identity, we construct a family of symmetric functions which deform the modular Hall-Littlewood functions. We also give a determinant formula for the Macdonald functions.
Using vertex operator we study Macdonald symmetric functions of rectangular shapes and their connection with the q-Dyson Laurent polynomial. We find a vertex operator realization of Macdonald functions and thus give a generalized Frobenius formula for them. As byproducts of the realization, we find a q-Dyson constant term orthogonality relation which generalizes a conjecture due to Kadell in 2000, and we generalize Matsumoto's hyperdeterminant formula for rectangular Jack functions to Macdonald functions.