On irreducible characters of the Iwahori-Hecke algebra in type $A$

We use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type $A$. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in lower degrees. Explicit formulas are derived for the irreducible characters labeled by hooks and two-row partitions. Using duality, we also formulate a determinant type Murnaghan-Nakayama formula and give another proof of Rams combinatorial Murnaghan-Nakayama formula. As applications, we study super-characters of the Iwahori-Hecke algebra as well as the bitrace of the regular representation and provide a simple proof of the Halverson-Luduc-Ram formula.

Lattice structure of modular vertex algebras

In this paper we study the integral form of the lattice vertex algebra $V_L$. Based on the lattice structure, we introduce and study the associated modular vertex algebras $V_p$ and the quotient algebra $overline{V}_p$ as well as their irreducible modules over $mathbb Z_p$. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant $mathbb Z$-forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras. We also prove that the irreducible modules (modulo Heisenberg generators of degree divisible by $p$) remain irreducible for the modular vertex algebra $overline{V}_p$.

The Green polynomials via vertex operators

An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood functions. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Greens polynomials associated with upper partitions of length $leq 3$ and the diagonal lengths $leq 3$; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group $G$ and the Iwahori-Hecke algebra of type $A$ on the permutation module of $G$ by its Borel subgroup.

Representations of Quantum Affine Algebras in their $R$-Matrix Realization

We use the isomorphisms between the $R$-matrix and Drinfeld presentations of the quantum affine algebras in types $B$, $C$ and $D$ produced in our previous work to describe finite-dimensional irreducible representations in the $R$-matrix realization. We also review the isomorphisms for the Yangians of these types and use Gauss decomposition to establish an equivalence of the descriptions of the representations in the $R$-matrix and Drinfeld presentations of the Yangians.

Level $-1/2$ realization of quantum N-toroidal algebras in type $C_n$

We construct a level $-frac{1}{2}$ vertex representation of the quantum N-toroidal algebra for type $C_n$, which is a natural generalization of the usual quantum toroidal algebra. The construction also provides a vertex representation of the quantum toroidal algebra for type $C_n$ as a by-product.

Uncertainty relations based on Wigner-Yanase skew information

In this paper, we use certain norm inequalities to obtain new uncertain relations based on the Wigner-Yanase skew information. First for an arbitrary finite number of observables we derive an uncertainty relation outperforming previous lower bounds. We then propose new weighted uncertainty relations for two noncompatible observables. Two separable criteria via skew information are also obtained.

Deformation of Cayleys hyperdeterminants

We introduce a deformation of Cayleys 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 Matsumotos formula for Jack functions.

Vertex representations of quantum N-toroidal algebras for type C

Quantum N-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper we construct a level-one vertex representation of the quantum N-toroidal algebra for type C. In particular, we also obtain a level-one module of the quantum toroidal algebra for type C as a special case.

Dynamical quantum determinants and Pfaffians

We introduce the dynamical quantum Pfaffian on the dynamical quantum general linear group and prove its fundamental transformation identity. Hyper quantum dynamical Pfaffian is also introduced and formulas connecting them are given.

Uncertainty relations in the product form

We study the uncertainty relation in the product form of variances and obtain some new uncertainty relations with weight, which are shown to be tighter than those derived from the Cauchy Schwarz inequality.