We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes A. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomology group, under some nonresonance conditions on the local system, ...

In this paper, we give a new realization of crystal bases for irreducible highest weight modules over Uq(G2) in terms of monomials. We also discuss the natural connection between the monomial realization and tableau realization. Introduction In 1985, the quantum groups Uq(g), which may be thought of as q-deformations of the universal enveloping algebras U(g) of Kac-Moody algebras g, were introd...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group on the sets of commuting and anticommuting variables. In this work, we present the superspace extension of the clas...

We describe (reduced) Gröbner bases of the ideal of polynomials over a field, which vanish on the set of characterisic vectors of the complete unifom families ([n] d ). An interesting feature of the results is that they are largely independent of the monomial order selected. The bases depend only on the ordering of the variables. We can thus use past results related to the lex order in the pres...

We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group on the sets of commuting and anticommuting variables. In this work, we present the superspace extension of the clas...