We study a ramification theory for a division algebra D of the following type: The center of D is a complete discrete valuation field K with the imperfect residue field F of certain type, and the residue algebra of D is commutative and purely inseparable over F. Using Swan conductors of the corresponding element of Br(K), we define Herbrand‘s ψ-function of D/K, and it describes the action of the reduced norm map on the filtration of D$^*$. We also gives a relation among the Swan conductors and the ’ramification number‘ of D, which is defined by the commutator group of D$^*$.

We study a ramification theory for a division algebra D of the following type: The center of D is a complete discrete valuation field K with the imperfect residue field F of certain type, and the residue algebra of D is commutative and purely inseparable over F. Using Swan conductors of the corresponding element of Br(K), we define ψ-function of D/K, and it describe the action of the reduced norm map on the filtration of D$^*$. We also gives a relation among the Swan conductors and the ’ramification number‘ of D, which is defined by the commutator group of D$^*$.

We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for these polynomials: a q-integral representation and a combinatorial formula. Our main tool is a q-integral representation for ordinary Macdonald polynomial. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials.

Let G be a complex semi-simple group, and X a compact Riemann surface. The moduli space of principal G-bundles on X, and in particular the holomorphic line bundles on this space and their global sections, play an important role in the recent applications of Conformal Field Theory to algebraic geometry. In this paper we determine the Picard group of this moduli space when G is of classical or G$_2$ type (we consider both the coarse moduli space and the moduli stack).

We show that, for certain types of rigid analytic varieties X and constructible l-adic sheaves (F$_n$)$_n$ on, one has H$^p_c$ (X,(F$_n$)$_n$) ${\stackrel{\sim}{\longrightarrow}} {\displaystyle\lim_{\stackrel{\scriptstyle{\longleftarrow}} {\scriptstyle{n}}}}$ H$^p_c$ (X,F$_n$). As an application we obtain that, for an algebraic variety X and associated rigid analytic variety X$^rig$, the l-adic cohomology of X and X$^rig$ agree.

We construct a correspondence between the set of partitions of a finite set M and the set of pairs of walks to the same vertex on a graph giving the Bratteli diagram of the partition algebra on M. This is the precise analogue of the correspondence between the set of permutations of a finite set and the set of pairs of Young tableaux of the same shape, called the Robinson–Schensted correspondence.