The classical Pieri formula is an explicit rule for determining the coefficients in the expansion s1m · sλ = ∑ c 1,λ sμ , where sν is the Schur polynomial indexed by the partition ν. Since the Schur polynomials represent Schubert classes in the cohomology of the complex Grassmannian, this gives a partial description of the cup product in this cohomology. Pieri’s formula was generalized to the c...

In this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  We relate the cohomology of a normal subgroup $N$ of a topological group $G$ and the quotient $G/N$ to the cohomology of $G$. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-a...

Consider the ring R := Q[τ, τ−1] of Laurent polynomials in the variable τ . The Artin’s Pure Braid Groups (or Generalized Pure Braid Groups) act over R, where the action of every standard generator is the multiplication by τ . In this paper we consider the cohomology of these groups with coefficients in the module R (it is well known that such cohomology is strictly related to the untwisted int...

Inspired by work of Enright andWillenbring [EW], we prove a generalized Littlewood’s restriction formula in terms of Dirac cohomology. Our approach is to use a character formula of irreducible unitary lowest weight modules instead of the Bernstein-Gelfand-Gelfand resolution, and the proof is much simpler. We also show that our branching formula is equivalent to the formula of Enright and Willen...

Introduction 2 1. A Guide to Duality 3 1.1. Local Duality 3 1.2. Dualizing Complexes and Some Vanishing Theorems 10 1.3. Cohomological Annihilators 17 2. A Few Applications of Local Cohomology 21 2.1. On Ideal Topologies 21 2.2. On Ideal Transforms 25 2.3. Asymptotic Prime Divisors 28 2.4. The Lichtenbaum-Hartshorne Vanishing Theorem 35 2.5. Connectedness results 37 3. Local Cohomology and Syzy...