Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

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Combinatorial Commutative Algebra

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Groupes et algébres de lie

Introduction to lie algebras and representation theory

(1.1) This paper concerns three aspects of the action of a compact group K on a space X . The ®rst is concrete and the others are rather abstract. (1) Equivariantly formal spaces. These have the property that their cohomology may be computed from the structure of the zero and one dimensional orbits of the action of a maximal torus in K. (2) Koszul duality. This enables one to translate facts about equivariant cohomology into facts about its ordinary cohomology, and back. (3) Equivariant derived category. Many of the results in this paper apply not only to equivariant cohomology, but also to equivariant intersection cohomology. The equivariant derived category provides a framework in both of these may be considered simultaneously, as examples of ``equivariant sheaves''. We treat singular spaces on an equal footing with nonsingular ones. Along the way, we give a description of equivariant homology and equivariant intersection homology in terms of equivariant geometric cycles. Most of the themes in this paper have been considered by other authors in some context. In Sect. 1.7 we sketch the precursors that we know about. For most of the constructions in this paper, we consider an action of a compact connected Lie group K on a space X , however for the purposes of the introduction we will take K  S1 to be a torus. Invent. math. 131, 25±83 (1998)

/pdf/equivariant-cohomology-koszul-duality-and-the-localization-2973jtaovy.pdf

Equivariant cohomology, Koszul duality, and the localization theorem

These notes are the written version of my lectures at the Banach Center mini-school "Schubert Varieties" in Warsaw, May 18-22, 2003. Their aim is to give a self-contained exposition of some geometric aspects of Schubert calculus.

/pdf/lectures-on-the-geometry-of-flag-varieties-4wrwwdh93x.pdf

Lectures on the Geometry of Flag Varieties

We study Edidin and Graham's equivariant Chow groups in the case of torus actions. Our main results are: (i) a presentation of equivariant Chow groups in terms of invariant cycles, which shows how to recover usual Chow groups from equivariant ones; (ii) a precise form of the localization theorem for torus actions on projective, nonsingular varieties; (iii) a construction of equivariant multiplicities, as functionals on equivariant Chow groups; (iv) a construction of the action of operators of divided differences on theT-equivariant Chow group of any scheme with an action of a reductive group with maximal torusT. We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations. In particular, we obtain a presentation of the Chow ring of any smooth, projective spherical variety.

Equivariant Chow groups for torus actions

We obtain residue formulae for certain functions of several vari- ables. As an application, we obtain closed formulae for vector partition func- tions and for their continuous analogs. They imply an Euler-MacLaurin sum- mation formula for vector partition functions, and for rational convex poly- topes as well: we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope. Institut Fourier, B.P. 74, 38402 Saint-Martin d’Heres Cedex, France E-mail address: mbrion@fourier.ujf-grenoble.fr Ecole Normale Superieure, 45 rue d’Ulm, 75005 Paris Cedex 05, France E-mail address: vergne@dmi.ens.fr

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Residue formulae, vector partition functions and lattice points in rational polytopes

This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham.

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Michel Brion

Papers

Lectures on the Geometry of Flag Varieties

Equivariant Chow groups for torus actions

Residue formulae, vector partition functions and lattice points in rational polytopes

Equivariant cohomology and equivariant intersection theory

The total coordinate ring of a wonderful variety