Table of basic properties Notation and basic definitions Preface 1. What is the integral closure 2. Integral closure of rings 3. Separability 4. Noetherian rings 5. Rees algebras 6. Valuations 7. Derivations 8. Reductions 9. Analytically unramified rings 10. Rees valuations 11. Multiplicity and integral closure 12. The conductor 13. The Briancon-Skoda theorem 14. Two-dimensional regular local rings 15. Computing the integral closure 16. Integral dependence of modules 17. Joint reductions 18. Adjoints of ideals 19. Normal homomorphisms Appendix A. Some background material Appendix B. Height and dimension formulas References Index.

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Integral closure of ideals, rings, and modules

Harmonic analysis on semi‐simple lie groups—i

Page 55, proof of Lemma 3.13. This proof is incorrect as it stands because it involves an interchange of limits that has not been justified. A naive attempt to fix the proof might involve assuming that the given representation is continuous into the uniform operator topology, but this assumption is not necessarily valid. Instead, a different approach is needed, and such an approach may be found in the paper by M. Welleda Baldoni, “General representation theory of real reductive groups,” Proc. Sumposia in Pure Math. 61 (1997), 61–72. The relevant proof is the proof of Proposition 1 on page 63. The argument is straightforward enough, but it makes use of the fact that g → π(g)v is of class C1 if and only if for each w in the Hilbert space, g → hπ(g)v, wi is of class C1. She attributes this result to Grothendieck and points to J. B. Neto, Trans. Amer. Math. Soc. 111 (1964), 381–391, for “discussion and results.” Reading Proposition 1 and its proof, one sees that by approaching the development in a slightly different order, one gets around the temptation to use the uniform operator topology and obtains a valid proof of Proposition 3.14.

REPRESENTATION THEORY OF SEMISIMPLE GROUPS: An Overview Based on Examples

The Tata Institute of Fundamental Research

Lie Groups and Lie Algebras

Generalities on algebraic varieties.- Generalities on algebraic groups.- Grassmannian.- Determinantal varieties.- Symplectic Grassmannian.- Orthogonal Grassmannian.- The standard monomial theoretic basis.- Review of GIT.- Invariant theory.- SLn(K)-action.- SOn(K)-action.- Applications of standard monomial theory.

Standard Monomial Theory: Invariant Theoretic Approach

The main result of the paper is a determinantal formula for the restriction to a torus fixed point of the equivariant class of a Schubert subva- riety in the torus equivariant integral cohomology ring of the Grassmannian. As a corollary, we obtain an equivariant version of the Giambelli formula. The (torus) equivariant cohomology rings of flag varieties in general and of the Grassmannian in particular have recently attracted much interest. Here we con- sider the equivariant integral cohomology ring of the Grassmannian. Just as the ordinary Schubert classes form a module basis over the ordinary cohomology ring of a point (namely the ring of integers) for the ordinary integral cohomology ring of the Grassmannian, so do the equivariant Schubert classes form a basis over the equivariant cohomology of a point (namely the ordinary cohomology ring of the classifying space of the torus) for the equivariant cohomology ring (this is true for any generalized flag variety of any type, not just the Grassmannian). Again as in the ordinary case, computing the structure constants of the multiplication with respect to this basis is an interesting problem that goes by the name of Schubert calculus. There is a forgetful functor from equivariant cohomology to ordinary cohomology so that results about the former specialize to those about the latter. Knutson-Tao-Woodward (5) and Knutson-Tao (6) show that the structure con- stants, both ordinary and equivariant, count solutions to certain jigsaw puzzles, thereby showing that they are "manifestly" positive. In the present paper we take a very different route to computing the equivariant structure constants. Namely, we try to extend to the equivariant case the classical approach by means of the Pieri and Giambelli formulas. Recall, from (3, Eq.(10), p.146) for example, that the Gi- ambelli formula expresses an arbitrary Schubert class as a polynomial with integral coefficients in certain "special" Schubert classes—the Chern classes of the tautolog- ical quotient bundle—and that the Pieri formula expresses as a linear combination of the Schubert classes the product of a special Schubert class with an arbitrary Schubert class. Together they can be used to compute the structure constants. We only partially succeed in our attempt: the first of the three theorems of this paper—see §2 below—is an equivariant Giambelli formula that specializes to the ordinary Giambelli formula as in (3, Eq.(10), p.146), but we still do not have a satisfactory equivariant Pieri formula—see, however, §7 below. The derivation in Fulton (2, §14.3) of the Giambelli formula can perhaps be extended to the equi- variant case, but this is not what we do. Instead, we deduce the Giambelli formula from our second theorem which gives a certain closed-form determinantal formula for the restriction to a torus fixed point of an equivariant Schubert class.

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Equivariant Giambelli and determinantal restriction formulas for the Grassmannian

Hilbert functions of points on Schubert varieties in Grassmannians

We give an explicit combinatorial description of the multiplicity as well as the Hilbert function of the tangent cone at any point on a Schubert variety in the symplectic Grassmannian

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Hilbert functions of points on Schubert varieties in the symplectic Grassmannian

Cohen Macaulay property of fiber cones of ideals is characterized in terms of its Hilbert series. Hilbert series of fiber cones of ideals with minimal mixed multiplicity is calculated. It is proved that the fiber cone of an m-primary ideal I with minimal mixed multiplicity is Cohen-Macaulay if and only if its reduction number is atmost one. Hilbert series of fiber cones of ideals generated by quadratic sequences in standard graded rings is computed by deforming it to a face ring of a simplicial complex. Applications are given to defining ideals of monomial projective space curves lying on the quadric xw-yz=0, straightening-closed ideals in graded algebras with straightening law and Huckaba-Huneke ideals of analytic spread 1 and 2.

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K. N. Raghavan

Papers

Standard Monomial Theory: Invariant Theoretic Approach

Equivariant Giambelli and determinantal restriction formulas for the Grassmannian

Hilbert functions of points on Schubert varieties in Grassmannians

Hilbert functions of points on Schubert varieties in the symplectic Grassmannian

Cohen-Macaulay fiber cones