Given a permutation w ?? Sn, we consider a determinantal ideal Iw whose generators are certain minors in the generic n ?~ n matrix (filled with independent variables). Using ?emultidegrees?f as simple algebraic substitutes for torus-equivariant cohomology classes on vector spaces, our main theorems describe, for each ideal Iw.

Gröbner geometry of Schubert polynomials

Preface 1. The KdV equation and its symmetries 2. The KdV hierarchy 3. The Hirota equation and vertex operators 4. The calculus of Fermions 5. The Boson-Fermion correspondence 6. Transformation groups and tau functions 7. The transformation group of the KdV equation 8. Finite dimensional Grassmannians and Plucker relations 9. Infinite dimensional Grassmannians 10. The bilinear identity revisited Solutions to exercises Bibliography Index.

Solitons : differential equations, symmetries and infinite dimensional algebras

We suggest a new combinatorial construction for the cohomology ring of the ag manifold. The degree 2 commutation relations satissed by the divided diierence operators corresponding to positive roots deene a quadratic associative algebra. In this algebra, the formal analogues of Dunkl operators generate a commuta-tive subring, which is shown to be canonically isomorphic to the cohomology of the ag manifold. This leads to yet another combinatorial version of the corresponding Schubert calculus. The paper contains numerous conjectures and open problems. We also discuss a generalization of the main construction to quantum cohomology. Contents

Quadratic Algebras, Dunkl Elements, and Schubert Calculus

Our aim is to alternate elements of the general theory with significant examples. We deal now with synmietric functions. In this chapter we will develop some of the very basic theorems on symmetric functions, in part as a way to give a look into 19* century invariant theory, but as well to establish some useful formulas which will show their full meaning only after developing the representation theory of the linear and symmetric groups.

On symmetric functions.

We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in GLn form a real semi-algebraic cell of dimension n 1. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of GLn(C) relying in particular on the positivity of the structure constants, which are enumerative Gromov-Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterson's which we explain with proofs in the type A case.

/pdf/totally-positive-toeplitz-matrices-and-quantum-cohomology-of-54bsa8ilwl.pdf

Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties

We prove the Lefschetz property for a certain class of finite-dimensional Gorenstein algebras associated to matroids. Our result implies the Sperner property of the vector space lattice. More generally, it is shown that the modular geometric lattice has the Sperner property. We also discuss the Gr\"obner fan of the defining ideal of our Gorenstein algebra.

/pdf/sperner-property-and-finite-dimensional-gorenstein-algebras-31gzhq9bz0.pdf

Sperner property and finite-dimensional Gorenstein algebras associated to matroids

We give a characterization of the Lefschetz elements in Artinian Gorenstein rings over a field of characteristic zero in terms of the higher Hessians. As an application, we give new examples of Artinian Gorenstein rings which do not have the strong Lefschetz property.

/pdf/lefschetz-elements-of-artinian-gorenstein-algebras-and-4nds06leno.pdf

Lefschetz elements of artinian gorenstein algebras and hessians of homogeneous polynomials

https://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1599.pdf

Nichols-Woronowicz model of coinvariant algebra of complex reflection groups

Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa–Intriligator formula

We introduce a new algebra associated with a hyperplane arrangement $\mathcal{A}$, called the Solomon-Terao algebra $\mbox{ST}(\mathcal{A},\eta)$, where $\eta$ is a homogeneous polynomial. It is shown by Solomon and Terao that $\mbox{ST}(\mathcal{A},\eta)$ is Artinian when $\eta$ is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon-Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that $\mbox{ST}(\mathcal{A},\eta)$ is a complete intersection if and only if $\mathcal{A}$ is free. We also give a factorization formula of the Hilbert polynomials when $\mathcal{A}$ is free, and pose several related questions, problems and conjectures.

Solomon-Terao algebra of hyperplane arrangements

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