This unique book provides the first introduction to crystal base theory from the combinatorial
point of view. Crystal base theory was developed by Kashiwara and Lusztig from the …

Let X= Gr (d, C n) be the Grassmann variety of d-dimensional subspaces of C n. The goal of
this paper is to give an explicit combinatorial description of the Grothendieck ring K~ of …

Equivariant cohomology has become an indispensable tool in algebraic geometry and in
related areas including representation theory, combinatorial and enumerative geometry, and …

In this paper we study Grothendieck polynomials indexed by Grassmannian permutations,
which are representatives for the classes corresponding to the structure sheaves of Schubert …

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the
Grassmannian, we study six combinatorial Hopf algebras. These Hopf algebras can be …

This paper presents a formula for products of Schubert classes in the quantum cohomology
ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for …

In this manuscript we study braid varieties, a class of affine algebraic varieties associated to
positive braids. Several geometric constructions are presented, including certain torus …

Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the
Schubert classes of the affine Grassmannian as the $ k $-Schur functions in homology and …

We define a new family [inline-graphic xmlns: xlink=" http://www. w3. org/1999/xlink" xlink:
href=" 01i"/] of generating functions for w∈[inline-graphic xmlns: xlink=" http://www. w3 …

A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of
its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic …