The saturation conjectureA very old and fundamental question about the representation theory of GL n (C) is the following:For which triples of dominant weights λ, µ, ν does the tensor product V λ ⊗V µ ⊗V ν of the irreducible representations with those high weights contain a GL n (C)-invariant vector?Another standard, if less symmetric, formulation of the problem above replaces V ν with its dual, and asks for which ν is V * ν a constituent of V λ ⊗V µ . In this formulation one can without essential loss of generality restrict to the case that λ, µ, and ν * are polynomial representations, and rephrase the question in the language of Littlewood-Richardson coefficients; it asks for which triple of partitions λ, µ, ν * is the Littlewood-Richardson coefficient c ν * λµ positive. It is not hard to prove (as we will see later in this introduction) that the set of such triples (λ, µ, ν) is closed under addition, so forms a monoid. In this paper we prove that this monoid is saturated, i.e. that for each triple of dominant weights (λ, µ, ν),This is of particular interest because Klyachko has recently given an answer 1 to the general question above, which in one direction was only asymptotic [Kl]:If V λ ⊗V µ ⊗V ν has a GL n (C)-invariant vector, then λ, µ, ν satisfy a certain system of linear inequalities derived from Schubert calculus (plus the evident linear equality that λ + µ + ν be in the root lattice; in the L-R context this asks that the number of boxes

Given a permutation w ∈ S n , we consider a determinantal ideal I w whose generators are certain minors in the generic n × n matrix (filled with independent variables). Using 'multidegrees' as simple algebraic substitutes for torus-equivariant cohomology classes on vector spaces, our main theorems describe, for each ideal I w :• variously graded multidegrees and Hilbert series in terms of ordinary and double Schubert and Grothendieck polynomials;• a Gröbner basis consisting of minors in the generic n × n matrix;• the Stanley-Reisner simplicial complex of the initial ideal in terms of known combinatorial diagrams [FK96], [BB93] associated to permutations in S n ; and• a procedure inductive on weak Bruhat order for listing the facets of this complex.We show that the initial ideal is Cohen-Macaulay, by identifying the StanleyReisner complex as a special kind of "subword complex in S n ", which we define generally for arbitrary Coxeter groups, and prove to be shellable by giving an explicit vertex decomposition. We also prove geometrically a general positivity statement for multidegrees of subschemes. Our main theorems provide a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. More precisely, we apply these theorems to:• define a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric reasons;*AK was partly supported by the Clay Mathematics Institute, Sloan Foundation, and NSF. EM was supported by the Sloan Foundation and NSF. ALLEN KNUTSON AND EZRA MILLER• rederive from a topological perspective Fulton's Schubert polynomial formula for universal cohomology classes of degeneracy loci of maps between flagged vector bundles;• supply new proofs that Schubert and Grothendieck polynomials represent cohomology and K-theory classes on the flag manifold; and• provide determinantal formulae for the multidegrees of ladder determinantal rings.The proofs of the main theorems introduce the technique of "Bruhat induction", consisting of a collection of geometric, algebraic, and combinatorial tools, based on divided and isobaric divided differences, that allow one to prove statements about determinantal ideals by induction on weak Bruhat order.

While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown-Goodearl-Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen-Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov's and Buch-Kresch-Tamvakis' approaches to quantum Schubert calculus.

The product of two Schubert cohomology classes on a Grassmannian Gr k (C n ) has long been known to be a positive combination of other Schubert classes, and many manifestly positive formulae are now available for computing such a product (e.g. the Littlewood-Richardson rule, or the more symmetric puzzle rule from [Hon2]). Recently in [G] it was shown, nonconstructively, that a similar positivity statement holds for Tequivariant cohomology (where the coefficients are polynomials). We give the first manifestly positive formula for these coefficients, in terms of puzzles using an "equivariant puzzle piece".The proof of the formula is mostly combinatorial, but requires no prior combinatorics, and only a modicum of equivariant cohomology (which we include). As a by-product the argument gives a new proof of the puzzle (or Littlewood-Richardson) rule in the ordinarycohomology case, but this proof requires the equivariant generalization in an essential way, as it inducts backwards from the "most equivariant" case.This formula is closely related to the one in [MS] for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [G]. We include a cohomological interpretation of this problem, and a puzzle formulation for it.