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62
Schubert polynomials for the affine Grassmannian
 in preparation, 2005. POLYNOMIALS FOR THE AFFINE GRASSMANNIAN 13
"... Abstract. Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the kSchur functions in homology and affine Schur functions in cohomology. Our results rely on Kostant and Kumar’s nilHecke ring, work of Peterson on th ..."
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Cited by 58 (16 self)
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Abstract. Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the kSchur functions in homology and affine Schur functions in cohomology. Our results rely on Kostant and Kumar’s nilHecke ring, work of Peterson on the homology of based loops on a compact group, and earlier work of ours on noncommutative kSchur functions. 1.
Affine insertion and Pieri rules for the affine Grassmannian
 Memoirs of the American Mathematical Society
, 2010
"... Abstract. We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n, C). Our main results are: • Pieri rules for the Schubert bases of H ∗ (Gr) and H∗(Gr), which expresses the product of a special Schubert class and an arbitrary Schubert class in term ..."
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Cited by 43 (13 self)
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Abstract. We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n, C). Our main results are: • Pieri rules for the Schubert bases of H ∗ (Gr) and H∗(Gr), which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. • A new combinatorial definition for kSchur functions, which represent the Schubert basis of H∗(Gr). • A combinatorial interpretation of the pairing H ∗ (Gr) ×H∗(Gr) → Z. These results are obtained by interpreting the Schubert bases of Gr combinatorially as generating functions of objects we call strong and weak tableaux, which are respectively defined using the strong and weak orders on the affine symmetric group. We define a bijection called affine insertion, generalizing the RobinsonSchensted Knuth correspondence,
QUANTUM COHOMOLOGY AND THE kSCHUR BASIS
, 2007
"... Abstract. We prove that structure constants related to Hecke algebras at roots of unity are special cases of kLittlewoodRichardson coefficients associated to a product of kSchur functions. As a consequence, both the 3point GromovWitten invariants appearing in the quantum cohomology of the Grassm ..."
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Cited by 29 (12 self)
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Abstract. We prove that structure constants related to Hecke algebras at roots of unity are special cases of kLittlewoodRichardson coefficients associated to a product of kSchur functions. As a consequence, both the 3point GromovWitten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to �su(ℓ) are shown to be kLittlewoodRichardson coefficients. From this, Mark Shimozono conjectured that the kSchur functions form the Schubert basis for the homology of the loop Grassmannian, whereas kSchur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual kSchur functions defined on weights of ktableaux that, given Shimozono’s conjecture, form the Schubert basis for the cohomology of the loop Grassmannian. We derive several properties of these functions that extend those of skew Schur functions. 1.
POSITROID VARIETIES I: JUGGLING AND GEOMETRY
, 2009
"... While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the cyclic shifts of one Bruhat decomposition turns out to have many of the good properties of the Bruhat and Richardson decompositions. This decomposition coi ..."
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Cited by 29 (4 self)
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While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the cyclic shifts of one Bruhat decomposition turns out to have 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, and BrownGoodearlYakimov. However, its cyclicinvariance is hidden in this description. Postnikov gave many cyclicinvariant 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 adopt his terminology and call the strata positroid varieties. We show that positroid varieties are normal and CohenMacaulay, and are defined as schemes by the vanishing of Plücker coordinates. We compute their Tequivariant Hilbert series, and show that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and BuchKreschTamvakis ’ approaches to quantum Schubert calculus. Our principal tools are the Frobenius splitting results for Richardson varieties as developed by Brion, Lakshmibai, and Littelmann, and the HodgeGröbner degeneration of the Grassmannian. We show that each positroid variety degenerates to the projective StanleyReisner scheme of a shellable ball.
Tamvakis : A Giambelli formula for isotropic Grassmannians, preprint (2008), available at arXiv:0811.2781
"... Abstract. Let X be a symplectic or odd orthogonal Grassmannian parametrizing isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in H ∗ (X, Z) as a polynomial in certain special Schubert c ..."
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Cited by 17 (4 self)
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Abstract. Let X be a symplectic or odd orthogonal Grassmannian parametrizing isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in H ∗ (X, Z) as a polynomial in certain special Schubert classes. We study theta polynomials, a family of polynomials defined using raising operators whose algebra agrees with the Schubert calculus on X. Furthermore, we prove that theta polynomials are special cases of BilleyHaiman Schubert polynomials and use this connection to express the former as positive linear combinations of products of Schur Qfunctions and Spolynomials. Let G = G(m,N) denote the Grassmannian of mdimensional subspaces of C N. To each integer partition λ = (λ1,...,λm) whose Young diagram is contained in an m × (N − m) rectangle, we associate a Schubert class σλ in the cohomology ring of G. The special Schubert classes σ1,...,σN−m are the Chern classes of the universal
Vexillary Elements in the Hyperoctahedral Group
 JOURNAL OF ALGEBRAIC COMBINATORICS
, 1998
"... We propose a definition for vexillary elements in the hyperoctahedral group using the Stanley symmetric functions. We show that the vexillary elements can again be determined by pattern avoidance conditions. These results can be extended to include the root systems of types A, B, C, and D. Finall ..."
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Cited by 14 (4 self)
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We propose a definition for vexillary elements in the hyperoctahedral group using the Stanley symmetric functions. We show that the vexillary elements can again be determined by pattern avoidance conditions. These results can be extended to include the root systems of types A, B, C, and D. Finally, we give an algorithm for multiplication of Schur Qfunctions with a superfied Schur function and a method for determining the shape of a signed permutation using jeu de taquin.
Total positivity for loop groups III: regular matrices and loop symmetric functions, in preparation
"... Abstract. This is the second in a series of papers developing a theory of total positivity for loop groups. In this paper, we study infinite products of Chevalley generators. We show that the combinatorics of infinite reduced words underlies the theory, and develop the formalism of infinite sequence ..."
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Cited by 12 (8 self)
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Abstract. This is the second in a series of papers developing a theory of total positivity for loop groups. In this paper, we study infinite products of Chevalley generators. We show that the combinatorics of infinite reduced words underlies the theory, and develop the formalism of infinite sequences of braid moves, called a braid limit. We relate this to a partial order, called the limit weak order, on infinite reduced words. The limit semigroup generated by Chevalley generators has a transfinite structure. We prove a form of unique factorization for its elements, in effect reducing their study to infinite products which have the order structure of N. For the latter infinite products, we show that one always has a factorization which matches an infinite Coxeter element. One of the technical tools we employ is a totally positive exchange lemma which appears to be of independent interest. This result states that the exchange lemma (in the context of Coxeter groups) is compatible with total positivity in the form of certain inequalities. Contents
Ktheory Schubert calculus of the affine Grassmannian
, 2009
"... We construct the Schubert basis of the torusequivariant Khomology of the affine Grassmannian of a simple algebraic group G, using the Ktheoretic NilHecke ring of Kostant and Kumar. This is the Ktheoretic analogue of a construction of Peterson in equivariant homology. For the case G = SLn, the K ..."
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Cited by 11 (3 self)
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We construct the Schubert basis of the torusequivariant Khomology of the affine Grassmannian of a simple algebraic group G, using the Ktheoretic NilHecke ring of Kostant and Kumar. This is the Ktheoretic analogue of a construction of Peterson in equivariant homology. For the case G = SLn, the Khomology of the affine Grassmannian is identified with a subHopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, called KkSchur functions, whose highest degree term is a kSchur function. The dual basis in Kcohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in Khomology. Many of our constructions have geometric interpretations using Kashiwara’s thick affine flag manifold.
The shifted plactic monoid
, 2008
"... We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman’s mixed insertion. Applications include: a new combinatorial derivation (and a new version of) th ..."
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Cited by 11 (0 self)
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We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman’s mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted LittlewoodRichardson Rule; similar results for the coefficients in the Schur expansion of a Schur Pfunction; a shifted counterpart of the LascouxSchützenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.