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673
Rigidity of quasiisometries for symmetric spaces and Euclidean buildings
 Inst. Hautes Études Sci. Publ. Math
, 1997
"... 1.1 Background and statement of results An (L, C) quasiisometry is a map Φ: X − → X ′ between metric spaces such that for all x1, x2 ∈ X ..."
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Cited by 189 (28 self)
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1.1 Background and statement of results An (L, C) quasiisometry is a map Φ: X − → X ′ between metric spaces such that for all x1, x2 ∈ X
NonCrossing Partitions For Classical Reflection Groups
 Discrete Math
, 1996
"... We introduce analogues of the lattice of noncrossing set partitions for the classical reflection groups of type B and D. The type B analogues (first considered by Montenegro in a different guise) turn out to be as wellbehaved as the original noncrossing set partitions, and the type D analogues ..."
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Cited by 138 (5 self)
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We introduce analogues of the lattice of noncrossing set partitions for the classical reflection groups of type B and D. The type B analogues (first considered by Montenegro in a different guise) turn out to be as wellbehaved as the original noncrossing set partitions, and the type D analogues almost as wellbehaved. In both cases, they are ELlabellable ranked lattices with symmetric chain decompositions (selfdual for type B), whose rankgenerating functions, zeta polynomials, rankselected chain numbers have simple closed forms.
CRITERIA FOR VIRTUAL FIBERING
"... Abstract. We prove that an irreducible 3manifold whose fundamental group satisfies a certain grouptheoretic property is virtually fibered. As a corollary, we show that 3dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. Moreover, we sh ..."
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Cited by 63 (1 self)
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Abstract. We prove that an irreducible 3manifold whose fundamental group satisfies a certain grouptheoretic property is virtually fibered. As a corollary, we show that 3dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. Moreover, we show that a taut sutured compression body has a finitesheeted cover with a taut orientable depth one foliation. 1.
Bn Stanley Symmetric Functions
 Amer. J. Math
, 1994
"... Abstract. We define a new family ˜ Fw(X) of generating functions for w ∈ ˜ Sn which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions such as their symmetry and conjecture certain positivity properties. As an application, we relate these functions ..."
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Cited by 57 (13 self)
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Abstract. We define a new family ˜ Fw(X) of generating functions for w ∈ ˜ Sn which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions such as their symmetry and conjecture certain positivity properties. As an application, we relate these functions to the kSchur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. In [Sta84], Stanley introduced a family {Fw(X)} of symmetric functions now known as Stanley symmetric functions. He used these functions to study the number of reduced decompositions of permutations w ∈ Sn. Later, the functions Fw(X) were found to be stable limits of Schubert polynomials. Another fundamental property of Stanley symmetric functions is the fact that they are Schurpositive ([EG, LS]). This extended abstract describes work in progress on an analogue of Stanley symmetric functions for the affine symmetric group ˜ Sn which we call affine Stanley symmetric functions. Our first main theorem is that these functions ˜ Fw(X) are indeed symmetric functions. Most of the other main properties of Stanley symmetric functions established in [Sta84] also have analogues in the affine setting.
Triplygraded link homology and Hochschild homology of Soergel bimodules
"... We trade matrix factorizations and Koszul complexes for Hochschild homology of Soergel bimodules to modify the construction of triplygraded link homology and relate it to KazhdanLusztig theory. Hochschild homology. Let R be a kalgebra, where k is a field, R e = R ⊗k R op be the enveloping algebra ..."
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Cited by 57 (5 self)
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We trade matrix factorizations and Koszul complexes for Hochschild homology of Soergel bimodules to modify the construction of triplygraded link homology and relate it to KazhdanLusztig theory. Hochschild homology. Let R be a kalgebra, where k is a field, R e = R ⊗k R op be the enveloping algebra of R, and M be an Rbimodule (equivalently, a left R emodule). The functor of Rcoinvariants associates to M the factorspace MR = M/[R, M], where [R, M] is the subspace of M spanned by vectors of the form rm − mr. We have MR = R ⊗Re M. The Rcoinvariants functor is right exact and its ith derived functor takes M to Tor Re i (R, M). The latter space is also denoted HHi(R, M) and called the ith Hochschild homology of M. The Hochschild homology of M is the direct sum HH(R, M) def = ⊕ HHi(R, M). i≥0 To compute Hochschild homology, we choose a resolution of the Rbimodule R by projective Rbimodules and tensor the resolution with M:
POSITIVITY OF DUNKL’S INTERTWINING OPERATOR
 VOL. 98, NO. 3 DUKE MATHEMATICAL JOURNAL
, 1999
"... ..."
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 53 (14 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.
Permutation Statistics of Indexed Permutations
, 1994
"... The definitions of descent, excedance, major index, inversion index and Denert's statistic for the elements of the symmetric group S d are generalized to indexed permutations, i.e. the elements of the group S n d := Z n o S d , where o is wreath product with respect to the usual action of S d ..."
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Cited by 51 (2 self)
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The definitions of descent, excedance, major index, inversion index and Denert's statistic for the elements of the symmetric group S d are generalized to indexed permutations, i.e. the elements of the group S n d := Z n o S d , where o is wreath product with respect to the usual action of S d by permutations of f1; 2; : : : ; dg. It is shown, bijectively, that excedances and descents are equidistributed, and the corresponding descent polynomial, analogous to the Eulerian polynomial, is computed as the feulerian polynomial of a simple polynomial. The descent polynomial is shown to equal the hpolynomial (essentially the hvector) of a certain triangulation of the unit dcube. This is proved by a bijection which exploits the fact that the hvector of the simplicial complex arising from the triangulation can be computed via a shelling of the complex. The famous formula P d0 E d x d d! = sec x + tan x, where E d is the number of alternating permutations in S d , is general...
On the quotient ring by diagonal invariants
 Invent. Math
"... Abstract. For a Weyl group, W, and its reflection representation h, we find the character and Hilbert series for a quotient ring of C[h ⊕ h ∗ ] by an ideal containing the W–invariant polynomials without constant term. This confirms conjectures of Haiman. 1. ..."
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Cited by 50 (4 self)
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Abstract. For a Weyl group, W, and its reflection representation h, we find the character and Hilbert series for a quotient ring of C[h ⊕ h ∗ ] by an ideal containing the W–invariant polynomials without constant term. This confirms conjectures of Haiman. 1.