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32
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.
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.
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 39 (12 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,
Tableaux on k + 1cores, reduced words for affine permutations, and kSchur expansions, accepted to J Combin Theory Ser A
, 2004
"... Abstract. The kYoung lattice Y k is a partial order on partitions with no part larger than k. This weak subposet of the Young lattice originated [9] from the study of the kSchur functions s (k) λ, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed b ..."
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Cited by 33 (7 self)
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Abstract. The kYoung lattice Y k is a partial order on partitions with no part larger than k. This weak subposet of the Young lattice originated [9] from the study of the kSchur functions s (k) λ, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by kbounded partitions. The chains in the kYoung lattice are induced by a Pieritype rule experimentally satisfied by the kSchur functions. Here, using a natural bijection between kbounded partitions and k + 1cores, we establish an algorithm for identifying chains in the kYoung lattice with certain tableaux on k+1 cores. This algorithm reveals that the kYoung lattice is isomorphic to the weak order on the quotient of the affine symmetric group ˜ Sk+1 by a maximal parabolic subgroup. From this, the conjectured kPieri rule implies that the kKostka matrix connecting the homogeneous basis {hλ} λ∈Y k to {s (k) λ} λ∈Y k may now be obtained by counting appropriate classes of tableaux on k + 1cores. This suggests that the conjecturally positive kSchur expansion coefficients for Macdonald polynomials (reducing to q, tKostka polynomials for large k) could be described by a q, tstatistic on these tableaux, or equivalently on reduced words for affine permutations. 1.
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 28 (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.
The Distribution Of Descents And Length In A Coxeter Group
 ELECTRONIC J. COMBINATORICS
, 1995
"... We give a method for computing the qEulerian distribution W (t; q) = X w2W t des(w) q l(w) as a rational function in t and q, where (W; S) is an arbitrary Coxeter system, l(w) is the length function in W , and des(w) is the number of simple reflections s 2 S for which l(ws) ! l(w). Using t ..."
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Cited by 26 (2 self)
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We give a method for computing the qEulerian distribution W (t; q) = X w2W t des(w) q l(w) as a rational function in t and q, where (W; S) is an arbitrary Coxeter system, l(w) is the length function in W , and des(w) is the number of simple reflections s 2 S for which l(ws) ! l(w). Using this we compute generating functions encompassing the qEulerian distributions of the classical infinite families of finite and affine Weyl groups.
Affine Weyl groups as infinite permutations
 Electronic J. Combinatorics
, 1998
"... We present a unified theory for permutation models of all the infinite families of finite and a#ne Weyl groups, including interpretations of the length function and the weak order. We also give new combinatorial proofs of Bott's formula (in the refined version of Macdonald) for the Poincareseri ..."
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Cited by 12 (1 self)
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We present a unified theory for permutation models of all the infinite families of finite and a#ne Weyl groups, including interpretations of the length function and the weak order. We also give new combinatorial proofs of Bott's formula (in the refined version of Macdonald) for the Poincareseriesofthese a#ne Weyl groups. 1991 Mathematics Subject Classification. primary 20B35; secondary 05A15. 1 Introduction The aim of this paper is to present a unified theory for permutation representations of the finite Weyl groups A n1 , B n , C n , D n , and the a#ne Weyl groups # A n1 , # B n , # C n , # D n . Our starting point is the symmetric group S n , the group of permutations of [1,...,n]. If S n is presented as the group generated by adjacent transpositions, it is isomorphic to the Weyl group A n1 , and we obtain wellknown interpretations of several Coxeter group concepts in permutation language: 1. The Coxeter generators are the adjacent transpositions. 2. Reflections corresp...
An improved tableau criterion for Bruhat order
 ELECTRON. J. COMBIN
, 1996
"... To decide whether two permutations are comparable in Bruhat order of S n with the wellknown tableau criterion requires \Gamma n 2 \Delta comparisons of entries in certain sorted arrays. We show that to decide whether x y only d 1 + d 2 + : : : + d k of these comparisons are needed, where fd 1 ..."
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Cited by 12 (0 self)
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To decide whether two permutations are comparable in Bruhat order of S n with the wellknown tableau criterion requires \Gamma n 2 \Delta comparisons of entries in certain sorted arrays. We show that to decide whether x y only d 1 + d 2 + : : : + d k of these comparisons are needed, where fd 1 ; d 2 ; : : : ; d k g = fijx(i) ? x(i + 1)g. This is obtained as a consequence of a sharper version of Deodhar's criterion, which is valid for all Coxeter groups.