Affine insertion and Pieri rules for the affine Grassmannian

103 Pages·2010·1.08 MB·English

by  Thomas Lam, Luc Lapointe, Jennifer Morse, Mark Shimozono

MEMOIRS
of the
American Mathematical Society
Number 977
Affine Insertion and Pieri Rules
for the Affine Grassmannian
Thomas Lam
Luc Lapointe
Jennifer Morse
Mark Shimozono
November 2010 • Volume 208 • Number 977 (second of 6 numbers) • ISSN 0065-9266
American Mathematical Society

Number 977
Affine Insertion and Pieri Rules
for the Affine Grassmannian
Thomas Lam
Luc Lapointe
Jennifer Morse
Mark Shimozono
November2010 • Volume208 • Number977(secondof6numbers) • ISSN0065-9266
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AffineinsertionandPierirulesfortheaffineGrassmannian/ThomasLam... [etal.].
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1.Geometry,Affine. 2.Combinatorialanalysis. I.Lam,Thomas1980-
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Contents
Introduction vii
Chapter 1. Schubert Bases of Gr and Symmetric Functions 1
1.1. Symmetric functions 1
1.2. Schubert bases of Gr 1
1.3. Schubert basis of the affine flag variety 3
Chapter 2. Strong Tableaux 5
2.1. S˜ as a Coxeter group 5
n
2.2. Fixing a maximal parabolic subgroup 6
2.3. Strong order and strong tableaux 6
2.4. Strong Schur functions 9
Chapter 3. Weak Tableaux 11
3.1. Cyclically decreasing permutations and weak tableaux 11
3.2. Weak Schur functions 12
3.3. Properties of weak strips 13
3.4. Commutation of weak strips and strong covers 14
Chapter 4. Affine Insertion and Affine Pieri 19
4.1. The local rule φ 19
u,v
4.2. The affine insertion bijection Φ 19
u,v
4.3. Pieri rules for the affine Grassmannian 24
4.4. Conjectured Pieri rule for the affine flag variety 25
4.5. Geometric interpretation of strong Schur functions 26
Chapter 5. The Local Rule φ 27
u,v
5.1. Internal insertion at a marked strong cover 27
5.2. Definition of φ 29
u,v
5.3. Proofs for the local rule 29
Chapter 6. Reverse Local Rule 39
6.1. Reverse insertion at a cover 39
6.2. The reverse local rule 41
6.3. Proofs for the reverse insertion 41
Chapter 7. Bijectivity 49
7.1. External insertion 50
7.2. Case A (commuting case) 51
7.3. Case B (bumping case) 51
7.4. Case C (replacement bump) 52
iii
iv CONTENTS
Chapter 8. Grassmannian Elements, Cores, and Bounded Partitions 55
8.1. Translation elements 55
8.2. The action of S˜ on partitions 58
n
8.3. Cores and the coroot lattice 58
8.4. Grassmannian elements and the coroot lattice 60
8.5. Bijection from cores to bounded partitions 60
8.6. k-conjugate 61
8.7. From Grassmannian elements to bounded partitions 61
Chapter 9. Strong and Weak Tableaux Using Cores 63
9.1. Weak tableaux on cores are k-tableaux 63
9.2. Strong tableaux on cores 64
9.3. Monomial expansion of t-dependent k-Schur functions 66
9.4. Enumeration of standard strong and weak tableaux 68
Chapter 10. Affine Insertion in Terms of Cores 73
10.1. Internal insertion for cores 73
10.2. External insertion for cores (Case X) 74
10.3. An example 74
10.4. Standard case 75
10.5. Coincidence with RSK as n→∞ 76
10.6. The bijection for n=3 and m=4 77
Bibliography 81
Abstract
We study combinatorial aspects of the Schubert calculus of the affine Grass-
mannian Gr associated with SL(n,C). Our main results are:
• PierirulesfortheSchubert basesofH∗(Gr)andH∗(Gr), whichexpresses
the product of a special Schubert class and an arbitrary Schubert class in
terms of Schubert classes.
• Anew combinatorial definitionfork-Schur functions, whichrepresent the
Schubert basis of H∗(Gr).
• A combinatorial interpretation of the pairing H∗(Gr)×H∗(Gr) → Z in-
duced by the cap product.
TheseresultsareobtainedbyinterpretingtheSchubertbasesofGrcombinatorially
as generating functions of objects we call strong and weak tableaux, which are
respectivelydefinedusingthestrongandweakordersontheaffinesymmetricgroup.
We define a bijection called affine insertion, generalizing the Robinson-Schensted
Knuthcorrespondence,whichsendscertainbiwordstopairsoftableauxofthesame
shape, one strong and one weak. Affine insertion offers a duality between the weak
and strong orders which does not seem to have been noticed previously.
Our cohomology Pieri rule conjecturally extends to the affine flag manifold,
and we give a series of related combinatorial conjectures.
ReceivedbytheeditorJanuary17,2007.
ArticleelectronicallypublishedonApril28,2010;S0065-9266(10)00576-4.
2000 MathematicsSubjectClassification. Primary05E05,14N15.
Key words and phrases. Tableaux, Robinson-Schensted insertion, Schubert calculus, Pieri
formula,affineGrassmannian.
This project was partially supported by NSF grants DMS-0652641, DMS-0652652, DMS-
0652668,andDMS-0652648.
ThefirstauthorwaspartiallysupportedbyNSFDMS–0600677.
ThesecondauthorwaspartiallysupportedbytheAnilloEcuacionesAsociadasaReticulados
financedbytheWorldBankthroughtheProgramaBicentenariodeCienciayTecnolog´ıa,andby
theProgramaReticuladosyEcuacionesoftheUniversidaddeTalca.
ThethirdauthorwaspartiallysupportedbyNSFDMS–0638625.
ThefourthauthorwaspartiallysupportedbyNSFDMS–0401012.
(cid:2)c2010 American Mathematical Society
v

Introduction
LetGr=G(C((t)))/G(C[[t]])denotetheaffineGrassmannianofG=SL(n,C),
where C[[t]] is the ring of formal power series and C((t)) = C[[t]][t−1] is the ring
of formal Laurent series. Since Gr∼=G/P for an affine Kac-Moody group G and a
maximal parabolic subgroup P, we may talk about the Schubert bases
{ξw ∈H∗(Gr,Z)|w ∈S˜0}
n
{ξw ∈H∗(Gr,Z)|w ∈S˜n0}
in the cohomology and homology of Gr, where S˜0 is the subset of the affine sym-
n
metricgroupS˜ consistingoftheaffineGrassmannianelements,whichbydefinition
n
aretheelementsofminimallengthintheircosetsinS˜ /S . Quillen(unpublished),
n n
and Garland and Raghunathan [6] showed that Gr is homotopy-equivalent to the
group ΩSU(n,C) of based loops into SU(n,C), and thus H∗(Gr) and H∗(Gr) ac-
quire structures of dual Hopf-algebras. In [3], Bott calculated H∗(Gr) and H∗(Gr)
explicitly–theycanbeidentifiedwithaquotientΛ(n) andasubringΛ ofthering
(n)
Λ of symmetric functions. Using an algebraic construction known as the nilHecke
ring, Kostant and Kumar [13] studied the Schubert bases of H∗(Gr) (in fact for
flag varieties of Kac-Moody groups) and Peterson [32] studied the Schubert bases
ofH∗(Gr). Lam[16],confirmingaconjectureofShimozono,identifiedtheSchubert
classes ξw and ξ explicitly as symmetric functions in Λ(n) and Λ .
w (n)
Incohomology,theSchubertclassesξw aregivenbythedual k-Schur functions
{F˜ |w ∈S˜0}⊆Λ(n),
w n
introduced in [22] by Lapointe and Morse. The dual k-Schur functions are gen-
erating functions of objects called k-tableaux [20]. In [15], this construction was
generalizedtothecaseofanarbitraryaffinepermutationw ∈S˜ ;k-tableauxarere-
n
placedbywhatwecallweak tableaux andthegeneratingfunctionofweaktableaux
U is called the affine Stanley symmetric function or weak Schur function
(cid:2)
F˜ (x)=Weak (x)= xwt(U).
w w
U
When w ∈ S ⊂ S˜ is a usual Grassmannian permutation (minimal length coset
n n
representative in Sn/(Sr ×Sn−r)), the affine Stanley symmetric function reduces
to a usual Schur function.
In homology, the Schubert classes ξ are given by the k-Schur functions
w
{s(k)(x)|λ <n}⊆Λ
λ 1 (n)
where the k in k-Schur or dual k-Schur function always means
k =n−1.
vii
viii INTRODUCTION
The k-Schur functions were first introduced by Lapointe, Lascoux, and Morse [18]
for the study of Macdonald polynomials [29], though so far a direct connection
between Macdonald polynomials and the affine Grassmannian has yet to be estab-
lished. A number of conjecturally equivalent definitions of k-Schur functions have
beenpresented(see[18, 19, 21, 22]). Inthisarticle,ak-Schurfunctionwillalways
refertothedefinitionof[21, 22]andwecanthusview{s(k)(x)|λ <n=k+1}as
λ 1
thebasis ofΛ dual tothebasis{F˜ |w ∈S˜0}ofΛ(n) withrespecttoabijection
(n) w n
{λ|λ <n}→S˜0
(0.1) 1 n
λ(cid:8)→w
(see Proposition 8.15). Given an interval [v,w] in the strong order (Bruhat order)
on S˜ , we introduce the notion of a strong tableau T of shape w/v; it is a certain
n
kind of labeled chain from v to w in the strong order. We define a strong Schur
function to be the generating function of strong tableaux T of shape w/v:
(cid:2)
Strong (x)= xwt(T).
w/v
T
One of our main results (Theorem 4.11) is that k-Schur functions are special cases
of strong Schur functions:
s(k)(x)=Strong (x)
λ w/id
where id ∈ S˜ is the identity and λ (cid:8)→ w ∈ S˜0 under the bijection (0.1). When
n n
v,w ∈ S ⊂ S˜ are usual Grassmannian permutations, strong Schur functions
n n
reduce to usual skew Schur functions. Strong tableaux, in the case of S ⊂S˜ , are
n n
closely related to chains in the k-Bruhat order (where here k is unrelated to n) of
Bergeron and Sottile [1]. An important difference is that our chains are marked,
reflecting the fact that affine Chevalley coefficients are not multiplicity-free (see
Remark 2.4).
Ourmainresult(Theorem4.2)istheconstructionofanalgorithmicallydefined
bijection called affine insertion. In its simplest case, affine insertion establishes a
bijectionbetweennonnegativeintegermatriceswithrowsumslessthann,andpairs
(P,Q) where P is a strong tableau, Q is a weak tableau, and both tableaux start
at id ∈ S˜ and end at the same v ∈ S˜0. This bijection reduces to the usual row-
n n
insertion Robinson-Schensted-Knuth (RSK) algorithm (see [5]) as n → ∞. Affine
insertionyieldsacombinatorialproofofthefollowingaffineCauchyidentity, which
is obtained from Theorem 4.4 by taking u=v =id:
(cid:2)
Ω (x,y)= Strong (x)Weak (y)
n w w
w∈S˜0
n
and the affine Cauchy kernel is given by
(cid:3)(cid:4) (cid:5)
Ωn(x,y)= 1+yih1(x)+yi2h2(x)+···+yin−1hn−1(x)
i(cid:2)
= h (x)m (y).
λ λ
λ:λ1<n
This provides a combinatorial description of the reproducing kernel of the per-
fect pairing
H∗(Gr)×H∗(Gr)→Z