Affine Stanley symmetric functions

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by  Thomas Lam

AFFINE STANLEY SYMMETRIC FUNCTIONS
THOMAS LAM
5
0
0 Abstract. WedefineanewfamilyF˜w(X)ofgeneratingfunctionsforw∈S˜nwhichareaffine
2 analogues of Stanley symmetric functions. We establish basic properties of these functions

n includingsymmetry,dominanceandconjugation. Weconjecturecertainpositivityproperties
a intermsofasubfamilyofsymmetricfunctionscalledaffineSchurfunctions. Asapplications,
J we show how affine Stanley symmetric functions generalise the (dual of the) k-Schur func-

1 tions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov.
2 Conjecturally, affineStanleysymmetricfunctionsshouldberelatedtothecohomology ofthe
affine flag variety.

]
O
C
.
h
1. Introduction
t
a
m In [Sta84], Stanley introduced a family {F (X)} of symmetric functions now known as
w
[ Stanley symmetric functions. He used these functions to study the number of reduced decom-

positionsofpermutationsw ∈ S . Later, thefunctionsF (X)werefoundtobeclosely related
1 n w
v to the Schubert polynomials of Lascoux and Schu¨tzenberger [LS82], which are well known to
5 be related to the geometry of flag varieties.
3 The aim of this paper is to define and study an analogue F˜ (X) of Stanley symmetric
3 w
functions for the affine symmetric group S˜ which we call affine Stanley symmetric functions.
1 n
0 Our definition of F˜ (X) is motivated by [FS] and [FG] and involve an algebra which we call
w
5
the affine nilCoxeter algebra. This algebra is an affine version of the nilCoxeter algebra used
0
in [FS]. When w ∈ S ⊂ S˜ , we have F˜ (X) = F (X). Our first main theorem is that these
/ n n w w
h functions F˜ (X) are indeed symmetric functions. Imitating [Sta84], we show basic properties
t w
a of these functions:
m
(1) the relation to reduced words:
:
v
i [x x ···x ]F˜ (X) = #{reduced words of w},
X 1 2 l(w) w
r (2) a skewing formula:
a
s⊥·F˜ = F˜
1 w v
w⋗v
X
where ⋗ denotes the covering relation in weak Bruhat order,
(3) a conjugacy formula:
F˜w∗ = ω+(F˜w)
where ∗: S˜ → S˜ and ω+ : Λ(n) → Λ(n) are involutions (Λ(n) = Chm | λ ≤ n−1i),
n n λ 1
(4) and the existence of a unique dominant monomial term m :
µ(w)
F˜ = m + b m .
w µ(w) wλ λ
λ≺µ(w)
X
Date: January,2005.
1
2 THOMASLAM
An important special case occurs when w is a Grassmannian permutation. A permutation
w ∈ S˜ is Grassmannian if it is a minimal length coset representative of a coset of S \S˜ .
n n n
In this case we obtain the affine Schur functions F˜ (X) = F˜ (X) which may be labelled
λ w
by partitions with no part greater than n − 1. We show that the affine Schur functions
{F˜ | λ ≤ n−1} form a basis of the space Λ(n) spanned by {m | λ ≤ n−1} where the m
λ 1 λ 1 λ
are monomial symmetric functions. Edelman and Greene [EG] and separately Lascoux and
Schu¨tzenberger[LS85]have shownthat Stanley symmetricfunctions F (X) expandpositively
w
in terms of Schur functions s (X). We conjecture that affine Stanley symmetric functions
λ
expand positively in terms of affine Schur functions. We prove that a unique maximal and
minimal “dominant” term exists in such an expansion.
Our definition of affine Stanley symmetric functions is motivated by relations with two
other classes of symmetric functions which have received attention lately. Lapointe, Lascoux
(k)
and Morse [LLM] initiated the study of k-Schur functions, denoted s (X), in their study of
λ
Macdonald polynomial positivity. It is conjectured that k-Schur functions form an “interme-
diate” basis between the Macdonald polynomials {H (X;q,t)} [Mac] and the Schur functions
µ
{s (X)} so that the transition coefficients are positive in both intermediate steps. Lapointe
λ
and Morse have more recently connected the multiplication of k-Schur functions with the
Verlinde algebra of U(m).
Our affine Schur functions had earlier been defined using “k-tableaux” by Lapointe and
Morse,whocalledthesefunctionsdual k-Schurfunctions. WorkofLapointeandMorse[LM04]
relatingk-Schurfunctionston-coresandtotheaffinesymmetricgroupshowthatourdefinition
of affineSchurfunctionsare indeeddualtok-Schur functions. In thiscontext thesymmetry of
affine Schur functions is not obvious, but follows from the symmetry of general affine Stanley
symmetricfunctions. Therelation withk-Schurfunctionsalsosuggest thestudyofskew affine
Schur functions F˜ (X), another special case of affine Stanley symmetric functions which we
λ/µ
study.
Separately, cylindric Schur functions were defined by Postnikov [Pos] (see also [GK]). He
showed that certain coefficients of the expansion of toric Schur functions (a special case
of cylindric Schur functions) in terms of Schur functions were equal to the 3-point genus 0
Gromov-Witten invariants oftheGrassmannianGr (whicharethemultiplicationconstants
m,n
of thequantum cohomology QH∗(Gr )of theGrassmannian). CylindricSchurfunctions are
m,n
definedas generating functions of cylindricsemi-standard tableaux, which are tableaux drawn
on a cylinder. We show that cylindric Schur functions are special cases of skew affine Schur
functions and that they are exactly equal to affine Stanley symmetric functions F˜ labelled
w
by affine permutations w which are “321-avoiding”. These results are affineanalogues of some
of the results in [BJS]. However, the affine case is significantly more difficult. For example,
any normal Stanley symmetric function F is equal to some skew affine Schur function.
w
We also show that our conjecture that affine Stanley symmetric functions expand positively
in terms of affine Schur functions implies both the Schur positivity of toric Schur polynomials
and the positivity of the multiplication of k-Schur functions (the latter is currently still a
conjecture). Our work also explains why cylindric Schur functions are not in general Schur
positive in infinitely many variables (see [McN]). The three families of symmetric functions:
affine Stanley, skew affine Schur and cylindric Schur can be thought of as arising from three
different representations of the affine nilCoxeter algebra U in a uniform manner. The repre-
n
sentation fromwhichaffineStanley symmetricfunctions ariseis theleftregular representation
of U and so leads to the most general symmetric functions which can arise in this manner.
n
The connections with k-Schur functions and cylindric Schur functions already indicate that
affineStanley symmetricfunctionsareimportantobjects. Infactourwork,combinedwiththe
AFFINE STANLEY SYMMETRIC FUNCTIONS 3
known connection between quantum cohomology and the Verlinde algebra, essentially implies
that the main results of [Pos] and [LM05] are equivalent. However, it seems the most exciting
direction to take is to extend our definition of affine Stanley symmetric functions F˜ to affine
w
Schubert polynomials S˜ and connect them with the affine flag variety G/B of type A˜ .
w n−1
Usual Stanley symmetric functions are certain stable “limits” of the Schubert polynomials
S , which are well known to represent Schubert varieties in the cohomology H∗(G/B) of the
w
flag variety and possess numerous remarkable properties. In the affine case, the cohomology
classes [Ω ] ∈ H∗(G/B) representing Schubert varieties are labelled by w ∈ S˜ and should
w n
conjecturally be related to affine Stanley symmetric functions.
In the Grassmannian case, Morse and Shimozono [MS] have conjectured that affine Schur
functions represent the Schubert classes in the cohomology of the affine Grassmannian. The
study of affine Schur functions should make explicit the relationship between the affine Grass-
mannianG/P, andtheVerlindealgebrasofU(m)withleveln−morthequantumcohomology
QH∗(Gr ) of the Grassmannian, which are already known to be connected; see for example
m,n
[Wit].
Finally,wemakethenaturalgeneralisationtoaffine stableGrothendieck polynomials G˜ (X)
w
which speculatively should be stable limits of the K-theory Schubert classes of the affine flag
variety.
Much work has also been done with a version of the Stanley symmetric functions for the
hyperoctahedral group; see [Kra, LTK, FK96]. We intend to generalise this to the affine case
and also investigate Stanley symmetric functions for general Coxeter groups in later work.
Overview. In sections 2 and 3, we establish some notation for affine permutations and
for symmetric functions. In section 4 we recall the definition of Stanley symmetric functions,
give their main properties and explain the relationship with Schubert polynomials. In section
5, we define the affine nilCoxeter algebra and affine Stanley symmetric functions and prove
that the latter are symmetric. In Section 6, we explain how affine Stanley symmetric func-
tions arise from different representations of the affine nilCoxeter algebra. In section 7, we
prove a coproduct formula for affine Stanley symmetric functions. In section 8, we show that
affine Stanley symmetric functions have a unique dominant monomial term. In section 9, we
prove a conjugacy formula, imitating [Sta84]. In section 10, we define and study affine Schur
functions. In section 11, we study the relationship between n-cores and the affine symmetric
group, following in part [Las]. In sections 12, 13 and 14 we define skew affine Schur functions
and relate them to k-Schur functions. In section 15, we recall the definition of a cylindric
Schur function and connect them with skew affine Schur functions. In section 16, we show
that cylindric Schur functions correspond exactly to 321-avoiding permutations. In section
17, we make a number of positivity conjectures concerning the expansion of affine Stanley
symmetric functions in terms of affine Schur functions. Finally, in section 18, we discuss some
furtherextensionsofourtheoryandinparticularageneralisation toaffinestableGrothendieck
polynomials.
A condensed preliminary version of this paper appeared as [Lam04b].
Acknowledgements. I thank Jennifer Morse for interesting discussions about k-Schur
functions and dualk-Schur functions. I also thank Mark Shimozonofor discussionsrelating to
the affine Grassmannian. I am grateful to Alex Postnikov for introducing cylindric and toric
Schur functions in his class. I am also indebted to my advisor, Richard Stanley for guidance
over the last couple of years.
4 THOMASLAM
2. Affine symmetric group
A positive integer n ≥ 3 will be fixed throughout the paper. Let S˜ denote the affine
n
symmetric group with simple generators s ,s ,...,s satisfying the relations
0 1 n−1
s s s = s s s for all i
i i+1 i i+1 i i+1
s2 = 1 for all i
i
s s = s s for |i−j| ≥ 2.
i j j i
Here and elsewhere, the indices will be taken modulo n without further mention. One may
realize S˜ as the set of all bijections w : Z → Z such that w(i+n) = w(i)+n for all i and
n
n w(i) = n i. In this realization, to specify an element w ∈ S˜ it suffices to give the
i=1 i=1 n
“window” [w(1),w(2),...,w(n)]. The product w · v of two affine permutations is then the
Pcomposed bijePction w◦v : Z → Z. Thus ws is obtained from w by swapping the values of
i
w(i+kn) and w(i+kn+1) for every k ∈ Z. See [BB] for more details.
The symmetric group S embeds in S˜ as the subgroup generated by s ,s ,...,s . Since
n n 1 2 n−1
there are many embeddings of the S into S˜ we will denote this particular embedding by
n n
S .
n♦
For an element w ∈ S˜ let R(w) denote the set of reduced words for w. A word ρ =
n
(ρ ρ ···ρ ) ∈ [0,n − 1]l is a reduced word for w if w = s s ···s and l is the smallest
1 2 l ρ1 ρ2 ρl
possible integer for which such a decomposition exists. Abusing notation slightly, we also
call s s ···s a reduced word for w. The integer l = l(w) is called the length of w. If
ρ1 ρ2 ρl
ρ,π ∈ R(w) for some w, then we write ρ ∼ π. If ρ is an arbitrary word with letters from
[0,n−1]then we writeρ ∼0 if it is not areduced wordof any affinepermutation. Ifw,u ∈ S˜
n
thenwesaythatw covers uifw = s ·uandl(w) = l(u)+1; andwewritew⋗u. Thetransitive
i
closure of ⋗ is called the weak Bruhat order and denoted >.
The code c(w) or affine inversion table [BB, Las] of an affine permutation w is a vector
c(w) = (c ,c ,...,c ) ∈ Nn −Pn of non-negative entries with at least one 0. The entries are
1 2 n
given by c = #{j ∈ Z | j > i and w(j) < w(i)}. It is shown in [BB] that there is a bijection
i
between codes and affine permutations and that l(w) = |c(w)| = n c . The right action of
i=1 i
the simple generator s on the code c = (c ,c ,...,c ) is given by
i 1 2 n
P
(c ,...,c ,c ,...,c )·s = (c ,...,c +1,c ,...,c )
1 i i+1 n i 1 i+1 i n
whenever c > c . Thus c(ws ) = c(w)·s .
i i−1 i i
3. Symmetric functions
A partition λ = (λ ≥ λ ≥ ··· ≥ λ > 0) is a weakly decreasing finite sequence of positive
1 2 l
integers. We use λ, µ and ν to denote partitions and will always draw them in the English
notation (top-left justified). The dominance order (cid:22) on partitions is given by λ (cid:22) µ if and
only if λ +λ +···+λ ≤ µ +µ +···+µ for every i. We will also assume the reader is
1 2 i 1 2 i
reasonably familiar with the usual notions of corners, conjugates and (semistandard) Young
tableaux.
We will follow mostly [Mac, Sta99] for our symmetric function notation. Let Λ denote the
ring of symmetric functions over C. Usually, our symmetric functions will have an infinite set
of variables x ,x ,... and will be written as f(x ,x ,...) or f(X). If we need to emphasize
1 2 1 2
the variables used, we write Λ .
X
We willusem , p , e , h and s to denotethemonomial, power sum, elementary, homoge-
λ λ λ λ λ
neous andSchur basesofΛ. Itiswellknownthat{e }and{h }arealgebraically independent
n n
generators of Λ. Let h.,.i denote the Hall inner product of Λ satisfying hh ,m i = hs ,s i =
λ µ λ µ
AFFINE STANLEY SYMMETRIC FUNCTIONS 5
δ . For f ∈ Λ, write f⊥ : Λ → Λ for the linear operator adjoint to multiplication by f with
λµ
respect to h.,.i. We let ω : Λ→ Λ denote the C-algebra involution of Λ sending h to e .
n n
If f(X) ∈ Λ then f(x ,x ,...,y ,y ,...) = f(X,Y) = f (X)⊗g (Y) ∈ Λ ⊗Λ for
1 2 1 2 i i i X Y
some f and g . This is the coproduct of f, written ∆f = f ⊗g ∈ Λ⊗Λ. We have the
i i Pi i i
following formula for the coproduct ([Mac]):
P
(1) ∆f = s⊥f ⊗s .
λ λ
λ
X
The ring of symmetric functions Λ is a self dual Hopf-algebra with respect to h.,.i, so that
(2) h∆f,g⊗hi = hf,ghi,
where hf ⊗f ,g ⊗g i:= hf ,g ihf ,g i.
1 2 1 2 1 1 2 2
Let Parn denote the set {λ |λ ≤ n−1} of partitions with no row longer than n−1. The
1
following two subspaces of Λ will be important to us:
Λ(n) := Chm | λ ∈ Parni
λ
Λ := Chh |λ ∈Parni= Che | λ ∈ Parni = Chp |λ ∈ Parni.
(n) λ λ λ
If f ∈ Λ and g ∈ Λ(n) then define hf,gi to be their usual Hall inner product within Λ.
(n)
Thus {h } and {m } with λ ∈ Parn form dual bases of Λ and Λ(n). Note that Λ is a
λ λ (n) (n)
subalgebra of Λ but Λ(n) is not closed under multiplication. Instead, Λ(n) is a coalgebra; it is
closed under comultiplication.
4. Stanley symmetric functions
Let w ∈ Sn with length l = l(w). Define the generating function Fw−1(X) by
Fw−1(x1,x2,...) = xb1xb2···xbl.
a1a2··X·al∈R(w) 1≤b1≤Xb2≤···≤bl
ai>ai+1⇒bi+1>bi
We have indexedtheFw−1(X) by theinversepermutation to agree with thedefinitionwe shall
give later. Note that the length l(w) is equal to the degree of F and the number |R(w)| of
w
reduced decompositions of w is given by the coefficient of x x ···x in F .
1 2 l w
Theorem 1 ([Sta84]). The following properties of the generating function F hold for each
w
w ∈ S :
n
(1) F (X) is a symmetric function in (x ,x ,...).
w 1 2
(2) Define a ∈ Z by F (X) = a s (X). Then there exists partitions λ(w) and
wλ w λ wλ λ
µ(w) so that a = a = 1 and
wλ(w) wµ(w)
P
F (X) = a s (X).
w wλ λ
λ(w)(cid:22)λ(cid:22)µ(w)
X
(3) Defineaninvolution∗ : S → S by∗ : w w ···w 7→ (n+1−w )(n+1−w )···(n+
n n 1 2 n n n−1
1−w ). Then
1
ω(Fw) = Fw∗.
(4) We have
s⊥·F = F .
1 w v
w⋗v
X
Edelman and Greene and separately Lascoux and Schu¨tzenberger showed the following
(significantly harder) result concerning the coefficients a .
wλ
6 THOMASLAM
Theorem 2 ([EG] and [LS85]). The coefficients a are non-negative.
wλ
We now give a different formulation of the definition in a manner similar to [FS]. Let C[S ]
n
denote the group algebra of the symmetric group equipped with a inner product hw,vi = δ .
wv
Define linear operators u :C[S ] → C[S ] for i ∈[1,n−1] by
i n n
s .w if l(s .w) > l(w),
i i
u .w =
i
(0 otherwise.
The operators satisfy the braid relations u u u = u u u together with u2 = 0 and
i i+1 i i+1 i i+1 i
u u = u u for |i−j| ≥ 2. They generate an algebra known as the nilCoxeter algebra. Note
i j j i
that the action on C[S ] is a faithful representation of these relations.
n
Let A (u) = u u ···u . Then the Stanley symmetric functions can be
k b1>b2>···>bk b1 b2 bk
written as
P
(3) F (X) = A (u)A (u)···A (u)·1,w xa1xa2···xat
w at at−1 a1 1 2 t
a=(a1X,a2,...,at)(cid:10) (cid:11)
where the sum is over all compositions a. The symmetry of F (X) is then a consequence of
w
the fact that the A (u) commute.
k
For completeness, we explain briefly the relationship between F (X) and the Schubert
w
polynomials of Lascoux and Schu¨tzenberger (see [BJS]). For w ∈ S , we have a Schubert
n
polynomialS ∈ C[x ,x ,...,x ]. Ifw ∈ S ,thenw×1s ∈ S denotesthecorresponding
w 1 2 n−1 n n+s
permutationof S acting on theelements [1,n] of[1,n+s]. Similarly, 1s×w ∈ S denotes
n+s n+s
the corresponding permutation acting on the elements [s + 1,n +s] of [1,n +s]. Schubert
polynomials have the important stability propertySw = Sw×1s. Stanley symmetric functions
Fw(X) are obtained by taking the other limit: Fw = lims→∞S1s×w. The limit is taken by
treating both sides as formal power series and taking the limit of each coefficient.
5. Affine Stanley symmetric functions
Our first definition of affine Stanley symmetric functions will imitate the definition (3)
above. LetU betheaffine nilCoxeter algebra generated over C by generators u ,u ,...,u
n 0 1 n−1
satisfying
u2 = 0 for all i ∈ [0,n−1],
i
u u u = u u u for all i ∈ [0,n−1],
i i+1 i i+1 i i+1
u u = u u for all i,j ∈ [0,n−1] satisfying |i−j| ≥ 2.
i j j i
Here and henceforth the indices are to be taken modulo n. A basis of U is given by the
n
elements u = u u ···u where ρ = (ρ ρ ···ρ ) is some reduced word for w (see [Hum,
w ρ1 ρ2 ρl 1 2 l
Chapter 7]). The element u ∈U does not depend on the choice of reduced word ρ.
w n
Let a = a a ···a be a word with letters from [0,n −1] so that a 6= a for i 6= j. Let
1 2 k i j
A = {a ,a ,...,a } ⊂ [0,n−1]. The word a is cyclically decreasing if for every i such that
1 2 k
i,i+1 ∈ A, theletter i+1 precedes iina. We will call anelement u∈ U cyclically decreasing
n
if u = u = u ···u for some cyclically decreasing word a. If u is cyclically decreasing and
a a1 ak
u = u ···u then necessarily a = a ···a will be cyclically decreasing. The element u
a1 ak 1 k
is completely determined by the set A = {a ,a ,...,a } ⊂ [0,n−1] and we write u = u .
1 2 k A
Replacing u by s we make similar definitions of cyclically decreasing affine permutations for
i i
the affine symmetric group.
AFFINE STANLEY SYMMETRIC FUNCTIONS 7
Define h (u) ∈ U for k ∈ [0,n−1] by
k n
h (u) = u
k A
A∈(X[0,n−1])
k
wherethesumisoversubsetsof[0,n−1]ofsizek. Forexampleifn = 9andA= {0,2,4,5,6,8}
then u = u u u u u u = u u u u u u = ···. A related formula was given in [Pos], in
A 0 8 2 6 5 4 2 6 5 4 0 8
the context of the affine nil-Temperley-Lieb algebra. The affine nil-Temperley-Lieb algebra
is a quotient of the affine nilCoxeter algebra given by the additional relations u u u =
i i+1 i
u u u = 0.
i+1 i i+1
Define a representation of U on C[S˜ ] by
n n
s .w if l(s .w) > l(w),
i i
u .w =
i
(0 otherwise.
It is easy to see that this is indeed a representation of U . If we identify w ∈ C[S˜ ] with
n n
u ∈ U then this is essentially the left regular representation of U . Equip C[S˜ ] with the
w n n n
inner product hw,vi = δ . The following definition was heavily influenced by [FG].
wv
Definition 3. Let w ∈ S˜ . Define the affine Stanley symmetric functions F˜ (X) by
n w
F˜ (X) = h (u)h (u)···h (u)·1,w xa1xa2···xat,
w at at−1 a1 1 2 t
a=(a1X,a2,...,at)(cid:10) (cid:11)
where the sum is over compositions of l(w) satisfying a ∈ [0,n−1].
i
The seemingly more general “skew” affine Stanley symmetric functions
F˜ (X) = h (u)h (u)···h (u)·v,w xa1xa2···xat
w/v at at−1 a1 1 2 t
a=(a1X,a2,...,at)(cid:10) (cid:11)
are actually equal to the usual affine Stanley symmetric functions F˜wv−1(X).
Two properties follow straight from the definition.
Proposition 4. Let w ∈ S˜ . Then the coefficient of x x ···x in F˜ (X) is equal to the
n 1 2 l(w) w
number of reduced words of w.
Proposition 5. Suppose w ∈S ⊂ S˜ . Then F˜ (X) = F (X).
n♦ n w w
The main theorem of this section is the following.
Theorem 6. The generating functions F˜ (X) ∈ Λ(n) are symmetric.
w
Theorem 6 follows immediately from Proposition 8. In the following, intervals [a,b] are
to be taken in the cyclic fashion within [0,n−1]. Also, max and min of a cyclic interval is
meant to be taken modulo n in the obvious manner. So if n = 6 then [4,1] = {4,5,0,1} and
max([4,1]) =1 and min([4,1]) = 4. We will need a technical lemma first.
Lemma 7. We have the following identities for reduced words.
(1) Let a,b ∈ [0,n−1] with a 6= b−1. Then
a(a−1)(a−2)···ba(a−1)(a−2)···b ∼ 0.
(2) Let a,b,c ∈ [0,n−1] satisfying a6= b−1; c 6= b and c∈ [b,a]. Then
a(a−1)(a−2)···bc∼ (c−1)a(a−1)(a−2)···b.
Proof. Both results can be calculated by induction. (cid:3)
8 THOMASLAM
Sofor example, the element (s s s )(s s s )is not reducedand we have (s s s s s s )s =
4 3 2 4 3 2 6 5 4 3 2 1 4
s (s s s s s s ).
3 6 5 4 3 2 1
Proposition 8. The elements h (u) for k ∈ [0,n−1] commute.
k
Proof. Foreach w ∈ S˜ satisfyingl(w) = x+y,wecalculate thecoefficientofu inh (u)h (u)
n w x y
and h (u)h (u). We assume that x and y are both not equal to 0 for otherwise the result is
y x
obvious. Let u = u u where |A| = x and |B| = y. We need to exhibit a bijection between
w A B
reduced decompositions of this form and those of the form u = u u with |C| = y and
w C D
|D|= x. We assumeforsimplicity (thoughitisnotcrucialtoourproof)thatA∪B = [0,n−1]
for otherwise we are in the non-affine case and the proposition follows from results of Stanley
[Sta84] or Fomin-Greene [FG]. Let A = A and B = B be minimal decompositions of
i i i i
A and B into cyclic intervals. If A ⊂ B for some pair (i,j) then we call A an inner interval
i j i
S S
and similarly for B ⊂ A . Otherwise the interval is called outer.
k l
Using Lemma 7 and our assumption that A ∪ B = [0,n − 1] we can describe the outer
intervals in an explicit manner. Each outer interval A touches an outer interval rn(A )= B
i i k
called the right neighbour of A , for a unique k, so that min(A ) = max(B )+1. Also A
i i k i
overlaps with an outer interval ln(A ) = B for a unique l, so that max(A ) ≥ min(B )−1
i l i l
called the left neighbour. If rn(A ) = B then we also write A = ln(B ) and similarly for
i k i k
rn(B ). Note that it is possible that rn(A ) = ln(A ) since we are working cyclically.
k i i
Our bijection will depend only locally on each pair of an outer interval A∗ and its right
neighbour B∗ = rn(A∗). We call the interval I = [min(B∗),min(ln(A∗)) − 1] a critical
interval. Critical intervals cover [0,n − 1] in a disjoint manner. For example, suppose
n = 10 and A = {1,2,3,6,7,8,9} and B = {0,2,4,5,7,9} (Figure 1), so that u u =
A B
u u u u u u u u u u u u u u . Then A = [1,3] and A = [6,9] are both outer intervals.
9 8 7 6 3 2 1 0 9 7 5 4 2 0 1 2
Also B = [2,5], B = {7} and B = [9,0]. Only B is an inner interval. The left neighbour
1 2 3 2
of A is ln(A ) = B and the right neighbour is rn(A ) = B . The critical intervals are [9,1]
1 1 1 1 3
and [2,8].
Leta = min(ln(A∗))−1andb = min(B∗). Letc =|[b,a]|, d= |A∩[b,a]|ande = |B∩[b,a]|.
Renaming for convenience, we let S ,S ,...,S be the inner intervals (of B) contained in A∗
1 2 r
and T ,...,T be those contained in B∗, arranged so that S > S for all k within [b,a] and
1 t k k+1
similarly T > T . We now define a subset U ⊂ [b,a] satisfying |U| = d. The algorithm
k k+1
beginswithU = [b,a]andachanging indexisettoi := atobeginwith. Theindexidecreases
from a to b and at each step the element i may be removed from U according to the rule:
(1) If i ∈ A∗ then we remove it from U unless i ∈ S for some k ∈[1,r].
k
(2) If i ∈ B∗ then we remove it from U unless i ∈ T +1 for some k ∈ [1,t].
k
(3) Otherwise we do not remove i from U and set i := i−1. Repeat.
When |U| = d we stop the algorithm. The algorithm always terminates with |U| = d since
there are at least c − d = |[b,a]| − (A ∩ [b,a]) elements to remove. In fact the algorithm
terminates before i = b since ∪ S 6= A∗ ∩I. We will denote the result of the algorithm by
i i
φ(A∗∪ T ,B∗∪ S ):= U. Note that min(U) = b.
i i i i
The bijection u u 7→ u u is obtained by letting D ⊂ [0,n−1] be the subset obtained
A B C D
from B by changing B ∩ I in each critical interval I to U. By the definition of U we see
that |D| = |A|. We claim that u u = u u or alternatively s s (s )−1 = s for some C
A B C D A B D C
satisfying|C|= |B|(hereitisslightlymoreconvenienttocalculatewithintheaffinesymmetric
group, which is legal since our words are all reduced). We can calculate this locally on each
critical interval since the s commute as I varies over critical intervals. Note that U always
D∩I
has the form of a disjoint union S1∪S2∪···∪Sr′∪[b,a′] for some r′ ≤ r where a′ > max(B∗)
or the form S1∪···∪Sr ∪{T1+1}∪{T2+1}∪···∪{Tt′ +1}∪[b,a′] where a′ ≤ max(B∗).
AFFINE STANLEY SYMMETRIC FUNCTIONS 9
LetusassumethatU hasthefirstform. FocusingonI = [b,a] = [min(B∗),min(ln(A∗))−1]
we are interested in
s= sA∗∩IsT1···sTtsS1···sSrsB∗(s[b,a′])−1(sSr′)−1···(sS1)−1.
Then we get
s = sA∗∩IsT1···sTtsSr′+1···sSr(s[max(B∗)+1,a′])−1
= sSr′+1−1···sSr−1sT1···sTtsA∗∩I(s[max(B∗)+1,a′])−1
= sSr′+1−1···sSr−1sT1···sTts[a′+1,a] using max(B∗)+1= min(A∗).
We used Lemma 7 repeatedly and also the fact that the certain intervals do not “touch” and
so commute. Let U′ be the disjoint union [a′+1,a]∪{Ss′+1−1}∪···∪{Ss−1}∪T1∪···Tt.
Note that it is always the case that max(U′) = a. The other form of U involves a similar
calculation. One checks that we can combine this argument for each critical interval showing
that s s (s )−1 is indeed equal to s for some C.
A B D C
Finally, we need to show that this map is a bijection. Again we work locally on a critical
intervalandassumethatU hasthefirstform. IfwereplaceA∗ (morepreciselyA∗∩I)byU′and
B∗ by U, then our internal intervals are S1′ = S1,...Sr′′ = Sr′ and T1′ = Sr′+1−1,...,Tr′−r′ =
Sr′ −1,Tr′−r′+1 = T1,...Tr′−r′+t = Tt. We now show that B∗ ∪i Si = φ(U′,U) from which
the bijectivity will follow. Note that since min(U) = b and max(U′) = a the critical intervals
of u u are the same as those of u u . By definition φ(U′,U) keeps S′,S′,... and keeps
C D A B 1 2
T′ + 1,T′ − 1,...,T′ + 1, removing all other values up to this point. At this point the
1 2 r−r′
algorithm stops sinceφ(U′,U)is of thecorrectsize. We seethat weobtain φ(U′,U) = B∗∪ S
i i
back in this way. A similar argument works for the second form of U. (cid:3)
7 8
rsb b
6 9
b brs
5 rs rs0
rs b
4 1
rsb brs
3 2
Figure 1. Dots represent elements of A. Squares represent elements of B.
Example 1. We illustrate the map U = φ(A∗ ∪ T ,B∗ ∪ S ) of the proof. Suppose [b,a] =
i i i i
[2,20] and A∗ = [14,20], B∗ = [2,13]. Let S = [16,18] and T = [8,11] and T = {5} be the
1 1 2
inner intervals. Then d = 12 and U = {2,3,4,5,6,9,10,11,12,16,17,18}. We can compute
that
sA∗s11s10s9s8s5sB∗s18s17s16s2s3s4s5s6s9s10s11s12s16s17s18 = sA∗s[7,13]s5
so that U′ = [7,20]∪{5}. Finally one checks that B∗∪ S = φ(U′,U).
i i
We end this section by giving two alternative descriptions of the affine Stanley symmetric
functions, the first one imitating the original definition of Stanley. Let w ∈ S˜ . Let a =
n
(a ,...,a ) ∈ R(w) be a reduced word and b = (b ≥ b ··· ≥ b ) be an positive integer
1 l 1 2 l
sequence. Then (a,b) is called a compatible pair for w if whenever b = b = ··· = b
i i+1 j
and {k,k+1} ⊂ {a ,a ,...,a } then we have that k +1 precedes k (for any i,j,k). Two
i i+1 j
compatible pairs (a,b) and (a′,b′) are equivalent if b = b′ and for any maximal interval [i,j] ⊂
10 THOMASLAM
[1,l] satisfying b = b = ··· = b we have that a a ···a and a′a′ ···a′ are reduced
i i+1 j i i+1 j i i+1 j
words for the same affine permutation. Thefollowing proposition is clear from the definitions.
Proposition 9 (Alternative Definition 1). The affine Stanley symmetric functions are given
by
F˜ (X) = x x ···x
w b1 b2 bl
(Xa,b)
where the sum is over equivalence classes (a,b) of compatible pairs for w.
Now let w ∈ S˜ of length l and suppose α = (α ,α ,...,α ) is a composition of l.
n 1 2 r
An α-decomposition of w is an ordered r-tuple of cyclically decreasing affine permutations
(w1,w2,...,wr) ∈ S˜r satisfying l(wi) = α and w = w1w2···wr. The following alternative
n i
definition is also immediate.
Proposition 10 (Alternative Definition 2). The affine Stanley symmetric function F˜ (X) is
w
given by
F˜ (X) = (number of α-decompositions of w)·xα
w
α
X
where the sum is over all compositions α of l.
6. Representations of the affine nilCoxeter algebra
Let V be a complex representation of U with a distinguished basis {v | p ∈ P} for some
n p
indexingsetP. Leth.,.i :V×V → Cbetheinnerproductdefinedbyhv ,v i = δ forp,q ∈ P.
p q pq
For any p,q ∈P one can define V-affine Stanley symmetric functions by F˜ (X) ∈ Λ(n) by
q/p
F˜ (X) = h (u)h (u)···h (u)·v ,v xa1xa2···xat,
q/p at at−1 a1 p q 1 2 t
a=(a1X,a2,...,at)(cid:10) (cid:11)
where the sum is over compositions of l(w) satisfying a ∈ [0,n−1]. By Proposition 8 these
i
functions are indeed symmetric functions.
Proposition 11. Suppose u ·v = v and w ∈ S˜ is the only affine permutation such that
w p q n
hu ·v ,v i =6 0. Then F˜ (X) = F˜ (X).
w p q q/p w
Proof. For each composition a = (a ,a ,...,a ) expand h (u)h (u)···h (u) in the basis
1 2 t at at−1 a1
{u } of U . Using the assumption, the proposition follows immediately uponcomparison with
v n
Definition 3. (cid:3)
More generally, for arbitrary v and v let c = hu ·v ,v i. Then F˜ = c F˜ .
p q w w p q q/p w∈S˜n w w
We have not found any interesting generating functions of this form.
If in addition u acts on the basis {v } with non-negative matrix coefficienPts, then F˜
i p p∈P p/q
will be monomial-positive. This will be the case for all the representations of U that we will
n
be considering.
7. Coproduct
We now give the analogue of part (4) of Theorem 1.
Theorem 12 (Coproduct formula). The following coproduct expansion holds:
F˜ (x ,x ,...,y ,y ,...)= F˜ (x ,x ,...)F˜ (y ,y ,...).
w 1 2 1 2 v 1 2 u 1 2
uv=w
X