q-Analogs of symmetric function operators

18 Pages·0.24 MB·

by  Mike Zabrocki

q-ANALOGS OF SYMMETRIC FUNCTION OPERATORS
MICHAEL ZABROCKI
2
0
Abstract. For any homomorphism V on the space of symmetric functions, we introduce an
0
operation that creates a q-analog of V. By giving several examples we demonstrate that this
2
quantization occurs naturally within the theory of symmetric functions. In particular, we show

n that the Hall-Littlewood symmetric functions are formed by taking this q-analog of the Schur
a symmetric functions and the Macdonald symmetric functions appear by taking the q-analog of
J theHall-Littlewoodsymmetricfunctions inthe parameter t. This relationisthen usedtoderive

9 recurrencesontheMacdonaldq,t-Kostkacoefficients.
1 R´esum´e. PourunhomomorphismeV surl’espacedesfonctionssym´etriques,nouspr´esentonsune
op´erationquicr´eeunq-analoguedeV. Endonnantplusieursexemplesnousd´emontronsquecette

] quantization se produit naturellement dans la th´eorie de fonctions sym´etriques. En particulier,
A nous prouvons que les fonctions sym´etriques de Hall-Littlewood sont constitu´ees en prenant ce
Q q-analogue des fonctions sym´etriques de Schur et les fonctions sym´etriques de Macdonald appa-
raissentenprenantleq-analoguedesfonctionssym´etriquesdeHall-Littlewooddansleparam`etre
.
h t. Cette relation est alors employ´ee pour d´eriver des r´ecurrence sur les coefficients Macdonald
t q,t-Kostka.
a
m
[

3
1. Introduction
v
8
The Hall-Littlewood and Macdonald symmetric functions are two examples of families of sym-
8
metric functions that depend on a parameter q such that setting this parameter q equal to 0 yields
1
8 one class of symmetric functions which is not a product of generators and setting the parameter q
0 equal to 1 yields a multiplicative basis. There are other other classes of symmetric functions with
0 thesameproperty,andinthisarticlewewillshowthatpracticallyanyofthesefamiliesareinstances
0
of the same q-twisting of the symmetric function found by setting q =0.
/
h This remarkable fact has lead to a completely elementary proof of the polynomiality of the q,t-
at Kostkacoefficients[GZ]andinthisarticleweusetheverysameobservationtoderiveacombinatorial
m recurrenceonthese coefficients as wellas algebraicformulasfor operatorsthat adda column to the
partition indexing a Macdonald symmetric function.
:
v The first section of this article will introduce some necessary notation and the definition of this
i
X q-analog. In the second section we give several examples where it arises. Some examples will be
nothing more than showing that V for some V is a formula that is well known in the literature.
r
a Other examples present some completely new equations, the most important of which will concern
the relation of the Hall-Littlewood symmetric functions to the Macdonald symmetric functions.
This sectionwillshow that this single q-analogappearsin the creationofseveraldifferent classesof
Schur positive symmetric functions.
In the third section we derive some formulas related to an operator that adds a column to the
Hall-Littlewoodsymmetricfunctions. Theq-analogofthisoperatoraddsacolumntotheMacdonald
symmetricfunctions. Inthefourthsectiontheseequationsareusedtogiveaformulafortheactionof
thisoperatorontheSchurbasisgivingacombinatorialruleforcomputingtheMacdonaldsymmetric
functions (i.e. a ‘Morris-like’ recurrence for the q,t-Kostka coefficients).
1
2 MICHAEL ZABROCKI
2. Notation
A partition of n is a sequence of non-negative integers λ = (λ ≥ λ ≥ λ ≥ ···) such that
1 2 3
λ = n. The length of a partition is the largest index i such that λ is nonzero, and it will be
i i i
denoted here by ℓ(λ). A partition will be drawn as a sequence of rows of boxes aligned at the left
Pedge with λ cells in the ith row. We will use the French convention and draw these diagrams with
i
the largest row on the bottom and the smallest row on the top. The conjugate partition λ′ is the
sequence whose ith entry is the number of cells in the ith column of the diagram for λ.
The partition will be sometimes be identified with its diagram in the sort of language that is
used. For instance, the operations of adding rows or columns to partitions indexing bases for the
symmetric functions are important here. the notation (m,µ) is used to represent the sequence µ
with a part of size m prepended, which will be a partition as long as µ ≤ m. The notation 1m|µ
1
will be used to represent the partition (µ +1,µ +1,... ,µ +1) (as long as ℓ(µ)≤m).
1 2 m
LetΛbethespaceofsymmetricfunctionswiththestandardbasesforthisspace,h homogeneous,
λ
e elementary,m monomial,f forgotten,p power,ands theSchursymmetricfunctionsdefined
λ λ λ λ λ
as they are in [M]. The involution ω that sends p to (−1)k−1p relates these bases by ω(h )=e ,
k k λ λ
ω(m )=f and ω(s )=s . The standard inner product on this space determines the dual bases
λ λ λ λ′
(1) hp ,p /z i=hh ,m i=he ,f i=hs ,s i=δ ,
λ µ µ λ µ λ µ λ µ λµ
where zλ = i>1ini(λ)ni(λ)! with ni(λ) equalto the number ofparts ofsizei inλ, andwe haveset
δ =1 and δ =0 if λ6=µ.
λλ λµ
Q
For any element f of Λ, let f⊥ be the operation that is dual to multiplication by f with respect
to the standard inner product. By definition we have that for any dual bases {a } and {b } , the
λ λ λ λ
action of f⊥ on another symmetric function g is given by the formula
(2) f⊥g = hg,fa ib .
λ λ
λ
X
‘Plethystic’notationisadeviceforexpressingthesubstitutionofthemonomialsofoneexpression
inasymmetricfunction. AssumethatE isaformalseriesinasetofvariablesx ,x ,...withpossible
1 2
specialparametersq andt. Fork≥1,setp [E]tobeE withx replacedbyxk andq andtreplaced
k i i
by qk and tk respectively, that is
(3) p [E(x ,x ,... ,q,t)]=E(xk,xk,... ;qk,tk)
k 1 2 1 2
ForanarbitrarysymmetricfunctionP,P[E]willrepresentthetheformalseriesfoundbyexpanding
P in terms of the power symmetric functions and then substituting p [E] for p . More precisely, if
k k
the power sum expansion of the symmetric function P is given by P = c p then P[E] is given
λ λ λ
by the formula
P
(4) P[E]= c p [E]p [E]···p [E].
λ λ1 λ2 λℓ(λ)
λ
X
The symmetric functions in the infinite set of variables x ,x ,x ,... will be denoted by ΛX. Λ
1 2 3
and ΛX are isomorphic and in this exposition we will identify the two spaces when it is convenient.
In plethystic notation, the isomorphism that identifies the two spaces is given by f 7→ f[X] where
X =x +x +x +··· since under this map p is sent to xk+xk+xk+···. Also use the notation
1 2 3 k 1 2 3
X = x +x +···+x to represent when a symmetric function being evaluated a finite set of
n 1 2 n
variables. For sets of variables using other letters we use a similar convention.
The symbol Ω = h will represent a special generating function, and we will use the
n≥0 n
plethysticnotationforsymmetricfunctionswiththisexpressionaswellwiththefollowingidentities.
P
(5) Ω[X +Y]=Ω[X]Ω[Y]
q-ANALOGS OF SYMMETRIC FUNCTION OPERATORS 3
1
(6) Ω[X]=
1−x
i
i
Y
(7) Ω[XY]= s [X]s [Y]
λ λ
λ
X
Operators that have the property that they add a row or a column to the partition indexing a
symmetric function will be known as ‘creation operators.’ The creation operators that will be used
repeatedly are those that add a row to the Schur symmetric functions (due to Bernstein, see [Ze,
p. 69], [M, p. 96]) and the Hall-Littlewood symmetric functions (due to Jing, see [J], [G] or [M,
p. 238]). In the third section, the operator introduced in [Za1] that adds a column to the partition
indexing a Hall-Littlewood symmetric function will be developed further.
DefinethefollowinginvolutiononthespaceHom(Λ,Λ)thatisausefultoolforderivingidentities
withinthetheoryofsymmetricfunctions. LetV beanelementofHom(Λ,Λ)andP ∈Λ. Wedefine
the flip of V by the formula
(8) VP[X]=VYP[X −Y] .
Y=X
(cid:12)
It seems to arise naturally when one considers the sorts of(cid:12)operators that will concern us here (see
(cid:12)
[Za1] and [Za2]).
ThedegreeofasymmetricfunctionP ∈Λisthehighestpowerofz inP[zX]andwillbedenoted
by deg(P). If P[zX]=zdeg(P)P[X] then we will say that P is of homogeneous degree.
Use this involution here to define a q-twisting of a symmetric function operator. Let V once
again be an element of Hom(Λ,Λ) and let Fq be defined by FqP[X]=P[X(1−q)]. Our q-analog
is defined when it acts of the symmetric function P ∈Λ by the formula
(9) VqP[X]=VYP[qX +(1−q)Y] =VFqP[X].
Y=X
(cid:12)
It is easily seen that tehis q-analog has the following fu(cid:12)ndamental propery.
(cid:12)
Remark 1. Let V be an element of Hom(Λ,Λ) and create the q-twisting of this operator from
formula (9), Vq, and act this new operator on a symmetric function P[X] to create an expression
such as
e
(10) VqP[X]
This q-analog has the property that when q =0, the expression becomes
e
(11) VP[X]
and if q =1, then it reduces to the product
(12) V(1)P[X]
This paper is concerned with generalizations of the standard bases, the Hall-Littlewood and
Macdonald symmetric functions, which depend on additional parameters q and t. There are two
important scalar products on the symmetric functions related to these bases. They are defined by
their values on the power symmetric basis.
ℓ(λ)
(13) hp ,p i =δ z 1−tλi
λ µ t λµ λ
i=1
Y
4 MICHAEL ZABROCKI
ℓ(λ)
(14) hp ,p i =δ z (1−qλi)(1−tλi)
λ µ qt λµ λ
i=1
Y
The Macdonald symmetric functions H [X;q,t] are defined by the following three conditions.
µ
1. hH [X;q,t],H [X;q,t]i =0 if λ6=µ.
λ µ qt
2. FtH [X;q,t]= c m [X] for suitable coefficients c and the sum is over all partitions λ
µ λ≤µ λµ λ λµ
that are smaller than µ in the standard dominance order.
P
3. hH [X],h [X]i=tn(µ) where n(µ)= (i−1)µ .
µ n i i
P
The expansion of the H [X;q,t] basis in the Schur basis for the symmetric functions defines the
µ
coefficients K (q,t), that is H [X;q,t]= K (q,t)s [X].
λµ µ λ⊢|µ| λµ λ
The Hall-Littlewood basis is defined similarly with respect to the h,i scalar product; simply
P t
stated H [X;t]=H [X;0,t].
µ µ
The symmetric functions H [X;q,t] and H [X;t] are the two families of symmetric functions
µ µ
thatwill interestus the mosthere. A symmetricfunction with the propertythatwhen expressedin
terms of the Schur basis their coefficients are polynomials in q and t with non-negative coefficients
will be called Schur positive. The Macdonald and Hall-Littlewood functions are just two examples
of families with this property.
3. Examples
3.1. Schur symmetric functions I. In [Ze] an operator attributed to Bernstein that adds a row
to the Schur function is given by S = (−1)ih e⊥. The formula has a very convenient
m i≥0 m+i i
form when expressed in terms of plethystic notation. Let P[X] be a symmetric function in the X
P
variables. Define a generating function of operators given by
1
(15) S(z)P[X]=P X− Ω[zX].
z
(cid:20) (cid:21)
Now for any m ∈ Z, set S P[X] = S(z)P[X] . If m ≥ µ , then it easily follows that
m zm 1
S s [X]=s [X]. S is a creation operator for the Schur basis since we have the formula
m µ (m,µ) m (cid:12)
(cid:12)
(16) S S ···S 1=s [X].
µ1 µ2 µℓ(µ) µ
q
Now if we set H(z)=S(z) and Hq =H(z) , then this is a q-analog of the operator S and
m zm m
we may calculate that
(cid:12)
g (cid:12)
H(z)P[X] = S(z)P[qX +(1−q)Y]
Y=X
(cid:12)
(cid:12)1
(17) = P qX +(1−q) X −(cid:12) Ω[zX]
z
(cid:20) (cid:18) (cid:19)(cid:21)
1−q
= P X − Ω[zX]
z
(cid:20) (cid:21)
Remarkably, this is the formula for the Hall-Littlewood creation operator of Jing [J] in the
notation used by Garsia [G]. These operators have the property that
q
Theorem 2. (Jing [J]) Let Hq =S . Then
m m
(18) Hq Hq ···Hq 1=H [X;q].
fµ1 µ1 µℓ(µ) µ
The fact that H [X;0]=s [X] and H [X;1]=h [X] follows from Remark 1.
µ µ µ µ
q-ANALOGS OF SYMMETRIC FUNCTION OPERATORS 5
3.2. Schur symmetric functions II. For any sequence of integers γ = (γ ,γ ,... ,γ ), let xγ
1 2 n
represent the monomial xγ11xγ22···xγnn. We say that γ is a dominant weight if γ1 ≥γ2 ≥···≥γn.
The symmetric groupS acts onany polynomialin the X variablesby permuting their indices.
n n
For anyσ ∈S , setε to be the signofthe permutation. Also setδ =(n−1,n−2,...,1,0). Then
n σ
defineforanypolynomialf inthe x variablesasymmetrizationoperatorπ (f)=J(f)/J(1)where
i n
(19) J(f)= ε σ(xδf).
σ
σX∈Sn
When λ is a partition, π sends xλ to the Schur function s [X ]. When λ is any dominant weight,
n λ n
set s (X )=π (xλ) to be the resulting Laurent polynomial.
λ n n
Let η be a sequence of positive integers whose sum is n (a composition) and define the set of
ordered pairs Roots = {(i,j) : 1 ≤ i ≤ η +η +···+η < j ≤ n for some r}. Consider the
η 1 2 r
following formal power series given by the formula
(20) H (X ;q)=π xµ (1−qx /x )−1 .
µη n n i j
 
(i,j)∈YRootsη
 
ThisformalserieshasanexpansionintermsoftheSchurfunctionsindexedbyalldominantintegral
weights. Define K (q) as the coefficient of s (X ) in H (X ;q), so that
λµη λ n µη n
(21) H (X ;q)= K (q)s (X ),
µη n λµη λ n
λ
X
where the sum is over all dominant integral weights λ. The K (q) are known as the generalized
λµη
or parabolic Kostka polynomials. For a more complete expositionof these polynomials we refer the
reader to [KS], [SW], or [K].
Now consider a compositionof the Bernstein Schur function operatorsS =S S ···S and
ν ν1 ν2 νℓ(ν)
q
define Hq =S . By a calculation similar to (17), one may show that
ν ν
f
(22) HqP[X]=P [X−(1−q)Z∗]Ω[ZX] 1−z /z ,
ν i j zν1zν2···zνℓ(ν)
1≤j<Yi≤ℓ(ν) (cid:12) 1 2 ℓ(ν)
(cid:12)
where Z∗ = ℓ(ν)1/z . (cid:12)
i=1 i
In work with Mark Shimozono [SZ], we demonstrated that Hq is an operator with interesting
P ν
propertiesrelatedtothegeneralizedKostkacoefficients. Inparticular,theycanbeusedasgenerating
functions for the generalized Kostka coefficients. They are also operators which act on symmetric
functions and can be used to build a family of symmetric functions.
Theorem 3. (Shimozono, Zabrocki [SZ]) Let η be a composition of k and µ ∈ Zk, set µ(i) =
(µ ,... ,µ ). For any ν ∈Zℓ set H =S q where S =S ···S , then we have
η1+···+ηi−1 η1+···+ηi ν ν ν ν1 νℓ(ν)
(23) Hq Hq ···Hq = K (q)Hq,
µ(1) µ(2) µ(ℓ) fλµη λ
λ
X
where the sum is over all dominant weights λ. In particular, when this operator is applied to the
symmetric function 1, we arrive at a class of symmetric functions and may set
(24) H [X;q]:=Hq Hq ···Hq 1= K (q)s [X],
µη µ(1) µ(2) µ(k) λµη λ
λ
X
where the sum is over all partitions λ.
6 MICHAEL ZABROCKI
Inaddition, these operatorsalsoseemto be fundamentally relatedto the new class ofsymmetric
functions referred to as ‘atoms’ A(k)[X;q] (see [LLM], [LM1], [LM2]). In particular, when the
λ
partitionindexingtheoperatorisarectangle,itwasconjecturedthatHq isacreationoperator
(ℓk+1−ℓ)
for this class of symmetric functions.
By Remark 1, the coefficients hH [X;q],s [X]i= K (q) have the property that when q =1,
µη λ λµη
they are the Littlewood-Richardson coefficients cλ and when q = 0, the K (0) = 1 if
µ(1)µ(2)···µ(k) λµη
µ = λ and 0 otherwise. It is conjectured that the coefficients K (q) are polynomials in q with
λµη
non-negative integer coefficients.
3.3. Homogeneous creation operator. Multiplication by h is an operator that adds a row to
k
the homogeneous symmetric functions. The q-analog of this operator is once again h and so this
k
is not particularly interesting. However, in [Za2] we gave a formula for an operator that adds a
column to the homogeneous symmetric functions and in [Za1] we gave a combinatorial description
of the action of this operator on the Schur function basis. Let H be a family of operators with
1m
the property that
(25) H H ···H 1=h [X].
1λ1 1λ2 1λℓ(λ) λ′
The q-twisting of this operator is another example where this q-analog appears naturally to
q
produce a family of Schur postitive symmetric functions. It develops that H is an operatorthat
1m
addsacolumntothethesymmetricfunctions(q;q) h X ,where(q;q) =(1−q)(1−q2)···(1−
λ λ 1−q k
g
qk) and (q;q) = (q;q) (q;q) ···(q;q) . This famhily ihas the property that when q = 1 the
λ λ1 λ2 λℓ(λ)
symmetric functions become h and when q =0 they become h .
1|λ| λ
Theorem 4. Let H be an operator with the property that H h [X]= h [X] for ℓ = ℓ(λ) ≤
1m 1m λ 1m|λ
m. Then we have
]q]q ]q X
(26) H H ···H 1=(q;q) h
1λ1 1λ2 1λℓ λ′ λ′ 1−q
(cid:20) (cid:21)
Proof:
q X qX +(1−q)Y
(27) H (q;q) h =(q;q) HY h
1m λ λ 1−q λ 1m λ 1−q Y=X
(cid:18) (cid:20) (cid:21)(cid:19) (cid:18) (cid:20) (cid:21)(cid:19)(cid:12)
Since the sugmmation formula for a homogeneous symmetric function in two(cid:12)sets of variables is
(cid:12)
givenash [X+Y]= m h [X]h [Y],thenforapartitionλ,h [X+Y]= ℓ(λ) λn h [X]h [Y].
m i=0 i m−i λ n=1 i=0 i λn−i
It follows that (27) reduces to
P Q P
m λn qX
= (q;q) h h [X]
λ i 1−q λn−i+1
n=1i=0 (cid:20) (cid:21)
YX
m λn+1 qX qX
= (q;q) h h [X]−h
λ i 1−q λn−i+1 λn+1 1−q
n=1 i=0 (cid:20) (cid:21) (cid:20) (cid:21)!
Y X
m
X X
(28) = (q;q) h −qλn+1h
λ λn+1 1−q λn+1 1−q
n=1 (cid:20) (cid:21) (cid:20) (cid:21)
Y
m
X
= (q;q) (1−qλn+1)h
λ λn+1 1−q
n=1 (cid:20) (cid:21)
Y
X
= (q;q) h .
1m|λ 1m|λ 1−q
(cid:20) (cid:21)
♦
q-ANALOGS OF SYMMETRIC FUNCTION OPERATORS 7
Thereareelementaryproofsthatthefunctions(q;q) h X areSchurpositive. Onceagainwe
λ λ 1−q
havea caseof a Schur positive q-analogarisingfroma Schuhr posiitive family ofsymmetric functions
h [X]. Remark 1 implies that the limit as q goes to 1 of these symmetric functions is h [X] when
λ 1n
λ is a partition of n and when q =0 we have that they reduce to h [X].
λ′
3.4. Macdonald’sOperators. MacdonaldintroducedoperatorsDr (see[M]p. 315)suchthatthe
n
Macdonald polynomials P (X;q,t) are characterized as eigenfunctions of this family of operators.
λ
What we will show is that the Macdonald operators are the q-twisting of the same operators with
q set equal to 0.
Let T be an operator on polynomials with the property T P[X ] = P[X −x ]. Consider the
xi xi n i
operator
(29) Dr(t)= A (X ;t) T
n I n xi
I⊆{X1,...,n} Yi∈I
where AI(Xn;t)=t(r2) i∈I,j∈/I txxii−−xxjj and the sum is over all subsets I of size r.
^q
Now define Dr(q,t)=QDr(t) , then calculate that
n n
Dr(q,t)P[X ] = A (Y ;t) T P[qX +(1−q)Y ]
n n I n yi n n
I⊆{X1,...,n} Yi∈I (cid:12)Yn=Xn
|I|=r (cid:12)
(cid:12)
(30) = A (Y ;t)P[X −(1−q)X ].
I n n I
I⊆{X1,...,n}
|I|=r
q
If we set T P[X ]=P[X −(1−q)x ]=T , then
q,xi n n i xi
(31) Dr(q,t)= A (X ;t) T ,
n f I n q,xi
I⊆{X1,...,n} Yi∈I
|I|=r
and these are the operators Dr as they are defined in [M]. As was presented in this reference, set
n
D (u;q,t)= n urDr(q,t).
n r=0 n
Consider again the Bernstein operators, S , as they were defined in equation (15) and also
m
considerS˜ =PωS ω =(−1)mS(−z) . Fromthese operatorscreateaq,tanalogbyapplyingthis
m m zm
parameter deformation twice. Set D˜m(cid:12)(cid:12) =S˜mq1/t and D˜m∗ =Smt1/q.
A simple calculation yields that
gf (1−q)(1−1/t)gf
(32) D˜ P[X]=P X+ Ω[−zX] ,
m
z zm
(cid:20) (cid:21) (cid:12)
and (cid:12)
(cid:12)
(1−1/q)(1−t)
(33) D˜∗ P[X]=P X − Ω[zX] .
m z zm
(cid:20) (cid:21) (cid:12)
These families of operators were studied in [GHT] and more ext(cid:12)ensively in [BGHT] to show
(cid:12)
the polynomiality of the q,t-Catalan numbers. In particular, when m = 0 it is known that these
operatorsarerelatedtotheoperatorsD1 andthattheyhavethefamilyH [X;q,t]aseigenfunctions.
n µ
Theorem 1.2 of [GHT] was the following result (translated on the H [X;q,t] basis).
µ
Theorem 5. (Garsia-Haiman-Tesler [GHT]) For µ a partition of n, we have
(34) D˜ H [X;q,t]= 1−(1−1/t) t−i(1−qµi) H [X;q,t],
0 µ µ
!
i≥1
Y
8 MICHAEL ZABROCKI
(35) D˜∗H [X;q,t]= 1−(1−t) ti(1−q−µi) H [X;q,t].
0 µ µ
!
i≥1
Y
3.5. Hall-Littlewood creation operator. Consider now an operator that adds a column to the
Hall-Littlewood symmetric functions H [X;t]. One such operator was introduced in [Za2] where
λ
the combinatorialactiononthe Schur function basis was discussedandsome explicit formulas were
presented. We will discuss some of these operators in more detail in the following section. For now
we present the following theorem.
Theorem 6. Let λ be a partition such that ℓ = ℓ(λ) ≤ m. Any operator Ht with the property
1m
Ht H [X;t]=H [X;t] satisfies the equation
1m µ 1m|λ
]q]q ]q
(36) Ht Ht ···Ht =H [X;q,t]
1λ1 1λ2 1λℓ λ′
The most elementary proof of this theorem can be seen in [GZ] and is self contained and uses
nothing more than the identities in the originalpaper ofMacdonald. We presenthere a shortproof
that follows by demonstrating that if the theoremis true for one such operatorHt , then it is true
1m
for all such operators.
Some of the first proofs of the polynomiality of the q,t-Kostka coefficients used operators of this
type to showthat the q,t-Kostkapolynomialssatisfiedrecurrencesthat did nothavedenominators.
Almost any of the operators given by Kirillov-Lapointe-Noumi-Vinet [KN1] [KN2] [LV1] [LV2] are
of this type. We need only one example and the following lemma.
Lemma 7. Let λ be a partition such that ℓ(λ) ≤ m. Assume that there exists some operator Ht
1m
such that
(37) Ht H [X;t]=H [X;t]
1m λ 1m|λ
and
q
(38) Ht H [X;q,t]=H [X;q,t].
1m λ 1m|λ
Then for any operator H′t that satisfies equation (37) will also satisfy equation (38).
1m g
Proof: It followsfromthe definition ofthe Macdonaldsymmetric functions andthe propertythat
H [X;t,q]=ωH [X;q,t]thatwehavethetriangularityrelationH [X(1−q);q,t]= a (q,t)s [X].
µ µ′ µ λ≥µ λµ λ
Since H [X;t] = K (t)s [X], then there exist coefficients b (q,t) such that H [(1 −
µ λ≥µ λµ λ λµ P µ
q)X;q,t]= b (q,t)H [X;t].
λ≥µ λPµ λ
Consider the expansion of H [X+Y;q,t]= H [X;q,t]H [Y;q,t] which may be seen as
P µ ν⊆µ µ/ν ν
a transformation of formula (7.9) on p. 345 of [M].
P q
Since our operator Ht adds a column of size m to H [X;t] and Ht adds a column to
1m ν 1m
H [X;q,t], then H [X;q,t] is given by the formula
µ 1m|µ
g
q
Ht H [X;q,t] = H [qX;q,t]Ht H [(1−q)X;q,t]
1m µ µ/λ 1m λ
λ⊆µ
X
g
(39) = H [qX;q,t] b (q,t)Ht H [X;t]
µ/λ λν 1m ν
λ⊆µ ν≥λ
X X
= H [qX;q,t] b (q,t)H [X;t].
µ/λ λν 1m|ν
λ⊆µ ν≥λ
X X
The right hand side of this expression is independent of the operator that is used to derive it,
thereforeabsolutelyanyoperatorH′t thathasthepropertythatH′t H [X;t]=H [X;t],also
1m 1m µ 1m|µ
]q
has the property that H′t adds a columnto the Macdonaldsymmetric functions H [X;q,t]. ♦
1m µ
q-ANALOGS OF SYMMETRIC FUNCTION OPERATORS 9
To prove the theorem it is necessary to produce at least one operator that satisfies the property
of the lemma. Fortunately this is relatively easy since practically any in the development in [KN1]
[KN2][LV1][LV2]suchthatonecansetq =0withoutneedingtotakealimitsatisfiestheproperties
of Lemma 7 (once transformed to the H [X;q,t] basis).
µ
Consider Expression3 of [LV1]. Let Dr(u;q,t) be the Macdonald operator of equation (31) that
I
acts onlyonthe variablesx fori inthe setI. Aformula foranoperatorthat addsa columnofsize
i
m onto the J [X;q,t]:=H [X(1−t);q,t] basis is given by
µ µ
m
x −x /t
(40) B(3)(q,t)= t−rx i j Dr(−t;q,t)
m I x −x I
i j
|IX|=mXr=0 Yji∈∈/II
where x here represents x . It is easy to demonstrate by acting on an arbitrary symmetric
I i∈I i
function that
Q
^ q
(41) B(3)(q,t)=B(3)(0,t)
m m
Let Ft be an operator that sends the symmetric function H [X;q,t] to the symmetric function
µ
J [X;q,t]. Moreprecisely,foranarbitrarysymmetric functionP[X]setFtP[X]=P[X(1−t)]and
µ
denote the inverse of this operator F−1. Since B(3)(q,t) adds a column of size m to the J [X;q,t]
t m µ
basis, F−1B(3)(q,t)Ft is an operator that adds a column to the H [X;q,t] basis.
t m µ
Whenq =0intheoperatorB(3)(q,t)itbecomesanoperatorthataddsacolumntotheH [X(1−
m µ
t);t] basis. That is, we have
(42) B(3)(0,t)H [X(1−t);t]=H [X(1−t);t].
m µ 1m|µ
Since FtH [X;t]=H [X(1−t);t], this implies F−1B(3)(0,t)Ft is an operator that adds a column
µ µ t m
to the H [X;t] basis.
µ
To demonstrate the theorem, it remains to show that the q-twist of F−1B(3)(0,t)Ft is exactly
t m
theoperatorF−1B(3)(q,t)Ft. Thisfollowsfromthe factthatconjugationbyFt commuteswiththe
t m
q-twisting for any symmetric function operator.
Lemma 8. For V ∈Hom(Λ,Λ) we have
^ q
(43) F−1VqFt =F−1VFt .
t t
Proof: Thisfollowsbyactingboththeleftandtherighthandsideofthisequationonanarbitrary
e
symmetric function. ♦
This leads us to several other formulas for operators with similar properties. Consider the fol-
lowing corollary.
Corollary 9. Define hλ(q) = s∈λ1−qaλ(s)+lλ(s)+1. Let H1qm be an operator with the property
Hq H [X;q]=H [X;q] for ℓ=ℓ(λ)≤m. Then
1m λ 1m|λ Q
]t]t ]t
(44) Hq Hq ···Hq 1=ωH [X;q,t],
1λ1 1λ2 1λℓ λ
and
]q]q ]q X
(45) Hq Hq ···Hq =h (q)s .
1λ1 1λ2 1λℓ λ′ λ′ 1−q
(cid:20) (cid:21)
Proof: ThisfollowsfromTheorem6andthefollowingtwoidentitiesaboutMacdonald’ssymmetric
functions.
10 MICHAEL ZABROCKI
(46) H [X;t,q]=ωH [X;q,t]
µ µ′
X
(47) H [X;q,q]=h (q)s
µ µ µ
1−q
(cid:20) (cid:21)
♦
In the next two sections we will develop this last example in more detail. The observation that
the Macdonald polynomials are built up from the q-twisting of the operators that build the Hall-
Littlewood symmetric functions allows us to derive severalinteresting formulas for these operators.
Remark 10. Unfortunately, the analog of section 3.2 does not seem to extend to these opera-
tors as a way of generalizing the Maconald symmetric functions. Consider the operator Ht :=
1λ
q
Ht Ht ...Ht . We would hope that a composition of Ht are Schur positive if reasonable
1λ1 1λ2 1λℓ(λ) 1λ
conditions are placed on λ. By calculating examples we begin to be encouraged by such a conjec-
ture, however for large enough examples it seems to break dogwn (for example if λ(1) = (4),λ(2) =
^q^q^q
(2,2),λ(3) =(1,1), then Ht Ht Ht 1 is not Schur positive).
1λ(1) 1λ(2) 1λ(3)
4. Ribbons and Hall-Littlewood symmetric functions
In[Za1]we gavea combinatorialformula forthe actionofanoperatorthatadds a columnto the
Hall-Littlewoodsymmetric functions. We willrecallsomeofthe definitions andtheoremsfromthat
work and use them to derive some useful formulas.
Thedefinitionofaribbonisaskewpartitionthatcontainsno2×2subdiagrams. Foranon-empty
partitionλ, define λrc =(λ −1,λ −1,... ,λ −1)(the rc indicates thatλ has the firstrowand
2 3 ℓ(λ)
firstcolumnremoved). IfRisaribbonofsizem(denotedbyR|=m)thenRwillbe equaltoλ/λrc
for some partition λ with λ +ℓ(λ)−1=m.
1
Set D(R) equal to the descent set of R, that is the set {i | i+1st cell lies below the ith cell in R
}whenthe cells arelabeledwiththe integers1ton fromleft torightandtopto bottom. Therefore
every ribbon can be identified with a subset of {1,... ,m−1}.
There is a natural statistic associated with a ribbon. Define the major index of a ribbon to be
maj(R)= i. Its complementary statistic will be comaj(R)= |R| −maj(R).
i∈D(R) 2
Fromformula(15),S isanoperatorthataddsarowtothepartitionindexingaSchursymmetric
P m (cid:0) (cid:1)
function. ByconjugatingS byω,oneobtainsanoperatorthataddsacolumn. DefineS˜ =ωS ω.
m m m
In plethystic notation, this operator is given as
1
(48) S˜ P[X]=(−1)mP X + Ω[−zX] .
m
z zm
(cid:20) (cid:21) (cid:12)
Now for each ribbon of size m, define an operator that raises th(cid:12)e degree of a symmetric function
(cid:12)
by m. For R=λ/λrc set
(49) SR =s⊥ S˜ S˜ ···S˜ ,
λrc λ′1 λ′2 λ′λ1
where λ′ is the length of the ith column in λ. This is a combinatorial operator in the sense that
i
all calculations can be computed on the Schur basis using the Littlewood-Richardson rule and the
commutationrelationsS˜ S˜ =−S˜ S˜ andS˜ S˜ =0 sothat the operatorSR canbe thought
a b b−1 a+1 a a+1
of as an operator that acts on s by adding the ribbon R to the left of λ.
λ
The main theorem in [Za1] was the following result.
Theorem 11. (Theorem 1.1 of [Za1]) The operator Hq = qcomaj(R)SR has the property
1m R|=m
that Hq H [X;q]=H [X;q] for ℓ(µ)≤m.
1m µ 1m|µ P