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SEQUENCES OF SYMMETRIC POLYNOMIALS ANDCOMBINATORIAL PROPERTIES OF TABLEAUX
RUDOLF WINKEL
Abstract. In 1977 G.P. Thomas has shown that the sequence of Schur polynomialsassociated to a partition λ can be comfortably generated from the sequence of variablesx = (x1, x2, x3, . . . ) by the application of mixed Baxter/multiplication operators, whichin turn can be easily computed from the set SY T (λ) of standard Young tableaux ofshape λ.
We generalize this construction, thereby making possible the explicit and effectivecomputation of the Hall-Littlewood, Jack, and Macdonald polynomials used in repre-sentation theory, combinatorics, multivariate statistics, and quantum algebra. Thesegeneralized formulas have a pleasing recursive structure with respect to the Young lat-tice and they can easily be specialized to yield ‘skew’ forms in all cases and ‘super’forms in the Schur case.
We introduce and investigate: (1) the ‘descent polynomial of a partition λ ’, whicharises naturally in the enumeration of semistandard Young tableaux of shape λ; (2)the Boolean lattice G(ζ) associated to any ζ ∈ SY T (λ), which is fundamental forthe ‘weighted’ generalization of Thomas’ approach to Schur polynomials; and (3) anaction of the symmetric groups on semistandard Young tableaux, which is connectedwith Knuth’s combinatorial proof of the symmetry of Schur functions. Moreover weargue that a generalization of Thomas’ approach is a natural starting point in searchof ‘universal weighted symmetric functions’.
Let s(m)λ (x) ∈ Z[x1, . . . , xm] denote the Schur polynomial in the variables x1, . . . , xm
associated to a partition λ ` N of the natural number N . It is well known (cf. [M1, S,K]) that these Schur polynomials can be defined algebraically by various determinantalformulas or combinatorially by the formula
s(m)λ ≡ s
(m)λ (x) :=
∑
η∈SSY T(m)(λ)
xη ,
where xη ≡ xρ(η) := xρ1(η)1 x
ρ2(η)2 x
ρ3(η)3 . . . is the monomial associated to the content ρ(η)
of η and SSY T(m)(λ) is the set of semistandard Young tableaux of shape λ with entries in{1, . . . , m}. (The exact definition of these and other no(ta)tions appearing subsequentlyhas been collected in an Appendix.)
The determinantal formulas are very compact and appropriate for many theoreticalpurposes, but difficult to evaluate: it is even hard to decide, which monomials in a given
s(m)λ (x) occur. To the contrary this is an easy task from the combinatorial definition,
but the latter is clearly not very compact due to the large number of possible SSYT
Date: November 1995, revised November 1996.1991 Mathematics Subject Classification. 05E05, 05E10.
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(semistandard Young tableaux). Therefore one seeks for a ‘systematization’ of all theSSYT of a given shape. It is well known that the partition µ ` N formed from thecontent vector ρ of any η ∈ SSY T (λ) precedes λ in the dominance order ‘≤’:
(0.1) µ � λ =⇒ [SSY T (λ, ρ) = ∅ ∀ρ ∈ FN(µ)] ,
and that the cardinality of the set SSY T (λ, ρ) of all η ∈ SSY T (λ) with content ρ isinvariant under arbitrary permutations of the components of ρ:
(0.2) ∀ρ ∈ FN(µ) : |SSY T (λ, ρ)| = |P (λ, µ)| =: Kλµ ,
where the non-negative numbers Kλµ are called Kostka numbers. These two facts to-gether with the obvious equivalence:
xη = xη′ ⇐⇒ ρ(η) = ρ(η′) ,
and the definition of monomial symmetric functions mµ(x) (cf. [M1,S]) yield the expan-sion of Schur functions into monomial symmetric functions:
(0.3) sλ(x) :=∑
η∈SSY T (λ)
xη =∑
µ≤λ
Kλµ mµ(x) .
An alternative way of collecting the xη’s has been described by G.P. Thomas in [T1,T2]:fix λ ` N , then the sequence of Schur polynomials
( s(1)λ , s
(2)λ , s
(3)λ , . . . )
can be generated comfortably by applying a λ-dependent sum of certain mixed shift/multi-plication operators to the sequence x = (x1, x2, x3, . . . ). Every single operator in such asum is easily computed from exactly one standard Young tableaux ζ ∈ SY T (λ), i.e. theproblem of handling the infinite set SSY T (λ) of semistandard Young tableaux is reducedto that of finding the finite set SY T (λ) of standard Young tableaux. We will speak ofτPx-formulas, because the operators appearing are build up from the shift operator ‘τ ’,the Baxter operator ‘P ’ and the multiplication operator ‘x’.
Clearly, SSY T(m)(λ) = ∅, if m < s = l(λ) (l(λ) being the length of λ), and SSY T(m)(λ) ⊂SSY T(m′)(λ), if m < m′. Consequently s
(m)λ = 0, if m < s, and s
(m′)λ = s
(m)λ +
‘positive terms’, if m < m′, which can be concisely expressed by saying that the Schurpolynomials are cumulative. Therefore it is enough to consider the sequence of differencepolynomials
(0.4) s[λ](x) := (s[1]λ (x), s
[2]λ (x), s
[3]λ (x), . . . )
with s[m]λ (x) :=
∑
η∈SSY T[m](λ)
xη = s[m]λ (x)− s
[m−1]λ (x) .
We call s[λ] the graded Schur function for λ, where the part of ‘degree’ m or m-part s[m]λ
is the sum of all monomials xη with η ∈ SSY T (λ), which contain xm, but no xν withν > m.
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The above definitions can now be extended from the Schur case to more generalweighted symmetric functions, polynomials, and graded functions:
(0.5)
{λ}w(x) :=∑
η∈SSY T (λ)
w(η) xη , {λ}(m)w (x) :=
∑
η∈SSY T(m)(λ)
w(η) xη , and
[λ]w(x) := ({λ}[1]w (x), {λ}[2]
w (x), {λ}[3]w (x), . . . ) with {λ}[m]
w (x) :=∑
η∈SSY T[m](λ)
w(η)xη ,
where w is a weight, which associates to every η ∈ SSY T (λ) an element of a ring.Obviously, in the Schur case one has the ‘trivial’ weight wS(η) ≡ 1; the well known ‘non-trivial’ weighted generalizations investigated in the present paper are Hall-Littlewood(HL) functions (wHL(η) ∈ Z[t]), the Jack functions (wJ(η) ∈ Q(α)), and the Macdon-ald functions (wM(η) ∈ Q(q, t)). HL and Jack functions are generalizations of Schurfunctions and Macdonald functions generalize HL as well as Jack functions. For moredetails on the relationship between the different families of symmetric functions as wellas their applications in representation theory, both in the classical and quantum case, incombinatorics, and in statistics the reader may consult the first paragraphs of sections5,6, and 7, respectively.
It is not hard to see, that {λ}w is symmetric exactly when
(0.6)
λ(ρ1) = λ(ρ2) =⇒
∑
η1∈SSY T (λ,ρ1)
w(η1) =∑
η2∈SSY T (λ,ρ2)
w(η2)
,
i.e. the sum of weights∑
w(η) over all η ∈ SSY T (λ) with content vector ρ is invari-ant under arbitrary permutations of the components of ρ. Therefore in the case of a‘symmetric weight w’ it makes sense to define the w-Kostka factors:
(0.7) wλ(µ) :=∑
η∈P (λ,µ)
w(η) for all µ ` N = |λ| ,
where P (λ, µ) is the set of all η ∈ SSY T (λ) with partition like content µ. Then (0.3)generalizes to
(0.8) {λ}w(x) =∑
µ≤λ
wλ(µ) mµ(x) .
The aim of the present paper is to present ‘weighted τPx-formulas’ in the HL case (Sec-tion 5), the Jack case (Section 6) and the Macdonald case (Section 7). These weightedτPx-formulas enable the effective generation of the respective graded functions and poly-nomials, for which there are explicit formulas only in very special cases. Moreover theyallow recursive computations with respect to the Young lattice Y and easy evaluationsof the ‘skew’ forms in all cases and the ‘super’ forms in the Schur case (Section 2).
In fact the w −Kostkafactors appearing in the expansion (0.8) are computed mosteffectively from weighted τPx-formulas: using (0.7) one has to find the sets P (λ, µ) for
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all µ ≤ λ, compute the weights in all cases, and sum up; using the weighted τPx-formulasone has to find only the the set SY T (λ) = P (λ, 1N), set up the weighted τPx-formula,which is roughly equivalent to finding the weights for the ζ ∈ SY T (λ), and finally to
compute {λ}(N)w (x), which is not very expansive and clearly sufficient for finding the
coefficients wλ(µ).The approach of G.P. Thomas to the τPx-expansions of graded Schur functions, of
which we give a simplified and concise account in Section 1, is to coarse to deal with theweighted case; therefore we give a new refined derivation in Section 5 (Thm.5.4). Centralto this new approach is the Boolean lattice G(ζ) of gapless SSYT associated to everySYT ζ (Def.3.3). Moreover we introduce the descent polynomial Dλ(τ) of a partitionλ in Section 3, which appears naturally in the counting of the sets SSY T[m](λ), anddescribe an action of permutations on SSYT in Section 4, which ‘improves’ the bijectionof Knuth appearing in the combinatorial proof of the symmetry of Schur polynomials.
It will turn out that a necessary condition for the existence of the τPx-representationis:
(0.9) w(η) = w(prG η) for all η ∈ SSY T (λ) .
This means that the weight w(η) does not depend on the absolute values of the numbersappearing in η, but only on its structure of horizontal stripes as represented by the gaplesselements G(λ) of SSY T (λ). In order to get a reasonably compact τPx-representationof a weighted symmetric function one needs moreover that the weight w is S-insected asexplained in Section 6 (Def.6.2) — indeed Schur, HL, Jack, and Macdonald functionshave S-insected weights.
To our understanding the central point in the combinatorial approach to symmetricfunctions is that weights encode combinatorial properties of semistandard Young tableaux:the Schur case expresses the mere fact that a multiset of numbers (or indices of variablesin a monomial) occurs as a semistandard numbering of a shape λ, whereas the moregeneral HL, Jack, and Macdonald weights encode additional facts about the distributionof horizontal stripes placed in this shape. A question which therefore naturally arises, butdoesn’t seem to have been addressed before, is the existence of combinatorially meaningfuluniversal symmetric functions Uλ, where the term ‘combinatorially meaningful universal’in accordance with the above ‘philosophy’ means:
For η ∈ SSY T (λ) the gapless representant prG η ∈ G(λ) or at least the set of horizontalstripes H(η) should be reconstructable from the given weight w(η), and Schur, HL, Jack,and Macdonald symmetric functions should be contained as special cases. In Section 7we briefly describe an important result of Kerov ([Ke]), which says that every essentialstep beyond the above mentioned special cases has to avoid the ‘superorthogonality’ ofthe weight.
Notice that the existence of an expansion (0.8) of a symmetric function into a weightedsum of monomial symmetric functions does not pose any restriction on the w-Kostka fac-tors wλ(µ) and can therefore not be regarded as ‘combinatorially meaningfull’. To the
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contrary the τPx-approach achieves this goal in a natural way (compare the above dis-cussion of (0.9) and S-insected weights); therefore the existence of weighted τPx-formulasappears to be a basic step in the construction of the universal symmetric functions Uλ.
Clearly, with some labor one can figure out weights w, which contain more or lesscomplete information about the elements η ∈ G(λ), but at present it seems very difficultto do this in a way, which makes {λ}w symmetric; the action of permutations on SSYTdescribed in Section 4 may be helpfull in this respect.
The construction and investigation the universal symmetric functions Uλ is intimatelyconnected with a unified treatment of the following problems:
(1) Combinatorial proofs of the symmetry of the weighted symmetric functions (HL,Jack, and Macdonald).
(2) Combinatorial proofs of ‘Cauchy identities’: the Robinson-Schenstedt-Knuth (RSK)correspondence ‘solves’ the Schur case (cf. [S, Sec.4.8]); and for super Schur func-tions a generalization has been given by Remmel ([Re]).
(3) The combinatorial treatment of Kostka-Foulkes polynomials (see Remarks 3.6and 5.9).
(4) The possibility of choosing the symmetric weight w in such a way that the w-Kostka factors wλ(µ) are polynomials with integer coefficients: for w = wS, wHL
this is trivially true, because all w(η) are already such polynomials; in the Jackcase w = wJ the corresponding conjecture by R.P. Stanley and I.G. Macdonald[St2,M3] has been proven recently by F. Knop and S. Sahi [KS], who gave inaddition a combinatorial formula for the computation of the wJ
λ(µ), and by L.Lapointe and L. Vinet [LV1, LV2]; in the Macdonald case w = wM proofs havebeen given independantely by L. Lapointe and L. Vinet [LV3], F. Knop [Kn],A.M. Garsia and J. Remmel [GR], and A.N. Kirillov and M. Noumi [KN].
An essential tool in proving the latter facts is the generation of symmetric functionsassociated to a partition λ ≡ λ1 . . . λs by a sequence of ‘creation operators’, say
(Ks)λs(Ks−1)
λs−1−λs . . . (K1)λ1−λ2(1) .
This approach has been pionered by J.N. Bernstein for Schur functions (cf.[Z, p.69]), andsubsequently extended for Hall-Littlewood functions by N. Jing [J], for Jack functions byL. Lapointe and L. Vinet [LV1], and for Macdonald functions by L. Lapointe and L. Vinet[LV4] and A.N. Kirillov and M. Noumi [KN]. With regard to explicit computations of therespective polynomials however, one sees that setting up the creation operator formulasis easier as in the τPx-case, but the evaluation usually will involve the massive occurenceof cancelations due to the appearence of divided difference operators.
We finally remark that the existence of τPx-formulas is not restricted to symmetricfunctions, but can be extended to sequences of (in general nonsymmetric) Schubertpolynomials, which contain the Schur polynomials as special cases. This is the subjectof the paper [W2] about ‘graded Schubert functions’.
5
1. Schur polynomials
In this section we mainly present Thomas’ results on τPx-formulas for Schur polyno-mials, but we carefully seperate the combinatorial basis of his construction in terms ofSSYT from its algebraic translation. The result of Prop.1.10 is new.
In [T1] Thomas introduced a partitioning of SSY T (λ) into equivalence classes ζ ≡SSY T (ζ), which have as canonical representatives exactly the standard Young tableauxζ ∈ SY T (λ). Every such ζ ⊂ SSY T (λ) is N-graded and cumulative:
SSY T[m](ζ) := ζ ∩ SSY T[m](λ) and SSY T(m)(ζ) := ζ ∩ SSY T(m)(λ) ,
whence SSY T (ζ) =⊎∞
m=1 SSY T[m](ζ) =⋃∞
m=1 SSY T(m)(ζ). The following definition,which seems to have been considered first by Schensted [Sch], introduces the basic conceptfor the whole paper:
Definition 1.1. To every η ∈ SSY T (λ) associate a numbering ζ(η) of the Ferrer dia-gram of shape λ by attaching the numbers 1, . . . , N to the boxes of the Ferrer diagram λin the following linear order w.r.t. η:1. the box with smaller η-label precedes the box with greater η-label,2. in case of the equal η-labels the lower box precedes, and3. in case of equal η-labels and equal rows the box more to the left precedes.The above linear order will be called the standard order on η.
Clearly ζ(η) ∈ SY T (λ), such that sets
ζ ≡ SSY T (ζ) := {η ∈ SSY T (λ) | ζ(η) = ζ}give the desired partition of SSY T (λ).
If on the other hand ζ ∈ SY T (λ) is given and ρ is a ‘suitable’ content vector, then onecan construct uniquely an η ∈ SSY T (λ, ρ), such that ζ(η) = ζ: just number the boxes
of λ in the order given by ζ with ρ1 one’s, ρ2 two’s, etc. . Hence for arbitrary ρ ∈ F[m]N
the set ζ ∩ SSY T (λ, ρ) has cardinality 0 or 1.
Example 1.2. For η =5 83 6 6 82 3 5 5 8
∈ SSY T[8]( 5 4 2 ) one gets ζ(η) =4 92 7 8 101 3 5 6 11
; η can
be recovered from ζ and the content vector ρ = (0, 1, 2, 0, 3, 2, 0, 3).
For fixed ζ ∈ SY T (λ) we speak of the box with ζ-label ν ∈ {1, . . . , N} in λ as the‘ζ-box ν’. If the row number of some ζ-box ν ′ is greater than the row number of anotherζ-box ν, we simply say ‘ν ′ is ζ-below ν’ or ‘ν is ζ-above ν ′’; similarly: ‘ν ′ is ζ-left to ν’,‘ν is ζ-right to ν ′ ’.
For later use we introduce a related notion applicable to an arbitrary η ∈ SSY T (λ):running through the boxes of η in standard order, the next box is reached by an S-step,if it is ‘below’, and by a P-step otherwise.
Lemmma 1.3. Fix ζ ∈ SY T (λ) and let (– only for this Lemma –) η be an arbitrarynumbering of λ. Let furthermore ‘η(ν)’ be a shorthand for ‘the number of the ζ(η)-box ν
6
in η’ and M(ζ) be the set of arbitrary numberings η of λ subjected to the two conditions:
1.) η(1) ≤ · · · ≤ η(N), and 2.) [ ν + 1 below ν =⇒ η(ν) < η(ν + 1) ] .
Then ζ = M(ζ).
Proof. First let η ∈ ζ, then the procedure given above for finding the representative ζfor a given η ∈ SSY T (λ) immediately implies the validity of conditions 1 and 2. Let onthe other hand η ∈ M(ζ) be given, then also immediately η ∈ ζ, provided η ∈ SSY T (λ);but conditions 1 and 2 already imply η ∈ SSY T (λ): let ν and ν ′ be two ζ-boxes andν and ν ′ in the same row with ν ′ right to ν, then ν < ν ′ and by condition 1: η(ν) ≤ η(ν ′);or ν and ν ′ in the same column with ν ′ below ν, then ν < ν ′ and by condition 2:η(ν) < η(ν ′). ¤
Now Lemma 1.3 and the definitions of sλ, s(m)λ and s
[m]λ immediately imply:
Proposition 1.4. Let λ ` N , then
sλ =∑
ζ∈SY T (λ)
∑
η∈M(ζ)
xη , s(m)λ =
∑
ζ∈SY T (λ)
∑
η∈M(ζ)η(N)≤m
xη ,
and s[m]λ =
∑
ζ∈SY T (λ)
∑
η∈M(ζ)η(N)=m
xη .
The definition of M(ζ) suggests another one: that of the descent set D(ζ) for anyζ ∈ SY T (λ):
D(ζ) := { i | i + 1 below i }.Then Prop 1.4 says, that the contribution of a ζ ∈ SY T (λ) to sλ resp. s
[m]λ is
∑1≤i1≤···≤iNiν∈D(ζ)=⇒iν<iν+1
xi1 . . . xiN resp.∑
1≤i1≤···≤iN=m
iν∈D(ζ)=⇒iν<iν+1
xi1 . . . xiN .
Let R be a commutative ring with unit and x = (x1, x2, x3, . . . ) a sequence of variables;then
A ≡ AR(x) := ( R[x1], R[x1, x2], R[x1, x2, x3], . . . )
is a R-algebra under componentwise operations. For every a = (a1, a2, . . . ) ∈ A wedenote the nth-component an by [a]n. The shift operator
τ : A −→ A, (a1, a2, a3, . . . ) 7→ (0, a1, a2, . . . ) ,
i.e. [τa]n := an−1, where an = 0 for n ≤ 0, and all its powers τ ν for ν ∈ N0 (τ 0 := id) arealgebra endomorphism of A; consequently the same is true for all operators f(τ) ∈ R[τ ]or f(τ) ∈ R[[τ ]], since [A]n is not affected by the τ ν with ν > n. One can calculate asusual in the rings R[τ ] and R[[τ ]]. Note that for x = (x1, x2, . . . ) ∈ A and all n ∈ N
7
the sets { [τ νx]n | ν ∈ N0 } generate the R-algebras [A]n = R[x1, . . . , xn]. Especiallyimportant is the ‘geometric shift operator’
P :=∞∑
ν=0
τ ν , P (a1, a2, a3, . . . ) = (a1, a1 + a2, a1 + a2 + a3, . . . ) .
and its companion S := τP .A Baxter operator on an arbitrary commutative R-algebra A is an R-linear mapping
B : A −→ A such that for some fixed θ ∈ R:
B(aB(b)) + B(bB(a)) = B(a)B(b) + B(θab) for all a, b ∈ A.
Indeed the above defined operators P and S are Baxter for θ = 1 and θ = −1, re-spectively. Baxter operators have been introduced by G. Baxter ([B]), and investigatedto some extend by G.-C. Rota [R1,R2,RS] and P. Cartier [C]. The basic result is theisomorphism between the standard Baxter algebra, containing the symmetric functionsand the above Baxter operators P and S, and the free Baxter algebra (in the sense ofuniversal algbra), which makes it possible to prove results for the free Baxter algebrawith the help of symmetric functions. One example in this direction is the remarkeblyshort proof of the Bohnenblust-Spitzer formula of fluctuation theory by Rota.
It is not hard to see (e.g. by induction) that a sequence of the form
(. . . , an, . . . ) ≡ ( . . . ,∑
1≤i1≤···≤iN=niν∈D=⇒iν<iν+1
xi1 . . . xiN , . . . ) ∈ AZ(x)
can be written as the Baxter sequence
B1 . . . BN−1(x) := xBN−1 . . . xB1x ,
where for ν = 1, . . . , N − 1: Bν ∈ {P, S} and Bν = S iff ν ∈ D. Together with Prop.1.4this shows
Theorem 1.5. (G.P. Thomas [T1]) For λ ` N and ζ ∈ SY T (λ) let B(ζ) := B1 . . . BN−1
denote a sequence of operators Bν ∈ {P, S}, where Bν = S iff ν ∈ D(ζ); then
(S) s[λ] =∑
ζ∈SY T (λ)
B(ζ)(x) .
Example 1.6. Let λ = 221. The columns below show ζ ∈ SY T (λ), D(ζ), B(ζ), and
ρ(ζ) (cf. Def.3.3).
(2,2,1)PSPS
{2, 4}531
42
(2,1,2)PSSP
{2, 3}431
52
(1,2,2)SPSP
{1, 3}421
53
(1,1,2,1)SSPS
{1, 2, 4}321
54
(1,2,1,1)SPSS
{1, 3, 4}521
43
Therefore s[221](x) = PSPS(x) + PSSP (x) + SPSP (x) + SSPS(x) + SPSS(x).
8
Proposition 1.7. Let λ′ be the conjugate partition to λ ` N and B′(ζ) the same asB(ζ) in Thm.1.5, but with P ↔ S (P exchanged by S and vice versa); then
a) D(ζ ′) = D′(ζ) := {1, . . . , N − 1} \D(ζ) ;
b) B(ζ ′) = B′(ζ) for all ζ ∈ SY T (λ) and s[λ′](x) =∑
ζ∈SY T (λ)
B′(ζ)(x) .
Proof. The conjugation of the Ferrer diagram λ induces a bijection ′ : SY T (λ) −→SY T (λ′). Fix ζ ∈ SY T (λ) and two consecutive ζ-boxes ν and ν + 1; then one has twopossibilities: either ν + 1 is below ν in ζ, then ν + 1 not right to ν and therefore in ζ ′ is(ν +1)′ not below ν ′, or ν +1 not below ν, then ν +1 right to ν and in ζ ′ is (ν +1)′ belowν ′. This shows (a) . Part (b) is then immediate by the definition of B(ζ) and B′(ζ). ¤
The elementary and complete symmetric polynomials have an especially simple rep-resentation in terms of Baxter sequences:
s[1N ](x) = (xS)N−1(x) and s[N ](x) = (xP )N−1(x) .
Remark 1.8. Thomas developed his arguments more generally for ‘numbered frames’= ‘columnstrictly numbered finite subsets of unit squares in the Z × Z-plane’. In factthe set D used for the definition of the (graded) functions of the type B(ζ) can originatein many different ways, for example as a set of ranks of a poset in [St1]. Functions of thetype B(ζ), now called ‘fundamental quasisymmetric functions’, have been investigatedmore systematically by I. Gessel [G] and subsequently many others.
Remark 1.9. A natural question that comes to mind is: what happens, if P and S inThm.1.5 are substituted more generally by shift operators f(τ) =
∑∞i=0 fiτ
i, (f0 6= 0)and g(τ) =
∑∞i=1 giτ
i, (g1 6= 0) ? Without loss of generality we can assume f0 = g1 = 1.The requirement that the generalized sequences “s[λ](x; f, g)” are sequences of symmetricpolynomials forces f(τ) = P and g(τ) = S, as can be seen easily by investigating the casesλ = 2 and λ = 12. Therefore the only possibility to generate more general symmetricfunctions by τPx-formulas is to introduce appropriate ‘weights’, i.e. extend AZ(x) to aring AR(x), where R ) Z.
If on the other hand one dispenses with symmetry and allows arbitrary scalars fi, gi
in the style of “umbral calculus”, then the resulting polynomials should share manyproperties of Schur polynomials.
Proposition 1.10. i) Let D(λ) := {D(ζ) | ζ ∈ SY T (λ)} and D∗(λ) := {D∗(ζ) | ζ ∈SY T (λ)} for D∗(ζ) := {|λ| − i | i ∈ D(ζ)}; then D∗(ζ) = D(ζ).ii) For λ ` N and ζ ∈ SY T (λ) associate to every B(ζ) = B1 . . . BN−1 as defined inThm.1.5 the ‘reversed Baxter sequence’ B∗(ζ) := BN−1 . . . B1; then
s[λ](x) =∑
ζ∈SY T (λ)
B∗(ζ)(x) .
Proof. Clearly i) and ii) are equivalent; we show i): every equivalence class ζ contains
a unique element ζ called the maximal identification of ζ, which originates from ζ by9
numbering the ζ-boxes 1, . . . , N consecutively with ν = 1, 2, . . . , where ν is increased by
1 iff the corresponding ζ-step is a S-step (cf. Def.3.3). Let ρ = ρ(ζ) the content vector of
ζ and consider the monomial xρ contained in some component of s[λ]; by symmetry thiscomponent contains also a monomial xρ∗ , where ρ∗ is ρ (without end zeroes) in reversed
order. This is possible only if there is some ξ ∈ SY T (λ) with ρ∗ = ρ(ξ), whence for thisξ one has D(ξ) = D∗(ζ). (As an illustration see Ex.1.6 .) ¤
2. Skew Schur and hook Schur polynomials
Let λ ` N, µ ` M, µ ⊂ λ (M ≤ N),and m ∈ N, then for the skew shape λ/µ onedefines the skew Schur functions ([M1, I (5.12)]) and polynomials as
sλ/µ :=∑
η∈SSY T (λ/µ)
xη and s(m)λ/µ :=
∑
η∈SSY T(m)(λ/µ)
xη .
It is not hard to see that Thm.1.5 and Prop.1.7 apply similarly in the ‘skew case’, i.e.
(sS) s[λ/µ] := (s[m]λ/µ)m∈N =
∑
ζ∈SY T (λ/µ)
Bλ/µ(ζ)(x)
and s[λ′/µ′] =∑
ζ∈SY T (λ′/µ′) B′λ/µ(ζ)(x), where SY T (λ/µ), D(ζ), Bλ/µ(ζ) and B′
λ/µ(ζ) are
defined completely analogous to the case µ = ∅. Now
(2.1) SY T (λ/µ) = M−{ζ ∈ SY T (λ) | λ(ζ(M)) = µ} ,
which should be understood as follows: the set SY T (λ/µ) can be obtained from the setSY T (λ) by first selecting all ζ ∈ SY T (λ) for which the numbers 1, . . . , M fill exactly asub-SYT of shape µ and second subtracting M from all entries and cancel all boxes withentries ≤ 0.
This procedure extends to an easy method for obtaining the ‘skew Baxter sequences’Bλ/µ(ζ) from the ‘full Baxter sequences’ Bλ(ζ) ≡ B(ζ): take the Baxter sequences Bλ(ζ)
of all elements in {ζ ∈ SY T (λ) | λ(ζ(M)) = µ} and cancel the first M symbols; this givesthe ‘skew Baxter sequences’ Bλ/µ(ζ) for ζ ∈ SY T (λ/µ).
Example 2.1. Let λ = 221 and µ = 12. Then in Ex.1.6 exactly the last three SYThave the property that the numbers 1 and 2 fill a sub-SYT of shape µ. Deletion of thefirst two entries of their Baxter sequences gives SP , PS and SS, whence: s[221/12](x) =SP (x) + PS(x) + SS(x) .
The above described procedure also yields an economic way to compute ‘extendedτPx-formulas’ for the hook Schur or super Schur polynomials, the definition ofwhich we recall next (for details see [BR,Re]):
For k, l ∈ N0 and λ ` N one defines the set SST(k,l)(λ) of (k, l)-semistandard tableauxof shape λ as the set of all numberings of the Ferrer diagram λ with numbers from{1, . . . , k}∪{1, . . . , l } (– the second set is the set of ‘overlined’ numbers 1, . . . , l –), suchthat there is some β, ∅ ⊂ β ⊂ λ with ‘β columnstrict with entries in {1, . . . , k}’ and ‘λ/β
10
rowstrict with entries in {1, . . . , l}’. Hence the set SST(k,l)(λ) can be more convenientlydescribed as a set of pairs of SSYT:
(2.2) SST(k,l)(λ) ∼=⊎
β: ∅⊂β⊂λ
SSY T(k)(β)× SSY T(l)(λ′/β′) .
The hook Schur polynomials HS(k,l)λ (x; y) are defined combinatorially as
HS(k,l)λ (x; y) ≡ HSλ(x1, . . . , xk; y1, . . . , yl) :=
∑
T∈SST(k,l)(λ)
w(T ) ,
where for T ≡ (η, η′) ∈ SSY T(k)(β)×SSY T(l)(λ′/β′) we forget the bars and set w(T ) :=
xηyη′ . The definition and the above bijection (2.2) immediately imply:
(2.3) HS(k,l)λ (x; y) =
∑
β: ∅⊂β⊂λ
s(k)β (x) s
(l)λ′/β′(y) .
Moreover ([BR, Thm.6.13]):
(2.4) HS(k,l)λ = HS
(l,k)λ′ .
Setting
SST[k,l](λ) :=⊎
β: ∅⊂β⊂λ
SSY T[k](β)× SSY T[l](λ′/β′)
gives us the possibility to express the HS(k,l)λ (x; y) for all pairs (k, l) ∈ N0 ×N0 simulta-
neously as
HS(k,l)λ (x; y) =
k∑i=0
l∑j=0
HS[i,j]λ (x; y) ,
where
(2.5) HS[i,j]λ (x; y) :=
∑
T∈SST[i,j](λ)
w(T ) =∑
β: ∅⊂β⊂λ
s[i]β (x) s
[j]λ′/β′(y) ≡
∑
β: ∅⊂β⊂λ
HS[i,j]λ,β (x; y) .
and(hS)
HS[λ](x; y) :=(HS
[i,j]λ (x; y)
)i,j≥0
=∑
β: ∅⊂β⊂λ
(HS
[i,j]λ,β (x; y)
)i,j≥0
≡∑
β: ∅⊂β⊂λ
HS[λ],β(x; y)
is an N0 × N0-grading of the product sβ(x) sλ′/β′(y). Note that SST[0,0](λ) = ∅ =⇒HS
[0,0]λ (x; y) = 0, HS
[k,0]λ (x; y) = s
[k]λ (x) and HS
[0,l]λ (x; y) = s
[l]λ′(y).
In analogy to Section 1 we define for any commutative unitary ring R and sequences ofvariables x = (x1, x2, x3, . . . ), y = (y1, y2, y3, . . . ) the N0 × N0-array of polynomial rings
AR(x; y) := (R[x1, . . . , xi; y1, . . . , yj])i,j≥0 ,
where [AR(x; y)]0,0 := R, [AR(x; y)]i,0 := R[x1, . . . , xi] and [AR(x; y)]0,j := R[y1, . . . , yj].AR(x; y) is a commutative R-algebra under componentwise operations. Furthermore we
11
define x, y ∈ AR(x; y) by [x]ij := δ0,j xi and [y]ij := δi,0 yj and the two commuting shiftoperators τ, τ by
[τ kτ la]ij := ai−k,j−l ( with ai−k,j−l := 0, if k > i or l > j, for all a = (aij)i,j≥0 ).
Let P and S as in Section 1 and in addition P :=∑∞
ν=0 τ ν , S := τP .We observe next that one has the isomorphism of algebras AR(x; y) ∼= AR(x)⊗AR(y),
where for f(x) = (f0, f1(x1), . . . , fi(x1, . . . , xi), . . . ) ∈ AR(x) and g(y) =(g0, g1(y1), . . . , gj(y1, . . . , yj), . . . ) ∈ AR(y) we define:
f(x)⊗ g(y) := (fi(x1, . . . , xi)⊗ gj(y1, . . . , yj))i,j≥0 ≡ (fi(x1, . . . , xi)gj(y1, . . . , yj))i,j≥0 .
Under the above isomorphism it follows from (2.5) and (hS) that
(2.6) HS[λ],β(x; y) = s[β](x)⊗ s[λ′/β′](y) .
Using the results of Section 1 and formula (2.6) it is easy now to compute the ‘extendedτPx-expansions’ of the summands HS[λ],β(x; y) from the set SY T (λ):
(1) Compute the (regular) ‘τPx-expansion’ of sβ(x) according to Thm.1.5 (cf. Rem.2.3below);
(2) Compute the (regular) ‘τPy-expansion’ of the skew Schur functions as describedin the beginning of this section;
(3) Take the ⊗-product over the two sums obtained in steps 1 and 2, where in the‘τPy-factor’ P ↔ S (P are substituted by S and vice versa) by Prop.1.7.
Example 2.2. Let λ = 221. According to (hS) we list all β, ∅ ⊂ β ⊂ λ together withthe ‘extended τPx-expansion’ of the HS[λ],β(x; y):
β HS[λ],β(x; y)221 (PSPS + PSSP + SPSP + SSPS + SPSS)(x)22 (PSP + SPS)(x)⊗ y211 (PSS + SPS + SSP )(x)⊗ y21 (PS + SP )(x)⊗ (S + P )(y)2 P (x)⊗ (PS + SP )(y)
111 SS(x)⊗ P (y)11 S(x)⊗ (PS + SP + PP )(y)1 x⊗ (PSP + PPS + SPS + PSP + SPP )(y)∅ (SPSP + SPPS + PSPS + PPSP + PSPP )(y)
Note that in accordance with formula (2.4) the sum∑
β: ∅⊂β⊂λ HS[λ],β(x; y) of theabove ‘extended τPx-expressions’ is invariant in a nontrivial way under the simultaneousexchanges x ↔ y, P ↔ S and S ↔ P .
Remark 2.3. A very pleasant property of the τPx-expansions of Schur functions is itsrecursive structure with respect to the Young lattice Y :
12
(1) Suppose the τPx-expansion of some s[λ](x) is available; then the τPx-expansions
of all s[λ](x) with λ ⊂ λ can be computed in a fashion similar to the skew case:
let λ ` N, λ ` M and M < N ; for every ζ ∈ SY T (λ) single out one ζ ∈SY T (λ), which contains ζ as sub-SYT; in the corresponding B(ζ)(x) delete thelast N −M symbols P or S, e.g. for λ = 221 and µ = 22 we (must) choose thefirst and the last SYT in Ex.1.6 to compute: s[22](x) = PSP (x) + SPS(x).
(2) Suppose that for all λ ` N the τPx-expansions of the s[λ](x) have been computed;
let λ ` N + 1 and
C(λ) := { λ ` N | λ covers λ in Y }be the λ-covered set in Y ; then s[λ](x) can be determined as the sum of all τPx-
expansions of the s[λ](x) with λ ∈ C(λ), where a symbol P [ or S ] is added to
the right of a B(ζ) iff the single box in λ/λ is right [ or below ] the ζ-box N . Asan example study again Ex.1.6 and observe that C(221) = {212, 22}.
Remark 2.4. The noncommutative Schur functions as pioneered by Lascaux andSchutzenberger and developed further e.g. by Fomin and Greene in [FG] (— not to beconfused with the noncommutative symmetric functions of [GKLLRT] —) are definedas sums over SSYT, where the entries of some η ∈ SSY T (λ) are to be read columnwisebottom-up and the columns from left to right in order to yield the sequence of non-commuting variables in the monomial xη. Since for every ζ ∈ SY T (λ) the sequence ofmultiplications by x in B(ζ)(x) is done in the ζ-standard order, it is not hard to accus-tom the evaluation of B(ζ)(x) in such a way that one gets the noncommutative gradedSchur functions.
3. The descent polynomial of a partition and the lattices G(ζ)
In this section we investigate for all partitions λ the sequences λ] := (λ]1, λ
]2, . . . ) of
numbers λ]m := |SSY T[m](λ)|. Obviously one has
(3.1) λ] :=∑
ζ∈SY T (λ)
B(ζ)(1) with 1 = (1, 1, . . . ) .
Since all factors x = 1 except the rightmost can be neglected and shift operators in R[τ ]commute, one has
(3.2) B(ζ)(1) = τ |D(ζ)| PN−1(1) = τ |D(ζ)| N ] for all ζ,
where we used the special τPx-formula (just in front of Rem.1.8) for the graded completesymmetric functions s[N ](x).
Definition 3.1. For any partition λ the descent polynomial Dλ ∈ R[τ ] of λ is definedas
Dλ(τ) :=∑
ζ∈SY T (λ)
τ |D(ζ)| .
13
An immediate consequence from the above discussion is:
(]) λ] := Dλ(τ) N ] for all λ ` N .
Example 1.6 shows: D221(τ) = 3τ 2 + 2τ 3 and D32(τ) = 2τ + 3τ 2. In the specialcase where λ = (n − k + 1)1k, i.e. λ is a (n, k)-hook with ‘arm length’ n − k and ‘leglength’ k, one has Dλ(τ) =
(nk
)τ k: each of the k boxes in the leg can be reached only
by descent steps, which can be done in(
nk
)ways. In general a simple formula for the
descent polynomial Dλ doesn’t seem to exist, so that one has explicitly to compute theset of descent sets
D(λ) := {D(ζ) | ζ ∈ SY T (λ)} .
For the numbers [PN(1)]m we have the recursion
[PN(1)]m =m∑
ν=1
[PN−1(1)]ν = [PN−1(1)]m + [PN(1)]m−1 ,
and the ‘initial conditions’ [PN(1)]1 = 1 = [P 0(1)]m for all m,N ∈ N, whence weconclude
N ]m = [PN−1(1)]m =
(m + N − 2
N − 1
)for all m,N ∈ N .
We have therefore proven
Proposition 3.2. For λ ` N one has:
(3.3) λ]m = Dλ(τ)N ]
m =∑
ζ∈SY T (λ)
(m− |D(ζ)|+ N − 2
N − 1
).
As a special case we note
(1N)]m = τN−1
(m + N − 2
N − 1
)=
(m− (N − 1) + N − 2
N − 1
)=
(m− 1
N − 1
).
Definition 3.3. For any partition λ and every ζ ∈ SY T (λ) define the sets of gaplesselements for ζ as
G(ζ) := SSY T (ζ) ∩G(λ) .
We introduce now further no(ta)tions yielding as a byproduct that the sets G(ζ) can beequipped with a partial order turning them into boolean lattices:
Label the step from the ζ-box ν to the ζ-box ν +1 with ν, and let I(ζ) ⊂ {1, . . . , N −1}be the subset of (ζ-relative) P-steps, i.e. I(ζ) = D(ζ ′) (cf. Prop.1.7 a)) as sets ofnumbers. For later use we subdivide the set of P-steps in ζ further into the set I0(ζ) ofP0-steps, where the ζ-boxes ν and ν + 1 are in consecutive columns, and the set I1(ζ) ofP1-steps, where the is at least one column between ζ-boxes ν and ν + 1.
Given any subset I ⊂ I(ζ) let ζ I ∈ G(ζ) denote the unique element, which originatesfrom ζ by the following procedure: label the upper left box with ‘1’ and run through theboxes 1, . . . , N of ζ, where the label remains the same exactly in the steps contained in I
and otherwise is increased by 1. Special cases are ζ = ζ ∅ and the maximal identification
ζ := ζ I(ζ). If on the other hand one begins with some η ∈ G(ζ), one can compute a set14
I(η) by running through the η-boxes in standard order, such that η = ζ I(η). We call theset I ≡ I(η) characterizing one element of G(ζ) the (ζ-relative) identification set of η.(Moreover, set I0(η) := I(η) ∩ I0(ζ) and I1(η) := I(η) ∩ I1(ζ).)
Therefore G(λ) can be equipped with the order structure induced by the boolean lattice
B(I(ζ)) of all subsets of I(ζ) ordered by inclusion; the top element is ζ and the bottomelement is ζ.
Example: Let η =4 62 2 4 51 1 2 3 5
∈ G(ζ) , where ζ =7 113 4 8 91 2 5 6 10
; then D(ζ) =
{2, 6, 10}, I(ζ) = {1, 3, 4, 5, 7, 8, 9}, I(η) = {1, 3, 4, 7, 9}, I0(ζ) = {1, 3, 4, 5, 8, 9}, I0(η) ={1, 3, 4, 9}, and I1(ζ) = I1(η) = {7}.
Proposition 3.4. For λ ` N one has:
Dλ′(τ) = τN−1Dλ(τ−1) ;(a)
|G(λ)| = Dλ′(2) = 2N−1Dλ(1
2) .(b)
Proof. a) Since |D(ζ ′)| = N − 1 − |D(ζ)| by Prop.1.7 a), the assertion follows fromthe definition of the descent polynomial. For b) observe G(λ) =
⊎ζ∈SY T (λ) G(ζ) and
|B(I(ζ))| = 2|I(ζ)| = 2|D(ζ′)|; hence:
|G(λ)| =∑
ζ∈SY T (λ)
|G(ζ)| =∑
ζ∈SY T (λ)
2|D(ζ′)| = Dλ′(2)a)= 2N−1Dλ(
1
2) .
¤
Remark 3.5. For fixed partition λ and ζ ∈ SY T (λ) the sets P (ζ) := ζ ∩ P (λ) aresublattices of the respective G(ζ):
On the set P (N) of partitions of N = |λ| define the refinement order, in which ‘ λ iscovered by λ ’ :iff ‘ λ originates from λ by subdivision of exactly one part of λ intotwo parts ’. (Clearly, the refinement order turns P (N) into a lattice, which is in generaldifferent from the dominance order lattice.)
In terms of the partitionlike content of a tableaux η ∈ G(λ) the above ‘subdivision’corresponds to ‘adding one more element to the identification set’. Let λ(ρ(η)) be thepartition of the unique element of P (ζ), which has the maximal possible identificationset; then P (ζ) is embedded into G(ζ) as a poset isomorphic to the principal order idealin P (N) generated by λ(ρ(η)), and is therefore itself a lattice.
Remark 3.6. Let λ, µ ` N , n(µ) :=∑
j≥1(j−1)µj, and DΣλ (t) :=
∑ζ∈SY T (λ) tD
Σ(ζ) with
DΣ(ζ) :=∑
ν∈D(ζ) ν. The Kostka-Foulkes polynomials Kλµ(t) appearing as coefficients in
the expansion of sλ(x) into the HL-polynomials Pµ(x; t) (cf. [M1, Sec. III 6] and [Bu])15
can be characterized combinatorially as
Kλµ(t) =∑
η∈P (λ,µ)
tc(η) or Kλµ(t) =∑
η∈P (λ,µ)
tcc(η) ,
where c(η) ∈ N0 is the charge of η, cc(η) ∈ N0 is the cocharge of η and Kλµ(t) =tn(µ)Kλµ(t−1). Note that the above combinatorial characterizations (found by Lascauxand Schutzenberger) imply:
(3.4) cc(η) = n(µ)− c(η) for all η ∈ P (λ, µ).
Now from [M1, III 6 Ex.2] and our above notations follows:
Kλ(1N )(t) =∑
ζ∈SY T (λ′)
tDΣ(ζ) = DΣ
λ′(t) .
An argument similar to that used in the proof of Prop.3.4 a) then shows:
Kλ(1N )(t) = DΣλ (t) .
This does not genera-Lise to arbitrary ζ ∈ SY T (λ): in general one has DΣ(ζ) 6= cc(ζ),which leads to the conclusion that the Kostka-Foulkes polynomials are associated to the(weighted) τPx-expansions of HL-functions (see Section 5) in a nontrivial way.
We introduce now the Lascaux-Schutzenberger (LS) order on tableaux (“lifting back”to tableaux the procedure given on words of tableaux in [M1,Bu]):
begin with the upper left 1 and label it with a ‘1’;assume that your “standpoint” is an η-box ν ≥ 1 with label ‘q’ (1 ≤ q ≤ µ1), then thereare several possibilities for the next step:
(1) there is a unlabeled η-box ν + 1 below: take the rightmost of the uppermostoccurrence below and label it with ‘q’;
(2) there is no unlabeled η-box ν + 1 below, but to the right: take the rightmost ofthe uppermost occurrence and label it with ‘q’;
(3) there is no unlabeled η-box ν + 1 at all: (by the partitionlike content of η thereare no unlabel η-boxes > ν) if q < µ1, begin with the rightmost unlabeled 1 inrow one and label it with ‘q + 1’; if q = µ1, every entry in η is labeled and theprocedure stops.
The ‘LS order’ is the linear order given by reading the labels from 1 to µ1 and for equallabels the boxes numbered in their natural order.
It is now easy to define c(η) [ cc(η) ] for η ∈ P (λ, µ): run through the boxes of η in LSorder and attach c-indices [ cc-indices ] to them according to the following rule: an η-box1 always gets the index ‘0’; if a box has c-[cc-]index ‘r’ and the next box is below, itsc-index is ‘r’ [ cc-index is ‘r+1’ ], otherwise (provided it is not a an η-box 1) its c-indexis ‘r+1’ [ cc-index is ‘r’ ]. Then c(η) [ cc(η) ] is the sum of all c-indices [ cc-indices ].
Note that for any η-box ν the sum of its c-index and cc-index equals ν−1 in accordancewith (3.4); note further that substituting all words ‘rightmost’ in the definition of LSorder by ‘leftmost’ does not change the distribution of c- and cc-indices.
16
Example: For η =3 52 2 3 4 41 1 1 2 3
∈ P (522, 3321) one has the labeling1 12 1 3 2 13 2 1 3 2
, the
c-indices0 10 0 1 1 10 0 0 1 1
, and the cc-indices2 31 1 1 2 20 0 0 0 1
. Hence c(η) = 6, cc(η) = 13, and
c(η) + cc(η) = 19 = n(3321).The complete definition of LS order on tableaux has been given here not only for the
sake of selfcontainedness, but mainly for comparison with the standard order: we suspectthat the “orthogonality” of the two orders will simplify the combinatorial approach toKostka-Foulkes polynomials.
4. An action of permutations on SSYT
Every (finite) permutation π can be decomposed as a product of the elementary trans-positions σν := (ν, ν + 1) (ν ∈ N), i.e. the symmetric groups Sn on n ‘letters’ aregenerated by the σν (1 ≤ ν ≤ n− 1) with relations:
σ2ν = id(i)
σνσν′ = σν′σν , if |ν − ν ′| > 1(ii)
σνσν+1σν = σν+1σνσν+1 .(iii)
The symmetric group S∞ :=⋃
n≥1 Sn (with the natural embedding of Sn into Sn′ forn < n′) acts on ρ ∈ FN by π(ρ) := (ρπ(1), ρπ(2), . . . ). We first discuss an extension of asingle σν to a mapping σK
ν : SSY T (λ, ρ) −→ SSY T (λ, σν(ρ)), which has been attributedto Knuth in [S, Prop.4.4.2]. The σK
ν enable an elegant combinatorial proof of the sym-metry of the Schur functions sλ and of formula (0.2), but since relation (iii) is in generalnot valid they do not extend to an action π : SSY T (λ, ρ) −→ SSY T (λ, π(ρ)) on thesets SSY T (λ). We introduce therefore in this section mappings σν : SSY T (λ, ρ) −→SSY T (λ, σν(ρ)), which have the desired properties, and which we suspect to play animportant role in — yet to be found! — combinatorial proofs of the symmetry of HL,Jack and more general functions {λ}w.
The mapping σKν : SSY T (λ, ρ) −→ SSY T (λ, σν(ρ)) for fixed η ∈ SSY T (λ, ρ) and
ν ∈ N is defined as follows:Suppose ρν+1 = 0, then of course changing all η-boxes ν into ν +1-boxes does the job;
similarly in the case ρν = 0. Assume therefore ρν , ρν+1 6= 0; changes occur in the doublestrip η[ν+1,ν] (cf. Appendix), and clearly columns with ν-paired boxes, i.e. an η-box νdirectly above a box ν + 1, can not be affected; apply now the following rule to all rowsin η[ν+1,ν] \ { ν-paired boxes }: suppose a certain row contains α boxes ν and β boxesν + 1 (necessarily directly to the right of the ν’s): change |α − β| boxes of these α + βboxes, such that the row contains first β boxes ν and then (continuing to the right) αboxes ν + 1.
17
Proposition 4.1. The mappings σKν defined above obey relations (i) and (ii).
Proof. (i) is immediate from the definition, and (ii) is a consequence of the fact thatη[ν+1,ν] ∩ η[ν′+1,ν′] = ∅, if |ν − ν ′| > 1. ¤
Corollary 4.2. [Formula (0.2)] ∀ρ ∈ FN(µ) : |SSY T (λ, ρ)| = |P (λ, µ)|.Proof. (i) implies that the σK
ν are bijections, whence |SSY T (λ, ρ)| = |SSY T (λ, σν(ρ))|and clearly there is a (finite) chain of σν , which transforms an arbitrary ρ ∈ FN(µ) intoa partitionlike ρ′ ∈ FN(µ) ∩ PFN . ¤
Corollary 4.3. The combinatorially defined Schur functions sλ (or Schur polynomials
s(m)λ ) are symmetric.
Proof. SSY T (λ) =⊎
µ`|λ| SSY T (λ, µ) and∑
η∈SSY T (λ,µ) xη = Kλµmµ(x) imply the as-sertion. The case of polynomials is analogous. ¤
Clearly every family of mappings σν : SSY T (λ, ρ) −→ SSY T (λ, σν(ρ)), which obeys(i) and (ii), yields the results of the above corollaries. On the other hand the mappingsσK
ν have some deficiencies: first, if for some η ∈ SSY T (λ) one has ρν = ρν+1, then
one should have σνη = η, but for example σK2 3
1 2= 2
1 3, and second, the σν should
obey also relation (iii), in order to have welldefined compositions π : SSY T (λ, ρ) −→SSY T (λ, π(ρ)), but for example:
σK1 σK
2 σK1 2 3
21 1 1= σK
1 σK2 2 3
21 1 2= σK
1 2 331 1 3
= 2 331 2 3
, and
σK2 σK
1 σK2 2 3
21 1 1= σK
2 σK1 2 3
31 1 1= σK
2 2 331 2 2
= 3 331 2 2
.
Therefore we introduce mappings σν : SSY T (λ, ρ) −→ SSY T (λ, σν(ρ)), which haveall desired properties:
Definition 4.4. Fix η ∈ SSY T (λ, ρ) and ν ∈ N; in case of ρν = 0 or ρν+1 = 0 thereis no difference between σν and σK
ν . Assume therefore ρν , ρν+1 6= 0; as before changesoccur only in the set η[ν+1,ν] \ { ν-paired boxes }, but now we introduce additional pairsof (ν-)fixed boxes as follows: examine the set η[ν+1,ν] \ { ν-paired boxes } for ‘(1st-order)fixed pairs’: by this we mean ‘a ν + 1-box , which has as the next neighbor to the righta ν-box’ (necessarily with lower row number); remove all 1st-order fixed pairs and searchagain for ‘(2nd-order) fixed pairs’, remove them, too, etc. until there are no further fixedpairs; we write for the set of remaining boxes
M(η, ν) := η[ν+1,ν] \ { ν-paired boxes, ν-fixed pairs (of all orders) } .
Suppose M(η, ν) contains α boxes ν and β boxes ν + 1; by construction the ‘ν’ arecontained in α columns, which are all to the left of the β columns containing the ‘ν + 1’;now change |α − β| boxes, such that the ‘new ν’ are contained in β columns, which areall to the left of the α columns containing the ‘new ν + 1’.
18
Example 4.5. We examine again the relation (iii), now with σν instead of σKν :
σ1σ2σ1 2 321 1 1
= σ1σ2 2 321 1 2
= σ1 3 331 1 2
= 3 331 2 2
, and
σ2σ1σ2 2 321 1 1
= σ2σ1 3 321 1 1
= σ2 3 321 2 2
= 3 331 2 2
.
Proposition 4.6. The mappings σν defined above obey the relations (i), (ii), and (iii),and for every η ∈ SSY T (λ) one has: σνη = η, if ρν = ρν+1.
Proof. (i) and (ii) are valid by the same arguments as in the proof of Prop.4.1; and alsothe last assertion is almost immediate: ρν = ρν+1 implies either M(η, ν) = ∅, so thereremains nothing to be changed, or M(η, ν) 6= ∅, but then ν- and ν + 1-boxes occur withthe same multiplicity.
For the proof of relation (iii) we use as a shortcut the combinatorial rule for thegeneration of Schubert polynomials proved in [W1].
Let η ∈ SSY T[m](λ) and 1 ≤ ν ≤ m; then η corresponds by [W1, Sec.4] to a unique box
diagram B, in signs: ‘η −→ B’. Further correspondences are: ‘η[ν+1,ν] −→ Mn−ν(B)’,‘ν-paired boxes −→ ν-paired boxes’, ‘ν-fixed boxes −→ ν-fixed boxes’, and ‘M(η, ν) −→f(n− ν,B)’. Now ‘σνη −→ the unique maximal (or minimal) element in the equivalenceclass [B]n−ν∼ ’ (cf. [W1, Prop.3.3]), which itself corresponds to an term in the algebraically
derived sum ∂n−νxB (we need not care about the ‘dummy box’ [1, n− ν]). The algebraic
divided difference operators ∂n−ν are easily checked to obey relation (iii), whence dotheir combinatorial counterparts; this finally translates back to: ‘the σν satisfy relation(iii)’. ¤Remark 4.7. While finishing the present paper we learned from [GKLLRT, Sec.7.3] thatLascoux and Schutzenberger have already introduced the above action of the symmetricgroups on words: reading the entries of some SSYT columnwise from bottom to top andthe columns from left to right one obtains a word and the translation of our action toan action on these words gives the rule of Lascoux and Schutzenberger. Neverthelesswe find it useful to give an independent exposition of this action with other connectionsmade. (In [GKLLRT, Sec.7.3] it is shown that the ‘ribbon Schur polynomials’ introducedthere, which are nonsymmetric in the ordinary sense, are ‘symmetric’ with respect to theLS-action.)
5. Hall-Littlewood polynomials
Hall-Littlewood (HL) functions Qλ(x; t) or Pλ(x; t) (cf. [Bu,M1,Mo]) are one-parameterextensions of Schur functions, i.e. they include Schur functions for the parameter valuet = 0 (and also the mλ(x) for t = 1). They appear in the enumeration of subgroupsof abelian p-groups, and in the theory of ordinary, projective (t = −1) and modularrepresentations of the symmetric and general linear groups. Moreover the HL functionsappear in [J] as a basis of a representation of the Virasoro algebra.
19
Our starting point is the combinatorial definition of Hall-Littlewood functions givenin [M1, Sec.III 5], which is of the form (0.5). The weight w(η) ∈ Z[t] for η ∈ SSY T (λ)occurs in two different versions: as ϕη(t) ≡ wQ(η) and as ψη(t) ≡ wP (η), i.e.
Qλ(x; t) :=∑
η∈SSY T (λ)
ϕη(t) xη and Pλ(x; t) :=∑
η∈SSY T (λ)
ψη(t) xη .
According to (0.5) the HL polynomials Q(m)λ and P
(m)λ are defined similarly with summa-
tion over SSY T(m)(λ). Recall from the Appendix that RH(η) and LH(η) are the rightand left boundary boxes, respectively, of the H-components of η and that mj(λ) is themultiplicity of j as a part of λ; let j(ν) denote the column number of an η-box1 ν andset L0H(η) := { ν ∈ LH(η) | j(ν) > 1 }. Then the characterization of the HL-weightsgiven in [M1, Sec.III 5] can be re-casted in our terminology as:
ϕη(t) :=∏
ν∈RH(η)
(1− tmj(ν)(η(ν))) and ψη(t) :=
∏
ν∈L0H(η)
(1− tmj(ν)−1(η(ν−1))) .
Obviously the weights ϕ and ψ do not depend on the absolute values of the entries of η,and it is therefore sufficient to investigate them only for η ∈ G(λ). Notice that from thecombinatorial definition of HL functions the equalities Qλ(x; 0) = Pλ(x; 0) = sλ(x) areimmediate.
Example 5.1. Let η =3 42 21 1 2 3
∈ G(422), where the boxes ν ∈ RH(η) are printed bold;
then ϕη(t) = (1− t)4(1− t2) and ψη(t) = (1− t)2.
The functions Qλ(x; t) and Pλ(x; t) are related by a multiplicative factor ([M1, III(2.11-12)]):
(5.1) Qλ(x; t) = bλ(t)Pλ(x; t) with bλ(t) :=∏j≥1
mj(λ)∏i=1
(1− ti) .
Note that bλ(t) = ϕλ(η), if η is the superstandard tableaux (see Appendix), and thatψλ(η) = 1 for this η; note further that bλ(t) by definition has
∑j≥1 mj(λ) = l(λ) factors
and that on the other hand |RH(η)| = |H(η)| = |LH(η)| and |LH(η)| = |L0H(η)|− l(λ)imply |RH(η)| = |L0H(η)| + l(λ). From formulas (0.7-8) one sees that Qλ = bλPλ is
equivalent to ∀µ ≤ λ : wQλ (µ) = bλ(t) wP
λ (µ), but we will prove the stronger result:ϕη(t) = bλ(t)ψη(t) for all η ∈ SSY T (λ) with the help of the following
Lemmma 5.2. Let ζ ∈ SY T (λ) and η1, η2 elements of the lattice G(ζ) (cf. Def.3.3).(‘a | b’ means ‘a divides b’.) Then:a) the relation ‘η1 covers η2 in G(ζ)’, i.e. I(η1) \ I(η2) = {s}, implies the followingalternative:
1Subsequently we write “ν” as an abbreviation for the tripel (ν, i, j), if (i, j) is the position of anη-box with label ν.
20
ϕη2(t) contains one more factor than ϕη1(t), if s ∈ I0(η1), orϕη2(t) = ϕη1(t), if s ∈ I1(η1);
b) η1 > η2 =⇒ ϕη2(t) | ϕη1(t), ϕη1∧η2(t) | gcd (ϕη1(t), ϕη2(t)) andlcm (ϕη1(t), ϕη2(t)) | ϕη1∨η2(t).a) and b) are valid also with ψ instead of ϕ.
Proof. a) η1 > η2 =⇒ I(η2) ⊂ I(η1) =⇒ RH(η1) ⊂ RH(η2). Assume now that η1 coversη2 in G(ζ), then there are two possibilities: the removal of s splits a H-component ofη1, i.e. s ∈ I0(η), then RH(η2) contains a new element and ϕη2(t) a new factor, or the
removal of s splits a certain stripe η[ν]1 without creating a new element of H(η1), i.e.
s ∈ I1(η), then ϕη2(t) = ϕη1(t).b) the first assertion is immediate from a); for the other two note: ϕη1∧η2(t) | ϕη1(t), ϕη2(t)and ϕη1(t), ϕη2(t) | ϕη1∨η2(t).The proof with ψ instead of ϕ is completely analogous; observe that the boxes in thefirst column can not be reached by P-steps. ¤Proposition 5.3. ϕη(t) = bλ(t)ψη(t) for all η ∈ SSY T (λ).
Proof. Of course it is sufficient to prove the assertion for all η ∈ G(λ). Assume thatη1 covers η2 in G(ζ), i.e. I(η1) \ I(η2) = {s}, and ϕη1(t) = bλ(t)ψη1(t). Then s ∈I1(η1) implies by Lemma 5.2, that ϕη2(t) = ϕη1(t) and ψη2(t) = ψη1(t), hence ϕη2(t) =bλ(t)ψη2(t). The case ‘s ∈ I0(η1)’ is only slightly more difficult: let θ ∈ H(η1) consist ofboxes ν and let s be the step from column j to column j + 1 in θ, then the new elementin RH(η2) is the box ν in column j and the new element in L0H(η2) is the box ν + 1 in
column j + 1; consequently in both cases the new factor generated is (1− tmj(η(ν)2 )). We
conclude that it is already sufficient to prove the assertion for all maximal elements inthe lattices G(ζ), i.e.
η ∈ max(λ) := {η = ζ | ζ ∈ SY T (λ)} .
Assume ϕη(t) = bλ(t)ψη(t) to be true for all λ of length r ≥ 1, i.e. λ = λ1 . . . λr; the
case r = 1 is trivial for η ∈ max(λ). Let now λ = λ1 . . . λrλr+1 and η ∈ max(λ). Incase of λr > λr+1 one has bλ = bλ(t) · (1 − t); the new row r + 1 in η consists of thesame kind of boxes, say ν, because η ∈ max(λ); the uniquely determined new element ofRH(η) in row r +1 and column λr+1 generates a new factor (1− t) in ϕη(t), which is notcontained in ψη(t), since L0H(η) = L0H(η); now ϕη(t) = bλ(t)ψη(t) follows by inductionhypothesis. In case of λr = λr+1 similarly there is a new factor (1 − tmλr (λ)+1) in ϕη(t),which does not appear in ψη(t). ¤
For the derivation of the weighted τPx-formulas we need a refinement of Thm.1.5,which is based on the notions introduced in Def.3.3; this gives also the promised alter-native proof of the τPx-expansions for graded Schur functions.
Theorem 5.4. For λ ` N , ζ ∈ SY T (λ), SSY T (ζ) ≡ ζ and ∅ ⊂ I, J ⊂ I(ζ) withI ∩ J = ∅, let (gr
∑stands for the grading as in (0.4) of the terms in
∑)
SSY T (ζ, I) := { η ∈ SSY T (ζ) | prG η = ζ I } , BI(ζ)(x) := gr∑
η∈SSY T (ζ,I)xη ,
21
BI|J(ζ)(x) :=∑
J ′⊂J
BI∪J ′(ζ)(x) , and B(ζ)(x) :=∑
I⊂I(ζ)
BI(ζ)(x) .
Then one has
(5.2) BI|J(ζ)(x) = xBI|JN−1x . . . B
I|J1 x with BI|J
ν =
S , if ν /∈ I ∪ J1 , if ν ∈ IP , if ν ∈ J
,
and as special cases: BI(x) = BI|∅(x), B(ζ)(x) = B∅|I(ζ)(x); moreover
s[λ](x) =∑
ζ∈SY T (λ)
B(ζ)(x) .
Proof. The special cases: BI(x) = BI|∅(x) and B(ζ)(x) = B∅|I(ζ)(x) are immediate fromthe definitions; and the τPx-expansion of s[λ](x) follows from SSY T (λ) =
⊎ζ∈SY T (λ) SSY T (ζ);
since D(ζ) = {1, . . . , N − 1} \ I(ζ), formula (5.2) gives for B(ζ)(x) the same expressionas already encountered in Thm.1.5 .
Hence it remains to show (5.2): in case of J = ∅ one sees directly from the definitionthat
BI(ζ)(x) = gr∑
1≤i1≤···≤iNxi1 . . . xiN ,
where iν = iν+1 ⇐⇒ ν ∈ I, and iν < iν+1 ⇐⇒ ν /∈ I; therefore BI(x) = xBIN−1x . . . BI
1xwith BI
ν = 1 ⇐⇒ ν ∈ I, and BIν = S ⇐⇒ ν /∈ I in accordance with (5.2). Assume
now that (5.2) is valid for some J ⊂ I(ζ) with 0 ≤ |J | < N and let J ⊂ I(ζ) be suchthat J \ J = {j}; then for every I ⊂ I(ζ) with I ∩ J = ∅ one computes (omitting thearguments ‘(ζ)(x)’)
BI|J =∑
J ′⊂J
BI∪J ′ =∑
J ′′⊂J
(BI∪J ′′ + BI∪J ′′∪{j}) = BI|J + BI∪{j}|J ;
by induction hypothesis BI|Jν = B
I∪{j}|Jν for all ν 6= j and B
I|Jj + B
I∪{j}|Jj = S + 1 = P ,
which gives the desired result. ¤
Remark 5.5. Theorem 5.4 shows that general weighted τPx-formulas are created bysumming up the ‘building blocks’ BI
w(ζ). In order to get the weighted form of Thm.5.4simply attach a subscript ‘w’ to every ‘B’ and define BI
w(ζ)(x) := gr∑
η∈SSY T (ζ,I)w(η)xη.
In case of the graded Schur function and in the HL case (cf. Thm.5.6 below) the summa-tion for a fixed ζ ∈ SY T (λ) leads to a complete collapse of the hypercube B(I(ζ)) (cf.Def.3.3) to a single chain of weighted operators P (and S). In Sections 6 and 7 we willsee that for the graded Jack and Macdonald functions this collapsing process leads only
to a chain of in general smaller hypercubes associated to H-strips in ζ or, equivalently,maximal sequences of consecutive P-steps.
For the moment we simply emphasize that property (0.9) is crucial for the existenceof reasonable ‘building blocks’ BI
w(ζ), namely
[w(η) = w(prGη) ∀η ∈ SSY T (ζ, I)] =⇒ BIw(ζ) = w(ζ I)BI(ζ) .
22
Theorem 5.6. For λ ` N and ζ ∈ SY T (λ) let ν ∈ {1, . . . , N − 1} denote the box ν orthe step ν in ζ (cf. Def.3.3) depending on the context; for fixed ζ ∈ SY T (λ) let j(ν) bethe column number of the ζ-box ν. Define Q(ζ; t) by the following rules:
(1) write down x QN−1x . . . Q1x, where Qν = P , if ν is a P-step or (equivalently)ν ∈ I(ζ), and Qν = S, if ν is a S-step or (equivalently) ν /∈ I(ζ);
(2) substitute Qν by
Suν(ζ), if ν /∈ I(ζ)Suν(ζ) + 1, if ν ∈ I0(ζ)Puν(ζ), if ν ∈ I1(ζ)
, where uν(ζ) = (1− tmj(ν)(ζ
(ν)));
(3) multiply with (1− tmj(N)(ζ)) on the left.
Then
(HL) Q[λ](x; t) =∑
ζ∈SY T (λ)
Q(ζ; t)(x),
where Q[λ](x; t) is the sequence of the Q[m]λ (x; t) :=
∑η∈SSY T[m]
ϕη(t) xη with m ∈ N.
Proof. The τPx-formula for Q[λ](x; t) is a weighted form of the one given for s[λ](x); weuse the notations of Thm.5.4 above with ‘Q’ instead of ‘B’ to indicate that weights ϕη(t)are involved.
For fixed I ⊂ I(ζ) let η := ζ I ∈ G(ζ). Then from the various definitions one sees
QI(ζ; t)(x) = xQIN−1x . . . QI
1x with QIν =
1 , if ν ∈ I0(η)
(1− tmj(ν)(η(n(ν)))) , if ν ∈ I1(η)
(1− tmj(ν)(η(n(ν))))S , if ν /∈ I(η)
,
where the ‘n(ν)’ in η(n(ν)) denotes the number of the ζ-box ν in η. In case of ν ∈ I0(η) theshape λ(η(n(ν))) contains properly λ(ζ(ν)); for ν /∈ I(η) one has λ(η(n(ν))) = λ(ζ(ν)), and
for ν ∈ I1(η) at least mj(ν)(η(n(ν))) = mj(ν)(ζ
(ν)); therefore the weights (1− tmj(ν)(η(n(ν))))
are in fact independent of η resp. I and can be abbreviated uν ≡ uν(ζ) := (1−tmj(ν)(ζ(ν))).
(Note that this independence explains combinatorially why it is nessecary to distinguishbetween P0- and P1-steps or in other words the choice of just the set RH(η) in thedefinition of ϕη(t).)
Now the summation∑
I⊂I(ζ) QI(ζ; t)(x) yields similarly as in the proof of Thm.5.4 the
terms Q(ζ; t)(x) as specified above and hence the τPx-formula for Q[λ](x; t). ¤Example 5.7. For λ = 3 2 we compute the Q(ζ; t)(x), the sum of which gives the(weighted) τPx-formula for Q[32](x; t):
4 51 2 3
: (1− t)x (S(1− t) + 1)x S(1− t)x (S(1− t) + 1)x (S(1− t) + 1)x
3 41 2 5
: (1− t)x (S(1− t2) + 1)x (S(1− t) + 1)x S(1− t)x (S(1− t) + 1)x
3 51 2 4
: (1− t)x S(1− t)x P (1− t)x S(1− t)x (S(1− t) + 1)x
2 51 3 4
: (1− t)x S(1− t)x (S(1− t) + 1)x (S(1− t2) + 1)x S(1− t)x23
2 41 3 5
: (1− t)x (S(1− t2) + 1)x S(1− t)x (S(1− t2) + 1)x S(1− t)x
Remark 5.8. The nice combinatorial argument of Cor.4.3 showing the symmetry of theSchur functions does not apply in the case of HL functions, because in general ϕη(t) 6=ϕσνη(t); for example η1 = 2 3
1 1 2and η2 = 2 2
1 1 3in SSY T (32, (2, 2, 1)) have ϕ-weights
(1−t)4 and (1−t)2(1−t2), respectively, but their images under σ2 in SSY T (32, (2, 1, 2)),
i.e. 3 31 1 2
and 2 31 1 3
, have both the ϕ-weights (1 − t)3. Of course the sum of weights
remains constant, but this is a non-combinatorial argument.
Remark 5.9. (Continuation of Rem.3.6) Let λ ` N , Kλµ the Kostka numbers andKλµ(t) the Kostka-Foulkes polynomials; let moreover
ψµλ(t) :=∑
η∈P (µ,λ)
ψη(t) ,
then
(5.3) sλ(x) =∑
µ≤λ
Kλµ(t) Pµ(x; t) ⇐⇒ ∀λ ≤ λ : Kλλ =∑
λ≤µ≤λ
Kλµ(t) ψµλ(t) .
Proof. (of (5.3)) Collecting in sλ(x) =∑
µ≤λ Kλµ(t) Pµ(x; t) all terms with a non-increasing gapless sequence of exponents gives
∑
η∈P (λ)
xη =∑
µ≤λ
Kλµ(t)∑
η∈P (µ)
ψη(t) xη .
Using P (λ) =⊎
λ≤λ P (λ, λ) and (0.1-2) one concludes
Kλλxλ =
∑
η∈P (λ,λ)
xη =∑
λ≤µ≤λ
Kλµ(t)∑
η∈P (µ,λ)
ψη(t) xλ for all λ ≤ λ.
For the other direction reverse the arguments and finally apply the symmetry of Schurand HL functions. ¤
From the charge definition of the Kostka Foulkes polynomials it is not hard to see thatKλλ(t) = 1 and that t divides Kλµ(t) for all µ < λ; consequently the r.h.s. of (5.3) showsthat the summand Kλλ(t) ψλλ(t) = ψλλ(t) = Kλλ ± . . . alone yields the Kostka numberKλλ and all subsequent summands with λ ≤ µ < λ are necessary to cancel the othercontributions to ψλλ(t).
Remark 5.10. The skew Hall-Littlewood functions are defined for skew shapes λ/µ (withλ ` N, µ ` M, µ ⊂ λ, M ≤ N) by ([M1, III (5.11)]):
Qλ/µ(x; t) :=∑
η∈SSY T (λ/µ)
ϕη(t) xη .
24
In complete analogy to Section 2 the sequences Qλ/µ(ζ; t)(x) of Q[λ/µ](x; t) can be com-puted from the sequences Qλ(ζ; t)(x) of Q[λ](x; t): single out all ζ ∈ SY T (λ) with
λ(ζ(M)) = µ and delete everything to the right of the (N − M)th x, e.g. for λ = 32and µ = 2 the first three SYT in Ex.5.7 are used to compute:
Q[32,2](x; t) = (1− t)x (S(1− t) + 1)x S(1− t)x
+ (1− t)x (S(1− t2) + 1)x (S(1− t) + 1)x + (1− t)x S(1− t)x P (1− t)x .
Remark 5.11. Generalizing Rem.2.3 to the HL case we discuss again the recursivestructure τPx-expansions with respect to the Young lattice Y :
(1) Suppose the τPx-expansion of some Q[λ](x; t) is available; then the τPx-expansions
of all Q[λ](x; t) with λ ⊂ λ can be computed similarly as in the Schur case:
let λ ` N, λ ` M and M < N ; single out a subset of SY T (λ) representingSY T (λ); in the corresponding Q(ζ; t)(x) delete everything to the left of the M th
x counted from the right except the factor uM(ζ), e.g. for λ = 32 and µ = 21 wechoose the second and the fourth SYT in Ex.5.7 to compute:Q[21](x; t) = (1−t)x S(1−t)x (S(1−t)+1)x + (1−t)x (S(1−t2)+1)x S(1−t)x .
(2) Suppose that for all λ ` N the τPx-expansions of Q[λ](x; t) have been computed;
then Q[λ](x; t) for λ ` N +1 can be determined as the sum of the τPx-expansions
of the Q[λ](x; t) with λ ∈ C(λ), where the leftmost factor uN of the Q(ζ; t)(x)can be reused in the execution of 2. and 3. of Thm.5.6 . As an example studyagain Ex.5.7 and observe that C(32) = {31, 22}.
6. Jack polynomials
Jack functions Jλ(x; α) (cf. [M2,St2]) are one-parameter extensions of Schur functions,i.e. they include Schur functions for the parameter value α = 1 (and also the λ′-productof elementary symmetric functions for α = 0 and the monomial symmetric functions forα −→∞). For α = 2 they specialize to the zonal polynomials used in multivariate statis-tics ([MPH]). Moreover every exited state in the Calogero-Sutherland model describingthe long-range interaction of n quantum particles on a circle can be written as a linearcombination of Jack polynomials and also the singular vectors of a conformal field theoryare given by Jack functions with rectangular Ferrer diagram (cf. [LV, AKOS1]).
Our starting point is the combinatorial definition of Jack functions given in [St2,Thm.6.3], which is of the form (0.5). The weight wJ(η) ∈ Q(α) for η ∈ SSY T (λ) canbe computed as follows:
Let λ ` N and s = (i, j) ∈ λ be the box in row i and column j of λ; then
Hλ(s) := { (i′, j) ∈ λ | i′ ≥ i } ∪ { (i, j′) ∈ λ | j′ ≥ j }25
is the hook based at s in λ with arm length aλ(i, j) := λi−j and leg length lλ(i, j) := λ′j−i.To every hook Hλ(s) one associates two important ‘linear factors’ in Z[α]:
the lower hooklength h+λ (i, j) := (lλ(i, j) + 1) + aλ(i, j)α, if (i, j) ∈ λ, and
= 1 otherwise; the upper hooklength h−λ (i, j) := lλ(i, j) + (1 + aλ(i, j)α),if (i, j) ∈ λ, and = 1 otherwise;
i.e. the ‘base box’ s is taken up to the leg in the ‘lower’ case (corresponding to thevertical line in ‘+’) or to the arm in the ‘upper’ case (corresponding to the horizontalline in ‘−’).
Let (λ| j) ≡ (λ| j1, . . . , jr) be a pair consisting of a partition λ and an r-tuple j with1 ≤ j1 < · · · < jr ≤ λ1 of ‘column numbers in λ’ and set {j} := {j1, . . . , jr}; then oneassociates a polynomial wJ(λ| j) ∈ Z[α] to such a pair by:
(6.1) wJ(λ| j) :=∏
(i,j)∈λj∈{j}
h+λ (i, j)
∏
(i,j)∈λj /∈{j}
h−λ (i, j) .
For arbitrary η ∈ SSY T (λ) set η[0] := ∅ and let j(η[ν]) be the ordered tuple of columnnumbers of the boxes in the H-strip η[ν]; then the ‘Jack weight’ wJ as a rational functionin Q(α) is defined by:
(6.2) wJ(η) :=∏ν≥1
wJ(η(ν); j(η[ν])) with wJ(η(ν); j(η[ν])) :=wJ(η(ν)| j(η[ν]) )
wJ(η(ν−1)| j(η[ν]) ).
In the case of η[ν] = ∅ for some ν one has j(η[ν]) = ∅ and η(ν) = η(ν−1) and consequentlywJ(η(ν); j(η[ν])) = 1; hence
(6.3) wJ(η) = wJ(prG η) for all η ∈ SSY T (λ).
The Jack functions Jλ(x; α) :=∑
η∈SSY T (λ) wJ(η) xη by (0.7-8) have expansions of theform:
(6.4) Jλ(x; α) =∑
µ≤λ
vλµ(α) mµ(x) with vλµ(α) ≡ wJλ(µ)(α) :=
∑
η∈P (λ,µ)
wJ(η) .
Stanley and Macdonald have conjectured ([M2,St2]) that the vλµ(α) are polynomialsZ[α] with non-negative coefficients (compare the Introduction).
Directly from the definition one computes J(1N )(x; α) = N ! eN(x): in the case of
λ = 1N one has P (λ) = SY T (λ) = {ζ} and v1N ,1N (α) = wJ(ζ) =∏N
i=1wJ (ζ(i)|1)
wJ (ζ(i−1)|1)=
wJ(ζ(N)|1) = wJ(ζ|1) =∏N
i=1 h+(1i)
(α) = N !. Slightly more involved combinatorial argu-
ments yield the formula for J(N)(x; α) given in [St2, Prop.2.2 a)].In case of α = 1 one has h+
λ (s) = h−λ (s) = |Hλ(s)| =: hλ(s) for s = (i, j) ∈ λ,i.e. upper and lower hooklength specialize to the ordinary hooklength; consequentlywJ(η) =
∏s∈λ hλ(s) ≡ hλ(λ) for arbitrary η ∈ SSY T (λ) and
Jλ(x; 1) = hλ(λ)sλ(x) .26
Signed Diagrams (for human calculations): The product of hooklength’s h−λ (s) resp.h+
λ (s) relative to a fixed shape λ can be represented diagrammatically by marking the‘base box’ s of every ‘hook factor’ in the Ferrer diagram λ with a ‘−’ resp. a ‘+’ ; boxesin λ for which no hook factor occurs are marked with ‘0’. We call Ferrer diagrams, inwhich every box has a ‘sign’∈ {+, 0,−}, signed (Ferrer) diagrams. For example:
0 0− − + − 00 0 + 0 +
= (2 + 2α) · 2 · (1 + 5α)(1 + 4α)(1 + 2α) · 2α .
Since it is possible to cancel ‘+’-hooks or ‘−’-hooks of the same shape occurring in aquotient of signed diagrams, one can comfortably compute wJ(η) for every η ∈ SSY T (λ).Example:
wJ(2 31 1 1 3 ) = + + + ·
++ − −
+ − −·− +− + − +
−− + −
= 0 + 0 ·0+ 0 0
0 0 0· 4
0 0− + 0 0
0− + 0
= 16(1 + α)2(1 + 4α)/(1 + 3α). Using the method of signed diagrams it not very hardto give combinatorial proofs of many of the formulas for the vλµ(α) contained in [St2];we mention the formulas for vNµ(α) [St2, Prop.2.2 a], vλλ(α) [St2, Thm.5.6], and vλµ(α)[St2, Prop.7.1], where λ = λ1 . . . λl and µ = µ1 . . . µj1
λj+1+···+λl for 1 ≤ j ≤ l. ¤
Proposition 6.1. (for computer calculations): Let η ∈ SSY T (λ) for some λ ` N ,ν ∈ N with |η[ν]| = r > 0, λ := λ(η(ν)), λ′ its conjugate defined by λ′i :=
∑j≥i mj(λ) and
set
η[ν] = { (iv, jv) ∈ λ | v = 1, . . . , r }with l(λ) ≥ i1 ≥ · · · ≥ ir ≥ 1 and 1 ≤ j1 < · · · < jr ≤ λ1. Then:
(6.5) wJ(η, ν) = ϕ0(η, ν) ϕ+(η, ν) ϕ−(η, ν) ≡r∏
v=1
ϕ0v ·
r∏v=1
ϕ+v ·
∏
v∈p(η,ν)
ϕ−v
with
p(η, ν) := { p′ | p′ = 1 or ip′−1 > ip′ } , ϕ0v = 1 + (λiv − jv)α ,
ϕ+v =
iv−1∏pv=1
pv + (λpv − jv)α
pv − 1 + (λpv − jv)α, ϕ−v =
jv−1∏qv=1
qv 6=j1,...,jv−1
(λ′qv− iv) + (jv − qv)α
(λ′qv− iv) + (λiv − qv + 1)α
.(6.6)
Proof. The above formulas are simply a recasting of the previous ‘diagrammatic method’.The main observation is that in general only some hook factors survive cancelation,namely the factors ϕ+
v based in columns j1, . . . , jr, because their leg length changes,and the factors ϕ−v based in rows i1, . . . , ir, because their arm length changes, and ofcourse the factors ϕ0
v of the boxes of η[ν]. Notice that p(η, ν) is the set of the leftmostboxes in every row of η[ν] and that the boxes with column numbers ∈ {j1, . . . , jr} do notcontribute to the ϕ−v . ¤
27
Definition 6.2. Let [λ]w be a graded weighted symmetric function (as in (0.5)) fulfilling(0.9), i.e. the basic ‘building blocks’
BIw(ζ)(x) := gr
∑η∈SSY T (ζ,I)w(η)xη
are well defined for all ζ ∈ SY T (λ) (see also Rem.5.5). Assume moreover that w(η) is aproduct of factors w(η, ν), where each factor w(η, ν) depends only on the shapes of η(ν)
and η(ν−1) or in other words: on the position of η[ν] in ζ(η). Then the weight w is calledS-insected and [λ]w has a S-insected τPx-expansion.
Clearly by (6.2-3) wJ is S-insected (and so is wM as we will see in Sec.7); but also wS
and wHL are S-insected: trivially wS(η) =∏
wS(η, ν) with wS(η, ν) = 1 for all η and ν,and wHL(η) ≡ ϕη(t) =
∏wHL(η, ν) with
wHL(η, ν) =∏
ν∈RH(η[ν])
(1− tmj(ν)(η(ν))) (see Section 5).
Proposition 6.3. Let w be a S-insected weight as in Def.6.2 above and fix ζ ∈ SY T (λ).Partition the set I(ζ) of P-steps for ζ into maximal subsets M1, . . . , Ms of consecutiveintegers, i.e. if Mk = {ν + 1, ν + 2, . . . , ν + r} ⊂ I(ζ) for appropriate k, ν, r ∈ N0, thenν, ν + r + 1 /∈ I(ζ). Let the notation ‘ (M ′
1, . . . ,M′s) ≺ M1 × · · · ×Ms ’ express the fact
that M ′k ⊂ Mk for all k ∈ {1, . . . , s} in the s-tuple.
Then with the notations of the ‘weighted Thm.5.4’ (cf. Rem.5.5) one has
Bw(ζ)(x) :=∑
I⊂I(ζ)
BIw(ζ)(x) =
∑
(M ′1,...,M ′
s)≺M1×···×Ms
BM ′1∪···∪M ′
sw (ζ)(x) .
Therefore Bw(ζ)(x) is represented by a chain of ‘ |Mk|-dimensional (k = 1, . . . , s) hyper-cubes’ of weighted operators separated by weighted S-operators.
Proof. Similar to Thm.5.4 . ¤
We describe now the weighted τPx-formulas for graded Jack functions J[λ](x; α).
Theorem 6.4. For λ ` N and ζ ∈ SY T (λ) let ν ∈ {1, . . . , N − 1} denote the box ν orthe step ν in ζ (cf. Def.3.3) depending on the context; for fixed ζ ∈ SY T (λ) let j(ν) bethe column number of the ζ-box ν and for a sequence of r consecutive natural numbers1 ≤ ν +1, ν +2, . . . , ν +r ≤ N−1 let j = (j1, . . . , jr) ≡ (j(ν +1), . . . , j(ν +r)). Note that
1 ≤ j1 < · · · < jr ≤ λ(ζ(ν+r)1 ), if {ν +1, ν +2, . . . , ν +r} ⊂ I(ζ), i.e. ν +1, ν +2, . . . , ν +r
are consecutive P-steps in ζ, whence for k with 0 ≤ k ≤ r − 1 and fixed ζ the followingnotation is well defined:
[ r; r − k, . . . , r]ν := wJ(λ(ζ(ν+r)); jr−k, . . . , jr) (cf. (6.2)).
One uses the following rules to compute J(ζ; t):
(1) write down xBN−1x . . . B1x for ζ as in Thm.1.5;(2) if Bν = S, then substitute Bν by [1; 1]ν S;
28
(3) assume {ν + 1, ν + 2, . . . , ν + r} ⊂ I(ζ) is maximal, i.e. ν, ν + r + 1 /∈ I(ζ), thensubstitute the expression xBν+rx . . . xBν by xP(r+1)(ζ, ν), where
P(2)(ζ, ν) = [2; 2]ν S +[2; 1, 2]ν[1; 1]ν
and recursively for r ≥ 3:
P(r)(ζ, ν) =r−3∑
k=0
xk[r; r − k, . . . , r]ν SxP(r−k−1)(ζ, ν)
+ xr−2
([r; 2, . . . , r]ν S +
[r; 1, . . . , r]ν[1; 1]ν
).
Then
(J) J[λ](x; α) =∑
ζ∈SY T (λ)
J(ζ; α)(x),
where J[λ](x; t) is the sequence of the J[m]λ (x; α) :=
∑η∈SSY T[m]
wJ(η) xη with m ∈ N.
Proof. The τPx-formula for J[λ](x; α) is a weighted form of the one given for s[λ](x); weuse Thm.5.4 with ‘J ’ instead of ‘B’ to indicate that weights wJ are involved.
It is easy to see that the sequence ‘. . . x’ of step 1 translates to . . . x [1; 1]0 = . . . x wJ(1; 1)= . . . x and that wJ(ζ(ν+1); j(ν + 1)) = [1; 1]ν gives the weight for an S-step. Thereforeit is only necessary to proof the validity of step 3. Since wJ is S-insected, Prop.6.3applies and it is sufficient to show the validity of step 3 in the case of a single setMν := {ν + 1, . . . , ν + r}. Note that such a set corresponds to a unique H-strip in the
maximal identification ζ of ζ; clearly the special values of the row and column numbers ofthe boxes in this strip are not important for the translation step 3, whence it is possibleto restrict to the case of ζr ∈ SY T (r), i.e. the study of the unique SYT of shape (r).
Subsequently let [λ; j] ≡ [λ; j]0 and P(r) ≡ P(r)(ζr, 0). We have to show:
J(ζr)(x) =∑
I⊂{1,...,r−1}J I(ζr)(x)
(!)= xP(r)x .
For r = 2 the assertion is true:
J(ζ2)(x) = J{1}(ζ2)(x) + J∅(ζ2)(x) = x [2; 2] S x + x [2; 1, 2]x = x([2; 2] S +[2; 1, 2]
[1; 1])x .
Now let r ≥ 3; then
J(ζr+1)(x) =∑
I⊂{1,...,r−1}J I(ζr+1)(x) +
∑
I⊂{1,...,r−1}J{r}∪I(ζr+1)(x) .
The first summand is by induction hypothesis
x [r + 1; r + 1]S∑
I⊂{1,...,r−1}J I(ζr)(x) = x [r + 1; r + 1]S xP(r)x
29
and the second equalsr−2∑
k=0
∑
I⊂{1,...,r−k−2}J{r−k,...,r}∪I(ζr+1)(x) + J{1,...,r}(ζr+1)(x) .
For 0 ≤ k ≤ r − 3 one has∑
I⊂{1,...,r−k−2}J{r−k,...,r}∪I(ζr+1)(x) =
xk+2 [r + 1; r − k, . . . , r + 1] S∑
I⊂{1,...,r−k−2}J I(ζr−k)(x) =
xk+2 [r + 1; r − k, . . . , r + 1] Sx P(r−k−1)x
and for k = r − 2: J{2,...,r}(ζr+1)(x) = xr [r + 1; 2, . . . , r + 1]Sx. Finally
J{1,...,r}(ζr+1)(x) = xr [r + 1; 1, . . . , r][r; 1, . . . , r]
[1; 1]x
and summation gives J(ζr+1)(x) = x P(r+1)x as desired. ¤Example 6.5. For λ = 3 2 we compute the J(ζ; α)(x), which sum up to the weightedτPx-expansion of J[32](x; α). (We omit the ‘wJ ’ and the ‘ζ’ in Pr(ζ, ν).):
For 4 51 2 3
step 1.) gives xPxSxPxPx, which translates to
xP(2)(3) x[1; 1]3 Sx P(3)(0)x =
x([2; 2]3 S + [2;1,2]3
[1;1]3
)x[1; 1]3 Sx
[[3; 3]0Sx
([2; 2]0 S + [2;1,2]0
[1;1]0
)+ x
([3; 2, 3]0 S + [3;1,2,3]0
[1;1]0
)]=
x((32; 2) S + (32;1,2)
(31;1)
)x(31; 1) Sx [(3; 3)Sx ((2; 2) S + (2; 1, 2)) + x ((3; 2, 3) S + (3; 1, 2, 3))];
Similarly one computes:
wJ( 3 41 2 5
) = xP(3)(2) x[1; 1]2 Sx P(2)(0)x =
x[(32; 3)Sx
((22; 2) S + (22;1,2)
(21;1)
)+ x
((32; 2, 3) S + (32;1,2,3)
(21;1)
)]x(21; 1)Sx ((2; 2) S + (2; 1, 2));
wJ( 3 51 2 4
) = x[1; 1]4 SxP(2)(2) x[1; 1]2 Sx P(2)(0)x =
x(32; 2)Sx((31; 3) S + (31;1,3)
(21;1)
)x(21; 1)Sx ((2; 2) S + (2; 1, 2));
wJ( 2 51 3 4
) = x[1; 1]4 SxP(3)(1) x[1; 1]1 Sx =
x(3; 2, 3) Sx[(31; 3)Sx
((21; 2) S + (21;1,2)
(12;1)
)+ x
((31; 2, 3) S + (31;1,2,3)
(12;1)
)]x(12; 1)Sx;
wJ( 2 41 3 5
) = xP(2)(3) x[1; 1]3 SxP(2)(1) x[1; 1]1 Sx =
x((32; 2, 3) S + (32;2,3)
(22;2)
)x(22; 2)Sx
((21; 2) S + (21;1,2)
(12;1)
)x(12; 1)Sx.
30
Remark 6.6. The Skew Jack funtions for skew shapes λ/µ (with λ ` N, µ ` M, µ ⊂ λ,M ≤ N) are combinatorially defined by ([St2, Thm.6.3, Thm.5.8])
Jλ/µ(x; α) := jµ
∑
η∈SSY T (λ/µ)
wJ(η) xη , where jµ :=∏s∈µ
h+(s) h−(s) .
The J[λ/µ](x; α) can be computed similarly to the Schur and HL case from the J(ζ; α)(x)
of J[λ](x; α): single out all ζ ∈ SY T (λ) with λ(ζ(M)) = µ and delete everything to theright of the (N−M)th x except an ‘appropriate weight’, i.e. if M /∈ I(ζ), then use [1; 1]M ,if M ∈ I(ζ), then one has to single out from the weight P(r)(ν) (ν < M) a specific partin accordance with the length r′ < r of the remaining sequence of consecutive numbersin “I(ζ/µ)”.
As an example consider λ = 32 and µ = 2 and the J(ζ)(x) for the first three SYTin Ex.6.5: jµ = 2α2(1 + α); the second and the third SYT give xP(3)(ζ, 2) x[1; 1]2 andx[1; 1]4 SxP(2)(ζ, 2) x[1; 1]2, respectively; the first SYT yields xP(2)(ζ, 3) x[1; 1]3 Sx [1; 1]2.
Remark 6.7. In analogy to the HL case the Jack functions have a recursive structurewith respect to the Young lattice Y (compare Rem.5.11); it is only necessary to adjustthe weights appropriately.
7. Macdonald polynomials and ‘Super-orthogonality’
Macdonald functions Qλ(x; q, t) (cf. [M3]) are two-parameter extensions of the pre-vious functions, i.e. they include Schur functions for q = t, HL functions for q = 0,and Jack functions for q = tα (α ∈ R, α > 0) and t → 1 (and also the λ′-product ofelementary symmetric functions for q = 1 and the monomial symmetric functions fort = 1). In the Macdonald case there are functions Pλ(x; q, t), too, which are related tothe Qλ(x; q, t) by a factor bλ(q, t) ([M3, (4.12), (5.9)]). Moreover Macdonald polynomi-als appear as singular vectors in representations of quantum deformations of both theVirasoro algebra ([AKOS1, AKOS2]) and the sln ([K2]). The Macdonald polynomialsdiscussed in this section are in fact ‘case A’ specializations of general orthogonal poly-nomials associated to root systems of irreducible Weyl groups ([M4]), which have beenextensively studied and further generalized for example by A.A. Kirillov Jr. ([K1, K2])and I. Cherednik ([C1, C2]).
Our starting point is the combinatorial definition of Macdonald functions2 given in[M3, (4.10-11), (5.10-13)], which is of the form (0.5). For every η ∈ SSY T (λ) the weightϕ(η) ≡ wM(η) ∈ Q(q, t) can be computed completely as in the Jack case (6.1-5), exceptthat the analog of (6.1) now reads:
(7.1) wM(λ| j) :=∏
s=(i,j)∈λj∈{j}
1− qaλ(s)tlλ(s)+1
1− qaλ(s)+1tlλ(s).
2The author is indebted to I.G. Macdonald for explaining him in a letter some intricacies of the paper[M3].
31
Therefore wM is S-insected and ‘signed diagrams’, Thm.6.4, Ex.6.5 and Rem.6.6-7 applyas in the Jack case: simply substitute wJ by wM .
We finish our short treatment of Macdonald functions by proving from the combina-torial definitions:
Qλ(x; 0, t) = Qλ(x; t) and(a)
limt→1
Qλ(x; tα, t) =1
h−λ (λ)Jλ(x; α) ,(b)
where h−λ (S) :=∏
s∈S h−λ (s) for S ⊂ N× N.
Proof. a) Set q = 0, fix η ∈ G(λ), ν ∈ N, j ∈ {j(η)[ν]} and let λ ≡ λ(η(ν)); then
wM(λ|j) =∏
s=(i,j)∈λj∈{j}
(1− qaλ(s)tlλ(s)+1
),
i.e. only (i, j) ∈ λ with armlength a(i, j) = 0 contribute to the product. Assume(i, j), (i−1, j), . . . , (i−r, j) ∈ η(ν) have all armlength zero relative to λ and (i+1, j), (i−r−1, j) /∈ η(ν), then the boxes (i, j), (i−1, j), . . . , (i−r, j) contribute a factor (1−t) . . . (1−tr+1) to the numerator of wM(η, ν). In case of j + 1 ∈ {j(η)[ν]} ⇐⇒ (i, j) /∈ RH(η[ν]) itfollows that (i−1, j), . . . , (i−r−1, j) are boxes with armlength zero relative to λ(η(ν−1))and numerator and denominator of wM(η, ν) cancel completely; if on the other handj + 1 /∈ {j(η)[ν]} ⇐⇒ (i, j) ∈ RH(η[ν]), then a(i− r − 1, j) 6= 0 relative to λ(η(ν−1)) and
there will ‘survive’ a factor (1− tr+1) = (1− tmj(η(ν))) as desired.
b) Let
bλ(s) ≡ bλ(s; q, t) :=1− qaλ(s)tlλ(s)+1
1− qaλ(s)+1tlλ(s), if s ∈ λ, and = 1 otherwise.
Clearly limt→1 bλ(s; tα, t) = h+
λ (s)/h−λ (s).Now fix η ∈ SSY T (λ) with maximal entry r. Using the notations
h+λ (S) :=
∏s∈S
h+λ (s) , bλ(S) :=
∏s∈S
bλ(s) for S ⊂ N× N and
Cν ≡ Cν(η) := {(i, j) ∈ η(ν) | j ∈ {j(η[ν])} } , Cν ≡ Cν(η) := {(i, j) ∈ η(ν) | j /∈ {j(η[ν])} }32
one computes:
limt→1
wM(η)|q=tα = limt→1
r∏ν=1
bη(ν)(Cν)
bη(ν−1)(Cν)
=r∏
ν=1
h+η(ν)(Cν)
h−η(ν)(Cν)
·h−
η(ν−1)(Cν)
h+η(ν−1)(Cν)
=r∏
ν=1
h+η(ν)(Cν) h−
η(ν)(Cν)
h−η(ν)(η(ν))
·h−
η(ν−1)(η(ν))
h+η(ν−1)(Cν) h−
η(ν−1)(Cν)
=r∏
ν=1
h−η(ν−1)(η
(ν−1))
h−η(ν)(η(ν))
·r∏
ν=1
wJ(η, ν) =1
h−λ (λ)wJ(η) ,
which implies the desired result. ¤
Definition 7.1. The weighted symmetric functions {λ}w(x) resp. the weight w is calledsuper-orthogonal, if for all partitions λ there exist factors cλ in the ring containing theweight such that
{λ}(n)w (x) = cλ x1 . . . xn {λ−}(n)
w ,
where n = l(λ) and λ− := λ1 − 1, . . . λn − 1.
Observing that the deletion of the first column defines a bijection from SSY T(n)(λ)to SSY T(n)(λ
−) it is not hard to see that Schur and HL polynomials are are super-orthogonal with all cλ = 1; using signed diagrams one similarly verifies super-orthogonalityin the Jack case with factors cλ(α) ∈ Z[α] as given in [St2, Prop.5.5]; and in case of Mac-donald functions Pλ(x; q, t) one has again all factors cλ = 1 (cf. [M3, (5.8)]). In factthis is a nearly complete list of super-orthogonal symmetric functions as S.V. Kerov hasshown using the theory of orthogonal polynomials:
Theorem 7.2. ([Ke, Thm.2]) Suppose the weighted symmetric functions {λ}w are super-orthogonal for n = 2, then these functions are (specializations of) Macdonald, Jack orslightly generalized Jack functions.
Appendix
Let N ∈ N and λ ≡ λ1 . . . λs with λ1 ≥ · · · ≥ λs ≥ 1 be a partition of N : λ ` N .Alternatively λ can be written as λ = 1m1(λ)2m2(λ) . . . , where mj(λ) denotes the numberof parts of λ with size j. Depending on the context the symbol λ will denote a partition,its (english style) Ferrer diagram, or a mapping, which associates in an obvious way apartition to an object. As usual let s = l(λ) be the lenght of λ, λ′ the conjugate of λ,λ/µ for µ ⊂ λ the skew shape of ‘λ without µ’, and for λ, µ ` N : µ ≤ λ :⇐⇒ ∀j :∑j
i=1 µi ≤∑j
i=1 λi the dominance order (cf. [M]).33
FN : = {ρ = (ρ1, ρ2, ρ3, . . . ) | ∀ν ∈ N : ρν ∈ N0 := N ∪ {0} , |ρ| :=∑∞
ν=1 ρν = N }is the set of all finite sequences of non-negative integers with sum of components= N (usually the sequence of end zeroes will be omitted);
F(m)N : = {ρ ∈ FN | ∀ν > m : ρν = 0 };
F[m]N : = {ρ ∈ FN | ∀ν > m : ρν = 0 , ρm 6= 0 } = {ρ ∈ F
(m)N | ρm 6= 0 } =
F(m)N \ F
(m−1)N ;
GFN : = {ρ ∈ FN | ρν = 0 =⇒ ρν+1 = 0 ∀ν ∈ N } the set of gapless elements of FN ;PFN : = {ρ ∈ FN | ρ1 ≥ ρ2 ≥ . . . } can be identified with the set of partitions of
N : P (N) := {λ ` N};prG: FN −→ GFN is the mapping, which ‘projects’ every ρ ∈ FN to an element of
GFN by deleting all ‘0’ between nonzero entries (without changing their order);prP : FN −→ PFN is the mapping, which ‘projects’ every ρ ∈ FN to an element of
PFN by ordering all nonzero entries in ρ nonincreasingly into the initial part ofthe sequence;
FN(µ): = {ρ ∈ FN | λ(ρ) ≡ λ(prP ρ) = µ } for some given µ ` N .
SSY T (λ) denotes the set of semistandard or columnstrict Young tableaux of shape λ,i.e. η ∈ SSY T (λ) is a numbering of the boxes of the Ferrer diagram λ with naturalnumbers, such that the numbers increase weakly in the rows from left to right andincrease strictly in the columns from top to bottom; then the content ρ ≡ ρ(η) of η isdefined as the sequence (ρ1, ρ2, . . . ) ∈ FN , where ρν is the multiplicity of the value ν inη.
SSY T(m)(λ): = {η ∈ SSY T (λ) | ρ(η) ∈ F(m)N };
SSY T[m](λ): = {η ∈ SSY T (λ) | ρ(η) ∈ F[m]N } = SSY T(m)(λ) \ SSY T(m−1)(λ);
SSY T (λ, ρ): = {η ∈ SSY T (λ) | ρ(η) = ρ }, where ‘ρ(η) = ρ’ should be understandas ‘the content of η is equal to the given ρ ∈ FN ’;
SSY T (λ, µ): = {η ∈ SSY T (λ) | λ(ρ(η)) = µ }, where ‘λ, µ ` N ’ and λ(ρ(η)) = µshould be read: ‘the partition prP ρ(η) is equal to the given µ ` N ’; clearly:SSY T (λ, µ) =
⊎ρ∈FN (µ) SSY T (λ, ρ), where
⊎stands for ‘disjoint union’;
G(λ): = {η ∈ SSY T (λ) | ρ(η) ∈ GFN } is the set of gapless SSY T (λ); G(λ, ρ) :=SSY T (λ, ρ) ∩G(λ), G(λ, µ) := SSY T (λ, µ) ∩G(λ);
P (λ): = {η ∈ SSY T (λ) | ρ(η) ∈ PFN } is the set of SSY T (λ) with partitionlikecontent; P (λ, µ) := SSY T (λ, µ) ∩ P (λ); the unique element of P (λ, λ) is calledthe superstandard tableaux;
prG: SSY T (λ) −→ G(λ) is the mapping, which ‘projects’ every η ∈ SSY T (λ) toan element of G(λ): let ρν1 , ρν2 , . . . with ν1 < ν2 < . . . be the subsequence ofnonzero entries in ρ(η), then prG(η) is the same as η, but with 1 instead of ν1, 2instead of ν2, etc. ;
prP : SSY T (λ) −→ P (λ) analogous to prG;SY T (λ): = G(λ, 1N) = P (λ, 1N) = SSY T[N ](λ, 1N) = SSY T(N)(λ, 1N) is the set of
standard Young tableaux of shape λ, i.e. the subset of all η ∈ SSY T (λ), whichtake every number from {1, . . . , N} exactly once.
34
η(ν): for some η ∈ SSY T (λ) is the sub-SSYT of η, which containes exactly theboxes with entries ≤ ν ;
η[ν]: = η(ν)− η(ν−1) is called the horizontal strip or H-strip of ν-boxes in η, becauseit contains at most one box per column (by the columnstrictness of SSYT);
η[ν,ν−k]: = η(ν) − η(ν−k−1) for 1 ≤ k ≤ ν − 2 is a multistrip; this includes as specialcase the double strip η[ν,ν−1].
The no(ta)tions for skew tableaux SSY T (λ/µ) are analogous.
Let Y denote the (distributive) Young lattice of all partitions ordered by inclusionof Ferrer diagrams with bottom element ∅ and rank function rk : Y −→ N0 given byrk(λ) := |λ|. We are interested in chains µ = λ(0) ⊂ λ(1) ⊂ · · · ⊂ λ(r) = λ, where inevery step is added at most one box per column, i.e. the growth of diagrams proceedsby adding horizontal stripes or H-stripes; these chains or multichains, i.e. chains withrepeated elements, in Y are called H-(multi)chains. In case of µ = ∅ one has a bijectionbetween the set of all H-multichains from ∅ to λ and SSY T (λ): all boxes added inone step ‘λ(ν−1) ⊂ λ(ν)’ are numbered with ‘ν’ in the corresponding η ∈ SSY T (λ) andconversely the sequence of shapes λ(η(ν)) is clearly a H-multichain. Similarly there arethe bijections: ‘H-chains←→ G(λ)’ and ‘saturated H-chains ( i.e. H-chains with minimalsteps) ←→ SY T (λ)’.
For ∅ 6= µ ⊂ λ the skew analogues are obtained. Using vertical instead of horizontalstripes gives essentially only SSY T (λ′), but the use of ‘rim hooks’ instead of H-stripesfor example yields important other informations: compare the ‘Murnaghan-NakayamaRule’ ([K,S]).
Let η[ν] be the H-strip of ν-boxes for some η ∈ SSY T (λ) as above, then the setH(η[ν]) of horizontal or H-components of η[ν] contains all subsets of boxes of η[ν], whichare horizontally connected, i.e. not separated by empty columns. Furthermore H(η) :=⊎
ν≥1 H(η[ν]) is the set of all H-components of η and RH(η) ≡ ⊎ν≥1 RH(η[ν]) the set of
all rightmost boxes in the H-components of η.Of course one can define similarly the sets: LH(η) of leftmost boxes in the H-components
of η, V (η[ν]) of vertical or V-components of η[ν] (no separation by empty rows), and C(η[ν])of complete or C-components of η[ν] (no separation by empty columns and rows), etc. .
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Institut fur Reine und Angewandte Mathematik, RWTH Aachen, D-52056 Aachen,Germany, papers/preprints: http://www.iram.rwth-aachen.de/ ∼winkel/
E-mail address: [email protected]
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