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Character Polynomials, their q-analogsand the Kronecker product A.. This paper may also be used as a tutorial for working with character polynomials in the computation of Kronecker coeff

Character Polynomials, their q-analogs

and the Kronecker product

A M Garsia* and A Goupil **

Department of MathematicsUniversity of California, San Diego, California, USA

garsia@math.ucsd.eduD´epartement de math´ematiques et d’informatiqueUniversit´e du Qu´ebec `a Trois-Rivi`eres, Trois-Rivi`eres, Qu´ebec, Canada

alain.goupil@uqtr.ca

Submitted: Sep 22, 2008; Accepted: Jul 25, 2009; Published: Jul 31, 2009

Mathematics Subject Classifications: 20C30, 20C08, 05E05, 05A18, 05A15

Dedicated to Anders Bj¨orner on the occasion of his sixtieth birthday

AbstractThe numerical calculation of character values as well as Kroneckercoefficients can efficently be carried out by means of character polynomi-

als Yet these polynomials do not seem to have been given a proper role in

present day literature or software To show their remarkable simplicity we

give here an “umbral” version and a recursive combinatorial construction

We also show that these polynomials have a natural counterpart in the

standard Hecke algebra Hn(q ) Their relation to Kronecker products is

brought to the fore, as well as special cases and applications This paper

may also be used as a tutorial for working with character polynomials in

the computation of Kronecker coefficients

I Introduction

We recall that the value χλ

α of the irreducible Sn character indexed by a partition

λ = (λ1, , λk) at a permutation of Sn with cycle structure α = 1a12a2· · · nan is given

by the Frobenius formula

χλα = ∆(x)pα

xλ1+n−11 xλ2+n−22 ···xλn+n−nn

I.1

* Work supported by a grant from NSF

** Work partially supported by a grant from NSERC

the electronic journal of combinatorics 16(2) (2009), #R19 1

where ∆(x) = ∆(x1, , xn) and pα = pα(x1, , xn) denote respectively the monde determinant and the power sums symmetric functions The character polynomial

Vander-qµ(x1, x2, , xn) is the unique polynomial in Q[x1, x2, , xn] with the property thatfor all partitions µ ⊢ k and λ = (n − k, µ) with n − k ≥ µ1 we have

χ(n−k,µ)1a1 2 a2 n an = qµ(a1, a2, , an) I.2Moreover, with an appropriate change of sign and rearrangements of the parts of (n −

k, µ) this can be shown to remain true even when n − k < µ1 The simplest case ofequation I.2 is the well known formula

which implies that for all n ≥ 2 the value of the character χ(n−1,1) at a permutation

σ ∈ Sn is simply equal to one less than the number x1 of fixed points of σ Characterpolynomials were implicitly used for the first time in the work of Murnaghan [Mu] andwere identified as such later by Specht in [Sp] where a determinantal form for characterpolynomials and a proof of equation I.2 are given Treatments of character polynomialsvary from the purely existential as in Kerber [Ke], to the very explicit as in Macdonald’sbook ([Ma] ex I.7.13 and I.7.14) What is quite surprising, as we see in I.3, is how littleinformation from the cycle structure may be needed to compute the whole sequence ofcharacter values χ(n−k,µ) We will show here that a slight addition to the computationcarried out in [Ma] yields a formula of utmost simplicity which brings to explicit evidencethis minimal dependence on the cycle structure

To state our formula let us denote by “↓” the “umbral ” operator that transforms

a monomial into a product of lower factorial polynomials To be precise we set

↓ xa = ↓ xa1

1 xa2

2 · · · xam

m = (x1)a1(x2)a2· · · (xm)am I.4with (x)a = x(x − 1)(x − 2) · · · (x − a + 1) This given we have

Proposition I.1 For all µ ⊢ k, the character polynomial qµ depends at most onthe first k variables More precisely

a) Expand the Schur function sµ in the power sums basis : sµ =P

α⊢k

χ

z αpα.b) Replace each power sum pi by ixi− 1

c) Expand each product Q

i(ixi− 1)ai as a sumP

gcgQixgi

i d) Replace each xgi

i by (xi)gi.Let us compute the polynomial q(3)(x1, x2, x3) by using the preceding steps :a) s3 = 16(p13 + 3p(21)+ 2p(3))

b) 16(p13+ 3p(21)+ 2p(3)) → 16((x1− 1)3+ 3(2x2− 1)(x1− 1) + 2(3x3− 1))c) 16((x1− 1)3+ 3(2x2− 1)(x1− 1) + 2(3x3− 1)) = 16x31− 12x21+ x1x2− x2+ x3

d) q(3)(x1, x2, x3) = 16(x1)3− 12(x1)2+ x1x2− x2+ x3

Note that when we set x1 = n in qµ and xi = 0 for all i > 1, we obtain the number

f(n−k,µ) of standard tableaux of shape (n − k, µ) In view of the classical hook formula,this must reduce to the identity

x 1 =n

I.6

∀ µ ⊢ k & n − k ≥ µ1, where µ′ = (µ′1, µ′2, ) is the conjugate partition of µ

An immediate consequence of equation I.5 is a recursive algorithm for the struction of the character polynomials that does not directly involve any of the sym-metric group characters

con-Corollary I.1 For a given partition µ ⊢ k, let ˜µ denote the partition obtained byremoving the first part from µ Then

where the inner sum is over all (j + 1)-tuples S of partitions µr such that each µr− µr+1

is a border strip of length i and ht(S) =Pj−1

r=0(height(µr− µr+1) − 1) The initial term

qµ(x1, 0, ) is computed via equation I.6

For instance, the polynomial q(3,1,1)(x1, x2, , x5) is recursively constructed as follows

f(3,1,1)5! + x2

x1− 13



− x2



x1(x1− 1) · · · (x1− 5)(x1− 2)(x1− 3)(x1− 4)

f(3)

3!



− 2x22

(x1− 1) + x5where the last equality follows from equation I.6

This paper is organized as follows In the first section we introduce our notation,make some definitions and prove some auxiliary facts In the second section we treatthe classical Sn case, prove our umbral formula for the character polynomials as well

as Theorem I.1 In the third section, striving to make our writing accessible to awider audience, we give a brief tutorial on Kronecker products including simple proofs

of some basic results of the theory The experts in symmetric function theory mayskip this section In the fourth section we use the pairing sµ→qµ to define a degreepreserving isomorphism that sends the vector space Λ of symmetric polynomials ontothe vector space of polynomials Q[x1, x2, x3, ] We then use this map to derive somewell known and some not so well known properties of Kronecker products The study ofthis map leads to another family of polynomials that we call “set polynomials ” whichenjoy properties akin to those of character polynomials and can also be used to computeKronecker products In the fifth section we treat the Hecke algebra case and derive ourq-analogs of character polynomials We present comparative tables of character andq-character polynomials In the sixth and last section we explore some consequences ofour techniques and give some applications In particular we obtain an explicit formulayielding a generating function for the occurence of s(n)in Kronecker powers of hrhn−rforevery fixed r ≥ 1 This generating function may be viewed as a solution to a problem firstformulated by Comtet in [Co], namely the enumeration of coverings of a set of cardinality

n by sets of cardinality r The corresponding generating function for r = 2 was firstgiven by Labelle in [La] A surprisingly simple argument yields the general result and

in particular the Labelle result The calculation of character polynomials also yield

unexpected results For instance, it comes out that the polynomialPk

s=0(−1)k−sn(n −1) · · · (n−s+1) enumerates, for n ≥ 2k, the number of permutations σ ∈ Snwith longestincreasing subsequence σ(1), σ(2), · · · , σ(n − k) = n A direct proof of this makes anamusing combinatorial exercise A second unexpected outcome is a formula for the wellknown Bell numbers that does not seem to appear in the literature We terminate thepaper with the computation of certain remarkable character polynomials

Acknowledgement The authors wish to thank Alain Lascoux for his helpfulsuggestions and remarks on an earlier version of this paper We also thank the refereesfor their excellent review

1 Definitions and basic concepts

To begin it will be convenient to write a partition α of n as a weakly decreasinglist of parts α = (α1 ≥ α2 ≥ αk ≥ 1) or by giving the list of multiplicities of itsparts : α = 1a 12a 2 na n We will use greek letters λ, µ, α, for the partitions andtheir parts and the corresponding roman letters ℓi, mi, ai for their multiplicities Thenumber k of parts of a partition α = (α1, α2, , αk) is called the length of α and isdenoted ℓ(α) The weight |α| of a partition α is the sum of its parts and we extend thisconvention to any vector a = (a1, , ak) The expression zα = 1a 12a 2· · · na na1! · · · an!will be used throughout the text Thus for a partition α of n we will use the notations

α = (α1, α2, , αk) = 1a12a2 nan , α ⊢ n, |α| = kak = 1a1+ 2a2+ · · · + nan = n;

ℓ(α) = k = |a| = a1+ a2+ + an

1.1

We will need to merge partitions and for α = 1a12a2 nan, β = 1b12b2 mbm and

n ≥ m we use the operation α ∨ β = 1a 1 +b 12a 2 +b 2 na n +b n As customary if µ is apartition, µ′ will denote the conjugate partition It will also be convenient to denote by

Λ, the space of symmetric functions and by Λ=k the subspace of symmetric functionsthat are homogeneous of degree k The number of variables in a symmetric functionwill always be assumed to be greater or equal to its degree The Hall scalar product ofsymmetric functions, with respect to which the Schur functions form an orthonormalsystem, will be denoted

In this paper we shall make extensive use of plethystic notation The first authorhas been a proponent of this device since the early 1980’s after Lascoux showed thatmany identities in Macdonald’s book could acquire a remarkable simplicity in terms

of it This notwithstanding, its use doesn’t seem to have yet achieved widespreadacceptance This work will give us one more opportunity to show the power of this

the electronic journal of combinatorics 16(2) (2009), #R19 5

notational device in the theory of symmetric functions To this end we recall that theplethystic substitution of a formal power series E(t1, t2, , ts, ) into a symmetricpolynomial A, denoted “A[E]” is obtained by setting

which is the generating function of the so-called “homogeneous ” symmetric functions

hm We should note that contrary to intuition the definition in 1.2 yields that

X =P

ixi we get

pk[− −X] = (−1)k−1pk[X] = ωpk[X] 1.4where “ω” is the fundamental involution on Λ that sends the homogeneous basis onto theelementary basis In particular from 1.4 we derive that for any symmetric funtion A[X]

we have, when the alphabet X has a sufficient number of variables, ω A[X] = A[−−X].Moreover, when A is homogeneous of degree k, we see that this is equivalent to

In the same vein we show that

Proposition 1.1 For any partition λ and any two alphabets X = P

Ω[t(X − Y )] = exp X

k≥1

tkpk[X − Y ]k

k≥1

−tkpk[Y ]k

and since 1.3 gives Ω[t(X − Y )] = P

n≥0tnhn[X − Y ], we obtain 1.7 by taking thecoefficient of tn

More generally, for any formal power series E we define, the “translation by E ” ator, “TE”, on symmetric functions A[X] by

oper-TEA[X] = A[X + E]

the electronic journal of combinatorics 16(2) (2009), #R19 7

Now for any symmetric function A it is customary to denote by A⊥ the adjoint of theoperator multiplication by A, with respect to the Hall scalar product It follows fromthe definition of skew Schur functions sλ/µ that we may write sλ/µ = s⊥µsλ Thus 1.6and 1.9 may be written in the form

An interesting generalization of 1.8 is the following

Proposition 1.2 For any partition µ = (µ1, µ2 , µk) we have

with the usual convention that hm = 0 when m < 0

Proof It is sufficient to illustrate the argument in a special case So let µ = (4, 3, 1)

In this case the Jacobi-Trudi identity gives

making the substitution X → X − 1 and using 1.8, we get

The general case of equation 1.12 can clearly be established in a similar manner

2 Proofs of the umbral and recursive formulas

After this foray into plethystic magic we are ready to play with character nomials Our point of departure, as in Macdonald ( [Ma]), is the Frobenius formula,equation I.1, in the variables x0, x1, , xn which for λ = (n − k, µ) with µ ⊢ k may bewritten in the form

where for convenience we have set µr = 0 for all r > ℓ(µ) As in [Ma] (ch 1 ex 14),

we note that the homogeneity in x0, x1, , xn of the polynomial in 2.1 allows us to set

x0 = 1 and reduce this identity to

Using the first identity in 1.11, this in turn can be rewritten as

Up to this point, albeit with some slight difference of notation, we have followed donald almost verbatim To obtain our umbral formula we only need to diverge slightlyfrom Macdonald’s path To begin we use the expansion

Mac-T−1sµ = X

β⊢k β=1 b1 2 b2 ···k bk

x i =a i,Expanding the first product in the right hand side of 2.5 we obtain

x i =a i

In other words, equation 2.4 yields that if we set

qµ(x1, x2, xk) = X

β⊢k β=1 b1 2 b2 ···k bk

Thus it follows, by recursive applications of equation 2.8, that for any partition λ =(λ1, λ2, , λk) we have

s(λ1,λ2, ,λk) = Hλ1Hλ2· · · Hλk1, 2.10which is the well known “Rodrigues” formula for Schur functions (see [WW]) What wefind surprising is that equation 2.9 as well as 2.10 could be such short distance away fromthe original Frobenius formula Even more surprising would be if the present derivation

of 2.9 did not previously appear in the extensive literature on Schur functions

Remark 2.1 It is easy to see that the action of Ha on any symmetric polynomialA(X) may be given the plethystic form

HaA(X) = Ω[zX]A[X − 1/z]

z a

Under this form, the operator P

aHaza has acquired the name of “vertex operator ”,(see [CT]) Using this notation, it is not difficult to derive the commutativity relation

HaHb = −Hb−1Ha+1from which it follows that equation 2.10 remains valid even when

λ1, λ2, , λk are not in weakly decreasing order

Remark 2.2 The operation that gave us the character polynomial can be extended

to a linear map that sends symmetric polynomials onto polynomials in the infinitealphabet X = {x1, x2, x3, } More precisely for each symmetric function A of degree

k we define the map q : Λ → Q[X] by setting

q(A) = qA[x1, x2, , xk] = yQA(p1, p2, , pk)

where QA gives the power sum expansion of A Clearly, using this notation we canalso write qµ = q(sµ) We shall see in the next section that the map A→q(A) has trulyremarkable properties But for the moment we will use it only as a convenient notation

Proof of Theorem I.1

Note that since multiplication of a Schur function by the power symmetric tion pi corresponds to the addition of a border strip of size i to its index, it follows thatapplying i∂pi = p⊥i to a Schur function corresponds to the removal of such a strip Insummary, we see that the combinatorial statement of theorem I.1 is simply equivalent

func-to the following polynomial identity

x i =x i+1 =···=x k =0 2.12

Now, recalling that zθ = 1t12t2· · · ktkt1!t2! · · · tk!, we have

s=1

tis

which is obviously true.

3 Basics on the Kronecker product

Let Cα denote the formal sum of the permutations of Sn with cycle type α andlet C(Sn) denote the center of the group algebra of Sn The map Fn : C(Sn) → Λ=n

defined by setting

usually called the “Frobenius map ”, was the tool used by Frobenius to identify thecharacters of Sn In fact this map is an isometry of C(Sn) onto Λ=n each endowedwith its natural scalar product The orthonormality of the Schur basis together withits integrality with respect to the homogeneous basis {hα}α⊢n yielded Frobenius thefundamental relation

and then equation I.1 followed from the bideterminantal formula for Schur functions

The Kronecker product in Λ=n is defined by setting for P, Q ∈ Λ=n

P ∗ Q = FnF−1

n (P ) × F−1n (Q)

3.4where the symbol ‘×” here represents the pointwise product in C(Sn) In particularfrom equation 3.2 it follows that

sµ∗ sν = X

α⊢n

χµαχναpα/zα (for all µ, ν ⊢ n) 3.5The orthonormality of the Schur functions then yields the expansion

α , pβ =  zα if α = β,

The cλ,µ,ν go by the name of “Kronecker coefficients” and they may be interpreted as themultiplicity of the irreducible representation of Sn indexed by λ in the tensor product ofthe irreducible representations indexed by µ and ν A fundamental open problem is togive a combinatorial interpretation to these integers akin to the Littlewood Richardsonrule for the coefficients gλ,µ,ν occuring in the expansion

to ordinary products and thereby ultimately express the cλ,µ,ν in terms of the gλ,µ,ν

To this end it is convenient to extend the Kronecker product to all of Λ by setting

P ∗ Q = 0 for all pairs P, Q with P ∈ Λ=r, Q ∈ Λ=s and r 6= s 3.9

This given, the following basic identities are immediate consequences of the definition

of Kronecker products given in equation 3.4

Proposition 3.1

1) pα ∗ pβ = χ(α = β)zαpα, for all pairs of partitions α, β

2) pα ∗ sλ = χλαpα, for all pairs of partitions α, λ ⊢ n

3) Ω ∗ f = hn∗ f = f, for all f ∈ Λ=n

3.10

where χ(α = β) is the Kronecker delta function

Proof Since conjugacy classes are disjoint, it follows that (“×” denoting pointwiseproduct) we have

Cα× Cβ =  Cα if α = β

0 if α 6= βThus 1) follows from equations 3.1 and 3.4 when α and β are partitions of the samenumber and from equation 3.9 when they are not This given, 2) follows by linearityfrom equation 3.2 Finally, 3) follows from equation 3.9 together with 1) and the wellknown expansion

The following basic identity of Littlewood ([Li]) provides an algorithm for thecomputation of Kronecker products.

Proposition 3.2 For any homogeneous symmetric functions f1, f2, , fk of spective degrees a1, a2, , akand any symmetric function A of degree a1+a2+· · ·+ak,

Proof We need only prove equation 3.12 for fi = pβ(i) with β(i) ⊢ ai and A = sλwith

λ ⊢ a1+ · · · + ak This given, using 3.10 2), the left hand side of equation 3.12 becomes

α (1)· · · sα(k), A α(1), s⊥

α (2)· · · s⊥

α (k)A This completes our proof

Remark 3.1 Murnaghan ([Mu]) noted that for any partition µ ⊢ k, the Kroneckermultiplication operator “s(n−k,µ)∗” does not depend on n in the sense that the value of

the Kronecker coefficients c(n−|λ|,λ),(n−|µ|,µ),(n−|ν|,ν) stabilizes when n exceeds a lowerbound depending on λ, µ, ν We can easily derive this from the identities in equations2.3 and 3.13 Indeed from equation 2.3 it follows that we can write

s(n−k,µ) = X

α⊢n (n−k,µ) , pαpα/zα

where “s(r)µ [X − 1]” denotes the homogeneous component of degree r in sµ[X − 1] Thisimplies that

where the sum is over all decompositions of D as a disjoint union of k skew diagrams

µ⊢a

X

ν⊢b γ/ν , sδsµsµsν

Substituting in this the expansion

and this is simply another way of writing equation 3.16 for k = 2 We may thus proceed

by induction on k So assume equation 3.16 true for some k ≥ 2 We then proceed asbefore and let D = γ/δ with γ ⊢ a + b + c and δ ⊢ c, and get from 3.12 for k = 2, f1= haand f2 = hα = hα1hα2· · · hαk

hahα ∗ sD = X

µ⊢a

X

ν⊢b γ/δ, sµsνsµ hα∗ sν

So the induction hypothesis for α = (α1, α2, , αk) gives

this proves equation 3.16 for k + 1 and completes the induction

It is worthwhile to close this section with an example illustrating a use of theseidentities in the computation of Kronecker products Let us compute the operator U21

defined in equation 3.15 To this end note first that equation 1.12 gives

sn−3,2,1∗ sn−3,3 = sn−1,1+ 2 sn−2,2+ 2 sn−2,12+ 2 sn−3,3+ 5 sn−3,2,1+ 2 sn−3,13

+ 2 sn−4,4+ 5 sn−4,3,1+ 3 sn−4,22 + 4 sn−4,2,12 + sn−4,14 + sn−5,5+ 3 sn−5,4,1+ 3 sn−5,3,2+ 3 sn−5,3,12 + 2 sn−5,22 1 + sn−5,2,13+ sn−6,3,2,1+ sn−6,5,1+ sn−6,4,2+ sn−6,4,1,1

4 Character polynomials and the Kronecker product.

We start with the definition of a degree in Q[x1, x2, ], different from the usualdegree and we call it the “partition degree ”:

where Hk(x1, x2, , xk) is the subspace of polynomials in Q[x1, x2, ] that are geneous of partition degree k Note that polynomials of partition degree ≤ k can onlycontain the variables x1, x2, , xk It is also convenient to set

µ, qν =  1 if µ = ν,

To show that this definition is natural, we need only prove that this scalar product may

be computed without using character polynomials This is an immediate consequence

of the following basic fact

Proposition 4.1 For any polynomials P (x1, x2, , xk), Q(x1, x2, , xk) of tition degree ≤ k we have

The cλ,µ,ν go by the name of ? ?Kronecker coefficients” and they may be interpreted as themultiplicity of the irreducible... used to computeKronecker products In the fifth section we treat the Hecke algebra case and derive ourq-analogs of character polynomials We present comparative tables of character andq -character polynomials...

notational device in the theory of symmetric functions To this end we recall that theplethystic substitution