Non-commutative standard polynomials applied to matrices

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Non-commutative standard polynomials applied to matrices

Denis Serre

To cite this version:

Denis Serre. Non-commutative standard polynomials applied to matrices. Linear Algebra and its Applications, Elsevier, 2016, 490, pp.202 - 223. �10.1016/j.laa.2015.11.003�. �ensl-01402364�

Non-commutative standard polynomials applied to matrices

Denis Serre^∗ November 3, 2015

Abstract

The Amitsur–Levitski Theorem tells us that the standard polynomial in 2nnon-commuting indeterminates vanishes identically over the matrix algebraMn(K). ForK=RorCand 2≤r≤ 2n−1, we investigate how bigSr(A₁, . . . ,Ar)can be whenA₁, . . . ,Arbelong to the unit ball. We privilegiate the Frobenius norm, for which the caser=2 was solved recently by several authors.

Our main result is a closed formula for the expectation of the square norm. We also describe the image of the unit ball whenr=2 or 3 andn=2.

MSC classification : 15A24, 15A27, 15A60

Key words : standard polynomial, random matrices.

1 The problem. First results

Let r≥2 be an integer. The standard polynomial in r non-commuting indeterminates x₁, . . . ,x_r is defined as usual by

S_r(x₁, . . . ,x_r):=

∑

whereS_r is the symmetric group inr symbols and εis the signature. Each monomial is a word in the letters x_j, affected by a sign ±1. Despite its superficial similarity with the determinant ofr×r matrices, S_r is a completely different object: on the one hand, its arguments are non-commuting indeterminates, on the other hand, there are onlyrindeterminates instead of ther²entries of a matrix.

We list here elementary properties ofS_r :

∗UMPA, UMR CNRS–ENS Lyon # 5669. ´Ecole Normale Sup´erieure de Lyon, 46, all´ee d’Italie, F–69364 Lyon, cedex 07.

1. Sris alternating.

2. Sr+1(x₁, . . . ,x_r+1) =∑_i(−1)ⁱ⁺¹x_iS_r(xˆ_i), where ˆx_i:= (x₁, . . . ,x_i−1,x_i+1, . . . ,x_r+1).

3. Ifr is even, and anx_i commutes with all otherx_j’s, then S_r(x₁, . . . ,x_r) =0. Mind that this is false ifris odd.

The first polynomial x₁x₂−x₂x₁ of the list is the commutator. When applied to the elements of an algebraA, it leads us to distinguish between commutative and non-commutative algebras. More generally, the polynomialsS_rmeasure somehow the degree of non-commutatitivity of a given algebra.

A classical theorem tells us that for a given matrixA∈M_n(C), the commutatorS2vanishes identically over the algebrahA,A^∗i(in other words,Ais normal) if and only ifAis unitarily diagonalizable. It is less known thatS<sub>2`</sub>vanishes identically over the algebrahA,A<sup>∗</sup>iif and only ifAis unitarily blockwise diagonalizable, where the diagonal blocks have at most size`×`; see Exercise 324 in [10].

In addition, we have the theorem of Amitsur and Levitski [2], of which an elegant proof has been given by Rosset [8].

Theorem 1.1 (Amitsur–Levitski.) Let K be a field (a commutative one, needless to say). The stan- dard polynomialS_2nof degree2n vanishes identically overM_n(K). However the standard polynomi- als of degree less than2n do not vanish identically.

In the sequel, we focus on the algebraM_n(K)(K=RorC) of real or complex matrices. A norm over M_n(K) is submultiplicative if it satisfies kABk ≤ kAk kBk. The main examples are operator norms

kAk:= sup

x∈Kⁿ,x6=0

|Ax|

|x| ,

where| · |is a given norm overKⁿ. One often says thatk · kisinducedby| · |. In particular, k · k₂ is the norm induced by the standard Euclidian/Hermitian norm. We are also interested in the Frobenius norm

kAk_F :=r

∑

|a_{i j}|²,

which is not induced, becausekI_nk_F =√

n>1. Nevertheless, it is submultiplicative. A more exotic norm is thenumerical radius

h(A):=sup

|x^∗Ax|

kxk²₂ : x∈Cⁿ\ {0}

.

This is not a submultiplicative norm but it satisfiesh(A^k)≤h(A)^k. The numerical radius is therefore asuper-stable norm.

The general question that we address is to find precise bounds of kS_r(A₁, . . . ,A_r)k

∏^r_j=1kA_jk .

This can be cast into two sub-problems. On the one hand, we are interested in the best constant C=C(r,n)satisfying

kS_r(A₁, . . . ,A_r)k ≤C

r

∏

kA_jk, ∀A₁, . . . ,A_r∈M_n(K).

On the other hand, we may ask what is a typical ratio.

For the first task, we look for the smallest ball containing the image when each argument is taken out of the unit ball ofM_n(K). We may even ask for an accurate description of this image. For instance, we shall prove that for 2×2 matrices (i.e. n=2) and the Frobenius norm, the image of the unit ball underS₂(the commutator) is the ball of radius√

2, while the image underS₃is an ellipsoid. In order to tackle the second problem, we shall compute the closed form of

E(kS_r(A₁, . . . ,A_r)k²_F) E(∏^r_j=1kA_jk²_F) ,

whenA₁, . . . ,A_rare chosen independently and uniformly in the Frobenius unit ball. Strangely enough,

our proof makes use of the Amitsur–Levitski Theorem.

When using a submultiplicative norm, the trivial bound kS_r(A₁, . . . ,A_r)k ≤r!

r

∏

kA_jk, ∀A₁, . . . ,A_r∈M_n(K), suggests to work with the normalized polynomial

T_r:= 1 r!S_r, which now satisfies

kT_r(A₁, . . . ,A_r)k ≤

r

∏

kA_jk, ∀A₁, . . . ,A_r∈M_n(K).

However this inequality ignores the cancellations that are likely to occur because of the signsε(σ)in the definition of Tr. For instance, the left-hand side vanishes wheneverr≥2n. For this reason, we are interested in thenormτ(r,n)ofT_r, defined by

τ(r,n):=sup

(kT_r(A₁, . . . ,A_r)k

∏^r_j=1kA_jk : A₁, . . . ,A_r ∈M_n(K)\ {0_n} )

.

Alternatively,

τ(r,n):=sup{kT_r(A₁, . . . ,A_r)k :A₁, . . . ,A_r∈M_n(K),kA₁k, . . . ,kA_rk ≤1}=C(r,n) r! .

Our definition above depends on the chosen norm on M_n(K), and our notation should indicate this dependance. In this sense, the Frobenius norm and the operator normk · k₂yield the numbersτ_F(r,n) andτ₂(r,n), respectively.

As said above, we always have τ(r,n)≤1. However this bound is very poor as for instance τ(2n,n) =0. Besides, one trivially has τ(1,n) =1. The first non-trivial case comes when r=2, whereS₂ is the commutator. Whenn=1, clearlyτ(2,1) =0. But forn≥2, the result depends on the norm we are considering. For instance, we know that

τF(2,n) =

√2 2 ,

the casen=2 being due to B¨ottcher and Wenzel [3], and the casen=3 to L´aszl´o [6]. The equality for everynwas conjectured in [3] and proved by [9], and independently by [7] forK=Rand [1] for K=C. See also [4]. The situation is significantly different with the standard operator norm, induced by the Hermitian norm. It is known ([4], Example 5.2) that

τ₂(2,n) =1, ∀n≥2.

All the subtlety in the upcoming analysis comes from cancellations. We shall prove several in- equalities that hold true for every submultiplicative norm. The simplest one,τ(r+s,n)≤τ(r,n)τ(s,n), is not sharp. But it suggests to extend our study to operators in infinite dimension. An interesting class in this respect is that of Hilbert–Schmidt operators, whose norm generalizes the Frobenius norm. We thus define the upper bound

τ_F(r):=sup

n≥1

τ_F(r,n) = lim

n→+∞τ_F(r,n).

Again, one hasτ_F(r+s)≤τ_F(r)τ_F(s). Since Theorem 1.1 is lost asn→+∞, we haveτ_F(r)>0 for everyr, and it becomes interesting to compute therate of cancellation

ρF := lim

r→+∞τF(r)^1/r =inf

r≥1τF(r)^1/r. Because of B¨ottcher–Wenzel’s inequality, we haveτ_F(2) =√

2/2 andρ_F ≤2^−1/4. One of our results from below is the improved bound

ρ_F ≤τ_F(2k+1)^1/2k, ∀k≥1.

2 The commutator in terms of the numerical radius

For the following result concerning the numerical radiush, we benefited of a fruitful disussion with Piotr Migdal.

Theorem 2.1 Let n≥2be given. Then

(1) h([A,B])≤4h(A)h(B), ∀A,B∈M_n(C).

The constant4is the smallest possible.

Proof IfM∈M_n(C), we define therealandimaginary partsofM by ℜM=1

2(M+M^∗)∈Hn ℑM= 1

2i(M−M^∗).

Then

h(M) = sup

x

sup

ϕ

ℜ(e^iϕx^∗Mx) kxk²₂ =sup

x

sup

ϕ

x^∗ℜ(e^iϕM)x kxk²₂

= sup

ϕ

sup

x

x^∗ℜ(e^iϕM)x

kxk²₂ = sup

ϕ∈R/2πZ

kℜ(e^−iϕM)k₂,

where the supremum is actually a maximum, becauseϕ7→ kℜ(e^−iϕM)k₂is continuous over the com- pact setR/2πZ.

LetA,B∈M_n(C)be given. We writeM= [A,B]and chose aϕwithh(M) =kℜ(e^−iϕM)k₂. Let us denoteX=ℜ(e^−iϕA),Y =ℑ(e^−iϕA),Z=ℜBandT =ℑB. From

e^−iϕM= [X+iY,Z+iT] = [X,Z] + [T,Y] +i([X,T] + [Y,Z]) and the fact that the commutator of Hermitian matrices is skew-Hermitian, we derive

ℜ(e^−iϕM) =i([X,T] + [Y,Z]).

We infer

h([A,B]) =k[X,T] + [Y,Z]k₂≤ k[X,T]k₂+k[Y,Z]k₂≤2(kXk₂· kTk₂+kYk₂· kZk₂).

This proves immediately (1).

The constant 4 is attained for the choice A=

0 1 0 0

⊕I_n−2, B= 0 0

1 0

⊕I_n−2, for which

[A,B] =

1 0 0 −1

⊕I_n−2. We have

h(A) =h(B) = 1

2, h([A,B]) =1.

This completes the proof.

Inequality (1) is of little interest; a similar proof yields the same optimal factor 4 if one replace the commutator byAB+BA:

h(AB+BA)≤4h(A)h(B).

Consequently the numerical radius does not detect the possible cancellations, in contrast to the Frobe- nius norm.

When considering polynomials of higher degree, the same proof provides the inequality h(Sr(A₁, . . . ,A_r))≤2^r−1r!

r

∏

h(A_j), which seems to be very far from optimal.

3 General inequalities

We assume for the remainder that the norm is submultiplicative:

kABk ≤ kAk · kBk.

3.1 Two formulæ about S_r

It is well-known that

S_r(x₁, . . . ,x_r) =

r i=1

∑

(−1)ⁱ⁻¹x_iS_r−1(xˆ_i),

where ˆx_iis obtained fromxby deletingx_i. The following formula generalizes the one above.

Proposition 3.1 Let r =r₁+· · ·+r_{`</sub> be a partition into positive integers (k 7→r<sub>k</sub> need not to be monotonous). Denote byP(r<sub>1</sub>, . . . ,r<sub>`})the set of ordered partitions of[[1,r]] ={1, . . . ,r}into subsets I₁, . . . ,I_{`</sub>such that|I<sub>k</sub>|=r<sub>k</sub>. If I= (I<sub>1</sub>, . . . ,I<sub>`})∈P(r₁, . . . ,r_`), and I_k={i_k,1<· · ·<i_k,rk}, let us denote α(I)the signature of the permutationρ_I defined by

ρ_I(j+r₁+· · ·+r_k−1) =i_k,j, 1≤ j≤r_k,1≤k≤`.

Then

(2) S_r(x₁, . . . ,x_r) =

∑

I∈P(r1,...,r`)

α(I)S_r1(X_I1)· · ·S_{r`</sub>(X<sub>I`}), where X_Ik = (x_k,1, . . . ,x_k,r(k)).

Proof Everyσ∈S_r can be factorized as

σ= (σ₁× · · · ×σ_`)◦ρ_I

withI ∈P(r₁, . . . ,r_`)andσ_k ∈S_rk. For this, takeI₁=σ({1, . . . ,r₁}), etc... This decomposition is unique; actually, one has

|S_r|=r!=r₁!· · ·r_`! r

r₁

r−r₁ r₂

· · · r_`

r_`

=|S_r1× · · · ×S_{r`</sub>| ×cardP(r<sub>1</sub>, . . . ,r<sub>`}).

The signature ofσis obviously the product of all the signatures ofσ1, . . . ,σ`,ρ_I. Therefore the right- hand side of (2) contains exactly once the monomialx_σ(1)· · ·x_σ(r), with the sign

α(I)ε(σ₁)· · ·ε(σ_`) =ε(σ).

Hence both sides are equal to each other. This proves the proposition.

Let 1≤k<m≤` be given, and denote ς the transposition (k,m). The ordered partition I^ς is defined by I_k^ς=I_m, I_m^ς =I_k and I^ς_j =I_j otherwise; mind that r_k and r_m have been flipped. We have α(I^ς) = (−1)^rkrmα(I). Hence we derive from (2) the following identities.

Proposition 3.2 Let r=r₁+· · ·+r_`. Then,

(3)

∑

I∈P(r1,...,r`)

α(I)S_{`</sub>(S<sub>r1</sub>(X<sub>I1</sub>), . . . ,S<sub>r`}(X_I`)) =

`!S_r(X), if every r_k is odd,

0, otherwise.

The caser=3=2+1 (hence`=2) of (3) is the Jacobi identity for the commutator.

Proof Let F(X) denote the left-hand side. The transposition i↔ j induces an involution S over P(r₁, . . . ,r_`)and we haveα(SI) =−α(I). Therefore

F(x₁, . . . ,x_i−1,x_j,x_i+1, . . . ,x_j−1,x_i,x_j+1, . . . ,x_r) =−F(X).

SinceFis a homogenenous polynomial, linear in each of the indeterminates, we deduce thatFequals a multiple ofS_r.

In order to determine the constant factor, we may focus on the monomial X^↑=x₁x₂· · ·x_r. This monomial occurs in the sum each time the interval[[1,r]]is split into consecutive intervals I_ν(1), . . . of respective lengthsr_ν(1), where ν∈S_` is arbitrary. It is accompanied by the signα(I_ν(1), . . .)ε(ν).

If all the r_k are odd, then α(I_ν(1), . . .)ε(ν) = +1 for every ν and all the terms have a positive sign, whence the factor`!.

Suppose on the contrary that somer_k is even, sayr₁, and let us denoteςthe transposition 1↔2.

ThenS_{`</sub>is the disjoint union ofA<sub>`}andςA_{`</sub>. Ifν∈A<sub>`}, then α(I_ςν(1), . . .) =α(I_ν(1), . . .),

whileε(ςν) =−ε(ν). Therefore half of the occurences ofX^↑have a positive sign, and half of them have a negative sign, whence the second formula and the proposition.

3.2 Submultiplicativity of τ(·,n)

The identity (2) allows us to derive a bound forτ(r,n)in terms ofτ(s,n)andτ(r−s,n). IfA₁, . . . ,A_r are in the unit ball, then each product in the sum above is bounded by

s!(r−s)!τ(s,n)τ(r−s,n)

r

∏

kA_jk.

Since there arer!/(s!(r−s)!)terms in the sum, we immediately obtain kTr(A₁, . . . ,A_r)k ≤τ(s,n)τ(r−s,n)

r

∏

kA_jk,

from which we derive the following estimate.

Proposition 3.3 If the norm is submultiplicative, then

τ(r,n)≤τ(s,n)τ(r−s,n), ∀1≤s<r.

In particular,τ_F(r)≤τ_F(s)τ_F(r−s). It is well known that for such a submultiplicative sequence, the sequenceu_r:=τ_F(r)^1/r converges to its lower bound. We call this limit therate of cancellation and denote it by

ρ_F = lim

r→+∞τ_F(r)^1/r=inf

r τ_F(r)^1/r. Because ofτ(r,n)≤1, we infer the next statement.

Corollary 3.1 As a function of its first argument r,τ(r,n)is non-increasing.

3.3 Improved submultiplicativity

The formula (2) has the drawback that it still involves the product of matrices, for which we have no gain in norm, since we cannot improvekABk ≤ kAk kBkin general. To go further, we use (3), which involves operatorsT`but no single matrix product. It can be recast as

(4)

∑

I∈P(r₁,...,r_`)

α(I)T`(Tr<sub>1</sub>(X<sub>I1</sub>), . . . ,Tr<sub>`</sub>(X_I`)) =

cardP(r₁, . . . ,r_`)·Tr(X), if everyr_kis odd,

0, otherwise.

For instance, we have

8T₄(A,B,C,D) = [A,T₃(B,C,D)] + [B,T₃(A,D,C)]

+[C,T₃(A,B,D)] + [D,T₃(A,C,B)], an identity which immediately gives

τ(4,n)≤τ(3,n)τ(2,n).

The latter inequality is tighter than τ(4,n)≤τ(2,n)² given by Proposition 3.3, since r7→τ(r,n) is non-increasing.

More generally, we have

(5) 4rT_2r(A₁, . . . ,A_2r) =

2r

∑

i=1

(−1)ⁱ⁺¹[A_i,T_2r−1(Aˆ_i)],

with the classical notation ˆA_ifor the listA₁, . . . ,A_i−1,A_i+1, . . . ,A_r in whichA_i has been omitted. We deduce immediately

τ(2r,n)≤τ(2r−1,n)τ(2,n).

Using again submultiplicativity, this yields

τ(2r,n)≤τ(2,n)τ(3,n)τ(2r−4,n).

By induction, we infer

τ(4k,n)≤τ(2,n)^kτ(3,n)^k, τ(4k+2,n)≤τ(2,n)^k+1τ(3,n)^k, where submultiplicativity alone only grantsτ(2`,n)≤τ(2,n)<sup>`</sup>.

More generally, (4) yields the inequality

(6) τ(r,n)≤τ(`,n)τ(r<sub>1</sub>,n)· · ·τ(r<sub>`</sub>,n).

This improves the submultiplicativity in the following way: let us define a new sequenceθby shifting the first argument

θ(s,n):=τ(s+1,n).

Then (6) becomes

(7) θ(s,n)≤θ(s₀,n)θ(s₁,n)· · ·θ(s_`,n),

fors=s₀+· · ·+s_{`</sub>, whenevers<sub>1</sub>, . . . ,s<sub>`}are even (s₀may be odd). This is exactly submultiplicativity, up to the restriction on parity. In particular, the sequenceµ(k,n):=θ(2k,n)is submultiplicative.

Likewise, let us denoteθ(s) =τ_F(s+1). Clearly,τ_F(r)^1/r andθ(s)^1/shave the same limit, which must be equal to the infimum of theθ(2k)^1/2k. Combined, this delivers the following result.

Proposition 3.4 We have

ρ_F ≤τ_F(2k+1)^1/2k, ∀k∈N.

The above bound improves, for odd arguments, the one we had before, namelyτ(r)^1/r.

4 The Frobenius norm

The Frobenius norm, which it is the one studied in [3], has several advantages for our study. First of all, it enjoys better bounds than either the operator norm, or the numerical radius: the limitn→+∞

seems non trivial. Next, it is a smooth, regular norm, and it is possible to use differential calculus when studyingr-uplets which realize the norm of T_r. At last, we have a duality principle, based on the inner product of two matrices

kMk_F =sup{ℜTr(M^∗N) :kNk_F ≤1}, whence

k[A,B]k_F =sup{ℜTr([A,B]C) :C∈B_F},

whereB_F denotes the unit ball for the Frobenius norm. Since 3Tr([A,B]C) =TrS3(A,B,C), we also have (say thatK=R)

τ_F(2,n) =sup{TrT₃(A,B,C) : A,B,C∈B_F}.

We warn the reader that this identity extends only to the even numbersr:

τ_F(2s,n) =sup{TrT_2s+1(A₁, . . . ,A_2s+1) : A₁, . . . ,A_2s+1∈B_F}, while

TrT_2s(A₁, . . . ,A_2s)≡0.

The latter identity expresses the fact thatT2s(A₁, . . . ,A_2s)can be written as a sum of commutators, an idea developed in Section 3.3. The vanishing of these traces is used in the proof by Rosset [8] of the Amitsur–Levitski Theorem.

4.1 Average estimate

Let us endowM_n(R)with the usual probability measure, where the entriesa_{i j} are Gaussian indepen- dent variables:

dµ(A) = 1

V_ne^−kAk2da₁₁· · ·da_nn, kAk²=Tr(A^TA).

HereV_nis a normalizing factor. For instance, we have E(kAk²) =n²

´ x²e^−x2dx

´ e^−x2dx =:n²m₂, wherem₂=¹₂ is the second moment of the Gaussian.

We wish to calculate the expectation of kS<sub>`</sub>(A<sup>1</sup>, . . . ,A<sup>`</sup>)k² whenA¹, . . . ,A<sup>`</sup> are independent ma- trices. This amounts to calculating the average of kS`(A¹, . . . ,A^{`</sup>)k<sup>2</sup> when A<sup>1</sup>, . . . ,A<sup>`} all have unit norm.

Lemma 4.1 As a function of the size n of the matrices, the expression n<sup>2`</sup>E[kS<sub>`</sub>(A¹, . . . ,A^`)k²]

is a polynomial.

Proof DenotingA^π =A^π(1)· · ·A^π(`), we have

E[kS`(A<sup>1</sup>, . . . ,A<sup>`</sup>)k²] = E[TrS`(A<sup>1</sup>, . . . ,A<sup>`</sup>)^TS`(A<sup>1</sup>, . . . ,A<sup>`</sup>)]

=

∑

π∈S`

E[ε(π)Tr(A^π)^TS`(A<sup>1</sup>, . . . ,A<sup>`</sup>)]

=

∑

π∈S`

E[ε(π)Tr(B^id)^TS<sub>`</sub>(B<sup>π−1(1)</sup>, . . . ,B<sup>π−1(`)</sup>)]

= `!E[Tr(B<sup>id</sup>)<sup>T</sup>S<sub>`</sub>(B¹, . . . ,B^`)]

= `!

∑

π∈S`

ε(π)E[Tr(B^id)^TB^π].

Givenπ∈S_`, we have

Tr(A^id)^TA^π=a^`_α

`−1α`· · ·a²_α

1α2a¹

β1α1a^π(1)

β1β2· · ·a^π(`)

β`α`,

with Einstein’s convention of summation over repeated indices. We stress thatβ1plays the role of an α₀, andα_{`</sub>plays the role of aβ<sub>`+1}. When taking the expectation of a given monomial, we obtain either (m₂)^`or 0, according to whether every entry shows up with a square, or not. A non-zero contribution happens when

(β_k,β_k+1) = (α_π(k)−1,α_π(k)), or equivalently if

(8) α_π(k)−1=α_π(k−1)=β_k

for everyk=1, . . . , `+1. Hereabove we have to extendπby

(9) π(0) =0, π(`+1) =`+1.

LetG^π be the graph whose vertices are the indices 0≤ j≤`for theα⁰s, and the indices 1≤k≤

`+1 for theβ<sup>0</sup>s. ThusG<sup>π</sup> has 2(`+1)vertices. The edges correspond to every equality of the form either j=π(k−1)or j=π(k)−1. This includes the edge between the vertices j=0 andk=1, and the edge between the vertices j=`andk=`+1. Notice that jandkmay be connected by two edges, in caseπ(k)−1=π(k−1) = j.

Given the permutationπ, many among the monomials a^`_α

`−1α`· · ·a²_α1α2a¹_β

1α1a^π(1)_β1β2· · ·a^π(`)

β`α`

3 3

1

4 2

1 1

2 2

3 0

3

4

0 1

2

Figure 1: The graphG^π when`=3. The α-indices, from 0 to 3, are outer; theβ-indices, from 1 to 4, are inner. Left: the cycle(123). Right: the transposition (12). In both cases, the graph has two connected components. For the identity, it should have four of them.

have zero expectation. The remaining ones have expectationm^`₂; they are parametrized by the maps G^{π (α,β)}−→[[1,n]]

that are constant on each connected component. The number of such maps (α,β) is n^N(π), where N(π)is the number of connected components ofG^π. Therefore, we obtain

E[Tr(A^id)^TA^π] =n^N(π)m^`₂ and

E[kS<sub>`</sub>(A<sup>1</sup>, . . . ,A<sup>`</sup>)k²] =`!m<sup>`</sup>₂

∑

π∈S`

ε(π)n^N(π)=:`!P<sub>`</sub>(n)m<sup>`</sup><sub>2</sub>, for someP<sub>`</sub>∈Z[X]. We infer

(10) E[kS_{`</sub>(A<sup>1</sup>, . . . ,A<sup>`</sup>)k²] =`!P<sub>`}(n)

n^{2`</sup> E[kA<sup>1</sup>k<sup>2</sup>· · · kA<sup>`}k²].

We are now going to express the polynomialP_{`</sub>in closed form. To begin with, we note that always N(π)≥1, and thereforeP<sub>`}(0) =0. Actually, the quantityN(π)can further be restricted.

Proposition 4.1 For everyπ∈S<sub>`</sub>, we have1≤N(π)≤`+1and N(π)≡`+1 mod 2.

In addition,

(N(π) =`+1)⇐⇒(π=id).