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Invariants in Non-Commutative Variables of the Symmetric and Hyperoctahedral Groups

Anouk Bergeron-Brlek

To cite this version:

Anouk Bergeron-Brlek. Invariants in Non-Commutative Variables of the Symmetric and Hyperoctahe-

dral Groups. 20th Annual International Conference on Formal Power Series and Algebraic Combina-

torics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.653-664, �10.46298/dmtcs.3609�. �hal-01185143�

Invariants in Non-Commutative Variables of the Symmetric and Hyperoctahedral Groups

Anouk Bergeron-Brlek

York University, Mathematics and Statistics, 4700 Keele Street, M3J 1P3, Toronto, Canada

Abstract.We consider the graded Hopf algebraN CSymof symmetric functions with non-commutative variables, which is analogous to the algebraSymof the ordinary symmetric functions in commutative variables. We give formulaes for the product and coproduct on some of the analogues of theSymbases and expressions for a shuffle product onN CSym. We also consider the invariants of the hyperoctahedral group in the non-commutative case and a state a few results.

R´esum´e.Nous consid´erons l’alg`ebre de Hopf gradu´eeN CSymdes fonctions sym´etriques en variables non-commu- tatives, qui est analogue `a l’alg`ebreSymdes fonctions sym´etriques en variables commutatives. Nous donnons des formules pour le produit et coproduit sur certaines des bases analogues `a celles deSym, ainsi qu’une expression pour le produitshufflesurN CSym. Nous consid´erons aussi les invariants du groupe hyperocta´edral dans le cas non-commutatif et ´enonc¸ons quelques r´esultats.

Keywords:invariants, symmetric function, non-commutative variables, Hopf algebra

1 Introduction

LetXn =x1, x2, . . . , xnbe a list of variables and denote byQhXnithe algebra of polynomials in non- commutative variables with rational coefficients. We consider the algebraQhXni^Sn of polynomials in QhXniwhich are invariant under the action of the symmetric groupSn. In Bergeronet al.(2), this algebra is extended to the graded Hopf algebraN CSymof symmetric functions in non-commutative variables (not to be confused with the algebra of non-commutative symmetric functions presented in Gelfandet al. (5)). The Hopf algebraN CSymis analogous to the algebra of the ordinary symmetric functions in commutative variables and appears in many recent works, see for instance (6; 7; 1; 3; 2). The analogues of the monomial, elementary, homogeneous and power sum bases are defined in Rosas and Sagan (10) and are indexed by set partitions.

In Section 2, we recalled some basic facts about combinatorics of set partitions. In Section 3, we consider the Hopf algebra structure of that algebra. Section 4 contains some formulaes for the product and coproduct, and expressions for a shuffle product onN CSym. One surprising thing that we found with this algebra is that all the bases defined by Rosas and Sagan (10) that are analogues multiplicative bases in the commutative algebra are also multiplicative and freely generated in the non-commutative version. We also found that the non-commutative homogeneous basis with the shuffle product is dual to

1365–8050 c2008 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

the non-commutative monomial basis with the usual coproduct, and the non-commutative monomial basis with the shuffle product is dual to the non-commutative homogeneous basis with the usual coproduct. This is very unusual and says that the bases defined by Rosas and Sagan (10) also behave naturally under the shuffle product.

Although this work is preliminary, at the end we introduce and state a few results on the invariants of the hyperoctahedral groupBn in non-commutative variablesXn. This algebra is indexed by the set partitions with at mostnparts whose size are even. This makes clear the connection with the work of Orellana (9) on the centralizer algebraEnd_Bn(V^⊗k)and the invariants ofB_n.

2 Definitions and notations

Let[n] ={1,2, . . . n}. Aset partition ofn, denoted byA`[n], is a family of disjoint nonempty subsets A1, A2, . . . , Ak ⊆[n]such thatA1∪A2∪. . .∪Ak = [n]. The subsetsAiare called thepartsofAand thelength`(A)ofAis the number of its parts. Thesizeof a set partitionAis denoted by|A|. To simplify the notation, we will write each part ofA ={A1, A₂, . . . , A_k}as a word. Moreover, the parts of every set partition will be arranged by increasing value of the smallest element in the subset.

Example. The set partitions of3are

{1,2,3} {12,3} {13,2} {1,23} {123}.

Assume that the parts of a set partitionAare listed in decreasing order of size, then toAcan be associated a partitionλ(A) = (|A1|,|A2|, . . . ,|Ak|)andA! =λ(A)! =λ1!λ2!· · ·λk!. The partitionλ(A, B)will denote the partition whoseith part is the number of partsAjsuch thatAj ⊆Bi, for1≤i≤`(B).

Example. ConsiderA={136,25,4,7}andB={12356,47}. Then λ(A) = (3,2,1,1), λ(A)! = 3!2!, λ(A, B) = (2,2).

Thesignof a permutationσ = σ(1)σ(2)· · ·σ(n)is defined bysgn(σ) = (−1)^inv(σ), whereinv(σ)is the number ofinversionsof σ, i.e. the number of pairs(σ(i), σ(j))such thati < j andσ(i) > σ(j).

Thesign(−1)^Aof a set partitionA is the sign of any permutation obtained in the following way: the cycles of the permutation will be formed by taking the integers in each part ofA. A cycle of lengthl has sign(−1)^l−1, so if the permutationσ=c₁. . . c_k ∈ S_nis written in cycle notation, thensgn(σ) = (−1)^l1−1. . .(−1)^lk−1= (−1)^n−k. Hence the sign of a set partitionA`[n]is(−1)<sup>n−`(A)</sup>.

Example. ConsiderA={1325,4}. Then

(−1)^A= (−1)⁵⁻²= sgn(35241) = (−1)^inv(35241)=−1.

GivenS⊆[`(A)]withS={s1, s2, . . . , sk}, we defineAS ={As₁, As₂, . . . , As_k}. Then thestandard- isationofAS,st(AS), is the set partition obtained by lowering the entries ofAS and keeping the values in relative order. We will also denote byA↓S the set partition A restricted to the entries which are inS andA↑S will mean raising the entries in A so that they remain in the same relative order, with the union of all parts equal toS. LetS^cdenotes thecomplementofS.

Example. ForS={1,3,4}we get

st({136,25,4,7}S) ={124,3,5}, st({136,25,4,7} ↓S) ={12,3},

{13,2} ↑_S ={14,3}.

For an arbitrary setS={s₁, s₂, . . . , s_k},S+nis defined to be{s₁+n, s₂+n, . . . , s_k+n}. Given two set partitionsA`[n]andB, we can build the set partitions

A|B ={A₁, A₂, . . . , A_{`(A)</sub>, B<sub>1</sub>+n, B<sub>2</sub>+n, . . . , B<sub>`(B)}+n}.

A set partitionA`[n]is calledsplittableif there exists non-empty set partitionsB`[k]andC`[n−k]

such thatA =B|C, and is callednon-splittableif it is non-empty and not splittable. IfAis splittable, thenA^! = (A⁽¹⁾, A⁽²⁾, . . . , A^(d)), where eachA⁽ⁱ⁾is non-splittable, will denote the split ofA. In that case,A^! =A⁽¹⁾|A⁽²⁾| · · · |A^(d).

Example. ConsiderA={13,25,4},B={13,2}andC={1}. Then A|B|C={13,25,4,68,7,9}.

There is an ordering (by refinement) on the set partitions defined as follow: We say thatA ≤ Bif and only if each part ofAis contained in some part ofB. Under this ordering, the set partitions ofnform a latticeΠn. The greatest lower bound and the least upper bound ofAandBwill be respectively denoted byA∧B={Ai∩Bj|1≤i≤`(A),1≤j≤`(B)}andA∨B. Each part inA∨Bis both a union only of parts ofAand a union only of parts ofB, with no proper subset having that property.

Example. We have

{136,25,4,7} ∧ {167,245,3}={16,3,25,4,7}, {136,25,4,7} ∨ {167,245,3}={1367,245}.

The set partition0_n ={1,2, . . . , n}is the minimal set partition and1_n ={12. . . n}is the maximal set partition. The M¨obius function ofΠnis given by

µ(A, B) =







1 ifA=B,

− X

A≤C<B

µ(A, C) otherwise.

TheKronecker deltaof two set partitions A and B is defined by δ_A,B=

(1 ifA=B, 0 otherwise.

3 Hopf algebra structure on NCSym

We will concentrate here on graded-connected Hopf algebras. These are defined as a graded algebra (H, µ)with unit, a graded coalgebra structure(H,∆)with counit, such that the coproduct is a morphism with respect to the algebra structure,i.e.forf, g∈H,τ(f⊗g) =g⊗f and

(µ⊗µ)◦(id⊗τ⊗id)◦(∆⊗∆) = ∆◦µ.

The symmetric groupS_nacts naturally onQhXniby

σf(x₁, x₂, . . . , x_n) =f(x_σ(1), x_σ(2),· · ·, x_σ(n)).

The algebraQhXni^Snof polynomials inQhXniwhich are invariant under this action can be extended to the graded algebra

N CSym=M

d≥0

N CSym_d

of symmetric functions in non-commutative variables, whereN CSym_dis the linear span of all invariants of homogeneous degreedin the infinitex₁, x₂, x₃, . . . non-commuting variables. This is done via an analogous technique to the one used in the commutative version (see (8) for technical references). The analogues of the bases of the symmetric functions in commutative variables are defined as follows. For A`[d], the monomial symmetric function in non-commutative variables is defined as

mA= X

(i1,i2,...,in)

xi₁xi₂. . . xi_d, (1)

where the sum is over all sequences(i1, i2, . . . , in)withia=ibif and only ifaandbare in the same part inA. We refer the reader to (10) for the definition of the elementary symmetric function, the complete homogeneous symmetric function and the power sum function in non-commutative variables and changes of variables. In the last section, there is a table summarizing those ones. Written in terms of the monomial symmetric function in non-commuting variables, we have:

eA=P

B∧A=0mB, hA=P

B(B∧A)!mB, pA=P

B≥AmB. Example.

m_{13,2}=x1x2x1+x2x1x2+x1x3x1+x3x1x3+x2x3x2+x3x2x3+· · · e_{13,2}=m_{1,2,3}+m_{12,3}+m_{1,23}

h_{13,2}=m_{1,2,3}+m_{12,3}+ 2m_{13,2}+m_{1,23}+ 2m_{123}

p_{13,2}=m_{13,2}+m_{123}.

In (2), they considerN CSym=⊕_d≥0N CSym_d, whereN CSym_dis the linear span of{m_A}_A`[d], as a graded Hopf algebra with product and coproduct given respectively by

µ:N CSym_d⊗N CSym_k −→N CSym_d+k, ∆ :N CSym_d→LN CSym_k⊗N CSym_d−k. µ(m_A⊗m_B) := X

C∈Πd+k C∧1d|1k=A|B

m_C ∆(m_A) = X

S⊆[`(A)]

m_AS⊗m_ASc

4 Results on NCSym

The first proposition shows that the non-commutative elementary basis is multiplicative.

Proposition 1 We have

eAeB =e_A|B.

Proof: We havep_Ap_B = p_A|B (see (1)) and forA ` [m],e_A = P

C≤Aµ(0_m, C)p_C (see Table 1).

Thus, for a set partitionB`[n],

eAeB= X

C≤A

µ(0m, C)pC

X

D≤B

µ(0n, D)pD

= X

C≤A

X

D≤B

µ(0m, C)µ(0n, D)p_C|D.

Now since{G:G≤A|B}={C|D :C|D ≤A|B} ={C|D :C ≤A|D ≤B}is isomorphic to the cartesian product{C:C≤A} × {D:D≤B}and the M¨obius functionµis multiplicative in the sense thatµ(A, C)µ(B, D) =µ(A|B, C|D), then

eAeB = X

C≤A

X

D≤B

µ(0m+n, C|D)p_C|D

= X

G≤A|B

µ(0m+n, G)pG=e_A|B.

2 From Proposition 1 we get the next corollary.

Corollary 1 N CSymis freely generated by the elementseA, whereAis non splittable.

Proof:Since thee_Aare multiplicative, we have that forA^! = (A⁽¹⁾, A⁽²⁾, . . . , A^(k)), e_A=e_A(1)|A⁽²⁾|...|A^(k)=e_A(1)e_A(2)· · ·e_A(k).

Since theeAare linearly independant, then{eA:Anon-splittable}must be algebraically independant.2 The next corollary also follows from Proposition 1 and use the involutionω : N CSym → N CSym defined in (10), that sendseAtohA, for all set partitionsAand linear extension.

Corollary 2 We have

hAhB=h_A|B.

Moreover,N CSymis freely generated by the elementshA, whereAis non-splittable.

Proof:This results from the sequence of equalities

hAhB=ω(eA)ω(eB) =ω(eA·eB) =ω(e_A|B) =h_A|B.

2 The coproduct on the non-commutative elementary basis and the non-commutative homogeneous basis is next given. See the Appendix for some examples.

Proposition 2 ForA`[n], we have

∆(eA) = X

S⊆[n]

e_st(A↓S)⊗e_st(A↓Sc).

Proof:ForA`[n],e_A=P

B∧A=0_nm_B(see Table 1). Therefore

∆(eA) = ∆

X

B∧A=0_n

mB

= X

B∧A=0_n

∆(mB) = X

B∧A=0_n

X

T⊆[`(B)]

m_st(BT)⊗m_{st(BT c)}.

Note that the parts of every set partition are arranged by increasing value of the smallest element in the subset. Let

S₁={(B, T)|B∧A=0_nandT ⊆[`(B)]},

S2={(C, D, S)|C∧st(A↓S) =0_|S|, D∧st(A↓S^c) =0_|Sc|andS⊆[n]}.

There is a bijectionϕ:S1−→S2defined byϕ((B, T)) = (st(BT), st(BT^c), BT₁∪BT₂∪ · · · ∪BT_|T|) with inverseϕ⁻¹((C, D, S)) = (C↑S∪D↑S^c,{i|(C↑S ∪D↑S^c)i⊆S}).Hence

∆(e_A) = X

S⊆[n]

X

C∧st(A↓_S)=0_|S|

X

D∧st(A↓_Sc)=0_|Sc|

m_C⊗m_D

= X

S⊆[n]

X

C∧st(A↓S)=0|S|

mC

⊗

X

D∧st(A↓Sc)=0|Sc|

mD

= X

S⊆[n]

e_st(A↓S)⊗e_st(A↓Sc).

2 The next lemma will be used in order to prove a formula for the coproduct on thehbasis.

Lemma 1 (ω⊗ω)◦∆ = ∆◦ω.

Proof:By applying∆and sinceω(p_A) = (−1^A)p_A(see Table 6), we get

(ω⊗ω)◦∆(pA) = (ω⊗ω) X

S⊆[`(A)]

p_st(AS)⊗p_st(ASc)

!

= X

S⊆[`(A)]

ω(p_st(AS))⊗ω(p_st(ASc))

= X

S⊆[`(A)]

(−1)^st(AS)p_st(AS)⊗(−1)^st(ASc)p_st(ASc).

Now observe that for anyS⊆[`(A)],(−1)^st(AS)(−1)^st(ASc)= (−1)^A. Hence (ω⊗ω)◦∆(pA) = X

S⊆[`(A)]

(−1)^Ap_st(AS)⊗p_st(ASc)

= (−1)^A∆(pA) = ∆((−1)^ApA) = ∆◦ω(pA).

2

Corollary 3 ForA`[n], one has

∆(h_A) = X

S⊆[n]

h_st(A↓S)⊗h_st(A↓Sc).

Proof:Let∆(e_A) =X

B,C

d^B,C_A e_B⊗e_C.Then

∆(hA) = ∆◦ω(eA) = (ω⊗ω)◦∆(eA) (by Lemma 1)

= (ω⊗ω)(X

B,C

d^B,C_A eB⊗eC) =X

B,C

d^B,C_A ω(eB)⊗ω(eC) =X

B,C

d^B,C_A hB⊗hC. 2 Letuandvbe monomials inQhXniand ^[|u|+|v|]_|u|

be the set of subsets of[|u|+|v|]with|u|elements.

Lets = {s1, s2, . . . , s_|u|} be an element of that set and lett = {t1, t2, . . . , t_|v|}be its complement in [|u|+|v|]. Thenu svis the monomialwsuch thatu=ws₁ws₂· · ·ws_|u|andv=wt₁wt₂· · ·wt_|v|. The shuffle of the monomialuwithvis then defined to be

u v= X

s∈(^[|u|+|v|]_|u| )

u sv.

Example.

x1x1 x1x2=x1x1 {1,2}x1x2+x1x1 {1,3}x1x2+x1x1 {1,4}x1x2

+x₁x_{1 {2,3}}x₁x₂+x₁x_{1 {2,4}}x₁x₂+x₁x_{1 {3,4}}x₁x₂

=x1x1x1x2+x1x1x1x2+x1x1x2x1+x1x1x1x2+x1x1x2x1+x1x2x1x1. If the monomial basis ofN CSymis expressed as a sum of words of non-commutative monomials as in equation (1), we can define

mA mB =X

C

α^C_A,BmC, (2)

where

α^C_A,B= #

S∈

[|A|+|B|]

|A|

st(C↓S) =Aandst(C↓S^c) =B

.

This shuffle product is commutative and associative and corresponds exactly to what happens when one consider the shuffle of monomials. Considering the pairing defined on the bases of N CSym by hh_A,m_Bi=δ_A,B, we have the following theorem.

Theorem 1 Letf, g, h∈N CSym. Thenh∆(f), g⊗hi=hf, g hi.

Proof:Sincehh_A⊗h_B,m_C⊗m_Di=hh_A,m_Cihh_B,m_Di, then h∆(h_A),m_B⊗m_Ci=h X

S⊆[|A|]

h_st(A↓S)⊗h_st(A↓Sc),m_B⊗m_Ci=hX

D,E

α^A_D,Eh_D⊗h_E,m_B⊗m_Ci

=X

D,E

α^A_D,EhhD⊗hE,mB⊗mCi=α^A_B,C.

On the other hand,hhA,m_B m_Ci = hhA,P

Dα^D_B,Cm_Di = P

Dα^D_B,ChhA,m_Di =α^A_B,C.so the

theorem follows. 2

Corollary 4 Letβ_A,B^C = #

S∈ [`(A)+`(B)]

`(A)

st(C_S) =Aandst(C_Sc) =B

. Then

hA hB =X

C

β_A,B^C hC.

Proof:Follows from Theorem 1 and the fact that

∆(m_C) = X

S⊆[`(C)]

m_st(CS)⊗m_st(CSc)=X

A,B

β_A,B^C m_A⊗m_B.

2 Conjecture 1 Letβ_A,B^C be the coefficients defined in Corollary 4. Then

pA pB =X

C

β_A,B^C pC.

Given the previous conjecture, the following proposition follows from a simple calculation.

Proposition 3 Letf, g∈N CSym. Thenω(f g) =ω(f) ω(g).

Proof:Sinceω(pC) = (−1)^CpC(see Table 6), then ω(pA pB) =ω

X

C

β_A,B^C pC

=X

C

β_A,B^C ω(pC) =X

C

β^C_A,B(−1)^CpC.

Note that when`(A) +`(B) =`(C)then(−1)<sup>A</sup>(−1)<sup>B</sup>= (−1)<sup>C</sup>. In the definition of the shuffle product on thepbasis, we certainly have`(A) +`(B) =`(C), so

ω(pA pB) =X

C

β_A,B^C (−1)^A(−1)^BpC= (−1)^A(−1)^BpA pB =ω(pA) ω(pB)

and the proposition follows. 2

The next corollary follows naturally from this proposition.

Corollary 5 Letβ_A,B^C be the coefficients defined in Corollary 4. Then

e_A e_B =X

C

β_A,B^C e_C.

Proof:e_A e_B =ω(h_A) ω(h_B) =ω(h_A h_B) =ω(P

Cβ^C_A,Bh_C) =P

Cβ_A,B^C e_C. 2

5 Results on the invariants of the hyperoctahedral group

In some preliminary work, we also consider the analogue of typeB_n forQhXni^Sn. We consider the algebraQhX_ni^Bn =L

d≥0QhX_ni^B_dⁿ of polynomials inQhX_niwhich are invariant under the action of the hyperoctahedral groupB_nof signed permutations of[n]. The action of a signed permutationσ∈B_n onQhX_niis characterized by sending a variablex_ito±x_|σ(i)|, where|σ(i)|is the absolute value ofσ(i).

One can show that a monomial basis ofQhXni^Bnis given by {m_A[X_n]}`(A)≤n

|Ai|even

,

whereA={A1, A2, . . . , A_`(A)}is a set partition andmA[Xn]is defined as in equation (1) with a finite number of variables. In other words, a monomial basis is indexed by the set partitions with at mostn parts of even cardinality. In (9), it has been proved that a basis for the centralizer algebraEndB_n(V^⊗d) of Bn is also indexed by the set partitions of2k with at mostnparts of even cardinality. We have a correspondence between the centralizer algebra and the invariants ofBnsince

EndB_n(V^⊗k)'(V^⊗2k)^Bn'QhXni^B_2kⁿ.

Lemma 2 Letα_d,kbe the number of set partitions ofdof lengthkwith even parts. Then

F_k(q) =X

d≥0

α_d,kq^d= 1·3·. . .·(2k−1)q^2k (1−q²) (1−4q²)· · ·(1−k²q²).

Proof:We haveF_k(q) =F_k−1(q)^(2k−1)q_1−k2q2², so

F_k(q) =k²q²F_k(q) +q²F_k−1(q) + 2(k−1)q²F_k−1(q).

Taking the coefficient ofqⁿon both sides yields the following recurrence:

αd,k =k²α_d−2,k+α_d−2,k−1+ 2(k−1)α_d−2,k−1.

We would like to show that the number of set partitions ofdof lengthkwith even parts satisfies this same recurrence. Let us first denote byCd,k the set of set partitions ofdof lengthkwith even parts. For a set partitionA={A1, A2, . . . , Ak}, denote bymiandm_i\{d}the greatest value of respectivelyAiand Ai\{d}. Order the parts ofAwith the ruleAi< Ajif and only ifmi< mj. ForA∈Cd,k, suppose that d−1∈Aiandd∈Aj. Let

E1={A∈Cd,k|i=j,|Ai|=|Aj|>2}, E2={A∈Cd,k|i < j,|Ai| ≥2,|Aj|>2}, E₃={A∈C_d,k|i > j,|A_i|>2,|A_j| ≥2}, E₄={A∈C_d,k|i=j,|A_i|=|A_j|= 2}, E5={A∈Cd,k|i < j,|Ai| ≥2,|Aj|= 2}, E6={A∈Cd,k|i > j,|Ai|= 2,|Aj| ≥2}.

These sets are clearly mutually disjoint, andC_d,k=S6 i=1E_i.

Define an injectionf :C_d,k →k²C_d−2,k∪C_d−2,k−1∪2(k−1)C_d−2,k−1by

f(A) =











{A1, A2, . . . , Ai\{d−1, d}, . . . , Ak} ifA∈E1, {A1, A2, . . . , Ai\{d−1} ∪ {m_j\{d}}, . . . , Aj\{d, m_j\{d}}, . . . , Ak} ifA∈E2∪E5, {A₁, A₂, . . . , A_j\{d} ∪ {m_i\{d−1}}, . . . , A_i\{d−1, m_i\{d−1}}, . . . , A_k} ifA∈E₃∪E₆,

A\{{d−1, d}} ifA∈E4.

We have thatf sendsE1∪E2∪E3tok²copies ofC_d−2,k,E4to one copie ofC_d−2,k−1andE5∪E6

to2(k−1)copies ofC_d−2,k−1. Denote byC_d−2,k^(i,j) the(i, j)-th copy ofC_d−2,kand byC_d−2,k−1⁽ⁱ⁾ thei-th copy ofC_d−2,k−1. The inverse offis defined as follows.

f⁻¹(A) =























{A1, A₂, . . . , A_i∪ {d−1, d}, . . . , Ak} ifA∈C_d−2,k^(i,i) , {A1, A2, . . . , Ai∪ {d−1}\{mi}, . . . , Aj∪ {d, mi}, . . . , Ak} ifA∈C_d−2,k^(i,j) , {A1, A2, . . . , Aj∪ {d}\{mj}, . . . , Ai∪ {d−1, mj}, . . . , Ak} ifA∈C_d−2,k^(j,i) , {A1, A2, . . . , A_k−1} ∪ {{d−1, d}} ifA∈C_d−2,k−1⁽¹⁾ , {A1, A2, . . . , Ai∪ {d−1}\{mi}, . . . , A_k−1,{d, mi}} ifA∈C_d−2,k−1⁽ⁱ⁾ , {A1, A2, . . . , Aj∪ {d}\{mj}, . . . , A_k−1,{d−1, mj}} ifA∈C_d−2,k−1^(j) .

So|C_d,k|=α_d,kbecause they satisfy the same recurrence. 2

From the previous lemma, and the fact that a basis ofQhX_ni^Bnis indexed by the set partitions with at mostnparts of even length, we get the following corollary.

Corollary 6 The Poincare series for the algebraQhXni^Bnis

X

d≥0

dim(QhXni^B_dⁿ)q^d= 1

|B_n| X

σ∈Bn

1

1−T r(σ)q = 1 +

n

X

k=1

1·3·. . .·(2k−1)q^2k (1−q²) (1−4q²)· · ·(1−k²q²).

Proof: The first equality is a non-commutative analogue of Molien’s Theorem, that has been proved by

Dicks and Formanek (4). 2

6 Appendix

The next two tables summarize the changes of bases and operations onN CSymbases. They are followed by some examples of the coproduct and shuffle product.

e h p e_A

X

B≤A

(−1)^Bλ(B, A)!hB

X

B≤A

µ(0, B)pB

hA

X

B≤A

(−1)^Bλ(B, A)!e_B X

B≤A

|µ(0, B)|pB

m_A

X

B≥A

µ(A, B) µ(0, B)

X

C≤B

µ(C, B)eC

X

B≥A

µ(A, B)

|µ(0, B)|

X

C≤B

µ(C, B)hC

X

B≥A

µ(A, B)pB

pA

1 µ(0, A)

X

B≤A

µ(B, A)e_B 1

|µ(0, A)|

X

B≤A

µ(B, A)h_B

Tab. 1:Changes of bases (see (10)). ForeA,hA,pAin terms ofm, see section 3.

f fA·fB ∆(fA) ω(fA) fA fB

e e_A|B

Proposition 1

X

S⊆[n]

e_st(A↓S)⊗e_st(A↓Sc)

Proposition 2

h_A

(10)

X

C

β_A,B^C eC

Corollary 5

h h_A|B

Corollary 2

X

S⊆[n]

h_st(A↓S)⊗h_st(A↓Sc)

Corollary 3

eA

(10)

X

C

β_A,B^C hC

Corollary 4

m

X

C∧([n]|[k])=A|B

mC

(2) Proposition 3.2

X

S⊆[`(A)]

m_st(AS)⊗m_st(ASc)

(2) (6)

?

X

C

α^C_A,BmC

Equation 2

p p_A|B

(1) Lemma 4.1

X

S⊆[`(A)]

p_st(AS)⊗p_st(ASc)

(1) Remark 4.4

(−1)^ApA

(10) Theorem 3.5

X

C

β_A,B^C p_C

Conjecture 1

Tab. 2:Operations on NCSym bases, whereA`[n]andB`[k].

f ∆(f_{124,3})

h 1⊗h_{124,3}+h_{1}⊗h_{12,3}+ 2h_{1}⊗h_{13,2}+h_{1}⊗h_{123}+ 3h_{12}⊗h_{1,2}

+3h_{1,2}⊗h_{12}+h_{123}⊗h_{1}+ 2h_{13,2}⊗h_{1}+h_{12,3}⊗h_{1}+h_{124,3}⊗1 m 1⊗m_{124,3}+m_{1}⊗m_{123}+m_{123}⊗m_{1}+m_{124,3}⊗1

f f_{12} f_{1,2}

h h_{1,2,34}+h_{13,2,4}+h_{1,23,4}+h_{14,2,3}+h_{1,24,3}+h_{12,3,4}

m m_{12,3,4}+m_{13,2,4}+m_{14,2,3}+m_{1,23,4}+m_{1,24,3}+m_{1,2,34}

+3m_{134,2}+ 3m_{123,4}+ 3m_{1,234}+ 3m_{124,3}

Tab. 3:Examples of coproduct and shuffle product.

Acknowledgements

The author is grateful to the anonymous referees for the useful and accurate comments provided.

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