# Bivariate Poincaré Series for  the Algebra of Covariants   of a Binary Form

Authors: Leonid Bedratyuk
Journal: ISRN Algebra
 (2011)

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## Abstract

A formula for computation of the bivariate Poincaré series𝒫d(z,t) for the
algebra of covariants of binary d-form is found.


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## Body

## 1. Introduction

LetVd be the complex vector space of binary forms of degree d endowed with the natural action of the special linear group G=SL(2,ℂ). Consider the corresponding action of the group G on the coordinate rings ℂ[Vd] and ℂ[Vd⊕ℂ2]. Denote by ℐd=ℂ[Vd]G and by 𝒞d=ℂ[Vd⊕ℂ2]G the subalgebras of G-invariant polynomial functions. In the language of classical invariant theory, the algebras ℐd and 𝒞d are called the algebra of invariants and the algebra of covariants for the binary form of degree d, respectively. The algebra 𝒞d is a finitely generated bigraded algebra: (1.1)Cd=(Cd)0,0+(Cd)1,0+⋯+(Cd)i,j+⋯,
where each subspace (𝒞d)i,j of covariants of degree i and order j is finite dimensional. The formal power series 𝒫d(z,t)∈ℤ[[z,t]],(1.2)Pd(z,t)=∑i,j=0∞dim⁡⁡((Cd)i,j)zitj,
is called the bivariate Poincaré series of the algebra of covariants 𝒞d. It is clear that the series 𝒫d(z,0) is the Poincaré series of the algebra ℐd and the series 𝒫d(z,1) is the Poincaré series of the algebra 𝒞d with respect to the usual grading of the algebras under degree. The finitely generation of the algebra of covariants implies that its bivariate Poincaré series is the power series expansion of a rational function of two variables z,t. We consider here the problem of computing efficiently this rational function.Calculating the Poincaré series of the algebras of invariants and covariants was an important object of research in invariant theory in the 19th century. For the casesd≤10, d=12 the series 𝒫d(z,t) were calculated by Sylvester, see in [1, 2] the big tables of 𝒫d(z,t), named them as generating functions for covariants, reduced form. All those calculations are correct up to d=6.Relatively recently, Springer [3] found an explicit formula for computing the Poincaré series of the algebra of invariants ℐd. In the paper we have proved a Cayley-Sylvester-type formula for calculating of dim(𝒞d)i,j and a Springer-type formula for calculation of 𝒫d(z,t). By using the formula, the bivariate Poincaré series 𝒫d(z,t) is calculated for d≤20.


## 2. Cayley-Sylvester-Type Formula fordim(𝒞d)i,j

To begin, we give a proof of the Cayley-Sylvester-type formula for the dimension of the graded component(𝒞d)i,j.LetVd=〈v0,v1,…,vd〉 and dimVd=d+1 be standard irreducible representation of the Lie algebra 𝔰𝔩2. The basis elements (0100),(0010), (100-1) of the algebra 𝔰𝔩2 act on Vd by the derivations D1,D2,E: (2.1)D1(vi)=ivi-1,D2(vi)=(d-i)vi+1,E(vi)=(d-2i)vi.
The action of 𝔰𝔩2 is extended to an action on the symmetrical algebra S(Vd) in the natural way.Let𝔲2 be the maximal unipotent subalgebra of 𝔰𝔩2. The algebra 𝒮d, defined by (2.2)Sd:=S(Vd)u2={v∈S(Vd)∣D1(v)=0},
is called the algebra of semi-invariants of the binary form of degree d. For any element v∈𝒮d, a natural number s is called the order of the element v if the number s is the smallest natural number such that (2.3)D2s(v)≠0,D2s+1(v)=0.
It is clear that any semi-invariant v∈𝒮d of order i is the highest weight vector for an irreducible 𝔰𝔩2-module of the dimension i+1 in S(Vd).The classical theorem [4] of Roberts implies an isomorphism of the algebra of covariants and the algebra of semi-invariants. Furthermore, the order is preserved through the isomorphism. Thus, it is enough to compute the Poincaré series of the algebra 𝒮d.The algebraS(Vd) is ℕ-graded (2.4)S(Vd)=S0(Vd)+S1(Vd)+⋯+Si(Vd)+⋯,
and each Si(Vd) is a completely reducible representation of the Lie algebra 𝔰𝔩2. Thus, the following decomposition holds (*)Si(Vd)≅γd(i,0)V0+γd(i,1)V1+⋯+γd(i,d⋅n)Vd⋅i,
here γd(i,j) is the multiplicity of the representation Vj in the decomposition of Si(Vd). On the other hand, the multiplicity γd(i,j) of the representation Vj is equal to the number of linearly independent homogeneous semi-invariants of degree i and order j for the binary d-form. This argument proves the following.Lemma 2.1.
(2.5)dim⁡(Cd)i,j=γd(i,j).The set of weights (eigenvalues of the operatorE) of a representation W denote by ΛW, in particular, ΛVd={-d,-d+2,…,d-2,d}.A formal sum(2.6)Char(W)=∑k∈ΛWnW(k)qk,
is called the character of a representation W, here nW(k) denotes the multiplicity of the weight k∈ΛW. Since, the multiplicity of any weight of the irreducible representation Vd is equal to 1, we have (2.7)Char(Vd)=q-d+q-d+2+⋯+qd-2+qd.The characterChar(Sn(Vd)) of the representation Sn(Vd) equals (2.8)Hn(q-d,q-d+2,…,qd),
(see [5]), where Hn(x0,x1,…,xd) is the complete symmetrical function (2.9)Hn(x0,x1,…,xd)=∑|α|=nx0α0x1α1⋯xdαd,|α|=∑iαi.By replacingxk with qd-2k, k=0,…,d, we obtain the specialized expression for Char(Sn(Vd)): (2.10)Char(Sn(Vd))=∑|α|=n(qd)α0(qd-2⋅1)α1⋯(qd-2d)αd=∑|α|=nqdn-2(α1+2α2+⋯+dαd)=∑k=-dndnωd(n,dn-k2)qk,
here ωd(n,(dn-k)/2) is the number nonnegative integer solutions of the equation (2.11)α1+2α2+⋯+dαd=dn-k2,
on the assumption that α0+α1+⋯+αd=n. In particular, the coefficient of q0 (the multiplicity of zero weight ) is equal to ωd(n,dn/2), and the coefficient of q1 is equal to ωd(n,(dn-1)/2).On the other hand, the decomposition (*) implies the equality for the characters: (2.12)Char(Sn(Vd))=γd(n,0)Char(V0)+γd(n,1)Char(V1)+⋯+γd(n,dn)Char(Vdn).
We can summarize what we have shown so far in the following.Theorem 2.2.
(2.13)dim⁡(Cd)i,j=ωd(i,di-j2)-ωd(i,di-(j+2)2).Proof.
The weightj appears once in any representation Vk, for k=jmod2,k≥j. Therefore
(2.14)ωd(i,di-j2)=γd(i,j)+γd(i,j+2)+⋯+γd(i,j+4)+⋯.
Similarly,
(2.15)ωd(i,di-(j+2)2)=γd(i,j+2)+γd(i,j+4)+⋯+γd(i,j+6)+⋯.
Thus,
(2.16)ωd(i,di-j2)-ωd(i,di-(j+2)2)=γd(i,j).
By using Lemma 2.1 we obtain
(2.17)dim⁡⁡(Cd)i,j=ωd(i,di-j2)-ωd(i,di-(j+2)2).For another proof of the formula see [3].Note that the original Cayley-Sylvester formula is(2.18)dim⁡⁡(Id)n=ωd(n,dn2)-ωd(n,dn2-1).
Also, in [6] we proved that (2.19)dim⁡⁡(Cd)n=ωd(n,dn2)+ωd(n,dn-12).
Here (ℐd)n,(𝒞d)n are the components of standard grading of the algebras ℐd, 𝒞d under degree.


## 3. Calculation ofdim(Cd)i,j

It is well known that the numberωd(i,(di-j)/2) of nonnegative integer solutions of the following system (3.1)α1+2α2+⋯+dαd=di-j2,α0+α1+⋯+αd=i,
is given by the coefficient of znt(di-j)/2 of the generating function(3.2)fd(z,t)=1(1-z)(1-zt)⋯(1-ztd).
We will use the notation [xk]F(x) to denote the coefficient of xk in the series expansion of F(x)∈ℂ[[x]]. Thus (3.3)ωd(i,di-j2)=[zit(di-j)/2]fd(z,t).
It is clear that (3.4)ωd(i,di-j2)=[zitdi-j]fd(z,t2)=[(ztd)i]tjfd(z,t2).Similarly, the numberωd(i,(di-(j+2))/2) of nonnegative integer solutions of the following system(3.5)α1+2α2+⋯+dαd=di-(j+2)2,α0+α1+⋯+αd=i,equals(3.6)[zit(di-(j+2))/2]fd(z,t)=[zitdi-(j+2)]fd(z,t2)=[(ztd)i]tj+2fd(z,t2).Therefore,(3.7)ωd(i,di-j2)-ωd(i,di-(j+2)2)=[(ztd)i]tjfd(z,t2)-[(ztd)i]tj+2fd(z,t2)=[(ztd)i](tj-tj+2)fd(z,t2)=[(ztd)i]tj(1-t2)fd(z,t2)=[zitdi-j](1-t2)fd(z,t2).
Thus, the following statement holds.Theorem 3.1.
The numberdim(𝒞d)i,j of linearly independent covariants of degree i and order j for the binary d- form is given by the formula
(3.8)dim⁡⁡(Cd)i,j=[zitdi-j](1-t2(1-z)(1-zt2)⋯(1-zt2d)).
It is clear that(3.9)[zitdi-j](1-t2(1-z)(1-zt2)⋯(1-zt2d))=[zit(di-j)/2](1-t(1-z)(1-zt)⋯(1-ztd)).
By using the decomposition
(3.10)1(1-z)(1-zt)⋯(1-ztd)=∑k=0∞[di]tzi,
where [dn]q is the q-binomial coefficient
(3.11)[dn]q:=(1-qd+1)(1-qd+2)⋯(1-qd+n)(1-q)(1-q2)⋯(1-qn),
one obtains the well-known formula
(3.12)dim⁡⁡(Cd)i,j=[t(di-j)/2](1-t)[di]t,
for instance, see [3].


## 4. Explicit Formula for𝒫d(z,t)

Let us prove Springer-type formula for the bivariate Poincaré series𝒫d(z,t) of the algebra covariants of the binary d-form. Consider the ℂ-algebra ℤ[[t,z]] of formal power series. For an integer d∈ℕ define the ℂ-linear function (4.1)Ψd:Z[[z,t]]⟶Z[[z,t]],
in the following way (4.2)Ψd(zitj)={zitdi-j,ifdi-j≥0,0,ifdi-j<0.The main idea of the ensuing calculations is that the Poincaré series𝒫d(z,t) can be expressed in terms of function Ψd. The following simple but important statement holds.Lemma 4.1.
(4.3)Pd(z,t)=Ψd(1-t2(1-zt)(1-zt2)⋯(1-zt2d)).Proof.
Theorem2.2 implies that dim(Cd)i,j=[zitdi-j]fd(z,t2). Then
(4.4)Pd(z,t)=∑i,j=0∞dim⁡⁡(Cd)i,jzitj=∑i,j=0∞([(zitdi-j)]fd(z,t2))zitj=Ψd(fd(z,t2)).Letψn:ℤ[[t]]→ℤ[[t,z]],n∈ℤ be a ℂ-linear function defined by (4.5)ψn(tm):=zitj,wherei:=min⁡{k′∣nk′-m≥0},j=ni-m,fori,j,m,n∈ℕ. Note that ψn(t0)=1 and ψ1(tm)=zm, ψ0(tm)=1. Also, put ψn(tm)=0 for n<0. It is clear that ψn(tni-j)=zitj if ni-j≥0, j<n.In important special cases, calculating the functionsΨ can be reduced to calculating the functions ψ. The following statements hold.Lemma 4.2.
(i) ForR(t),H(t)∈ℂ[[t]] holds ψn(R(tn)H(t))=R(z)ψn(H(t)).
(ii) ForR(t)∈ℂ[[t]] and for n,k∈ℕ holds
(4.6)Ψn(R(t)1-ztk)={ψn-k(R(t))1-ztn-k,n≥k,0,ifn<k.Proof.
(i) The statement follows from the linearity of the functionψn and from the following simple observation:
(4.7)ψn(tnktni-j)=ψn(tn(k+i)-j)=zk+itj=zkzitj=zkψn(tni-j),
for ni-j≥0 and j<n.
(ii) LetR(t)=∑m=0∞amtm. Then for k<n we have
(4.8)Ψn(R(t)1-ztk)=Ψn(∑m,s=0∞amtm(ztk)s)=Ψn(∑m,s=0∞amzstks+m)=∑(n-k)s-m≥0amzst(n-k)s-m=∑m,s=0∞amψn-k(tm)(ztn-k)s=∑m=0∞amψn-k(tm)11-ztn-k=ψn-k(R(t))1-ztn-k.Now we can present Springer-type formula for calculating of the bivariate Poincaré series𝒫d(z,t).Theorem 4.3.
(4.9)Pd(z,t)=∑0≤k<d/2ψd-2k((-1)ktk(k+1)(1-t2)(t2,t2)k(t2,t2)d-k)11-ztd-2k,
here (a,q)n=(1-a)(1-aq)⋯(1-aqn-1) is q-shifted factorial.Proof.
Consider the partial fraction decomposition of the rational functionfd(z,t2):
(4.10)fd(z,t2)=∑k=0dRk(z)1-tz2k.
It is easy to see, that
(4.11)Rk(t)=lim⁡z→t-2k(fd(z,t2)(1-zt2k))=lim⁡z→t-2k((1-t2)(z,t)d+1(1-zt2k))=1-t2(1-t-2k)(1-t2-2k)⋯(1-t2(k-1)-2k)(1-t2(k+1)-2k)⋯(1-t2d-2k)=t2k+(2k-2)+⋯+2(1-t2)(t2k-1)(t2k-2-1)⋯(t2-1)(1-t2)⋯(1-t2d-2k)=(-1)ktk(k+1)(1-t2)(t2,t2)k(t2,t2)d-k.
Using the above Lemmas we obtain
(4.12)Pd(z,t)=Ψd(fd(z,t2))=Ψd(∑k=0nRk(t2)1-zt2k)=∑0≤k<d/2φd-2k((-1)ktk(k+1)(1-t2)(t2,t2)k(t2,t2)d-k)11-ztd-2k.Corollary 4.4.
A denominator of the bivariate Poincaré series𝒫d(z,t),  d>2 can be written in the form
(4.13)∏k=0[(d+1)/2](1-ztd-2k)∏i=1d-2(1-zki),
where k1,k2,…,kd-2 are the degrees of elements of homogeneous system of parameters for the algebra of invariants ℐd.Proof.
The formula of Theorem4.3 implies that the bivariate Poincaré series has the form
(4.14)Pd(z,t)=Pd(z,t)∏k=0[(d+1)/2](1-ztd-2k)Rd(z),
for some polynomials Pd(z,t),Rd(z). Thus, the Poincaré series for the algebra of invariants ℐd has the form
(4.15)Pd(z,0)=Pd(z,0)Rd(z).
The algebra of invariants ℐd is Cohen-Macaulay, and its transcendence degree for d>2 equals d-2, see [3]. Therefore it has a homogeneous system of d-2 parameters and the denominator Rd(z) of its Poincaré series can be written in the following way Rd(z)=(1-zk1)(1-zk2)⋯(1-zkd-2), where k1,k2,…,kd-2 are the degrees of elements of this homogeneous system of parameters.


## 5. Examples

For direct computations we use the following technical lemma.Lemma 5.1.
ForR(t)∈ℂ[[t]] one has
(5.1)ψn(R(t)(1-tk1)(1-tk2)⋯(1-tkm))=ψn(R(z)Qn(tk1)Qn(tk2)Qn(tkm))(1-zk1)(1-zk2)⋯(1-zkm),
here Qn(t)=1+t+t2+⋯+tn-1, and ki are natural numbers.Proof.
Taking into account Lemma4.2 we get
(5.2)ψn(g(t)1-tm)=ψn(g(t)1-tnm1-tnm1-tm)=11-tmψn(g(t)1-tnm1-tm)=11-tmψn(g(t)(1+tm+(tm)2+⋯+(tm)n-1))=11-tmψn(g(t)Qn(tm)).
In a similar fashion we prove the general case.By using Lemma5.1 the bivariate Poincaré series 𝒫d(z,t) for d≤20 are found. All these results agree with Sylvester's calculations up to d=6, see [1, 2].Below is the list of several series:(5.3)P1(z,t)=11-zt,P2(z,t)=1(1-zt2)(1-z2),P3(z,t):=z2t2-zt+1(1-zt)(1-zt3)(1-z4),P4(z,t)=z2t4-zt2+1(1-zt2)(1-t4z)(1-z2)(1-z3),P5(z,t)=p5(z,t)(1-zt)(1-t3z)(1-zt5)(1-z8)(1-z6)(1-z4),p5(z,t)=1+z7t3-z6t4+z2t2+2z7t-z5t5-z8t2-2z8t6-z8t4+z5t3+z5t+z9t7-z10t6+z10t2-z10t4-z11t3+z9t3-t3z-z6+z4t4-zt+z2t6+z2t4+z12+z14t6-z13t-z13t5-z13t3-z15t7+z14t4-z3t7+z7t5.



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