´ SERIES. THE MAPLE PACKAGE FOR CALCULATING POINCARE

arXiv:submit/0176264 [math.AG] 8 Jan 2011

LEONID BEDRATYUK Abstract. We offer a Maple package Poincare_Series for calculating the Poincar´e series for the algebras of invariants/covariants of binary forms, for the algebras of joint invariants/covariants of several binary forms, for the kernel of Weitzenb¨ock derivations,for the bivariate Poincar´e series of algebra of covariants of binary d-form and for the multivariate Poincar´e series of the algebras of joint invariants/covariants of several binary forms.

1. Introduction The Poincar´ P e series of ai graded algebra A = ⊕i (A)i is defined as formal power series P(A, z) := ∞ e series is i=0 dim(A)i z . If an algebra is finitely generated then its Poincar´ the power series expansions of certain rational functions. The present package implements results of the following papers: • Leonid Bedratyuk, The Poincar´e series of the covariants of binary forms, Int. Journal of Algebra,2010 • Leonid Bedratyuk,The Poincar´e series of the joint invariants and covariants of the two binary forms, Linear and Multilinear algebra, 2010 • Leonid Bedratyuk, Linear locally nilpotent derivations and the classical invariant theory, I: The Poincare series, Serdica Math. J.,2010 • Leonid Bedratyuk, Bivariate Poincar´e series for the algebra of covariants of a binary form, preprint arXiv:1006.1974 • Leonid Bedratyuk, Multivariate Poincar´e series for the algebras of joint invariants and covariants of several binary forms, in preparation.

2. Instalation. The package can be downloaded from the web http://sites.google.com/site/bedratyuklp/. (1) download the file Poincare_Series.mpl and save it into your Maple directory; (2) download the Xin’s file (see a link at the web page) Ell2.mpl and save it into your Maple directory; (3) run Maple; (4) > read "Poincare_Series.mpl": read "Ell2.mpl": (5) If necessary use > Help();

3. Formulas for the Poincare´ series. Below are the list of main formulas. 1

2

LEONID BEDRATYUK

3.1. Invariants and covariants of binary form. Let Id , Cd be algebras of invariants and covariants of binary d-form graded under degree. We have   X (−1)k z k(k+1) (1 − z 2 ) (1) , (Springer’s formula), P(Id , z) = ϕd−2 k (z 2 , z 2 )k (z 2 , z 2 )d−k 0≤k
here (a, q)n = (1 − a)(1 − a q) · · · (1 − a q n−1) denotes the q-shifted factorial and the function ϕn : C[[z]] → C[[z]] defined by ! ∞ ∞ X X i ain z i . ϕn ai z = i=0

i=0

3.2. Joint invariants and covariants of binary form. Let Id , Cd , d = (d1 , d2 , . . . , dn ) be algebras of joint invariants and joint covariants of n binary forms of degrees d1 , d2 , . . . , dn . Then ∗

(3)

(4)

βi d X X

 dk−1 z k−1 ϕd∗ −k ((1 − z 2 ) Ai,k (z)) 1 P(Id , z) = , (k − 1)! dz k−1 i=0 k=1  βi d∗ X X dk−1 z k−1 ϕd∗ −k ((1 + z) Ai,k (z)) 1 P(Cd , z) = , k−1 (k − 1)! dz i=0 k=1

 ∂ βi −k (−1)βi −k d∗ i βi lim f (tz , z)(1 − tz ) . d (βi − k)! (z i )βi −k t→z −i ∂tβi −k The integer numbers βi , i = 0, . . . , 2d∗ , d∗ := max(d1 , d2 , . . . , dn ), are defined from the decomposition −1 ∗ ∗ fd (tz d , z) = (1 − t)β0 (1 − tz)β1 (1 − tz 2 )β2 . . . (1 − tz 2 d )β2 d∗ , Ai,k (z) =

where

fd (t, z) =

s Y

(tz − dk , z 2 )dk +1

k=1

!−1

.

3.3. Joint invariants and covariants of linear and quadratic binary forms. Let d1 = d2 = . . . = dn = 1, i.e. d = (1, 1, . . . , 1). Then  2n−k−1 ! n X (−1)n−k (n)n−k dk−1 Nn−2 (z 2 ) z = (5) P(Id , z) = , (k − 1)! (n − k)! dz k−1 1 − z2 (1 − z 2 )2n−3 k=1 where Nn (z) is the n-th Narayama polynomial    n X n 1 n−1 z k−1 . Nn (z) = k − 1 k − 1 k k=1

and (n)m := n(n + 1) · · · (n + m − 1), (n)0 := 1 denotes the shifted factorial.   n X (−1)n−k (n)n−k dk−1 (1 + z)z 2n−k−1 P(Cd , z) = (6) , (k − 1)! (n − k)! dz k−1 (1 − z 2 )2n−k k=1

3

Let d1 = d2 = . . . = dn = 2, d = (2, 2, . . . , 2), then !  n−k  n 2n−k−i−1 k−1 n−k X X d n−k (n)i (n)n−k−i (1−z)z (−1) (7) P(Id , z)= , k−1 (n−k)(k−1)! dz i (1−z)n+i (1−z 2 )2n−k−i i=0 k=1 !  n n−k  X (−1)n−k dk−1 X n − k (n)i (n)n−k−i z 2n−k−i−1 (8) P(Cd , z) = = (n − k)!(k − 1)! dz k−1 i=0 i (1 − z)n+i (1 − z 2 )2n−k−i k=1 2 n−1  X n−1 (z 2 )i i = i=0 n (9) . (1 − z) (1 − z 2 )2n−1 3.4. Kernel of Weitzenb¨ ok derivation. Denote by Dd the Weitzenb¨ok derivation (linear locally nilpotent derivation) with its matrix consisting of n Jordan blocks of size d1 + 1, d2 + 1, . . . , ds + 1, respectively. Since ker Dd ∼ = Cd and the isomorphism preserve degrees then have that P(ker Dd , z) = P(Cd , z). 3.5. Bivariate Poincare series for covariants of binary form. The algebra Cd of covariants is a finitely generated bigraded algebra: 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. We have

(10)

P(Cd , z, t) =

∞ X i=0

i j

(Cd )i,j z t =

X

ψd−2 k

0≤k


(−1)k tk(k+1) (1 − t2 ) (t2 , t2 )k (t2 , t2 )d−k



1 , 1 − ztd−2 k

where ψn : Z[[t]] → Z[[t, z]], n ∈ Z+ be a C-linear function defined by  i j z t , if m = n i − j, j < n, m ψn (t ) := 0, otherwise. Note that P(Cd , z, 0) = P(Id , z) and P(Cd , z, 1) = P(Cd , z). 3.6. Multivariate Poincar´ e series. The algebra Cd is a finitely generated multigraded algebra under the multidegree-order: Cd = (Cd )m,0 + (Cd )m,1 + · · · + (Cd )m,j + · · · , where each subspace (Cd )d,j of covariants of multidegree m := (m1 , m2 , . . . , mn ) and order j is finite-dimensional. The formal power series P(Cd , z1 , z2 , . . . , zn , t) =

∞ X

dim((Cd )m,j )z1m1 z2m2 · · · znmn tj ,

m,j=0

is called the multivariariate Poincar´e series of the algebra of join covariants Cd . The following formula holds:   d1 d2 ds 1 P(Cd , z1 , z2 , . . . , zn , t) = Ω fd z1 (tλ) , z2 (tλ) , . . . , zs (tλ) , , ≥0 tλ

4

LEONID BEDRATYUK

where fd (z1 , z2 , . . . , zn , t) =

1 dk n Y Y

,

(1 − zk tdk −2 j )

k=1 j=0

For the multivariariate Poincar´e series of the algebra of join invariants Id we have   d1 d2 ds 1 P(Id , z1 , z2 , . . . , zn , t) = Ω fd z1 (tλ) , z2 (tλ) , . . . , zs (tλ) , . =0 tλ

Here Ω and Ω are the MacMahon’s Omega operators. ≥0

=0

4. Package Commands and Syntax Command name: INVARIANTS_SERIES Feature: Computes the Poincare series for the algebras of joint invariants for the binary forms of degrees d1 , d2 , . . . , dn . Calling sequence: INVARIANTS_SERIES([d1 , d2, . . . , dn ]) ; Parameters: [d1 , d2, . . . , dn ] - a list of degrees of n binary forms. n - an integer, n ≥ 1. Command name: COVARIANTS_SERIES Feature: Computes the Poincare series for the algebras of joint covariants for the binary forms of degrees d1 , d2 , . . . , dn . Calling sequence: COVARIANTS_SERIES([d1 , d2, . . . , dn ]) ; Parameters: [d1 , d2, . . . , dn ] - a list of degrees of n binary forms. n - an integer, n ≥ 1. Command name: KERNEL_SERIES Feature: Computes the Poincare series for the kernel of Weitzenb¨ock derivation defined by n Jordan block of sizes d1 + 1, d2 + 1, . . . , dn . Calling sequence: KERNEL_SERIES([d1 , d2 , . . . , dn ]) ; Parameters: [d1 , d2, . . . , dn ] - a list of sizes of the n Jordan blocks. n - an integer, n ≥ 1. Command name: BIVARIATE_SERIES Feature: Computes the bivariate Poincare series for the algebra of covariants of binary form of degree d. Also, computes the bivariate Poincare series for the kernel of the basic Weitzenb¨ock derivation. Calling sequence: BIVARIATE_SERIES([d]); Parameters: d - the degree of binary form. Command name: MULTIVAR_COVARIANTS Feature: Computes the multivariate Poincar´e series for the algebra of joint covariants for n binary forms of degrees d1 , d2, . . . , dn . Calling sequence: MULTIVAR_COVARIANTS([d1 , d2 , . . . , dn ]) ;

5

Parameters: [d1 , d2, . . . , dn ] - a list of degrees of n binary forms. n - an integer, n ≥ 1. Command name: MULTIVAR_INVARIANTS Feature: Computes the multivariate Poincar´e series for the algebra of joint invariants for n binary forms of degrees d1 , d2, . . . , dn . Calling sequence: MULTIVAR_INVARIANTS([d1 , d2 , . . . , dn ]) ; Parameters: [d1 , d2, . . . , dn ] - a list of degrees of n binary forms. n - an integer, n ≥ 1. 5. Examples 5.1. Compute P(I6 , z). Use the command > INVARIANTS_SERIES([6]); z8 + z7 − z5 − z4 − z3 + z + 1 (z 6 + z 5 + z 4 − z 2 − z − 1) (z 6 + z 5 − z − 1) (−1 + z 2 ) (−1 + z) 5.2. Compute P(C6 , z). Use the command > COVARIANTS_SERIES([6]); z 10 + z 8 + 3 z 7 + 4 z 6 + 4 z 5 + 4 z 4 + 3 z 3 + z 2 + 1 (z 6 + z 5 + z 4 − z 2 − z − 1) (z 6 + z 5 − z − 1) (−1 + z 2 ) (−1 + z)3 5.3. Compute P(I(1,2,3) , z). Use the command > INVARIANTS_SERIES([1,2,3]); z 12 + z 9 + 2 z 8 + 3 z 7 + 3 z 6 + 3 z 5 + 2 z 4 + z 3 + 1 (−1 + z 4 )2 (−1 + z 3 )2 (−1 + z) (−1 + z 2 ) (z 4 + z 3 + z 2 + z + 1) 5.4. Compute P(C(2,2,2) , z). Use the command > COVARIANTS_SERIES([2,2,2]); z4 + 4 z2 + 1 (−1 + z)3 (−1 + z 2 )5 5.5. Compute P(ker D(4) , z). Use the command > KERNEL_SERIES([4]); z2 − z + 1 (−1 + z 2 ) (−1 + z 3 ) (−1 + z)2

6

LEONID BEDRATYUK

5.6. Compute P(ker D(1,1,1,2) , z). Use the command > KERNEL_SERIES([1,1,1,2]); z 8 + 2 z 7 + 7 z 6 + 11 z 5 + 11 z 4 + 11 z 3 + 7 z 2 + 2 z + 1 (−1 + z 2 )3 (−1 + z 3 )3 (−1 + z)2 5.7. Compute P(C4 , z, t). Use the command > BIVARIATE_SERIES([4]); t4 z 2 − zt2 + 1 (−1 + zt2 ) (−1 + zt4 ) (−1 + z 2 ) (−1 + z 3 ) 5.8. Compute P(C(1,1,2) , z1 , z2 , z3 , t). Use the command > dd:=[1,1,2]:MULTIVAR_COVARIANTS(dd); z2 2 z1 2 z3 2 t2 + tz3 z2 2 z1 − tz3 z2 − z2 z1 z3 + z2 z1 t2 z3 + tz3 z1 2 z2 − z1 tz3 − 1 (−1 + z3 t2 ) (−1 + z3 2 ) (−1 + tz2 ) (−1 + z3 z2 2 ) (−1 + z1 t) (−1 + z1 2 z3 ) (−1 + z2 z1 ) 5.9. Compute P(I(4,4) , z1 , z2 , t). Use the command > dd:=[4,4]:MULTIVAR_INVARIANTS(dd); −

z1 4 z2 4 + z2 2 z1 2 + 1 (−1 + z2 2 ) (−1 + z2 3 ) (−1 + z1 2 z2 ) (−1 + z1 z2 ) (−1 + z1 z2 2 ) (−1 + z1 2 ) (−1 + z1 3 )

arXiv:submit/0176264 [math.AG] 8 Jan 2011

form of degree d. Also, computes the bivariate Poincare series for the kernel of the basic Weitzenböck derivation. Calling sequence: BIVARIATE_SERIES([d]);.

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