The Poincare Series for the Algebra of Covariants of a Binary Form

Transcription

1 International Journal of Algebra, Vol 4, 010, no 5, The Poincare Series for the Algebra of Covariants of a Binary Form Leonid Bedratyuk Khmelnytsky National University Instituts ka st 11, Khmelnytsky 9016, Ukraine leoniduk@gmailcom Abstract A formula for computation of the Poincaré series P d z for the algebra of covariants of binary d-form is found Mathematics Subject Classification: 13N15, 13A50, 13F0 Keywords: Classical invariant theory, Covariants, binary forms, Poincaré series 1 Introduction Let V d be the complex vector space of binary forms of degree d endowed with the natural action of the special linear group G = SL, C Consider the corresponding action of the group G on the coordinate rings C[V d ] and C[V d C ] Denote by I d = C[V d ] G and by C d = C[V d C ] G the subalgebras of G- invariant polynomial functions In the language of classical invariant theory the algebras I d and C d are called the algebra of invariants and the algebra of covariants for the binary form of degree d, respectively The algebra C d is a finitely generated N-graded algebra: C d =C d 0 +C d 1 + +C d i +, where each subspace C d i of covariants of degree i is finite-dimensional The formal power series P d z = i=0 dimc d i z i, is called the Poincaré series of the algebra of covariants C d The finitely generation of the algebra of covariants implies that its Poincaré series is the power series expansion of a rational function 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

2 10 L Bedratyuk For the cases d 10, d= 1 the series P d z were calculated by Sylvester and Franklin, see [9, 10] In [3, 6] the Poincaré series P d z for d 6 is recalculated Relatively recently, Springer [8] found an explicit formula for computing the Poincaré series of the algebra of invariants I d This formula has been used by Brouwer and Cohen [1] for d 17 and also by Littelmann and Procesi [5] In the paper we have found a Cayley-Sylvester type formula for calculating of dimc d i and a Springer type formula for calculation of P d z By using the formulas, the series P d z is calculated for d 30 Cayley-Sylvester type formula for dimc d i To begin, we give a proof of the Cayley-Sylvester type formula for covariants of binary form Let V d = v 0,v 1,, v d, dim V d = d + 1 be standard irreducible representation of the Lie algebra sl The basis elements of the algebra sl act on V d by the derivations D 1,D,E : D 1 v i =iv i 1,D v i =d i v i+1,ev i =d i v i The action of sl is extended to an action on the symmetrical algebra SV d in the natural way Let u be the maximal unipotent subalgebra of sl The algebra S d, defined by S d := SV d u = {v SV d D 1 v =0}, is called the algebra of semi-invariants of the binary form of degree d For any element v S d a natural number s is called the order of the element v if the number s is the smallest natural number such that D s v 0,Ds+1 v =0 It is clear that any semi-invariant v S d of order i is the highest weight vector for an irreducible sl -module of the dimension i +1inSV d The classical theorem [7] of Roberts implies an isomorphism of the algebra of covariants and the algebra of semi-invariants Thus, it is enough to compute the Poincaré series of the algebra S d The algebra SV d isn-graded SV d =S 0 V d +S 1 V d + +S n V d +, and each S n V d is a completely reducible representation of the Lie algebra sl Thus, the following decomposition holds S n V d = γ d n, 0V 0 + γ d n, 1V γ d n, d nv d n, here γ d n, k is the multiplicity of the representation V k in the decomposition of S n V d On the other hand, the multiplicity γ d n, i of the representation V i is equal to the number of linearly independent homogeneous semi-invariants of degree n and order i for the binary d-form This argument proves Lemma 1 dimc d n = γ d n, 0 + γ d n, γ d n, d n

3 Poincaré series for covariants 103 The set of weights eigenvalues of the operator E of a representation W denote by Λ W, in particular, Λ Vd = { d, d +,,d,d} A formal sum CharW = k Λ W n W kq k, is called the character of a representation W, here n W k denotes the multiplicity of the weight k Λ W Since, the multiplicity of any weight of the irreducible representation V d is equal to 1, we have CharV d =q d + q d+ + + q d + q d The character CharS n V d of the representation S n V d equals H n q d,q d+,,q d, see [4], where H n x 0,x 1,,x d is the complete symmetrical function H n x 0,x 1,,x d = x α 0 0 x α 1 1 x α d d, α = α i i α =n By replacing x k with q d k,k=0,,d, we obtain the expression for CharS n V d : CharS n V d = q d α 0 q d 1 α 1 q d d α d = α =n = q α 1+α + +dα d = ω d n, kq k, α =n here ω d n, k is the number non-negative integer solutions of the equation k=0 α 1 +α + + dα d = k on the assumption that α 0 + α α d = n In particular, the coefficient of q 0 the multiplicity of zero weight is equal to ω d n,, and the coefficient of q 1 is equal to ω d n, 1 On the other hand, the decomposition implies the equality for the characters: CharS n V d = γ d n, 0CharV 0 +γ d n, 1CharV 1 + +γ d n, d ncharv We can summarize what we have shown so far in Theorem dimc d n = ω d n, + ω d n, 1

4 104 L Bedratyuk Proof The zero weight appears once in any representation V k, for even k, therefore ω d n, = γ d n, 0 + γ d n, + + γ d n, 4 + The weight 1 appears once in any representation V k, for odd k, therefore ω d n, 1 = γ d n, 1 + γ d n, γ d n, 5 + Thus, ω d n, + ω d n, 1 = γ d n, 0 + γ d n, γ d n, d n By using Lemma 1 we obtain dimc d n = ω d n, + ωd n, 1 Note, that the original Cayley-Sylvester formula is dimi d n = ω d n, ω d n, 1 It is well known that the number ω d n, of non-negative integer so- lutions of the following system { α 1 +α + + dα d =, α 0 + α α d = n is given by the coefficient of t n z of the generating function 1 f d t, z = 1 t1 tz 1 tz d We will use the notation [x k ]F x to denote the coefficient of x k in the series expansion of F x C[[x]] Thus ω d n, [ ] = t n z f d t, z [ It is clear that ω d n, = t n z ] f d t, z = [ tz d n] f d t, z Similarly, the number ω d n, 1 of non-negative integer solutions of the following system { α 1 +α + + dα d = 1, α 0 + α α d = n [ ] equals t n z 1 f d t, z = [ t n z 1] f d t, z = [ tz d n] zf d t, z Thus, the following statement holds

5 Poincaré series for covariants 105 Theorem 3 The number dimc d n of linearly independent covariants of degree n for the binary d-form is given by the formula dimc d n = [ tz d n] 1+z 1 t1 tz 1 tz d 3 Explicit formula for P d z Let us prove Springer-type formula for the Poincaré series P d z of the algebra covariants of the binary d-form Consider the C-algebra Z[[t, z]] of formal power series For an integer m, n N define the C-linear function Ψ m,n : Z[[t, z]] Z[[z]], in the following way Ψ m,n t m 1 z z s, if m 1 n 1 m = n 1 n = s N, = 1, if n 1 = m 1 =0, 0, otherwise Then for arbitrary series A Z[[t, z]] we get Ψ m,n A =a 0,0 +a m,n z+a m, n z + Define by ϕ n the restriction of Ψ m,n to Z[[z]], namely ϕ z m n, if m = 0 mod n, n z m := 0, if m 0 mod n, 1, for m =0 It is clear that for arbitrary series A = a 0 + a 1 z + a z +, we obtain ϕ n A =a 0 + a n z + a n z + + a sn z s + In some cases calculating of the functions Ψ can be reduced to calculating of the functions ϕ The following statements hold: Lemma 31 For Rz C[[z]] and for m, n, k N holds Proof Let Rz = Rz Ψ 1,n = 1 tz k ϕ n k Rz,n>k, R0, if n = k, 0, if n<k f j z j Then for k<nwe have j=0 Rz Ψ m,n =Ψ 1 t m z k m,n f j z j t m z k s = f sn k z s j,s 0 s 0 On other hand, ϕ n k Rz = ϕ n k prove the cases n = k and n<k j=0 f j z j = f sn k z s Similarly, we s 0

6 106 L Bedratyuk The main idea of these calculations is that the Poincaré series P d z can be expressed in terms of functions Ψ The following simple but important statement holds Lemma 3 1+z P d z =Ψ 1,d 1 t1 tz 1 tz d Proof Theorem implies that dimc d n =[tz d n ]f d t, z Then P d z = dimc,d n z n = [tz d n ]f d t, z z n =Ψ 1,d f d t, z n=0 n=0 Now we can present Springer-type formula for the Poincaré series P d z Theorem 33 P d z = 1 k z kk z ϕ d k, z,z k z,z d k 0 k<d/ here a, q n =1 a1 aq 1 aq n 1 is q-shifted factorial Proof Consider the partial fraction decomposition of the rational function f d t, z : d f d t, z R k z = 1 tz k k=0 It is easy to see, that R k z = lim fd t, z 1 tz k 1 + z = lim 1 tz k = t z k t z k t, z d+1 = = 1+z 1 z k 1 z k 1 z k 1 k 1 z k+1 k 1 z d k = z k+k z z k 1z k 1 z 11 z 1 z d k = = 1k z kk z z,z k z,z d k Using the above lemmas we obtain P d z=ψ 1,d n k=0 R k z 1 tz k = 0 k<d/ 1 k z kk z ϕ d k z,z k z,z d k

7 Poincaré series for covariants 107 We have implemented the formula in a Maple package The package can be downloaded from the web Using this package the Poincaré series P d z for d 30 are found All these results agree with calculations of the papers [3, 9] and [6] References [1] A Brouwer, A Cohen, The Poincare series of the polynomial invariants under SU in its irreducible representation of degree 17, preprint of the Mathematisch Centrum, Amsterdam, 1979 [] H Derksen, G Kemper, Computational Invariant Theory, Springer- Verlag, New York, 00 [3] V Drensky, GK Genov, Multiplicities of Schur functions with applications to invariant theory and PI-algebras C R Acad Bulg Sci, 57, No 3, 004, 5 10 [4] WFulton, J Harris, Representation theory: a first course, Graduate Texts in Mathematics, 19, New York etc: Springer-Verlag, 1991 [5] P Littelmann, C Procesi, On the Poincaré series of the invariants of binary forms, J Algebra, 133,, 1990, [6] N Onoda, Linear action of G a on polynomial rings, Proc 5th Symp Ring Theory, Matsumoto,199,11 16 [7] M Roberts, The covariants of a binary quantic of the n-th degree, Quarterly J Math, 4,1861, [8] TA Springer, On the invariant theory of SU, Indag Math, 4, [9] JJ Sylvester, F Franklin, Tables of the generating functions and groundforms for the binary quantic of the first ten orders, Amer J Math, II, 1879, 3 51 [10] JJ Sylvester, Tables of the generating functions and groundforms of the binary duodecimic, with some general remarks, and tables of the irreducible syzigies of certain quantics Amer J Math, IV, 1881,41 6 Received: June, 010