Separating invariants for the basic G_a-actions
aa r X i v : . [ m a t h . A C ] N ov SEPARATING INVARIANTS FOR THE BASIC G a -ACTIONS JONATHAN ELMER AND MARTIN KOHLS
Abstract.
We explicitly construct a finite set of separating invariants forthe basic G a -actions. These are the finite dimensional indecomposable ra-tional linear representations of the additive group G a of a field of character-istic zero, and their invariants are the kernel of the Weitzenb¨ock derivation D n = x ∂∂x + . . . + x n − ∂∂x n . Keywords:
Invariant theory, separating invariants, binary forms, locallynilpotent derivations, basic G a -actions, generalized hypergeometric series. AMS Classification: Introduction
A great many mathematical problems are special cases of the following: let K be a field of arbitrary characterstic and let G be any group. Suppose G actson the K -vector space V and that v and w are points of V . Is there a g ∈ G satisfying gv = w ? In other words, are v and w contained in the same G -orbit?Important examples include the case where G = GL n ( K ) acts on the vector space V of n × n matrices by conjugation, and the case where G = SL n ( K ) acts on thespace V of binary forms of degree n . The classical approach to these problems isto construct “invariant polynomials”. These are polynomial functions f : V → K which satisfy f ( v ) = f ( gv ) for all g ∈ G and v ∈ V , and so are constant on G -orbits. In fact, one can define an action of G on the set of polynomial functions K [ V ] via ( g · f )( v ) := f ( g − v ) for which the invariant polynomials are the fixedpoints, K [ V ] G , and these form a subalgebra of K [ V ] called the algebra of invariants .Ideally, one would like to find a complete set of algebra generators of K [ V ] G , thenuse this set to distinguish as many orbits as possible.This approach is not without its difficulties. For instance, it is not always possibleto distinguish all the orbits using invariant polynomials. As an example, consideronce more the case where GL n ( K ) acts on the vector space of n × n matrices over afield K by conjugation. Provided K is an infinite field, the invariants are generatedby the coefficients of the characteristic polynomial [5, Example 2.1.3], but it iswell known that a pair of matrices with the same characteristic polynomial are notnecessarily conjugate. More problematically, if G is not reductive then the algebra K [ V ] G may not even be finitely generated. Even if it is, finding a set of generatorscan be a very difficult problem. If, however, one is only interested in invariants fromthe point of view of separating orbits, then finding a complete set of generators isnot always necessary. It is perhaps surprising, then, that Derksen and Kempermade the following defininition [5, Definition 2.3.8] as recently as 2002. Definition 1.1. A separating set for the ring of invariants K [ V ] G is a subset S ⊆ K [ V ] G with the following property: given v, w ∈ V , if there exists an invariant f satisfying f ( v ) = f ( w ), then there also exists s ∈ S satisfying s ( v ) = s ( w ).There are many instances in which separating sets can be seen to have “nicer”properties than generating sets. For example, it is well known that if G is finite and Date : June 7, 2018. the characteristic of K does not divide | G | , then K [ V ] G is generated by elementsof degree ≤ | G | , see [10, 11], but this is not necessarily true in the modular case.On the other hand, the analogue for separating invariants holds in arbitary charac-teristic [5, Theorem 3.9.13]. Meanwhile, even if K [ V ] G is not finitely generated, itis guaranteed to contain a finite separating set [5, Theorem 2.3.15]. The existenceproof is non-constructive, which raises the question how to actually construct sepa-rating sets. Kemper [15] gives an algorithm for reductive groups, but using Gr¨obnerbases it is only effective for “small” cases. An example of a finite separating set fora non finitely generated invariant ring is given in [8]. For finite groups, a separatingset can always be obtained as the coefficients of a rather large polynomial [5, The-orem 3.9.13]. With refined methods, “nicer” separating sets have been obtainedfor several classes of finite groups and representations, see for example [20]. Thispaper goes in the same direction: for the basic actions of the additive group incharacteristic zero, we present a rather small separating set. See also [6, 7, 9, 16]for a small selection of other recent publications in the area.From this point onwards, k denotes a field of characterstic zero. In this articlewe will concentrate on linear actions of the additive group G a of the ground field k .The finite dimensional indecomposable rational linear representations of G a arecalled the basic G a -actions. There is one such action in each dimension, and theseare described below: Definition 1.2.
Let X n := h x , . . . , x n i k be a vector space of dimension n + 1.Then G a is said to act basically on X n (with respect to the given basis) if theaction of G a on X n is given by the formula a ∗ x i = i X j =0 a j j ! x i − j , for all a ∈ G a , i = 0 , . . . , n. Note the isomorphisms X n ∼ = X ∗ n ∼ = S n ( X ) for all n , where S n denotes the n th symmetric power. Let { x , x , . . . , x n } be the set of coordinate functions on a n + 1 dimensional vector space V n , so we consider X n = V ∗ n . As k is infinite, k [ V n ]can be viewed as the polynomial ring R n := S ( X n ) = k [ x , x , . . . , x n ]. If G a actsbasically on X n , one can then check that the induced action of G a on R n is givenby the formula a ∗ f = exp( aD n ) f for all a ∈ G a , f ∈ R n , where D n is the Weitzenb¨ock derivation D n := x ∂∂x + . . . + x n − ∂∂x n on R n . Furthermore, the algebra of invariants k [ V n ] G a is precisely the kernel of D n . Wedenote this by A n . The algebras A n have been objects of intensive study for wellover a hundred years, owing to their connection with the classical invariants andcovariants of binary forms. While they are known to be finitely generated by theMaurer-Weitzenb¨ock Theorem [24], the number of generators appears to increaserapidly with dimension, and explicit generating sets are (reliably!) known only for n ≤ all values of n .This article is organised as follows: in Section 2 we state our main results, andexplain briefly the connection between the algebras A n and the covariants of binaryforms. In Section 3 we prove a crucial lemma on the radical of the Hilbert idealof A n which may be of independent interest. Section 4 contains the main body ofthe proof of our result, while Section 5 is devoted to the proof of a technical lemmawhich is required in order to construct a separating set for A n when n ≡ ASIC G a -ACTIONS 3 Most of this work was completed during a visit of the first author to TU M¨unchenin July 2010. We would like to thank Gregor Kemper for making this visit possible.2.
Background and statement of results
Let U n denote the k -vector space of binary forms of degree n , which are homo-geneous polynomials of the form P ni =0 a i X i Y n − i in the variables X and Y , a i ∈ k .This is a vector space of dimension n + 1 with basis the set of monomials in X and Y of degree n . The natural action of the group G := SL ( k ) on a two dimen-sional vector space with basis { X, Y } induces an action of G on the vector space U n . Classically speaking, an invariant is a polynomial in the coefficients a i which isunchanged under the action of G - in modern notation, an element of k [ U n ] G . Notethat the additive group G a is embedded in G as the subgroup of matrices of the form (cid:18) ∗ (cid:19) , and the subgroup G a acts basically on U n (with respect to the basis { k ! X n − k Y k : k = 0 , . . . , n } ). The pioneers of invariant theory also studied “covari-ants”, which are polynomials in both the coefficients a i and the variables X and Y themselves which are fixed under the action of G . In modern notation, the algebraof covariants is k [ U n ⊕ U ∗ ] G . There is, in fact, an even stronger connection betweencovariants and the basic actions of G a : the algebras k [ U n ⊕ U ∗ ] G and k [ U n ] G a areactually isomorphic. Let us identify the algebra k [ U n ⊕ U ∗ ] with the polynomialring k [ a , a , · · · , a n , X, Y ] (we abuse notation by using the same letters a i for co-ordinates and coordinate functions). Define a mapping Φ : k [ U n ⊕ U ∗ ] G → k [ U n ] G a by(1) Φ( f ( a , a , a , . . . , a n , X, Y )) := f ( a , a , a , . . . , a n , , . The theorem of Roberts [19] states that Φ is an isomorphism. In classical invarianttheory one often studies the basic actions of G a in order to get a handle on thecovariants of binary forms using Roberts’ isomorphism. One word of caution isneeded at the point. While [16, Proposition 1] implies that a separating set for k [ U n ⊕ U ∗ ] SL ( k ) must be mapped under Φ to a separating set for k [ U n ] G a , theconverse is not necessarily true, so the separating sets we construct in this papermost likely do not lift to give separating sets for the covariants of binary forms (cf.[16, Remark 3]).We now state our main results. For any real number x , the symbol [ x ] denotesthe largest integer less than or equal to x . We begin by definining some importantinvariants, namely(2) f m := m − X k =0 ( − k x k x m − k + 12 ( − m x m ∈ ker D n for m = 1 , . . . , [ n f := x . Further, we define the elements(3) s m := m X k =0 ( − k m + 1 − k x k x m +1 − k ∈ R n for m = 1 , . . . , [ n −
12 ]and s := x , which satisfy D n s m = f m , for all m, and in particular D n s m ∈ A n \ { } . Elements with this property are called localslices. For any a ∈ R n \ { } , let ν ( a ) denote the nilpotency index ν ( a ) := min { m ∈ N : D m +1 n ( a ) = 0 } , and ν (0) := −∞ . If s ∈ R n is a local slice, then for any a ∈ R n we JONATHAN ELMER AND MARTIN KOHLS define ǫ s ( a ) := (exp( tD n ) a ) | t := − s/D n s · ( D n s ) ν ( a ) = ν ( a ) X k =0 ( − k k ! ( D kn a ) s k ( D n s ) ν ( a ) − k ∈ A n . By the Slice Theorem [12, Corollary 1.22], we have A n ⊆ k [ ǫ s ( x ) , . . . , ǫ s ( x n )] D n s . When s = x and so D n s = x , this is the first stage in Lin Tan’s (and van denEssen’s) algorithm for producing a generating set for A n [22, 23]. Theorem 2.1.
Given n , we define a set E n consisting of the following elements: f , f , . . . , f [ n ] ,ǫ s ( x ) , . . . , ǫ s ( x n ) ,ǫ s ( x ) , . . . , ǫ s ( x n ) ,ǫ s ( x ) , . . . , ǫ s ( x n ) ,ǫ s ( x ) , . . . , ǫ s ( x n ) , ... ǫ s [ n −
12 ] ( x [ n − ] ) , . . . , ǫ s [ n −
12 ] ( x n ) . If n ≡ we also append to E n an extra invariant w which is defined inLemma 5.4. Then the set E n is a separating set for A n . Note that ǫ s ( x ) = f and ǫ s ( x ) = 0. The size of this separating set is about n . The following table shows its exact size for some values of n. The lower linegives the size c n of a minimal generating set for A n , see Olver [17, p. 40]. Olver saysthis list can not be trusted for n ≥
7. For c , we use Bedratyuk’s value c = 147[3], while in Olver’s list values c = 124 or c = 130 are offered, depending onthe source. We also want to remark that for n ≥
5, we could save 5 elements byreplacing the 10 elements of E \ { w } appearing in E n by the 5 generators of A .(For n = 4 we could save 6 elements). Note that generators for n ≤ n 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | E n |
11 16 20 28 34 43 49 61 69 82 90 106 116 133 143 163 175 c n It is also worth noting that our separating set consists of invariants whose degreeis at most 2 n + 1. 3. The radical of the Hilbert ideal
Let R n , D n and A n be as in the introduction. For any m < n we have thealgebra homomorphism π m,n : R n → R m , f ( x , x , . . . , x n ) f ( 0 , . . . , | {z } n − m times , x , . . . , x m )which satisfies π m,n ◦ D n = D m ◦ π m,n and thus induces a map A n → A m .Consider the Hilbert ideal I n := A n, + R n ✂ R n . With the invariants f m definedin (2), we get the following inclusion for its radical:(4) ( x , . . . , x [ n ] ) R n = q ( f , f , . . . , f [ n ] ) R n ⊆ p I n . The main purpose of this section is to prove that the reverse inclusion holds too.
ASIC G a -ACTIONS 5 Proposition 3.1. (a)
The radical of the Hilbert ideal is given by p I n = ( x , . . . , x [ n ] ) R n . (b) π n − [ n ] − ,n ( A n ) = k . (c) π m, m ( A m ) = k [ x ] for m odd. (d) π m, m ( A m ) = k [ x , x ] for m even.Proof. We will make use of Roberts’ isomorphism as defined in the previous section,with the only difference that we will choose variables so that the G a -actions becomebasic, using the notations of the introduction. Additionally, let G a act basically on h y , y i k . The action of G a on ˜ R n := R n [ y , y ] extends to an action of SL ( k ) on˜ R n such that the following holds:(1) for any a ∈ k \{ } and µ a := (cid:18) a − a (cid:19) ∈ SL we have µ a ( x k ) = a k − n x k for k = 0 , . . . , n and µ a ( y k ) = a k − y k for k = 0 , τ := (cid:18) −
11 0 (cid:19) ∈ SL we have τ ( x k ) = ( − k ( n − k )! k ! x n − k for k =0 , . . . , n and τ ( y k ) = ( − k y − k for k = 0 , R n → R n , f ( x , . . . , x n , y , y ) f ( x , . . . , x n , , R SL n → R G a n = A n . Now let f ∈ A n and F ∈ ˜ R SL n with Φ( F ) = f . Then we also have f = Φ( µ a · F ) for all a ∈ k \ { } , i.e. f ( x , . . . , x n ) = F ( a − n x , a − n +2 x , . . . , a n x n , , a ) for all a ∈ k \ { } . Thus, π n − [ n ] − ,n ( f ) = F (0 , . . . , , a n ]+1) − n x , . . . , a n x n − [ n ] − , , a ) for all a ∈ k \{ } , and since this equation is polynomial in a and k is an infinite field, it also holdsfor a = 0. Therefore, π n − [ n ] − ,n ( f ) = F (0 , . . . , ∈ k , which proves (a) and (b).Similarly, for n = 2 m we find π m, m ( f ) = F (0 , . . . , , x , a x , . . . , a m x m , , a ) for all a ∈ k \ { } , which is again polynomial in a . When a = 0, we get π m, m ( f ) = F (0 , . . . , , x , , . . . ,
0) = p ( x )for some polynomial p ( x ) ∈ k [ x ], so π m, m ( A m ) ⊆ k [ x ]. Since π m, m ( f m ) = ( − m x , we get the inclusion k [ x ] ⊆ π m, m ( A m ). Using that F is also invariantunder τ , we find in the same manner as before π m, m ( f ) = π m, m (Φ( τ µ a − F )) | a =0 = F (0 , . . . , , ( − m x , , . . . ,
0) = p (( − m x ) . Therefore, for m odd we get π m, m ( f ) = p ( x ) = p ( − x ) ∈ k [ x ], which proves(c). For m even, to prove (d), we refer to Lemma 5.4, which gives a w ∈ A m with π m, m ( w ) = x , so k [ x , x ] ⊆ π m, m ( A m ) ⊆ k [ x ]. Since x generates the degreeone elements of A m and π m, m ( x ) = 0, we are done. ✷ We want to mention here that the method of proof for Proposition 5.4 (a) alsoworks for decomposable actions. Consider R := k [ x , , . . . , x n , , . . . , x ,k , . . . , x n k ,k ]and D = D n + . . . + D n k with D n i = x ,i ∂∂x ,i + . . . + x n i − ,i ∂∂x ni,i . Using analgebra homomorphism π which behaves on each subalgebra k [ x ,i , . . . , x n i ,i ] as π n i − [ ni ] − ,n i , we get with the same proof JONATHAN ELMER AND MARTIN KOHLS
Theorem 3.2.
The radical of the Hilbert ideal of ker D is given by ( x , , . . . , x [ n ] , , . . . , x ,k , . . . , x [ nk ] ,k ) R. Construction of a separating set
In this section, we prove our main result.
Proof of Theorem 2.1.
For V n = k n +1 with k [ V n ] = R n , assume there are twoelements a = ( a , . . . , a n ) and b = ( b , . . . , b n ) of V n such that f ( a ) = f ( b ) for all f ∈ E n . We have to show that f ( a ) = f ( b ) for all f ∈ A n . As x ∈ E n , wehave a = b . Assume first a = b = 0. By the Slice Theorem, A n ⊆ k [ E n ] x .Therefore, f ∈ A n can be written as f = px l with p ∈ k [ E n ] and l ≥
0. Byassumption, p ( a ) = p ( b ) and a l = b l = 0, so f ( a ) = p ( a ) /a l = p ( b ) /b l = f ( b ).Now assume a = b = 0 and let m be maximal such that a = a = . . . = a m = 0,so a m +1 = 0 (if m < n ). By induction on k , we shall show that b k = 0 for k = 0 , . . . , min { m, [ n ] } . By assumption this holds for k = 0, so assume it holds forsome k < min { m, [ n ] } . Then(5) ( − k +1 a k +1 = f k +1 ( a ) = f k +1 ( b ) = ( − k +1 b k +1 , so b k +1 = 0 since a k +1 = 0. If m ≥ [ n ], then f ( a ) = f (0) = f ( b ) for any f ∈ A n by Proposition 3.1, so now assume 0 ≤ m < [ n ]. Equation (5) for k = m shows0 = a m +1 = b m +1 . We now distinguish different cases. m < [ n − ]. Then s m +1 is defined, and by the Slice Theorem A n ⊆ k [ ǫ s m +1 ( x ) , . . . , ǫ s m +1 ( x n )] f m +1 . Applying π := π n − m − ,n on both sides yields π ( A n ) ⊆ k [ π ( ǫ s m +1 ( x m +1 )) , . . . , π ( ǫ s m +1 ( x n ))] π ( f m +1 ) , where we used that π n − m − ,n ( ǫ s m +1 ( x k )) = 0 for k = 0 , . . . , m . The right handside is included in k [ π ( E n )] π ( f m +1 ) . Therefore, for any f ∈ A n there is p ∈ k [ E n ]and l ≥ π ( f ) = π ( p ) π ( f m +1 ) l . Let γ : V n → V n − m − , ( c , . . . , c n ) ( c m +1 , . . . , c n ) . Then f ( a ) = π ( f )( γ ( a )) = π ( p ) π ( f m +1 ) l ( γ ( a )) = π ( p )( γ ( a ))( π ( f m +1 )( γ ( a ))) l = p ( a ) f m +1 ( a ) l = p ( b ) f m +1 ( b ) l = π ( p )( γ ( b ))( π ( f m +1 )( γ ( b ))) l = π ( f )( γ ( b )) = f ( b ) . Here we used that the elements p and f m +1 of E n take the same value on a, b byassumption, and f m +1 ( a ) = f m +1 ( b ) = 0. [ n − ] = m < [ n ]. In this case, n has to be even, and n = 2 m ′ with m ′ = m + 1. Let π and γ be as before, so π = π m ′ , m ′ and γ : V m ′ → V m ′ . Firstassume m ′ is odd. By Proposition 3.1 (c) we have π ( A n ) = k [ x ] = k [ π ( f m ′ )] . If m ′ is even, by Proposition 3.1 (d) we have π ( A n ) = k [ x , x ] = k [ π ( f m ′ ) , π ( w )] , with w the element of Lemma 5.4. In both cases, π ( A n ) = k [ π ( E n )], so for any f ∈ A n , there exists p ∈ k [ E n ] such that π ( f ) = π ( p ). Therefore, f ( a ) = π ( f )( γ ( a )) = π ( p )( γ ( a )) = p ( a ) = p ( b ) = π ( p )( γ ( b )) = π ( f )( γ ( b )) = f ( b ) . We have shown: for any f ∈ A n we have f ( a ) = f ( b ), and so we are done. ✷ ASIC G a -ACTIONS 7 The existence of w . In this section we prove Lemma 5.4, which requires some more machinery. Notethat we need this Lemma in order to construct a separating set only in the casewhere n ≡ w is not contained in our separatingset. We will make use of semitransvectants, which are the classical transvectantstransformed under Roberts’ isomorphism, see for example [3, 4, 17]. Recall that fora covariant F ∈ ˜ R SL n = R n [ y , y ] SL , its total degree in y , y is called the order of F . For covariants F and G of orders l and m respectively, we can construct newcovariants given by h F, G i ( r ) := r X k =0 ( − k (cid:18) rk (cid:19) ∂ r F∂y r − k ∂y k ∂ r G∂y k ∂y r − k r ≤ min( l, m ) , which is called the r th transvectant of F and G (see [17, p. 88]). Transvectants playa key role in Gordan’s famous proof of the finite generation of covariants of binaryforms [13]. The transformation of this construction under Roberts’ isomorphismleads the following definition (see also [2]). Definition 5.1.
Let Φ : R n [ y , y ] SL → R G a n = A n be Roberts’ isomorphism,given by substituting y := 0 , y := 1. Let f and g be a pair of invariants in A n .Then for r ≤ min( l, m ), where l and m are the orders of Φ − ( f ) and Φ − ( g ) asabove, we define the r th semitransvectant of f and g by[ f, g ] ( r ) := Φ( h Φ − ( f ) , Φ − ( g ) i ( r ) ) . In order to get an explicit expression for the semitransvectant, we introduce asecond derivation on R n , which is somewhat inverse to D n :∆ n := n X k =0 ( n − k )( k + 1) x k +1 ∂∂x k . This derivation comes from the other canonical embedding of G a in SL , namelyfor f ∈ R n , a ∈ k we have (cid:18) a (cid:19) ∗ f = exp( a ∆ n ) f . Let ord( f ) denote thenilpotency index of f with respect to ∆ n . Assume F ∈ R n [ y , y ] SL is homogeneousof degree m in the variables y , y , so it can be written in the form F = f y m + y · ( . . . ) with f ∈ R n . Then Φ( F ) = f , and invariance of F under the torusaction implies all terms x a . . . x a n n in f satisfy m = P nk =0 ( n − k ) a k . A polynomial f ∈ R G a n with this property is called isobaric of weight m , and then we have m = ord( f ). For an isobaric f ∈ R G a n , by [14, p. 43] the inverse of Roberts’isomorphism is given byΦ − ( f ) = ord( f ) X i =0 ( − i ∆ in ( f ) i ! y i y ord( f ) − i . Proposition 5.2.
Let f, g ∈ R G a n be isobaric. Then for r ≤ min(ord( f ) , ord( g )) ,the r th semitransvectant of f and g is given by the formula [ f, g ] ( r ) = r X k =0 ( − k (cid:18) rk (cid:19) ∆ kn ( f ) (ord( f ) − k )!(ord( f ) − r )! ∆ r − kn ( g ) (ord( g ) − r + k )!(ord( g ) − r )! Proof.
Let ∆ in ( f ) i ! := λ i and ∆ in ( g ) i ! := µ i . Then ∂ r Φ − ( f ) ∂y r − k ∂y k = ord( f ) − k X i = r − k ( − i λ i i !( i − r + k )! (ord( f ) − i )!(ord( f ) − i − k )! y i − r + k y ord( f ) − i − k and ∂ r Φ − ( g ) ∂y k ∂y r − k = ord( g ) − r + k X i = k ( − i µ i i !( i − k )! (ord( g ) − i )!(ord( g ) − i − r + k )! y i − k y ord( g ) − i − r + k , therefore Φ (cid:18) ∂ r Φ − ( f ) ∂y r − k ∂y k (cid:19) = ( − r − k λ r − k ( r − k )!(ord( f ) − r + k )!(ord( f ) − r )!and Φ (cid:18) ∂ r Φ − ( g ) ∂y k ∂y r − k (cid:19) = ( − k µ k k !(ord( g ) − k )!(ord( g ) − r )! . Using the fact that Φ is an algebra homomorphism we have[ f, g ] ( r ) = r X k =0 ( − k + r (cid:18) rk (cid:19) λ r − k ( r − k )!(ord( f ) − r + k )!(ord( f ) − r )! µ k k !(ord( g ) − k )!(ord( g ) − r )! , = r X k =0 ( − k + r (cid:18) rk (cid:19) ∆ r − kn ( f ) (ord( f ) − r + k )!(ord( f ) − r )! ∆ kn ( g ) (ord( g ) − k )!(ord( g ) − r )!= r X k =0 ( − k (cid:18) rk (cid:19) ∆ kn ( f ) (ord( f ) − k )!(ord( f ) − r )! ∆ r − kn ( g ) (ord( g ) − r + k )!(ord( g ) − r )!as required. ✷ Remark . This is analogous to [2, Lemma 1], using a different basis.The formula shows that, up to some scalar factor, f m (from (2)) equals [ x , x ] (2 m ) (while [ x , x ] ( r ) = 0 for r odd), and ǫ s m ( x ) equals [ x , f m ] (1) . We wonder whetherthere is also a connection between ǫ s m ( x j ) and [ x , f jm ] ( j ) . Lemma 5.4.
Suppose n is divisible by 4, so n = 2 m = 4 p . Then there is aninvariant w ∈ A n satisfying π m,n ( w ) = x .Proof. Throughout the proof we use the shorthand π := π m,n , and we set f := f p = P mi =0 ( − i x i x m − i (which is proportional to [ x , x ] ( m ) ). Obviously, f is isobaricof weight ( n − i ) + ( n − m − i )) = 2 n − m = n = ord( f ). Thus we may define¯ w := [ x , f ] ( n ) = n X k =0 ( − k n ! k !( n − k )! x k ∆ n − kn ( f ) = n X k =0 ( − k n ! ( n − k )! k ! x n − k ∆ kn ( f ) , where we used Proposition 5.2. Thus, π ( ¯ w ) = m X k =0 ( − k n ! ( n − k )! k ! x m − k π (∆ kn ( f )) . Using Leibniz’s formula for iterated differentiation of products, we have π (∆ kn ( x i x m − i )) = k X j =0 (cid:16) kj (cid:17) π (∆ jn x i ) π (∆ k − jn x m − i )= k − i X j = m − i (cid:16) kj (cid:17) ( i + j )!( n − i )! i !( n − i − j )! π ( x i + j ) ( m − i + k − j )!( m + i )!( m − i )!( m + i − k + j )! π ( x m − i + k − j )= k − m X j =0 (cid:16) kj + m − i (cid:17) ( m + j )!( n − i )! i !( m − j )! π ( x m + j ) ( k − j )!( m + i )!( m − i )!( n − k + j )! π ( x k − j ) . ASIC G a -ACTIONS 9 In particular, π (∆ kn ( x i x m − i )) = 0 for all k < m , and since f is a linear combinationof terms of the form x i x m − i , we have π (∆ kn ( f )) = 0 for all k < m . From this,remembering m is even, it follows that π ( ¯ w ) = n ! x π (∆ mn ( f )). Therefore, since π (∆ mn ( x i x m − i )) = (cid:18) mm − i (cid:19) x ( n − i )! i ! ( m + i )!( m − i )! , and f = P mi =0 ( − i x i x m − i , we obtain π (∆ mn ( f )) = 12 x m X i =0 ( − i (cid:18) mi (cid:19) ( n − i )! i ! ( m + i )!( m − i )!= ((2 p )!) x
20 2 p X i =0 ( − i (cid:18) pi (cid:19) (cid:18) p − i p (cid:19) (cid:18) p + ii (cid:19) . Thus, π ( ¯ w ) = n ! x π (∆ mn ( f )) is a nonzero multiple of x if the sum above isnonzero. This follows from Lemma 5.6. ✷ Remark . With g := ∆ nn ( f ), we have ¯ w = c · P nk =0 ( − k x k D k g with c ∈ k . Lemma 5.6.
For all p ≥ we have p X k =0 ( − k (cid:18) pk (cid:19) (cid:18) p − k p (cid:19) (cid:18) p + kk (cid:19) = ( − p (3 p )!( p !) . Proof.
The argument which follows was produced using the implementation of Zeil-berger’s algorithm [25] in the remarkable EKHAD package for Maple [18]. Let F ( p, k ) := ( − k (cid:18) pk (cid:19) (cid:18) p − k p (cid:19) (cid:18) p + kk (cid:19) ≤ k ≤ p , and let S ( p ) := P pk =0 F ( p, k ). We show that the following recurrence relation holds:(6) 6(3 p + 2)(3 p + 1) S ( p ) + 2( p + 1) S ( p + 1) = 0 . To do this, we consider the function G ( p, k ) := 12 k (180 − k + 1036 p + 59 k + 2192 p − pk − p k + 168 pk +2024 p − k + 688 p + k − p k + 120 p k − pk )( − p + k − F ( p, k ) R ( p, k )where R ( p, k ) = (2 p + 1)( − p − k ) ( − p − k ) . For 0 ≤ k ≤ p − G ( p, k + 1) − G ( p, k ) = 6(3 p + 2)(3 p + 1) F ( p, k ) + 2( p + 1) F ( p + 1 , k ) . Summing both sides over 0 ≤ k ≤ p − S ( p ) = ( − p (3 p )!( p !) . ✷ Remark . The sum S ( p ) is the well-poised hypergeometric series (cid:18) p p (cid:19) ∞ X k =0 ( − p ) k (2 p + 1) k ( − p ) k (1) k ( − p ) k k ! = (cid:18) p p (cid:19) F [ − p, p + 1 , − p ; 1 , − p ; 1] . Surprisingly, the series is not summable by any classical hypergeometric sum the-orem (e.g. Dixon’s theorem, Watson’s theorem) because the series F [ − p, p +1 , − p ; 1 , − p ; z ], p not an integer, does not converge when z = 1, see [21, Chap-ter 2]. For this reason, we have to apply Zeilberger’s algorithm for partial sums inorder to sum the series. In the language of WZ-theory, ¯ F , G is a WZ-pair, where¯ F ( p, k ) := ( − p ( p !) F ( p,k )(3 p )! , and R ( p, k ) is the corresponding WZ-proof certificate. References [1] Leonid Bedratyuk. Casimir elements and kernel of weitzenb¨ock derivation. arXiv:math/0512520 , 2005.[2] Leonid Bedratyuk. On complete system of invariants for the binary form of degree 7.
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