Báo cáo toán học: "Some new methods in the Theory of m-Quasi-Invariants" pps

This leads to an interesting free module of triplets of polynomials in the elementary symmetric functionse1, e2, e3 which explains certainobserved properties of S3 m-quasi-invariants.. W

Some new methods in the Theory of

m-Quasi-Invariants

J Bell, A.M Garsia and N Wallach

Department of MathematicsUniversity of California, San Diego, USA

agarsia@math.ucsd.eduSubmitted: Jan 29, 2005; Accepted: Jul 15, 2005; Published: Aug 30, 2005

Abstract

We introduce here a new approach to the study of m-quasi-invariants This

approach consists in representingm-quasi-invariants as N tuples of invariants Thenconditions are sought which characterize such N tuples We study here the case

of S3 m-quasi-invariants This leads to an interesting free module of triplets of

polynomials in the elementary symmetric functionse1, e2, e3 which explains certainobserved properties of S3 m-quasi-invariants We also use basic results on finitely

generated graded algebras to derive some general facts about regular sequences of

Sn m-quasi-invariants

The ring of polynomials in x1, x2, , x n with rational coefficients will be denotedQ[X n]

For P ∈ Q[X n ] we will write P (x) for P (x1, x2, , x n)

Let us denote by s ij the transposition which interchanges x i with x j Note that for

any pair i, j and exponents a, b we have the identities

x a

i x b

j − x a

j x b i

This shows that the ratio in (1.1) is always a polynomial that is symmetric in x i , x j It

immediately follows from (1.1) that the so-called “divided difference”operator

It follows from this that for any P ∈ Q[X n ] the highest power of (x i − x j) that dividesthe difference (1− s ij )P must necessarily be odd This given, a polynomial P ∈ Q[X n]

is said to be “m-quasi-invariant” if and only if, for all pairs 1 ≤ i < j ≤ n, the difference

(1− s ij )P (x)

is divisible by (x i − x j)2m+1 The space of m-quasi-invariant polynomials in x1, x2, , x n

will here and after be denoted “QI m [X n]” or briefly “QI m” Clearly QI m is a vectorspace over Q, moreover since the operators δ ij satisfy the “Leibnitz” formula

we see that QI m is also a ring Note that we have the inclusions

Q[X n] = QI0[X n]⊃ QI1[X n]⊃ QI2[X n]⊃ · · · ⊃ QI m [X n]⊃ · · · ⊃ QI ∞ [X n]

=SYM[X n ]

where SYM[X n ] here denotes the ring of symmetric polynomials in x1, x2, , x n

It was recently shown by Etingof and Ginzburg [4] that eachQI m [X n] is a free moduleover SYM[X n ] of rank n! In fact, this is only the S n case of a general result that isproved in [4] for all Coxeter groups There is an extensive literature (see [1], [3], [5], [7],[9]) covering several aspects of quasi-invariants These spaces appear to possess a rich

combinatorial underpinning resulting in truly surprising identities The S n case deservesspecial attention since the results in this case extend in a remarkable manner many wellknown classical results that hold true for the familiar polynomial ring Q[X n] To be

precise note that for each m we have the direct sum decomposition

generated in QI m [X n ] by the elementary symmetric functions e1, e2, , e n is also S ninvariant, it follows from the Etingov-Ginsburg result that the polynomial on the righthand side of (1.4) is none other than the graded Frobenius characteristic of the quotient

-QI m [X n ]/(e1, e2, , e n)QI m [X n]. (1.5)Unfortunately, the literature on quasi-invariants makes use of such formidable machinerythat presently the theory is accessible only to a few This given, the above examples

should provide sufficient motivation for a further study of S n m-quasi-invariants from a

more elementary point of view

In this vein we find particularly intriguing in (1.4) the degree shift of each isotypiccomponent of QI m expressed by the presence of the factor

q m (n2)−c λ

.

This shift pops out almost magically from manipulations involving a certain

Knizhnik-Zamolodchikov connection used in [6] to compute the graded character of QI m

The present work results from an effort to understand the underlining mechanism that

produces this degree shift In this paper we only deal with the S3 case but the methods

we introduce should provide a new approach to the general study of m-quasi-invariants The idea is to start with what is known when m = 0 and determine the deformations

that are needed to obtain QI m More precisely our point of departure is the followingwell known result

Theorem 1.6 Every polynomial P (x) ∈ Q[X n ] has a unique expansion in the form

It follows from this that each P (x) ∈ Q[X n ] may be uniquely represented by a n! tuple

of symmetric polynomials The question then naturally arises as to what conditions these

symmetric polynomials must satisfy so that P (x) lies in QI m In this work we give a

complete answer for S3 Remarkably, we shall see that, even in this very special case, theanswer stems from a variety of interesting developments We should mention that Feigin

and Veselov in [7] prove the freeness result of the m-quasi-invariants for all Dihedral

groups They do this by exhibiting a completely explicit basis for the quotients analogous

to (1.5) Of course, since the S3 invariants are easily obtained from the

m-quasi-invariants of the dihedral groupd D3, in principle, the results in [7] should have a bearing

on what we do here However, as we shall see in the first section, the freeness result for

m-quasi-invariants is quite immediate whenever the invariants form a polynomial ring on

two generators Moreover, the methods used in [7] are quite distinct from ours and don’treveal the origin of the observed degree shift

This paper is divided in to three sections In the first section we start with a review

of some basic facts and definitions concerning finitely generated graded algebras Twonoteworthy developments in this section are a very simple completely elementary proof

of the freeness result for dihedral groups m-quasi-invariants and the remarkable fact that the freeness result for all m-quasi-invariants follows in a completely elementary manner from one single inequality Namely that the quotient of the ring m-quasi-invariants by the ideal generated by the G-invariants has dimension bounded by the order of G In

the second section we determine the conditions that 6tuples of symmetric functions give

an element of QI m [X3] It develops that the trivial and alternating representations areimmediately dealt with In the third section we show how that these conditions, for the

2-dimensional irreducible of S3, lead to the construction of an interesting free module oftriplets over the ring Q[e1, e2, e3] which is at the root of the observed degree shift for S3

Before we can proceed with our arguments we need to introduce notation and state a fewbasic facts To begin let us recall that the Hilbert series of a finitely generated, gradedalgebraA is given by the formal sum

F A (t) =X

m ≥0

where H m(A) denotes the subspace spanned by the elements of A that are homogeneous

of degree m It is well known that F A (t) is a rational function of the form

F A (t) = P (t)

(1− t) k with P (t) a polynomial The minimum k for which this is possible characterizes the growth

of dimH m(A) as m → ∞ This integer is customarily called the “Krull dimension” of A

and is denoted “dimK A” It is easily shown that we can always find in A homogeneous

elements θ1, θ2, , θ k such that the quotient of A by the ideal generated by θ1, θ2, , θ k

is a finite dimensional vector space In symbols

It is shown that dimK A is also equal to the minimum k for which this is possible When

(2.2) holds true and k = dim K A then {θ1, θ2, , θ k } is called a ”homogeneous system

of parameters”, HSOP in brief.

It follows from (2.2) that if η1, η2, , η N are a basis for the quotient in (2.2) thenevery element of A has an expansion of the form

with coefficients P i (θ1, θ2, , θ k) polynomials in their arguments The algebra A is said

to be Cohen-Macaulay, when the coefficients P i (θ1, θ2, , θ k) are uniquely determined by

P This amounts to the requirement that the collection

is a basis for A as a vector space Note that when this happens and θ1, θ2, , θ k;

η1, η2, , η N are homogeneous of degrees d1, d2, , d k ; r1, r2, , r N then we must sarily have

neces-F A (t) =

PN

i=1t r i

(1− t d1)(1− t d2)· · · (1 − t d k) (2.5)

from which it follows that k = dim K A It develops that this identity implies that,

for any i = 1, 2, , k the element θ i is not a zero a zero divisor of the quotient

A/(θ1, θ2, , θ i −1)A

We call such sequences θ1, θ2, , θ k “regular” Conversely, if A has an HSOP θ1, θ2, , θ k that is a regular sequence, then (2.5) must hold true for any basis η1, η2, , η N of thequotient A/(θ1, θ2, , θ k)A and the uniqueness in the expansions (2.4) must necessarilyfollow yielding the Cohen-Macauliness of A However, for our applications to m-Quasi-

Invariants we need to make use of the following stronger criterion

Proposition 2.6 Let A be finitely generated graded algebra and θ1, θ2, , θ k be an HSOP with d i = degree(θ i ), then A is Cohen-Macaulay and θ1, θ2, , θ k is a regular sequence if and only if

lim

t → −1(1− t d1)(1− t d2)· · · (1 − t d k )F A (t) = dim A/(θ1, θ2, , θ k)A (2.7)This result is known An elemtary proof of it may be found in [8]

A particular example which plays a role here is when A = Q[x1, x2, , x n] is theordinary polynomial ring and the HSOP is the sequence e1, e2, , e n of elementarysymmetric functions As we mentioned in the introduction following result is well knownbut for sake of completeness we give a sketch of the proof

Theorem 2.8 Every polynomial P (x) ∈ Q[x1, x2, , x n ] has a unique expansion of the

Proof It is easily seen that we have

It is easily seen that (2.10) and (2.11) yield an algorithm for expressing, modulo the ideal

(e1, e2, , e n), every monomial as a linear combination of monomials in ART (n) This

implies that the collection

FQ[x1,x2, ,x n](t) = 1

(1− t) n

equality must hold in (2.13), but that implies that the collection in (2.12) has the correctnumber of elements in each degree and must therefore be a basis, proving uniqueness forthe expansions in (2.18)

We can now apply these observations to the study of m-quasi-invariants To begin

note that, we have the following useful fact

Theorem 2.14 To prove that e1, e2, , e n is a regular sequence in QI m [X n ] we need

only construct a spanning set of n! elements for the quotient

QI m [X n ]/(e1, e2, , e n)QI m [X n] (2.15)

In particular the Cohen-Macauliness of QI m [X n ] is equivalent to the statement that this

quotient has n! dimensions.

Proof Let Π(x) denote the Vandermonde determinant

is an injection of Q[x1, x2, , x n] into QI m [X n] This fact combined with the inclusion

QI m [X n]⊆ Q[x1, x2, , x n] yields the coefficient-wise Hilbert series inequalities

F QI m [X n](t) <<

PN

i=1t r i

(1− t)(1 − t2)· · · (1 − t n)combined with (2.16) yields that

n! ≤ N.

On the other hand if we have a spanning set of n! elements for the quotient in (2.17) we

must also have

N ≤ n!

This forces the equality

lim

t → −1(1− t)(1 − t2)· · · (1 − t n )F QI m [X n](t) = dim QI m [X n ]/(e1, e2, , e n)QI m [X n].

Thus we can apply Proposition 2.6 and derive that e1, e2, , e n is a regular sequence in

QI m [X n] This completes our argument

It develops that the regularity of e1, e2, e3, can be shown in a very elementary fashion

for all n This of course implies the Cohen-Macauliness of QI[X3] But before we give

the general argument it will be good to go over the case of e1, e2, e3 in QI[X3] In fact,

we can proceed a bit more generally and work in the Dihedral group setting

Let us recall that the Dihedral group D n is the group of transformations of the x, y plane generated by the reflection T across the x-axis and a rotation R n by 2π/n In

complex notation we may write

It follows from this that the two fundamental invariants of D n are

Note that if n = 2k and we set

(−1) r t k −r

then we may write

with Q(t) a polynomial of degree k − 1 Now setting t = x2/y2 in (2.20) and mutiplying

both sides by y 2k we get, since P ( −1) = (−1) k22k−1

g n (x, y) = ( −1) k22k−1 y 2k + p2(x, y) y 2k−2 Q(x2/y2) This shows that y 2k lies in the ideal (p2, g n)Q[x,y] In particular, under the total order

x > y we derive that x2 and y 2k lie in the upper set of leading monomials of the elements

of this ideal It follows that the monomials

1, y, y2, , y 2k−1 ; x, xy, xy2, , xy 2k−1 (2.21)span the quotient

This forces the Hilbert series inequality

FQ[x,y] (t) <<

(1 + t) 1 + t + · · · + t 2k−1
(1− t2)(1− t 2k) =

1(1− t)2

since we also have

FQ[x,y] (t) = (1− t)1 2

It follows that the monomials in (2.21) are in fact a basis for the quotient in (2.22) An

analogous argument yields a similar result when n = 2k + 1 We need only observe that

in this case we use the polynomial

(−1) r t r

and the total order y > x to obtain that y2 and x 2k+1 are in the upper set of leading

monomials of the ideal (p2, g n)Q[x,y] This implies that

1, x, x2, , x 2k ; y, yx, yx2, , yx 2k

are a basis of the quotient in (2.22) Thus in either case we obtain that and that p2, g n

are a regular sequence in Q[x, y].

It develops that this immediately implies the Cohen Macauliness the ring QI m (D n)

of m-quasi-invariants of D n More precisely we have

Theorem 2.23 The D n invariants p2, g n are a regular sequence in QI m (D n ).

Proof By definition, a polynomial P (x, y) ∈ Q[x, y] is said to be D n m-quasi-invariant

if and only if for any reflection s of D n we have

(1− s)P (x, y) = α s (x, y) 2m+1 P 0 (x, y) (P 0 (x, y) ∈ Q[x, y])

where α s (x, y) denotes the equation of the line accross which s reflects This given, since

QI m (D n)⊆ Q[x, y], we clearly see that p2 itself is not a zero divisor in QI m (D n) So we

need only show that g n is not a zero divisor modulo (p2)QI m (D n) Now suppose that for

some H ∈ QI m (D n) we have

H g n = p2K (with K ∈ QI m (D n )) Then since p2, g n are regular in Q[x, y] it follows that for some K 0 ∈ Q[x, y] we have

H = p2K 0

applying 1− s to both sides the invariance of p2 gives

(1− s)H(x, y) = (x2+ y2)(1− s)K 0 (x, y) and the m-quasi-invariance of H yields that α s (x, y) 2m+1divides the right hand side Since

x2+ y2 has no real factor, the polynomial (1− s)K 0 (x, y) must be divisible by (x, y) 2m+1.

This shows that K 0 ∈ QI m (D n ) proving that g n in not a zero divisor in (p2)QI m (D n) andour argument is complete

Our next step is to use the fact that the Weyl group of A2 is D3 to derive the Macauliness of QI m [X3] To this end set

3x

Since the vector

Theorem 2.28 The elementary symmetric functions e1, e2, e3 are a regular sequence in

QI m [X3].

Proof Clearly, e1 is not a zero divisor inQI m [X3] Likewise, we can show that e2 is not

a zero divisor in QI m [X3]/(e1)QI m [X3] in exactly the same way we showed that g n is not

a zero divisor inQI m (D n )/(p2)QI m (D n) The only remaining step is to show that

Since ψ(C) is an S3 m-quasi-invariant and 23e21 − 2 e2 is invariant, this relation forces D

to be S3 m-quasi-invariant as well and our argument is complete.

We terminate this section by showing that the mechanism we have used for passing

from the Weyl group of A2 to S3 can be extended to all n More precisely we can show

that

Theorem 2.35 For any 1 < i2 < i3 ≤ n the elementary symmetric functions e1, e i2, e i3

are a regular sequence in QI m [X n ].

We start by noting that the same argument we used for D n yields that for any 1 <

i ≤ n the two elementary symmetric functions e1, e i2 are a regular sequence in QI m [X n]

So the extension of the previous argument consists in deriving from this that e i3 is not

a zero divisor in QI m (X n ]/(e1, e i2)QI m [X n] Since e i2, e i3 are basic S n-invariants for the

polynomials on the space V = {x1+x2+· · ·+x n= 0}, this particular step is a consequence

of the following general result To state it we need some definitions

Let λ(x) = a1x1 +· · · + a n x n be a nonzero homogeneous polynomial in n variables and let u be such that λ(u) = 1 Let R be a subalgebra of the algebra of polynomials

on V If f is a polynomial on V we extend f to Qn by setting f (v + tu) = f (v). If

g ∈ Q[x1, , x n ] then we write g for the restiction g |V of g to V Let S be the subalgebra

of Q[x1, x2, , x n ] generated by the extensions of the elements of R and λ This given

with g, g1, , g r ∈ R Restricting to V we have gf1 = 0 Since f1 is not a zero divisor

in R this implies that g = 0 Hence g = λ(g1+· · · + g r λ r −1 ) = λh with h ∈ S Assume

that we have shown that λ, f1, , f j −1 is a regular sequence in S Suppose that we have

gf j = h0λ + h1f1+· · · + h j −1 f j −1 with h l ∈ S for l = 0, , j − 1 Restricting both sides of this equation to V we get

gf j = h1f1+· · · + h j −1 f j −1 . Here h l ∈ R for l = 1, , j − 1 and since f1, , f j is a regular sequence in R this implies

This completes the proof

To apply this result to m-quasi-invariants We take λ(x) = e1 = x1 + · · · + x n,

u = (1, , 1)/n and V the zero set of e1 Finally we take R be the S n m-quasi-invariants

polynomials on V and let S = QI M [X n] The only missing ingredient is given by thefollowing

Lemma 2.37 QI m [X n ] is the subalgebra of Q[x1, , x n ] generated by R and e1.

Proof First observe that if g = he1 and g ∈ QI m [X n ] then h ∈ QI m [X n ] Indeed, if α

is a root of A n −1then (1− s α )g = ((1 − s α )h)e1 Now (1− s α )g = α 2m+1 w Thus have

with f i polynomials on V then f i ∈ R for all i Note that the assertion is trivially true

for r = 0 We can thus proceed by induction on r and assume the assertion true up to

r − 1 To prove it for r note that if we restrict both sides of (2.38) to V we have f = f0.

Since f ∈ QI m [X n ], f ∈ R Thus f0 ∈ R Now f − f0 = e1(f1+· · · + f r e r1−1) From the

observation at the beginning of the proof we derive that f1+· · · + f r e r1−1 ∈ QI m [X3] andthe induction hypothesis completes the argument

Using Theorem 2.8 we will start by writing every element P (x) ∈ QI m [X3] in the form

P (x) = A000+ A010x2+ A001x3+ A011x2x3 + A002x22+ A012x2x23. (3.1)

Our goal is to see what conditions the coefficients A ijk must satisfy to assure that

P (x) ∈ QI m [X3] The idea is to use the fact that the spaces QI m [X n ] are S n modules

to gain information about these kinds of expansions This given, our point of departure

is the following identity in the algebra of S3

id = S3 +13(1− s12)(1 + s23) + 13(1− s23)(1 + s12) +A3 (3.2)

where

S3 = 16 1 + s12+ s13+ s23+ (1, 2, 3) + (3, 2, 1)
and

A3 = 16 1− s12− s13− s23+ (1, 2, 3) + (3, 2, 1)
Note that, since the operator A3 kills all the monomials 1, x2, x3, x2x3, x23, applying it to

However, note that it is an immediate consequence of the definition that any alternant in

QI m [X n ] must be a multiple of Π(x) 2m+1 by a symmetric polynomial This implies that

the symmetric polynomial A012 must necessarily be a multiple of Π3(x) 2m Note furtherthat a multiple of Π3(x) 2m by any polynomal in x1, x2, x3 lies inQI m [X3] This given, we

see that A012 may here and after be assumed to be of the form A012 = B(x)∆3(x) 2m with

B(x) an arbitrary symmetric polynomial It is also clear that A000 can also be arbitrarilychosen This reduces our study to the elements of QI m [X3] which are of the form

P (x) = A010x2+ A001x3+ A011x2x3+ A002x23. (3.3)When we apply the identity in (3.2) to this expansion we derive that

P (x) = A(x) + 13(1− s12)(1 + s23)P (x) + 13(1− s23)(1 + s12)P (x)

with A(x) a suitable symmetric polynomial This is because A3 kills every monomial in

(3.3) and S3 sends every monomial into a symmetric function

Now we see that

(1 + s23)P (x) = A010(x2+ x3) + A001(x2 + x3) + 2A011x2x3+ A002(x22+ x23) (3.4)but we can easily check that we have

together with their images s12P1 and s23P2

Now it develops that we have the following remarkably simple criterion

Theorem 3.6 The polynomials P1 = A1(x2+ x3) + B1x2x3 and P2 = A2x3+ B2x23, with

A1, A2, B1, B2 symmetric, are m-quasi-invariant if and if only we have

Proof We begin by proving necessity To this end note that for P1(x) to be

m-quasi-invariant we must have

Finally, multiplying both sides by x2+ x3 and applying δ12 we get

A2 = δ12(x2+ x3)(x1− x3)2m θ2(x) (3.16)

This proves (3.7) (b) To complete our proof of necessity, we are only left to show that

θ1 and θ2 must satisfy (3.8) (b) It turns out that (3.8) (b) is all we need to assure the

symmetry of A1, A2, B1, B2 To show this it is convenient to set

so that (3.12), (3.13), (3.18) and (3.19) become

A1 = δ12x1H1, A2 =−δ12(x2+ x3)H2,

Clearly, A1, A2, B1, B2 are symmetric if and only if they are invariant under the action

of s12 and s23 However, since all of them are images of δ12 there are automatically s12

-invariant Thus we only need to assure that they are also s23-invariant Note that since,

It develops that the first equation here is a consequence of the second To see this note

that since (3.21) (b) implies that δ12H1 is symmetric in particular it is left unchanged by