Toric geometry and equivariant bifurcations

Physica D 143 (2000) 235–261

Toric geometry and equivariant bifurcations

Ian Stewarta, Ana Paula S. Diasb,∗a Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

b Departamento de Matemática Pura, Centro de Matemática Aplicada, Universidade do Porto, 4050 Porto, Portugal

To the memory of John David Crawford

Abstract

Many problems in equivariant bifurcation theory involve the computation of invariant functions and equivariant mappingsfor the action of a torus group. We discuss general methods for finding these based on some elementary considerationsrelated to toric geometry, a powerful technique in algebraic geometry. This approach leads to interesting combinatorialquestions about cones in lattices, which lead to explicit calculations of minimal generating sets of invariants, from whichthe equivariants are easily deduced. We also describe the computation of Hilbert series for torus invariants and equivariantswithin the same combinatorial framework. As an example, we apply these methods to the interaction of two linear modesof a Euclidean-invariant PDE on a rectangular domain with periodic boundary conditions. © 2000 Elsevier Science B.V. Allrights reserved.

Keywords:Toric geometry; Equivariant bifurcation; Euclidean-invariant

1. Introduction

Equivariant bifurcation theory is the study of parametrised families of ordinary differential equations (ODEs)

dx

dt= f (x, λ) (1)

when the vector fieldf is equivariant under a group action. More specifically, we shall work in the following context:x ∈ Rm, λ ∈ Rp is a bifurcation parameter, andf is a smooth vector field onRm. Let 0 be a compact Lie groupacting linearly onV = Rm. We say that a vector fieldh onV is 0-equivariantif

h(γ x) = γ h(x) ∀γ ∈ 0,

and a functiong : V → R is 0-invariant if

g(γ x) = g(x) ∀γ ∈ 0.

In equivariant bifurcation theory we usually assume that0 acts trivially on the bifurcation parameterλ and thatfis equivariant inx, so thatf (γ x, λ) = γf (x, λ).

∗ Corresponding author.E-mail address:[email protected] (A.P.S. Dias)

0167-2789/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved.PII: S0167-2789(00)00104-4

236 I. Stewart, A.P.S. Dias / Physica D 143 (2000) 235–261

The analysis of (1) depends upon obtaining fairly detailed information about equivariant mapsf . For the followingdiscussion we temporarily suppress the parameterλ to simplify the notation. The spaceEEV (0) of smooth equivariantsis a module over the ringEV (0) of smooth invariants. Moreover, the structure ofEEV (0) as a module overEV (0)

can be inferred easily from their analoguesEPV (0) andPV (0) for polynomial mappings, by virtue of well-knowntheorems of Poénaru and Schwartz (see [17]). These results place the problem within the ambit of classical invarianttheory, so that in essence it becomes an algebraic (or algebro-geometric) question.

It is usually fairly difficult to determinePV (0) and EPV (0), however, except in simple cases. For example, when0 = SO(3) andRm is the seven-dimensional space of spherical harmonics of degree 3, this calculation is alreadyquite complicated and technical (see, e.g. [30]).

>From the point of view of invariant theory, one of the best-behaved classes of groups comprises thetorus groups

Tk ∼= SSS1 × · · · × SSS1︸︷︷︸k

,

whereS1 = R/2πZ is the circle group. Any torus action can be diagonalised over the complex numbers. Eachinvariant polynomial (underTk) is a linear combination of invariant monomials. In particular, there are minimalgenerating sets for the ring of invariants that contain only monomials. Moreover, the invariant monomials are inone-to-one correspondence with the elements of a semigroupS (see, e.g. [29,41]). This semigroup was studied byGordan [22] (see also [12]), where the ring ofTk-invariant polynomials is proved to be a finitely generated algebraby showing thatS is finitely generated as a semigroup. This fact permits the introduction of combinatorial methodswhen determining invariants and equivariants. Moreover, the equivariants can easily be deduced from the invariants[20]. The invariant theory of torus actions falls within the area of algebraic geometry known astoric geometry, thestudy oftoric varieties, surveyed in [12].

Torus actions are natural in a variety of equivariant bifurcation problems arising from applications. Typically,the appropriate symmetry group is not a torus as such, but a finite extension of a torus. That is, the connectedcomponent00 of the identity of0 is a torus, andQ = 0/00 is finite (as it must be for compact0). The invariantsand equivariants for0 can therefore be found by first restricting to00, and then symmetrising overQ. Circumstancesin which such groups arise include:1. Bifurcating waves on a crystallographic lattice. The torus is generated by translations modulo the dual lattice,

andQ is the holohedry.2. Problems with circular or cylindrical geometry. Here there is a rotationalT1 symmetry, and in the cylindrical case

with periodic boundary conditions, translations modulo the period convert this toT2. See, e.g. the discussion ofCouette–Taylor flow in [16,17].

3. Say that a PDE posed on a domain� ⊆ Rk is Euclidean-invariantif the image of any solution under a Euclideantransformation ofRk (i.e., a rigid motion) is again a solution (see, e.g. [33]). Euclidean-invariant PDEs onmultidimensional rectangular domains with periodic boundary conditions have torus symmetry. This fact hasbeen extensively exploited in connection with ‘hidden symmetries’ in analogous problems with Neumann orDirichlet boundary conditions, references include [1–11,18–21,23–28,31,32,36].In this paper, we make a start on putting together the powerful techniques of toric geometry and the prevalence

of torus group symmetry in equivariant bifurcation theory. Most of our discussion takes place within one family ofexamples: steadystate mode interactions between solutions of a Euclidean-invariant PDE on a rectangle� ⊆ R2,subject to periodic boundary conditions. In Section 2, we introduce some elementary ideas and terminology fromtoric geometry. In Section 3, we discuss invariants and equivariants for a special class of torus actions arisingin connection with mode interactions in a rectangular domain. Section 4 develops intuition on an example, the(3, 2)–(1, 3) mode interaction. Section 5 extends the analysis to a general(k1, `1)–(k2, `2) mode interaction. InSection 6, we compute the Hilbert series of torus invariants for a(k1, `1)–(k2, `2) mode interaction. Section 7 relates

I. Stewart, A.P.S. Dias / Physica D 143 (2000) 235–261 237

volumes of fundamental lattice parallelotopes with the number of candidates for the generators of theT2-invariants.In Section 8, we compute the Hilbert series of torus equivariants for a(k1, `1)–(k2, `2) mode interaction. Finally,Section 9 sets up an analogous discussion of general torus actions.

2. Toric geometry

In the algebro-geometric treatment of the topic, it is usual to work over the fieldQ of rational numbers. Bifurcationtheorists, on the other hand, will prefer to work overR ⊇ Q. The elementary parts of the theory (which are all weneed here) work equally well in both contexts, so bifurcation theorists should readQ but thinkR, especially whenvisualising the geometry.

Let V be a finite-dimensional vector space overQ. Let L : V → Q be aQ-linear map, so that (for nonzeroL)the zero-set

L0 = {v ∈ V : L(v) = 0}is a hyperplane. The complement ofL0 decomposes into positive and negativeopen linear half-spaces

L+ = {v ∈ V : L(v) > 0}, L− = {v ∈ V : L(v) < 0},and of course(−L)± = L∓. Our main interest is in theclosed linear half-space

L+0 = {v ∈ V : L(v) ≥ 0} = L+ ∪ L0.

A coneσ ⊆ V is the intersection of finitely many closed linear half-spaces

σ =k⋂

i=1

L+0i .

A faceof σ is a subset of the formσ ∩ L0i . A face of a cone is a cone, and the intersection of finitely many faces is

a face. Afan in V is a collection6 of cones such that: (a) every cone of6 has a vertex, (b) ifτ is a face of a coneσ ∈ 6, thenτ ∈ 6, (c) if σ, σ ′ ∈ 6, thenσ ∩ σ ′ is a face both ofσ andσ ′.

For any set of vectors{v1, . . . , v`} ⊆ V , we let〈v1, . . . , v`〉 be the smallest cone containing{v1, . . . , v`}. Everycone can be written in this form. If thevj are linearly-independent, then the cone〈v1, . . . , v`〉 is said to besimplicial.Thedimensionof a coneσ is the dimension overQ of theQ-vector space spanned byσ . Similarly, a fan is said tobesimplicial if it consists of simplicial cones.

A latticeM ⊆ V is a free Abelian group of finite rank: its rank is called itsdimension. A cone inM is a conein theQ-vector space spanned byM. If σ is a simplicial cone inM andσ = 〈v1, . . . , vk〉, where thevj belong toM (and are linearly-independent), callPM

σ = {∑ki=1µivi : 0 ≤ µi ≤ 1} its fundamental parallelotope. If σ is a

cone inM thenσ ∩M is a (commutative) subsemigroup. The following lemma goes back to Gordan (see [12,14]).

Lemma 2.1(Gordan).The semigroupσ ∩M is finitely generated.

Proof. The proof is straightforward, but for completeness we give it here. By breakingσ up into a union of simplicialcones, we may without loss of generality assume thatσ is simplicial so thatσ = 〈v1, . . . , vk〉, where thevj belongtoM and are linearly-independent. Form the fundamental parallelotopePM

σ = {∑ki=1µivi : 0 ≤ µi ≤ 1}. Since

PMσ is compact andM is discrete, the intersectionPM

σ ∩M is finite. Clearly any elementv of σ ∩M can berepresented (uniquely) in the form

∑ki=1rivi with ri ≥ 0, sori = mi + ti with mi a nonnegative integer and


0 ≤ ti ≤ 1. Then,v = ∑mivi + v′ with eachvi andv′ = ∑

tivi in PMσ ∩M. Therefore, the finite setPM

σ ∩Mgeneratesσ ∩M as a semigroup. �

3. Mode interactions in a rectangle

Let � ⊆ R2 be a rectangular domain in the plane, say� = [0, A] × [0, B] (A 6= B), and consider a parametrisedfamily of Euclidean-invariant PDEs(∂u/∂t)+P(u, λ) = 0 subject to periodic boundary conditions (PBCs). Steadystate solutions correspond to zeros of the partial differential operatorP. Because of Euclidean invariance and PBC,this problem is invariant under a torus groupT2 consisting of translations inR2 modulo the lattice generated by(A, 0) and (0, B). It is also invariant under reflections in the two coordinate directions, leading to a symmetrygroup O(2) × O(2). SinceA 6= B there are no further obvious domain symmetries (although in some casesthere may be ‘hidden rotations’, see [7]). This symmetry implies that linearised solutions are superpositions ofeigenfunctions

sin

(2kπx

A

)sin

(2`πy

B

), sin

(2kπx

A

)cos

(2`πy

B

),

cos

(2kπx

A

)sin

(2`πy

B

), cos

(2kπx

A

)cos

(2`πy

B

),

wherek, ` ∈ N are known asmode numbers.If the parameterλ can be chosen so that modes(k1, `1) and(k2, `2) bifurcate simultaneously, then the equation

undergoes amode interactionat such a value (see [20] for details). At such a mode interaction the bifurcation can bereduced to a finite-dimensional problem, the so-called (Liapunov–Schmidt) reduced bifurcation equation, see [17].The reduced bifurcation equation inherits the symmetries of the original problem (provided the reduction procedureis carried out using group-invariant subspaces, which is always possible) and hence is equivariant under a torusgroup

T2 = {(θ1, θ2) : θ1, θ2 ∈ S1},and reflectionsρ1, ρ2. The group action ofO(2)×O(2) for this mode interaction is shown in Table 1. The amplitudesz1, z2 belong to the(k1, `1) mode andz3, z4 to the(k2, `2) mode.

We focus on the subgroupT2. Because theT2-action is diagonal, allT2-invariants are generated by monomials

zα11 z

β11 z

α22 z

β22 z

α33 z

β33 z

α44 z

β44 , αj , βj ∈ N.

In order to be invariant, these monomials must satisfy the conditions

k1(γ1 + γ2) + k2(γ3 + γ4) = 0, (2)

`1(γ1 − γ2) + `2(γ3 − γ4) = 0, (3)

Table 1Group action for the(k1, `1)–(k2, `2) mode interaction

z1 z2 z3 z4

θ1 eik1θ1z1 eik1θ1z2 eik2θ1z3 eik2θ1z4

θ2 ei`1θ2z1 e−i`1θ2z2 ei`2θ2z3 e−i`2θ2z4

ρ1 z2 z1 z4 z3

ρ2 z2 z1 z4 z3


where

γj = αj − βj (j = 1, 2, 3, 4).

Eqs. (2) and (3) define a two-dimensional latticeL ⊆ Z4. We study this lattice further in Section 5.

Lemma 3.1. TheT2-invariants onC4 are (nonminimally) generated by|zj |2 (j = 1, 2, 3, 4) and all monomials of

the formw|γ1|1 w

|γ2|2 w

|γ3|3 w

|γ4|4 , wherewj = zj if γj ≥ 0, wj = zj if γj < 0, and theγj satisfy(2) and(3).

Proof. Clearly|zj |2 = zj zj is invariant. Letm = zα11 z

β11 z

α22 z

β22 z

α33 z

β33 z

α44 z

β44 be an invariant monomial. Let

δj = min(αj , βj ), j = 1, . . . , 4,

and writem in the form

m = (|z1|2)δ1(|z2|2)δ2(|z3|2)δ3(|z4|2)δ4m′,

wherem′ = zα′

11 z

β ′1

1 zα′

22 z

β ′2

2 zα′

33 z

β ′3

3 zα′

44 z

β ′4

4 . Thenm′ is also invariant, and for eachj = 1, . . . , 4 eitherα′j = 0 or

β ′j = 0. Moreover,γ ′

j = α′j − β ′

j = γj , som′ = w|γ1|1 w

|γ2|2 w

|γ3|3 w

|γ4|4 . �

The following theorem, presumably well known, reduces the computation of equivariants to that for invariants:

Theorem 3.2. LetTk act diagonally onCk with coordinateszj , zj for j = 1, . . . , k.1. If I1, . . . , Is generate theC-valued invariants, then the equivariants are generated over the invariants by the

mappings

rowj →

0...

0

∂Ig

∂zj

0...

0

(4)

for 1 ≤ j ≤ k and1 ≤ g ≤ s.2. If I1, . . . , Ip is a set of invariant monomials that spans the space of invariants of degreed + 1, then the maps

(4), f or 1 ≤ j ≤ k and1 ≤ i ≤ p, span the equivariants of degree d.

Proof. Part 1 is proved in [20]. Exactly the same calculations prove part 2. �

A monomial of the formw|ε1|1 w

|ε2|2 w

|ε3|3 w

|ε4|4 , wherewj is either equal tozj or zj , is said to bereduced. Equiv-

alently, a reduced monomial is one that is not divisible by anyzj zj . So Lemma 3.1 states that the invariants are

generated by the|zj |2 together with certain reduced monomials. The form ofw|γ1|1 w

|γ2|2 w

|γ3|3 w

|γ4|4 as a polynomial

in thezj andzj depends on the signs of theγj . It is this dependence that introduces cones into the analysis, as wesee below.


We next describeL explicitly, making the following simplifying assumptions:

hcf(k1, k2) = 1, hcf(`1, `2) = 1. (5)

If these assumptions do not hold we can factor out the kernel of the group action and thereby ensure that they dohold. This procedure does not change the invariants or equivariants: all it does is replacekj by kj /hcf(k1, k2) and`j by `j /hcf(`1, `2).

Lemma 3.3. With the assumptions(5), we have thatL is generated(as a lattice) by the following vectors.

TypeA :

k2

k2

−k1

−k1

`2

−`2

−`1

`1

12(k2 + `2)

12(k2 − `2)

12(−k1 − `1)

12(−k1 + `1)

if one pair of mode numbers is of type(odd,odd) and the other pair is either(odd,odd) or (even,even). Here eitherthe first or second generator can be omitted(the third generator is their mean).

TypeB :

k2

k2

−k1

−k1

`2

−`2

−`1

`1

if at least one pair of mode numbers has distinct parities.

Proof. We can solve (2) and (3). By (2),k2 dividesγ1 + γ2 andk1 dividesγ3 + γ4. Hence there existsa ∈ Z suchthat

γ1 + γ2 = ak2, γ3 + γ4 = −ak1. (6)

Similarly, there existsb ∈ Z such that

γ1 − γ2 = b`2, γ3 − γ4 = −b`1. (7)

Solving (6) and (7), we get

γ1 = 12(ak2 + b`2), γ2 = 1

2(ak2 − b`2), γ3 = 12(−ak1 − b`1), γ4 = 1

2(−ak1 + b`1). (8)

In addition, we requireγj ∈ Z, which leads to the two distinct cases as we now show. From (8) we have the followingconstraints on the parities ofa andb, see Table 2. A similar table holds for(k1, l1). Note that both tables must besatisfied simultaneously by the samea andb.

Table 2Parities ofa andb according to the parities ofk2 andl2

k2 l2 a b

Odd Odd Same ParityEven Odd Any EvenOdd Even Even AnyEven Even Any Any


Table 3Parities ofk1, l1, k2, l2 versus parities ofa andb

Case k1 l1 k2 l2 a andb

1 Odd Odd Odd Odd Same parity2 Odd Odd Even Odd Both even3 Odd Odd Odd Even Both even4 Odd Odd Even Even Same parity5 Even Odd Even Odd Violates (5)6 Even Odd Odd Even Both even7 Even Odd Even Even Violates (5)8 Odd Even Odd Even Violates (5)9 Odd Even Even Even Violates (5)

10 Even Even Even Even Violates (5)

Interchanging(k1, l1) with (k2, l2) if necessary we have ten possibilities for the parities ofk1, l1, k2, l2, five ofeach violate assumptions (5) and so we do not consider. Using Table 2, we get information on the parities ofa andb for the remaining cases (see Table 3). Finally, it follows that for the cases 1 and 4 we can take generators forLthe vectors of Type A. For the cases 2, 3 and 6, we have the generators forL of the Type B. �

4. An example

To motivate the algebra, we consider a typical example, the(3, 2)–(1, 3) mode interaction. That is,

k1 = 3, `1 = 2, k2 = 1, `2 = 3.

This is Type B of Lemma 3.3. Eqs. (2) and (3) become

3(γ1 + γ2) + (γ3 + γ4) = 0, 2(γ1 − γ2) + 3(γ3 − γ4) = 0

which imply that

γ3 = −16(11γ1 + 7γ2), γ4 = −1

6(7γ1 + 11γ2). (9)

Thus, we can parametrise solutions by pairs(γ1, γ2) in the latticeM ⊆ Z2 defined to be the projection ofL ontothe first two coordinates ofZ4.

Theγj are integers if and only ifγ1 ≡ γ2 (mod 6). The interpretation of a solutionγ = (γ1, γ2, γ3, γ4) changeswhen the sign of anyγj changes, i.e., on crossing any of the four linesγj = 0 inM. These lines have equations

γ1 = 0, γ2 = 0, γ1 = − 711γ2, γ1 = −11

7 γ2,

respectively.Fig. 1 shows the half-spaceγ1 ≥ 0 in the(γ1, γ2)-plane, the latticeM, and the linesγj = 0. Points inM are

shown as black or open dots. The half-spaceγ1 ≤ 0 decomposes in the same manner, but rotated by 180◦, whichhas the effect of complex conjugation on the corresponding reduced monomials. The figure divides into four cones(eight including conjugates). The set of these cones and their faces form a fan. Within each cone the choicewj = zj

or zj is fixed, so the product of two reduced monomials in the same cone is again reduced. We may therefore obtaina (minimal) generating set of reduced monomials, under multiplication, by finding a (minimal) generating set forσ ∩M for each coneσ , under semigroup addition, and taking the union of these sets.

Each coneσ has afundamental parallelotopePMσ , as defined in the proof of Lemma 2.1, whose sides are

determined by the vectors 0, v1, v2, wherev1 andv2 are the nonzero elements ofM of minimal length subject


Fig. 1. Lattice geometry for the(3, 2)–(1, 3) mode interaction.

to lying on the faces of the cone. The fundamental parallelotopes are shown shaded in Fig. 1. As in the proof ofGordan’s lemma, any minimal generating set lies inside the fundamental parallelotope.

Say that a nonzero element ofσ ∩M is σ -irreducible if it is not the sum of two nonzero elements ofσ ∩M.Clearly theσ -irreducibleelements lie inside the fundamental parallelotopePM

σ . It is easy to see that eachσ ∩Mhas auniqueminimal generating set, which consists of theσ -irreducibleelements of the finite setPM

σ ∩M. Theseelements can be found by inspection: in Fig. 1 they are marked with a black dot. Reducible elements are markedwith an open dot.

In summary, the torus invariants for the(3, 2)–(1, 3) mode interaction are minimally generated by the monomialslisted in Table 4, together with the complex conjugates of all but the first four entries (which equal their complexconjugates). The table also lists the degrees of these generating monomials. There are 28 generators, whose degreesrange from 2 to 30.

One implication of this table for equivariant bifurcation theory is worth emphasising. The table lists the fourobvious invariantszj zj . Apart from these, the lowest degree term that appears has degree 8. Taking Theorem 3.2into account, we see that the lowest degree equivariant that is not generated from thezj zj is one of degree 7.The equivariants generated from thezj zj are equivariant under any of the torus actions in Table 1, independent


Table 4Minimal generators for torus invariants for the(3, 2)–(1, 3) mode interaction

Monomial Degree

z1z1 2

z2z2 2

z3z3 2

z4z4 2

z62z

73z

114 24

z61z

113 z7

4 24

z1z2z33z

34 8

z51z2z

83z

44 18

z41z

22z

53z4 12

z31z

32z

23z

24 10

z21z

42z3z

54 12

z1z52z

43z

84 18

z71z

52z

73z4 20

z51z

72z3z

74 20

z111 z7

2z123 30

z71z

112 z12

4 30

of the mode numbers. Indeed, they are equivariant under the orthogonal groupO(8) acting onR8 = C4, whichhas dimension 28, whereas the dimension ofT2 is 2. Thus any analysis that does not take degree 7 terms intoaccount will find a profusion of spuriousO(8) group-orbits of solutions. Indeed, any special qualitative features ofthe(3, 2)–(1, 3) mode interaction, such as the detailed geometry of regions in parameter space for which solutionbranches exist, cannot be detected by truncating Taylor series at a degree less than 7. So it is necessary to retainterms of relatively high degree in order to obtain qualitatively accurate bifurcation diagrams. This phenomenon isquite subtle and deserves further study; the most appropriate framework is singularity theory, as in [15].

In saying this we acknowledge that because torus groups are Abelian, the complex structure of Table 1 implies thatthere are no axial subgroups, so the Equivariant Branching Lemma [17] cannot be applied here. For torus-equivariantproblems, we expect Hopf bifurcations to travelling waves. When the torus action is extended by some finite group,especially one containing reflections, axial subgroups can sometimes occur, and the above remarks remain valid.We have chosen the action of Table 1 because it is convenient to illustrate the general ideas of this paper.

5. The general mode interaction

The analysis of the general(k1, `1)–(k2, `2) mode interactions follows similar lines. We solve Eqs. (2) and (3)for γ3, γ4 in terms ofγ1, γ2, leading to

γ3 = − 1

2k2`2[(k1`2 + k2`1)γ1 + (k1`2 − k2`1)γ2], (10)

γ4 = − 1

2k2`2[(k1`2 − k2`1)γ1 + (k1`2 + k2`1)γ2]. (11)


Define

1 = k2`1 + k1`2

k2`1 − k1`2.

Then,γ3 = 0 whenγ2 = 1γ1 andγ4 = 0 whenγ2 = (1/1)γ1.When we projectZ4 with coordinates(γ1, γ2, γ3, γ4) onto Z2 with coordinates(γ1, γ2), the hyperplanes

in Z4 defined by (10) and (11) project to the lines inZ2 defined by the same equations, since those equationsdo not depend explicitly onγ3, γ4. Therefore we can decomposeL ⊆ Z4, and simultaneouslyM ⊆ Z2, intocones on which the signs of theγj remain constant. So addition in each cone corresponds to multiplication ofmonomials. On the boundaries of those cones, someγj = 0. Moreover, we can draw the picture in the(γ1, γ2)-plane.

The analysis now proceeds as in Section 4. For each coneσ we define a fundamental parallelotopePMσ , and

σ -irreducible elements, in exactly the same way as for the example. The key computational result is the followinglemma.

Lemma 5.1. Each lattice coneσ ∩M has a unique minimal generating set, which consists of theσ -irreducibleelements of the finite setPM

σ ∩M.

Again these elements can be found by inspection.

Theorem 5.2. The ring of theT2-invariants onC4 has a unique minimal basis consisting of monomials. This basisis formed by|zj |2 (j = 1, 2, 3, 4) and the union of the sets of theT2-invariant reduced monomials correspondingto theσ -irreducible generators inPM

σ ∩M of σ ∩M for each coneσ .

Proof. Clearly, by Lemma 3.1, the unionU of the |zj |2 together with theT2-invariant reduced monomials corre-sponding to theσ -irreducible generators inPM

σ ∩M of σ ∩M for each coneσ generate the ring of theT2-invariantpolynomials.

Suppose thatU is not minimal. Then there is a reduced monomialm in U that is the product of at least two otherreduced monomials, saym1, m2, that are inU . By Lemma 5.1 the monomialsm1, m2 have to correspond to twoirreducible lattice points ofM that lie in two distinct cones.

The reduced monomials are of the type: (a)w|ε1|1 w

|ε2|2 w

|ε3|3 w

|ε4|4 or (b)w

|εi1 |i1

w|εi2 |i2

w|εi3 |i3

, wherewj is either equalto zj or zj . We have three possibilities form1, m2. Case 1: Both are of type (a). Case 2: Both are of type (b). Case3: One is of type (a) and the other of type (b).

Case1. Sincem1, m2 are in distinct cones, then at least one of thewj is zj in one cone andzj in the other cone.Thus a power of|zj |2 appears inm = m1m2, which contradicts the hypothesis ofm being a reduced monomial.

Case2. For this case,m1 andm2 correspond to lattice points ofM that lie in two faces of two distinct cones.Therefore at least two of thewi of m1 are distinct from two of thewi of m2. Again, for this case, a power of some|zj |2 appears in the productm1m2 andm is not reduced. For example, in Fig. 1 an element in faceσ1 is of type

zγ22 z

γ33 z

γ44 and in faceσ3 is of typez

γ ′1

1 zγ ′

22 z

γ ′3

3 . The product of two elements lying in facesσ1 andσ3 contains a powerof |z2|2.

Case3. Similar to Case 1. �

Example 5.3. We consider the(1, 3)–(3, 1) mode interaction, which is of Type A. Here

k1 = 1, `1 = 3, k2 = 3, `2 = 1.


Fig. 2. Lattice geometry for the(1, 3)–(3, 1) mode interaction.

The latticeL is generated by

33

−1−1

1−1−33

21

−21

.

The cone boundaries are theγ1-axis, theγ2-axis, and the linesγ3 = 0, γ4 = 0, whose equations areγ2 = 54γ1 and

γ2 = 45γ1, respectively, since1 = 5

4. Furthermore,

γ3 = 13(−5γ1 + 4γ2), γ4 = 1

3(4γ1 − 5γ2),

and theγj are integers if and only ifγ1 + γ2 ≡ 0 (mod 3).The appropriate picture is shown in Fig. 2. We can read off a minimal generating system of reduced monomials

(omitting complex conjugates), see Table 5. This time there are 20 monomials in a minimal generating set withdegrees ranging from 2 to 12.

6. Hilbert series

The Hilbert series for a group0 acting on a vector spaceV is the generating function for the dimensionsof the spaces of invariants of given degree. We compute now Hilbert series for torus actions corresponding to(k1, l1)–(k2, l2) mode interactions using the elementary part of toric geometry considered in this paper. Hilbertseries for general torus actions are discussed by Stanley [38] and Renner [35], and the relation between theirmethods and our results deserves further investigation (but not in this paper).


Table 5Minimal generators for torus-invariants for the(1, 3)–(3, 1) mode interaction

Monomial Degree

z1z1 2

z2z2 2

z3z3 2

z4z4 2

z1z2z33z

34 8

z31z

53z

44 12

z21z2z

23z4 6

z51z

42z

33 12

z31z

32z3z4 8

z1z22z3z

24 6

z41z

52z

34 12

z32z

43z

54 12

Let PdV (0) be the space of homogeneous polynomial invariants of degreed for the action of0 on V . Then the

Hilbert seriesfor this action is the formal power series

80(t) =∞∑

d=0

dim(PdV (0))td

in the indeterminatet . For a compact Lie group action there is an explicit integral formula,Molien’s theorem:

80(t) =∫

0

1

det(1 − γ t)dµ0,

whereµ0 is the normalised Haar measure on0 andγ ∈ 0. See [34] or [40] for the original proof of the finite case,and [37] for the extension to a compact group. It is difficult (though not impossible) to use this formula to computethe Hilbert series, but it is often better to proceed by other means — as is the case here.

Let 0 = T2 in the action of Section 3, the(k1, l1)–(k2, l2) mode interaction fork1, l1, k2, l2 ∈ N. Letm = w

|γ1|1 w

|γ2|2 w

|γ3|3 w

|γ4|4 be a reduced invariant monomial. Then the integersγ1, γ2, γ3, γ4 satisfy the equations

k1(γ1 + γ2) + k2(γ3 + γ4) = 0, `1(γ1 − γ2) + `2(γ3 − γ4) = 0 (12)

that define a two-dimensional latticeL ⊆ Z4 (Lemma 3.3). These solutions can be parametrised by pairs(γ1, γ2)

in a latticeM ⊆ Z2 defined to be the projection ofL onto the first two coordinatesγ1, γ2 of Z4:

γ3 = − 1

2k2`2[(k1`2 + k2`1)γ1 + (k1`2 − k2`1)γ2], γ4 = − 1

2k2`2[(k1`2 − k2`1)γ1 + (k1`2 + k2`1)γ2].

(13)

Moreover, for the description of the reduced invariant monomials we just need to consider the half-spaceγ1 ≥ 0 inthe(γ1, γ2)-plane. The linesγ1 = 0, γ2 = 0, γ2 = 1γ1 (whenγ3 = 0) andγ2 = (1/1)γ1 (whenγ4 = 0) with

1 = k2`1 + k1`2

k2`1 − k1`2,


Fig. 3. Pictures: (a)1 > 1; (b)1 < −1.

divide this half-plane into four cones, say A, B, C, D, with facesσ1, σ2, σ3, σ4, σ5, defined by these lines. Weconsider here the case when

k2`1 − k1`2 6= 0. (14)

(Whenk2`1 = k1`2, in terms of Hilbert series, we just need to consider, e.g. the cone corresponding to the quadrantγ1, γ2 ≥ 0. The theory developed in this section also applies to this simplified case.) Within each cone the choicewj = zj or zj is fixed(j = 1, . . . , 4). On crossing any of the four linesγj = 0 inM someγj changes sign. Fromthe half-spaceγ1 ≤ 0, we obtain the conjugates of the reduced invariant monomials of the half-spaceγ1 ≥ 0. Thusin terms of the Hilbert series for the action ofT2 onC4 considered here, the study of the half-spaceγ1 ≥ 0 will beenough. Note that sincek1, l1, k2, l2 are natural numbers,

|1| > 1.

We have two kinds of pictures for the disposition of the half-spaceγ1 ≥ 0 into cones, according to whether1 > 1(picture (a)) or1 < −1 (picture (b)). We show these two cases in Fig. 3 to fix notation for the rest of this section.

The degree of a reduced invariant monomialm = w|γ1|1 w

|γ2|2 w

|γ3|3 w

|γ4|4 is ∂m = |γ1| + |γ2| + |γ3| + |γ4|. On each

cone A, B, C, D, the formula for∂m is linear in(γ1, γ2). Figs. 5 and 6 show the lines∂m = c in the four cones forthe(3, 2)–(1, 3) and(1, 3)–(3, 1) mode interactions, respectively. For the first case the numberc takes values thatare integer multiples of 8, 6, 10, 6. For example, on cone B in Fig. 5 we haveγ1 ≥ 0, γ2 ≤ 0, γ3 < 0, γ4 ≤ 0, so

∂m = γ1 − γ2 − γ3 − γ4 = 4γ1 + 2γ2

using (9). For the second case,c is a multiple of 4, 6, 4, 6, respectively.LetQd be the space ofreducedinvariant monomials of degreed, and define thereduced Hilbert series

8L0 (t) =

∞∑d=0

dim(Qd)td


which is a Hilbert series for the latticeL graded by degree in each cone. Lemma 3.1 lets us express80(t) in termsof 8L

0 (t). In fact, since the|zj |2 (j = 1, . . . , 4) are algebraically independent over the reduced monomials, wehave

80(t) = 1

(1 − t2)48L

0 (t).

For any coneσ , we define

8σ0(t) =

∞∑d=0

dim(Qdσ )td ,

whereQdσ is the subspace ofQd spanned by monomials corresponding to lattice points inσ .

Applying the inclusion–exclusion principle (and recalling that from the regionγ1 ≤ 0 we get the complexconjugates of the invariant monomials obtained in the regionγ1 ≥ 0), using notation of Fig. 3, we get the followinglemma.

Lemma 6.1.

8L0 (t) = 2[8A

0(t) + 8B0(t) + 8C

0(t) + 8D0(t)] − 2[8σ1

0 (t) + 8σ20 (t) + 8

σ30 (t) + 8

σ40 (t)] + 1, (15)

whereσj is either of the two half-rays along whichγj = 0, origin included.

Formula (15) converts the computation of8L0 to a series of analogous computations within cones (the regions A,

B, C, D and their common boundaries, the facesσj for j = 1, . . . , 4). We next show that the cones B and D (andcorresponding faces) have the same Hilbert series, as the apparent symmetry in the figure suggests.

Lemma 6.2.

8B0(t) = 8D

0(t)

and

8σ10 (t) = 8

σ20 (t), 8

σ30 (t) = 8

σ40 (t).

Proof. Letm = w|γ1|1 w

|γ2|2 w

|γ3|3 w

|γ4|4 be a reduced invariant monomial determined byγ = (γ1, γ2, γ3, γ4). Consider

first the case1 > 1, corresponding to picture (a) of Fig. 3. Thenγ belongs to cone B if and only if it satisfiesEqs. (12) andγ1 ≥ 0, γ2 ≥ 0, γ3 < 0, γ4 ≥ 0. Similarly,γ belongs to cone D if and only if it satisfies (12) andγ1 ≥ 0, γ2 ≥ 0, γ3 ≥ 0, γ4 < 0. It follows thatγ belongs to cone B if and only ifγ ′ = (γ2, γ1, γ4, γ3) belongs tocone D. Moreover,|γ | = |γ ′|.

Similarly for the case1 < −1 (picture (b) of Fig. 3),γ satisfying (12) belongs to cone B if and only ifγ1 ≥ 0,γ2 ≤ 0, γ3 < 0, γ4 ≤ 0, and it belongs to cone D if and only ifγ1 ≥ 0, γ2 ≤ 0, γ3 ≥ 0, γ4 > 0. Nowγ belongs tocone B if and only ifγ ′ = (−γ2, −γ1, −γ4, −γ3) belongs to cone D, and again|γ | = |γ ′|. Thus,8B

0(t) = 8D0(t).

Restricting to the boundaries of the cones B and D, it follows that8σ10 (t) = 8

σ20 (t) and8

σ30 (t) = 8

σ40 (t). �

Thus using Lemma 6.2, expression (15) now becomes

8L0 (t) = 2[8A

0(t) + 28B0(t) + 8C

0(t)] − 4[8σ10 (t) + 8

σ30 (t)] + 1. (16)


Definition 6.3. Letσ be a simplicial cone of dimension 2 in the latticeM ⊆ Z2 (projection ofL ⊆ Z4 on(γ1, γ2)).Denote the faces ofσ by σ1, σ2. Letv1, v2 be the two nonzero elements ofM of minimal length subject to lying onthe facesσ1, σ2 of the cone, and letu1, u2 be the corresponding elements inL ⊆ Z4 (and so satisfying Eqs. (12)).Define thefundamental parallelotopePM

σ of the latticeM relative to the coneσ to be the polytope with vertices0, v1, v2, v1 + v2 ∈M ⊆ Z2. (It is actually a parallelogram in this case, but it is useful to set up terminology for ageneral setting.) Define thefundamental parallelotope polynomialPL

σ to be

PLσ (t) = 1 +

∑γ∈ILσ

t |γ |, (17)

where

ILσ = {(γ1, γ2, γ3, γ4) ∈ L : (γ1, γ2) ∈ int(PMσ ∩M)}.

Here|γ | = |γ1| + · · · + |γ4|.

We prove now thatPLσ and the degrees of the reduced invariant monomials with exponentsu1, u2 determine8σ

0

and8σi

0 .

Lemma 6.4. Letσ be any of the simplicial conesA, B, C or D (of dimension2) in the latticeM ⊆ Z2 (projectionof L ⊆ Z4 on (γ1, γ2)), and let the faces ofσ beσ1, σ2. Let v1, v2 be the two nonzero elements ofM of minimallength subject to lying on the faces, and letu1, u2 be the corresponding elements inL ⊆ Z4. Then

8σ0(t) = PL

σ (t)

(1 − t |u1|)(1 − t |u2|)(18)

and

8σi

0 (t) = 1

(1 − t |ui |)(i = 1, 2).

Proof. We haveσ = 〈v1, v2〉, sincev1, v2 lie in the two distinct faces of the cone. Forv ∈ σ ∩M define

σv = {v + v′ : v′ ∈ σ }.We can think ofσv as a cone equal toσ but with apex atv (instead of the origin).

Consideru1, u2 ∈ L ⊆ Z4 the corresponding elements tov1, v2 ∈ M ⊆ Z2. Let m be a reduced invariantmonomial determined byγ ′ = (γ ′

1, γ′2, γ

′3, γ

′4) ∈ L such that(γ ′

1, γ′2) ∈ σv1 ∩M. Then(γ ′

1, γ′2) = v1 + (γ1, γ2) for

some(γ1, γ2) ∈ σ ∩M. Sincev1 and(γ1, γ2) are inσ ∩M, letm be the reduced invariant monomial correspondingto γ ′ = u1 + γ , whereγ = (γ1, γ2, γ3, γ4), is the lattice point ofL corresponding to the lattice point(γ1, γ2) ofσ ∩M. Then the degree ofm is given by

∂m = |γ ′| = |u1| + |γ |,sinceu1 andγ have components with same signs. Therefore, if we define an analogous8

σv10 , then

8σv10 (t) = t |u1|8σ

0(t).

Similarly,

8σv20 (t) = t |u2|8σ

0(t) and 8σv1+v20 (t) = t |u2|+|u2|8σ

0(t).


Fig. 4. Geometry for the conesσ , σv1, σv2 andσv1+v2.

Note also that

σv1+v2 = σv1 ∩ σv2

(see Fig. 4).Thus,

8σ0(t) = PL

σ (t) + (t |u1| + t |u2| − t |u1|+|u2|)8σ0(t)

and so

8σ0(t) = PL

σ (t)

(1 − t |u1|)(1 − t |u2|). �

Recalling again the notation of Fig. 3, it is now straightforward to obtain a formula for8L0 (and80) depending

on the fundamental parallelotope polynomials for each of the cones A, B, C.

Theorem 6.5. The Hilbert series for the action of0 = T2 onC4 of Section3 is

80(t) = 1

(1 − t2)48L

0 (t),


where

8L0 (t) = 2PL

A (t)

(1 − t |u1|)2+ 4PL

B (t)

(1 − t |u1|)(1 − t |u3|)+ 2PL

C (t)

(1 − t |u3|)2− 4

(1 − t |u1|)− 4

(1 − t |u3|)+ 1. (19)

Proof. Use Lemmas 6.1, 6.2 and 6.4. �

Remark 6.6. For the particular case

k1l2 − k2l1 = 0

it follows from(13) that γ3, γ4 are integers if and only ifγ1 ≡ 0 (modl2) andγ2 ≡ 0 (modl2). If we consider theconeA defined byγ1, γ2 ≥ 0, then

80(t) = 1

(1 − t2)4(48A

0(t) − 48σ10 (t) + 1) = 1

(1 − t2)4

(4

(1 − t l1+l2)2− 4

(1 − t l1+l2)+ 1

).

6.1. Examples

For illustration, we consider the Hilbert series of theT2-invariants for the(3, 2)–(1, 3) mode interaction of Section4, and for the(3, 1)–(1, 3) mode interaction of Section 5.

Example 6.7. Let 0 = T2 in the action of Section 4, the(3, 2)–(1, 3) mode interaction. Fig. 5 shows the lines∂m = c in the four cones for various values ofc. In cones A, B, C, D, the numberc takes values that are integermultiples of 8, 6, 10, 6, respectively. The figure also shows the fundamental parallelotope of the latticeM for eachof the cones. Applying Lemma 6.4 to each cone, we have

8A0(t) = 1 + t8 + t16 + t24 + t32 + t40

(1 − t24)2, 8B

0(t) = 8D0(t) = 1 + t12 + t18 + t24 + t30 + t36 + t42

(1 − t24)(1 − t30),

8C0(t) = 1 + t10 + 3t20 + 3t30 + 3t40 + t50

(1 − t30)2, 8

σ10 (t) = 8

σ20 (t) = 1

1 − t24,

8σ30 (t) = 8

σ40 (t) = 1

1 − t30.

By Theorem 6.5, we obtain an explicit formula for80

80(t) = F(t)

(1 − t2)4(1 − t24)2(1 − t30)2

= 1 + 4t2 + 10t4 + 20t6 + 37t8 + 66t10 + 116t12 + 196t14 + 317t16 + 494t18 + · · ·Here,

F(t) = 1 + 2t8 + 2t10 + 4t12 + 2t16 + 4t18 + 6t20 + 8t24 + 12t30 + 2t32 − 4t34 − 4t38 + 8t40 − 4t42

−12t44 − 4t46 − 9t48 + 2t50 − 32t54 + 2t58 − 9t60 − 4t62 − 12t64 − 4t66 + 8t68 − 4t70

−4t74 + 2t76 + 12t78 + 8t84 + 6t88 + 4t90 + 2t92 + 4t96 + 2t98 + 2t100 + t108.

Example 6.8. Let 0 = T2 in the action of Section 5, the(1, 3)–(3, 1) mode interaction. Fig. 6 shows the lines∂m = c in the four cones, for various values ofc. In cones A, B, C, D, the numberc takes values that are integermultiples of 4, 6, 4, 6, respectively. Applying Lemma 6.4 to each cone, we compute


Fig. 5. Hilbert series geometry for the(3, 2)–(1, 3) mode interaction.

8A0(t) = 8C

0(t) = 1 + t8 + t16

(1 − t12)2, 8B

0(t) = 8D0(t) = 1 + t6 + t12 + t18

(1 − t12)2,

8σi

0 (t) = 1

1 − t12(i = 1, 2, 3, 4).

By combining the above equations (using Theorem 6.5), an explicit formula for80 is the following:

80(t) = 1 + 4t6 + 4t8 + 10t12 + 4t16 + 4t18 + t24

(1 − t2)4(1 − t12)2

= 1 + 4t2 + 10t4 + 24t6 + 55t8 + 112t10 + 216t12 + 388t14 + 653t16 + 1048t18 + · · ·


Fig. 6. Hilbert series geometry for the(1, 3)–(3, 1) mode interaction.

7. Volume, lattice points and generators

As before let0 = T2 in the action of Section 3, the(k1, l1)–(k2, l2) mode interaction, fork1, l1, k2, l2 ∈ N. Inthis section we calculate the number of lattice pointsNILσ of the latticeM that lie in the interior of each fundamentalparallelotopePM

σ , whereσ is any of the cones A, B, C or D. This is derived from the volume of the fundamentalparallelotope normalised by the volume of one unit cell ofM. Recalling Theorem 5.2 this number estimates themaximum number of points that we need to check for finding the generators of the reduced monomials for eachcone. Moreover,NILσ + 1 = PL

σ (1), wherePLσ is the fundamental parallelotope polynomial of the latticeL relative

to the coneσ . Since the coefficients ofPLσ are nonnegative integers,NILσ +1 provides a measure of the ‘complexity’

of PLσ , namely, the sum of its coefficients. It is therefore of some interest to computeNILσ explicitly.

We also calculate the maximum degree of the generators of the ring of theT2-invariants (generators given bymonomials), which gives another way of quantifying how complicated the invariants are.

It is well known that ifL is a full sublattice ofZk andD is a fundamental domain forL, then the number ofpoints ofZk that lie insideD is equal to the volume vol(L) ≡ µ(D), whereµ is ak-dimensional Lebesgue measure.(Proof (sketch). The quotient mapφ : Rk → Rk/Zk = Tk is volume preserving. The images underφ of translatesof a fundamental domain forZk by elements ofZk ∩ D coverTk disjointly. Each such translate has measure 1.)

It follows easily that ifL1 ⊆ L ⊆ Zk are full lattices, andD is a fundamental domain ofL1, then

#{x : x ∈ L ∩ D} = vol(L1)

vol(L). (20)

(Proof (sketch). MapL linearly ontoZk and note that volumes are scaled according to the determinant ofL.)


Let PMσ be a fundamental parallelotope of the latticeL relative to the coneσ as defined in Definition 6.3 (so that

σ is one of the cones A, B, C or D). LetNILσ be the number of interior points ofPMσ ∩M. By Lemma 6.2

NILB = NILD .

Note that ifPLσ is the fundamental parallelotope polynomial as in Definition 6.3, then

PLσ (1) = 1 + NILσ .

By Lemma 3.3, the volume of one lattice cell (ofM) is k2l2 if the mode interaction is of Type A, and 2k2l2 if themode interaction is of Type B. Note thatM (projection ofL onto the first two coordinatesγ1, γ2) is generated bythe vectors(k2, k2) and(1

2(k2 + l2),12(k2 − l2)) in the first case, and by(k2, k2) and(l2, −l2) in the second case.

If v1, v2 are the two nonzero elements ofM of minimal length subject to lying on the facesσ1, σ2 of the coneσ ,then

vol(PMσ ) =

∣∣∣∣det

(v1

v2

)∣∣∣∣ ,and so by (20) we have the following lemma.

Lemma 7.1.

PLσ (1) =

∣∣∣∣det

(v1

v2

)∣∣∣∣k2l2

if the mode interaction is of Type A, and

PLσ (1) =

∣∣∣∣det

(v1

v2

)∣∣∣∣2k2l2

if the mode interaction is of Type B.

We calculate now the elementsvi for each of the cones A, B, C and D. By Lemma 6.2 fromv1 (orv5) andv3 on thefacesσ1 (or σ5) andσ3 (recall Fig. 3), we obtain the othervi . Specifically, ifv1 = (0, a) thenv2 = (−sgn(1)a, 0),and ifv3 = (b, c) thenv4 = (sgn(1)c, sgn(1)b).

Lemma 7.2. Letv1 be the nonzero element ofσ1 ∩M with minimal length(recall Fig. 3). Then

v1 = (0, −sgn(1)γ ′2),

where

γ ′2 =

{lcm′(k2, l2) if TypeA mode interaction,2 lcm(k2, l2) if TypeB mode interaction.

Herelcm(a, b) denotes the lowest(positive) common multiple of a and b, andlcm′(a, b) the lowest common multipleof a and b from the common(positive) multiples of a and b, say m, such that m/a and m/b have the same parity.

Proof. Elements inσ1 ∩M satisfyγ1 = 0 and so from (13)

(k1`2 − k2`1)γ2 ≡ 0 (mod 2k2`2), (k1`2 + k2`1)γ2 ≡ 0 (mod 2k2`2). (21)


Thus,

2k2`1γ2 ≡ 0 (mod 2k2`2), (22)

(k1`2 − k2`1)γ2 ≡ 0 (mod 2k2`2). (23)

From (22), it follows thatγ2 ≡ 0 (mod`2) since hcf(`1, `2) = 1. From (23), it follows thatγ2 ≡ 0 (modk2)

since hcf(k1, k2) = 1. Thus there aren1, n2 ∈ Z such thatγ2 = n1k2 = n2`2. Substituting in (23) it follows thatk1n1 − `1n2 ≡ 0 (mod 2). Now if n1, n2 are integers such thatγ2 = n1k2 = n2`2 andk1n1 − `1n2 ≡ 0 (mod 2),then Eq. (21) are satisfied.

Note that ifk1, `1 have the same parity and alsok2, `2 have the same parity, thenn1, n2 also have the same parity.If k1, `1 or k2, `2 have different parity, thenn1, n2 have to be both even. �

Lemma 7.3. Letv3 be the nonzero element ofσ3 ∩M with minimal length(recall Fig. 3). Then

v3 =(

γ ′1,

1

1γ ′

1

),

where

γ ′1 = m′ k2`1 + k1`2

2k1`1, m′ = min

m∈ZZZ+

{m : m

k2`1 + k1`2

2k1`1∈ Z+

}, 1 = k2`1 + k1`2

k2`1 − k1`2.

Proof. Elements inσ3 ∩M are of type(γ1, 1/1γ1), where from (13) the integerγ1 satisfies

(k1`2 + k2`1)γ1 + (k1`2 − k2`1)1

1γ1 ≡ 0 (mod 2k2`2),

i.e.,

2k2`12k1`2

k2`1 + k1`2γ1 ≡ 0 (mod 2k2`2),

and so

2k1`1

k2`1 + k1`2γ1 ≡ 0 (mod 1).

Note that‖(γ1, 1/1γ1)‖2 = γ 21 (1 + |1/1|2) and so‖(γ1, 1/1γ1)‖2 is minimum ifγ 2

1 is minimum. �

Proposition 7.4. With the notation of Lemmas7.2 and 7.3,and Fig.3

PLA (1) =

lcm′(k2, l2)2

k2`2if TypeA mode interaction,

2 lcm(k2, l2)2

k2`2if TypeB mode interaction.

PLB (1) = PL

D (1) =

m′; lcm′(k2, l2)|k2`1 − k1`2|2k1`1k2`2

if TypeA mode interaction,

m′; lcm(k2, l2)|k2`1 − k1`2|2k1`1k2`2

if TypeB mode interaction.


PLC (1) =

m′2

k1`1if TypeA mode interaction,

m′2

2k1`1if TypeB mode interaction.

Proof. Use Lemmas 7.1–7.3. �

Let m be a reduced monomialw|γ1|1 w

|γ2|2 w

|γ3|3 w

|γ4|4 , wherewi = zi if γi ≥ 0 andwi = zi if γi < 0, and where

γ = (γ1, γ2, γ3, γ4) ∈ L. The degree ofm is given by

∂m = |γ | = |γ1| + |γ2| + |γ3| + |γ4|.Using Lemmas 7.2 and 7.3, we calculate now the degrees of the reduced monomials determined by theui ∈ L

corresponding the lattice pointsvi ∈ σi ∩M.

Proposition 7.5.

|u1| = |u2| =

γ ′2

(1 + `1

`2

)if 1 > 0,

γ ′2

(1 + k1

k2

)if 1 < 0,

where

γ ′2 =

lcm′(k2, l2) if TypeA mode interaction,

2 lcm(k2, l2) if TypeB mode interaction.

|u3| = |u4| =

m′ k1 + k2

k1if 1 > 0,

m′ `1 + `2

`1if 1 < 0,

where

m′ = minm∈ZZZ+

{m : m

k2`1 + k1`2

2k1`1∈ Z+

}.

8. Equivariants for the general mode interaction

In Section 6, we computed the Hilbert series for the rings of invariants under the torus actions corresponding to(k1, l1)–(k2, l2) mode interactions. In this section we calculate the Hilbert series for the modules of equivariants forthe same torus actions.

Consider a group0 acting on a vector spaceV . Let EPV (0) be the space of equivariants with polynomial compo-nents for the action of0 onV . This is a graded module over the ringPV (0) of the invariants and itsHilbert seriesis the generating function

90(t) =∞∑

d=0

dim( EPdV (0))td ,


where EPdV (0) is the space of0-equivariants with polynomial components that are homogeneous of degreed (see

[13,39,42]). For a compact Lie group action there is an explicit integral formula, that generalises the Molien theoremfor the equivariants, theEquivariant Molien theorem:

90(t) =∫

0

trace(γ −1)

det(1 − γ t)dµ0, (24)

whereµ0 is again the normalised Haar measure on0 andγ ∈ 0. For the proof see [37,42].Using the notation of Section 6, we prove the following theorem.

Theorem 8.1. The Hilbert series for the equivariantsEPV (0) for the action of0 = T2 onC4 of Section3 is

90(t) = 2

t (1 − t2)4F0(t), (25)

where

F0(t) = 4(1 + t2)(8A0(t) + 8C

0(t)) + 8(1 + t2)8B0(t) − (10+ 6t2)(8

σ10 (t) + 8

σ30 (t)) + 4.

Proof. By Theorem 3.2 (part 2) everyT2-equivariant (with polynomial components) of degreed can be written asa real linear combination of equivariants of the type (4) for 1≤ j ≤ 4 and whereIg is aT2-invariant monomial ofdegreed + 1. Thus, the number of distinct equivariants of degreed of type (4) (forj = 1, . . . , 4) is equal to thenumber of distinct invariantsIg of degreed + 1 such thatzj dividesIg (for j = 1, . . . , 4), up to a real constant.

By Lemma 3.1 any suchIg can be written uniquely as

Ig = kr, (26)

wherek is a product of termszizi , andr is an invariant reduced monomial.Suppose that

∂m1

∂zj

= ∂m2

∂zj

(up to a constant multiple) for invariant monomialsm1, m2. Thenm1 = m2. Since there is a unique represen-tation

m1 = k1r1, m2 = k2r2,

wherekj is a product of termszizi , andrj is an invariant reduced monomial,k1 = k2 andr1 = r2. Note thatr1, r2

correspond to points of the latticeM. Thus differentrj give rise to differentmj (since the representation (26) isunique). Eachrj corresponds to a unique point of the lattice. Therefore, the calculation of the number of distinctequivariants of degreed reduces to the calculation of the number of distinct invariant monomialskr of degreed +1,such thatzj divideskr, and wherer belongs to one of the cones or one of the faces, forj = 1, . . . , 4.

We start with invariantskr, wherer is a reduced invariant monomial corresponding to a lattice point in the interiorof a cone. For easy of explanation, consider the interior of cone A and1 > 1. The corresponding reduced monomialsare of typezγ1

1 zγ22 z

γ33 z

γ44 . Thus invariantsm = k(z)z

γ11 z

γ22 z

γ33 z

γ44 are divided byz2 andz3, and ifk(z) = |z1|2k′(z),

where againk′(z) is a product of termszizi , then it is divided byz1. Similarly, if k(z) = |z4|2k′′(z), thenm isdivided byz4. Now the conjugatesk(z)z

γ11 z

γ22 z

γ33 z

γ44 are also invariant and are divided byz1 andz4, and also byz2

or z3 in casek(z) contains the factor|z2|2 or |z3|2. Thus from equivariants of type (4), whereIg = k(z)r(z) andr


is a reduced monomial corresponding to a lattice point in the interior of cone A (or conjugate to it), the contributionto 90 is given by

2

t (1 − t2)44(1 + t2)(8A

0(t) − 8σ10 (t) − 8

σ20 (t) + 1).

For the interior of the other cones (and for1 < −1) the same formula is obtained (with the corresponding faces).Consider now the faces, e.g.σ1, and again suppose that1 > 1. Reduced monomials are of the typez

γ22 z

γ33 z

γ44

(and conjugateszγ22 z

γ33 z

γ44 ). Thus invariantsk(z)z

γ22 z

γ33 z

γ44 are divided byz2, z3, and byz1, z4 if k(z) contains a

factor |z1|2 and |z4|2, respectively. For the invariantsk(z)zγ22 z

γ33 z

γ44 , these are divided byz4, and byz1, z2, z3 if

|z1|2, |z2|2, |z3|2 are factors ofk(z). From these, the contribution to90 is given by

2

t (1 − t2)4(3 + 5t2)(8

σ10 (t) − 1).

Repeat the same reasoning for the other cones and faces and use Lemma 6.2.Finally, invariant monomials of typek(z) give rise to equivariants of the type

k1(z)

0...

0

zj

0...

0

,

wherek1(z) is again a product ofzizi ’s. These equivariants contribute to90 with

24t

(1 − t2)4.

Summing we get (25). �

8.1. Examples

For illustration, we consider the Hilbert series for theT2-equivariants again for the(3, 2)–(1, 3) mode interactionand the(3, 1)–(1, 3) mode interaction as in Examples 6.7 and 6.7 of Section 6.1.

Example 8.2. Let 0 = T2 in the action of Section 4, the(3, 2)–(1, 3) mode interaction. Using Theorem 8.1 andrecalling Example 6.7,

90(t) = 8t + 32t3 + 80t5 + 168t7 + 328t9 + 616t11 + 1104t13 + 1872t15 + 3024t17 + · · ·

Example 8.3. Let 0 = T2 in the action of Section 5, the(1, 3)–(3, 1) mode interaction. Using Theorem 8.1 andrecalling Example 6.8,

90(t) = 8t + 32t3 + 96t5 + 256t7 + 584t9 + 1192t11 + 2248t13 + 3936t15 + 6504t17 + · · ·

For both examples we have also computed90 to degree 11 using Maple and the Molien formula (24), as a check,and we obtain the coefficients stated above to that degree.


9. General torus actions

We end by describing the first few steps in extending the above analysis to an arbitrary torus action. Let(θ1, . . . , θk)

be coordinates onTk, whereθj ∈ R/2πZ ∼= S1. Let (z1, . . . , zr ) be coordinates onCr . The general torus actiontakes the form

θizj = enij θi zj , θi z̄j = e−nij θi z̄j ,

whereN = [nij ]j=1,... ,r

i=1,... ,k is an integer matrix. Let

m =r∏

j=1

zαj

j z̄βj

j

be a monomial. Thenm is invariant under the action ofθi if and only if∑r

j=1nij γj = 0 (i = 1, . . . , k) forintegersγj , whereγj = αj − βj . Solutions depend only on the differencesαj − βj , reflecting the obvious fact thateach|zj |2 is invariant. We may therefore writem uniquely in the formm = (|z1|2)τ1 · · · (|zr |2)τr m′, wherem′ isreduced.

The above equations define a latticeL whose dimensiond equals the rank ofN . The possiblem′ are inone-to-one correspondence with elements ofL, as before. DecomposeL into cones on which theγj have constantsign.

Each coneσ has a fundamental parallelotopePLσ , and theσ -irreducible elements (which lie insidePL

σ ) form afinite generating set for the semigroupσ ∩ L.

Supposeσ is simplicial and it has dimensiond. Then it is generated byv1, . . . , vd for somevi ∈ L. Considerthe rays consisting of positive multiples of eachvi . Takeu1, . . . , ud the elements inL of minimal length subject tolying on these rays. Then formula (18) of Lemma 6.4 generalises to

8σ0(t) = PL

σ (t)

(1 − t |u1|) · · · (1 − t |ud |),

wherePLσ is the fundamental parallelotope polynomial as in (17) taking nowPL

σ , the fundamental parallelotope ofthe latticeL relative to the coneσ , the polytope with vertices 0, u1, . . . , ud and all sums of distinctuj . SimilarlyTheorem 8.1 generalises, though we do not attempt to state the general formula here since it involves tediousdefinitions.

Thus the basic set-up generalises to an arbitrary torus action — which is not surprising, given the original sourceof toric geometry. It seems plausible that more detailed analysis of torus-equivariant bifurcation problems will beable to exploit deeper features of toric geometry, e.g. to organise singularity-theoretic classifications [15,31,32].Potentially, toric geometry is a rich area for equivariant bifurcation theory.

Acknowledgements

We are grateful to Gabriela Gomes for several discussions. APSD thanks the Departamento de MatemáticaPura da Faculdade de Ciências da Universidade do Porto, for granting leave, and the Mathematics Institute of theUniversity of Warwick, where this work was carried out, for the hospitality. The research of APSD was supportedby Sub-Programa Ciência e Tecnologia do 2oQuadro Comunitário de Apoio through Fundação para a Ciência e aTecnologia.


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Toric geometry and equivariant bifurcations