3. Main theorem and consequences

DOI:10.1051/cocv/2015007 www.esaim-cocv.org

EXTERIOR CONVEXITY AND CLASSICAL CALCULUS OF VARIATIONS

Saugata Bandyopadhyay

and Swarnendu Sil

Abstract. We study the relation between various notions of exterior convexity introduced in [S.

Bandyopadhyay, B. Dacorogna and S. Sil,J. Eur. Math. Soc.17(2015) 1009–1039.] with the classical notions of rank one convexity, quasiconvexity and polyconvexity. To this end, we introduce a projection map, which generalizes the alternating projection for two-tensors in a new way and study the algebraic properties of this map. We conclude with a few simple consequences of this relation which yields new proofs for some of the results discussed in [S. Bandyopadhyay, B. Dacorogna and S. Sil,J. Eur. Math.

Soc.17(2015) 1009–1039.].

Mathematics Subject Classification. 49-XX.

Received July 20, 2014.

Published online March 4, 2016.

1. Introduction

The notion of exterior convexity introduced in [1] is of fundamental importance in calculus of variations on exterior spaces, playing a role similar to what is played by the usual notions of convexity in classical vectorial calculus of variations.

However, the precise connection between these two sets of notions of convexity is a question of somewhat delicate balance. In this article, we explore this connection through the introduction of an appropriate projection map. While this projection map coincides with the canonical alternating projection of the two-tensor fields onto the exterior two-forms, it is non-trivial in the context of higher order forms. Furthermore, the projection map has the crucial property that it projects the tensor product to the exterior product and the gradient to the exterior derivative. It also allows us to express the connection between the notions of exterior convexity and classical notions of convexity in a crisp and explicit way, which is the content of our main theorem stated as follows.

Keywords and phrases. Calculus of variations, rank one convexity, quasiconvexity, polyconvexity, exterior convexity, exterior form, differential form.

1 Department of Mathematics & Statistics, IISER Kolkata, Mohanpur-741246, India.saugata.bandyopadhyay@iiserkol.ac.in

2 Section de Math´ematiques, Station 8, EPFL, 1015 Lausanne, Switzerland.swarnendu.sil@epfl.ch

Article published by EDP Sciences c EDP Sciences, SMAI 2016

EXTERIOR CONVEXITY AND CLASSICAL CALCULUS OF VARIATIONS

Theorem 1.1. Let 2≤k≤n,f :Λ^k →Rand π:R _n

k−1

×n →Λ^k be the projection map. Then the following

equivalences hold

f ext. one convex ⇔ f◦πrank one convex.

f ext. quasiconvex ⇔ f◦πquasiconvex.

f ext. polyconvex ⇔ f◦πpolyconvex.

The aforementioned result essentially situates the circle of ideas discussed in Bandyopadhyay−Dacorogna–

Sil [1] in its proper place with respect to classical calculus of variations, which is well-developed and by now, standard (cf. Dacorogna [3]). It allows us to do calculus of variations back and forth between exterior spaces and the space of matrices. In particular, some results which were directly proved in Bandyopadhya–Dacorogna–

Sil [1] turn out be easy corollaries of the theorem aforementioned above, in conjunction with classical results of vectorial calculus of variations. Notable among them is the characterization theorem for ext. quasiaffine functions (compare the proof of Thm. 3.3 in [1] with that of Thm. 3.4). While in this process we do sacrifice the intrinsic character and the co-ordinate free advantage of a direct proof in exterior spaces, a simple proof is obtained nonetheless provided we are ready to assume the results of classical calculus of variations which are non-trivial and technical in their own right.

In this article, our main goal is to prove the aforementioned theorem. While proving the first two equivalences in Theorem1.1 is easy from the definition of the projection map, proving the third one turns out to be sur- prizingly difficult and is of our principal concern in this article. One of the obstacles to the proof is the burden of heavy notations. To clarify presentation and to facilitate bookkeeping, we employed a system of notations, which is explained in detail in Section7at the end of the article. However, once the cloud of heavy notations is cleared, the proof highlights many intricacies of the algebraic structure of alternating multilinear maps, namely the algebraic structure of determinants and minors and their interrelationship with the algebra of the wedge products which we believe should be of independent interest.

The rest of the paper is organized as follows: in Section 2, we recall the definitions of exterior convexity and introduce the projection map. Section 3 states the main theorem and presents the consequences along with a characterization theorem and a weak lower semicontinuity result. Section 4 explores the algebraic structure of the projection map is greater detail and Section 5 is devoted to the proof of an instrumental lemma, which singles out the crux of the matter. We conclude the proof of the main theorem in Section 6. Finally, the notations used throughout the article is explained in Section 7.

2. Preliminaries

2.1. Notions of exterior convexity

We start by recalling the notions of exterior convexity as introduced in [1].

Definition 2.1. Let 1≤k≤nandf :Λ^k→R.

(i) We say thatf isext. one convex, if the function

g:t→g(t) =f(ξ+t α∧β)

is convex for everyξ∈Λ^k, α∈Λ^k−1 andβ∈Λ¹.If the function gis affine we say thatf isext. one affine.

(ii) A Borel measurable and locally bounded functionf is said to beext. quasiconvex, if the inequality

Ωf(ξ+ dω)≥f(ξ) measΩ holds for every bounded open setΩ⊂Rⁿ,ξ∈Λ^k andω∈W₀^1,∞

Ω;Λ^k−1

. If equality holds, we say thatf is ext. quasiaffine.

(iii) We say thatf isext. polyconvex, if there exists a convex function F :Λ^k× · · · ×Λ^[n/k]k →R such that

f(ξ) =F

ξ,· · · , ξ^[n/k]

, for allξ∈Λ^k. IfF is affine, we say thatf isext. polyaffine.

There are analogous notions of interior convexity (cf.[1]). In what follows, we will discuss the case of exterior convexity only. The case of interior convexity can be derived from the case for exterior convexity by means of Hodge duality.

2.2. Projection maps

To study the relationship between the notions introduced in [1] and the classical notions of the vectorial calculus of variations, namely rank one convexity, quasiconvexity and polyconvexity (see [3]), we will introduce a projection map. We first introduce some notations. As usual, by abuse of notations, we identify Λ^k(Rⁿ) withR_n

k

.

Definition 2.2 (Projection map). Let 2≤k≤n. We write a matrix Ξ ∈R _n

k−1

×n, the upper indices being ordered alphabetically, as

Ξ=

⎛

⎜⎝

Ξ₁^{1···(k−1)} · · · Ξ_n^{1···(k−1)} ... . .. ... Ξ(n−k+2)···n

1 · · · Ξ(n−k+2)···n n

⎞

⎟⎠

=

Ξ_i^I_I∈T^k−1

i∈{1,···,n}=

⎛

⎜⎝

Ξ^{1···(k−1)} ... Ξ(n−k+2)···n

⎞

⎟⎠= (Ξ₁,· · ·, Ξ_n).

We define a linear mapπ:R _n

k−1

×n→Λ^k(Rⁿ) in the following way

π(Ξ) = n i=1

Ξ_i∧eⁱ, where

Ξ_i=

1≤i1<···<ik−1≤n

Ξ_i^{i1···ik−1}eⁱ¹∧ · · · ∧e^ik−1 =

I∈T^k−1

Ξ_i^Ie^I.

Remark 2.3. Observe that this projection map can also be written as,

π(Ξ) =

I∈T^k

⎛

⎝

j∈I

sgn(j, I_j)Ξ_j^Ij

⎞

⎠e^I,

see 3(vii) in Section7 for the notations.

Remark 2.4.

1. Note that the mapπ:R _n

k−1

×n →Λ^k(Rⁿ) is onto.

EXTERIOR CONVEXITY AND CLASSICAL CALCULUS OF VARIATIONS

2. It is easy to see that π:R^n×n→Λ²(Rⁿ) is given by π(ξ) =

n i=1

ξ_i∧eⁱ=

1≤i<j≤n

ξⁱ_j−ξ_i^j

eⁱ∧e^j,

so that, with abuse of notation,

π(ξ) =ξ−ξ^T = 2

ξ−ξ^T 2

·

So fork= 2,πis just twice the alternating projection for 2-tensors (or twice the skew-symmetric projection for square matrices).

3. Main theorem and consequences

3.1. Main theorem

The main result of the article is the following:

Theorem 3.1. Let 2≤k≤n,f :Λ^k →Rand π:R _n

k−1

×n →Λ^k be the projection map. Then the following

equivalences hold

f ext. one convex ⇔ f◦πrank one convex.

f ext. quasiconvex ⇔ f◦πquasiconvex.

f ext. polyconvex ⇔ f◦πpolyconvex.

Remark 3.2.

(i) Note that the theorem does not say that any quasiconvex or rank one convex functionφ:R _n

k−1

×n→Ris of the form f◦πwithf ext. quasiconvexorext. one convexas the following example shows. We letn=k= 2 and

φ(Ξ) = detΞ

which is clearly polyconvex (and thus quasiconvex and rank one convex). But there is no function f :Λ^k →R such thatφ=f◦π.Indeed if such anf exists, we arrive at a contradiction, since setting

X=

1 0 0 1

and Y = 0 0

0 0

, we haveπ(X) =π(Y) = 0 and thus

1 =φ(X) =f(π(X)) =f(π(Y)) =φ(Y) = 0.

(ii) The following equivalence is, of course, trivially true

f convex ⇔ f◦πconvex.

3.2. Relations between notions of exterior convexity We now list a few simple consequences of the main theorem.

Theorem 3.3. Let 1≤k≤nandf :Λ^k→R. Then

f convex ⇒ f ext. polyconvex ⇒ f ext. quasiconvex ⇒ f ext. one convex.

Proof. The result is immediate from Theorem3.1and the classical results (cf.[3]). Another, more direct proof,

without using the classical results, can be found in [1].

Theorem 3.4. Let 1≤k≤nandf :Λ^k→R. The following statements are then equivalent.

(i) f is ext. polyaffine.

(ii) f is ext. quasiaffine.

(iii) f is ext. one affine.

(iv) For every0≤s≤[n/k], there existc_s∈Λ^ks such that,

f(ξ) =

[n/k]

s=0

cs;ξ^s, for every ξ∈Λ^k.

Proof. From the definitions of ext. polyaffine functions, it is clear that (i) ⇔ (iv).

The statements

(i) ⇔ (ii) ⇔ (iii)

follow at once from classical results (cf.Thm. 5.20 in [3]) by virtue of Theorem3.1.

For a more direct proof of the above, see [1]. See also [2] for yet another proof.

Theorem 3.5. Let 1 ≤ k≤ n, 1 < p < ∞, Ω ⊂Rⁿ be a bounded smooth open set and f : Λ^k →R be ext.

quasiconvex verifying, for everyξ∈Λ^k,

c₁(|ξ|^p−1)≤f(ξ)≤c₂(|ξ|^p+ 1) for somec₁, c₂>0. if

α_s α inW^1,p

Ω;Λ^k−1 then

lim inf

s→∞

Ωf(dα_s)≥

Ωf(dα).

Proof. According to Theorem3.1, we have thatf◦πis quasiconvex. Then classical results (see Thm. 8.4 in [3]) show that

lim inf

s→∞

Ωf(dα_s) = lim inf

s→∞

Ωf(π(∇α_s))≥

Ωf(π(∇α)) =

Ωf(dα)

as wished.

4. Algebraic properties of the projection

We now start exploring the algebraic structure of the projection map in greater detail. The following properties are easily obtained. See [4] for a proof.

Proposition 4.1. Let 2≤k≤nandπ:R _n

k−1

×n →Λ^k(Rⁿ)be the projection map.

(i) If α∈Λ^k−1(Rⁿ)∼R _n

k−1

andβ ∈Λ¹(Rⁿ)∼Rⁿ,then, π(α⊗β) =α∧β.

EXTERIOR CONVEXITY AND CLASSICAL CALCULUS OF VARIATIONS

(ii) Letω∈C¹

Ω;Λ^k−1

,then, by abuse of notations,

π(∇ω) = dω.

The following result is crucial to establish the main theorem in the case of polyconvexity. See Section 5.4 of [3] for the definition of adjugates and Section7 for the notations.

Proposition 4.2(Adjugate formula). Ifk is even, then for2≤s≤[n/k],

[π(Ξ)]^s= (s!)

I∈T^sk

I

ssgn(J; ˜I)(adj_sΞ)^I_J^˜

e^I,

and

[π(Ξ)]^s= 0, for [n/k]< s≤min

n, _n

k−1

.

If kis odd,

[π(Ξ)]^s= 0, for alls, 2≤s≤min

n, _n

k−1

.

Proof. We prove only the first equality. Everything else follows by properties of the wedge power. So we prove the case when kis even and 2≤s≤[n/k]. We prove it by induction.

Step 1.To start the induction, we first prove the case whens= 2.

We have,

π(Ξ) =

I∈T^k

⎛

⎝

j∈I

sgn(j, I_j)Ξ_j^Ij

⎞

⎠e^I.

So,

[π(Ξ)]²=π(Ξ)∧π(Ξ)

=

I∈T^2k

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎝

I¹, I² I¹∪I²=I I¹∩I²=∅

sgn

I¹, I²⎛

⎝

j1∈I¹

sgn j₁, I_j¹₁

Ξ^I

j11

j1

⎛

⎝

j2∈I²

sgn j₂, I_j²₂

Ξ^I

j22

j2

⎞

⎠

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎠ e^I

Now, sincekis even, we have, [π(Ξ)]²

= 2

I∈T^2k

⎛

⎝_I

2( sgn([j₁, I_j¹₁],[j₂, I_j²₂]) sgn(j₁, I_j¹₁) sgn(j₂, I_j²₂)Ξ^I

1j1

j1 Ξ^I

j22

j2

+ sgn([j₁, I_j²₂],[j₂, I_j¹₁]) sgn(j₁, I_j²₂) sgn(j₂, I_j¹₁)Ξ^I

j11

j2 Ξ^I

j22

j1 )

⎞

⎠e^I

= 2

I∈T^2k

I

2(sgn(j₁, I_j¹₁, j₂, I_j²₂)Ξ^I

j11

j1 Ξ^I

j22

j2 + sgn(j₁, I_j²₂, j₂, I_j¹₁)Ξ^I

j11

j2 Ξ^I

j22

j1 )

e^I

= 2

I∈T^2k

I

2sgn(j₁, I_j¹₁, j₂, I_j²₂)(Ξ^I

j11

j1 Ξ^I

2j2

j2 −Ξ^I

j11

j2 Ξ^I

j22

j1 )

e^I

= 2

I∈T^2k

_I

2sgn(j₁, I_j¹₁, j₂, I_j²₂)(adj₂Ξ)^I

1j1I_j²₂ j1j2

e^I,

which proves the case fors= 2.

Step 2.We assume the result to be true for somes≥2 and show that it holds fors+ 1, thus completing the induction. Now we know, by Laplace expansion formula for the determinants,

(adj_s+1Ξ)^I_j1¹_...j^...I_s+1^s+1=

s+1

m=1

Ξ_j^I_l^m(−1)^l+m(adj_sΞ)^I_j^{1...Im...Is+1}

1...jl...js+1 , for any 1≤l≤s+ 1

= 1

s+ 1

s+1

l=1 s+1

m=1

Ξ_j^I_l^m(−1)^l+m(adj_sΞ)^I_j^{1...Im...Is+1}

1...jl...js+1 . Note that, for any 1≤l, m≤s+ 1,

sgn(j₁, I¹, . . . , j_s+1, I^s+1) = (−1){(l−1)+(m−1)(k−1)}sgn(j_l, I^m,I˜^l,m), where ˜I^l,m is a shorthand for the permutation (˜j₁,I˜¹, . . . ,˜j_s,I˜^s) and

• ˜j₁< . . . <˜j_sand{˜j₁, . . . ,˜j_s}={j1, . . . ,j_l, . . . , j_s+1}.

• I˜¹< . . . <I˜^sand{I˜¹, . . . ,I˜^s}={I¹, . . . ,I^m, . . . , I^s+1}.

Note that this means ˜j_r =j_r for 1≤r < l and ˜j_r=j_r+1 forl ≤r≤s. Similarly, ˜I^r=I^r for 1≤r < mand I˜^r=I^r+1 form≤r≤s. Now sincekis even, for any 1≤l, m≤s+ 1,

sgn(j₁, I¹, . . . , j_s+1, I^s+1) = (−1)^l+msgn(j_l, I^m,I˜^l,m)

= (−1)^l+msgn(j_l, I^m) sgn( ˜I^lm) sgn([j_l, I^m],[ ˜I^l,m]).

Thus,

sgn(j₁, I¹, . . . , j_s+1, I^s+1)(adj_s+1Ξ)^I_j1¹_...j^...I_s+1^s+1

= 1

(s+ 1) s+1 l,m=1

sgn([j_l, I^m],[ ˜I^l,m]) sgn(j_l, I^m)Ξ_j^I_l^msgn( ˜I^lm)(adj_sΞ)^I_j^{1...Im...Is+1}

1...jl...js+1 .

EXTERIOR CONVEXITY AND CLASSICAL CALCULUS OF VARIATIONS

Hence,

(s+ 1)!

I∈T^(s+1)k

I

s+1sgn(j₁, I¹, . . . , j_s+1, I^s+1)(adj_s+1Ξ)^I_j1¹_...j^...I_s+1^s+1

e^I

=(s+ 1)!

(s+ 1)

I∈T^(s+1)k

⎛

⎜⎜

⎜⎜

⎜⎜

⎝ _I

s+1 s+1

l,m=1

sgn([j_l, I^m],[ ˜I^l,m]) sgn(j_l, I^m)Ξ_j^I_l^m sgn( ˜I^lm)(adj_sΞ)^I_j^{1...Im...Is+1}

1...jl...js+1

⎞

⎟⎟

⎟⎟

⎟⎟

⎠ e^I

= (s!)

I∈T^(s+1)k

⎛

⎜⎜

⎜⎜

⎜⎜

⎝ _I

s+1

s+1 l,m=1

sgn([j_l, I^m],[ ˜I^l,m]) sgn(j_l, I^m)Ξ_j^I_l^m sgn( ˜I^lm)(adj_sΞ)^I_j^{1...Im...Is+1}

1...jl...js+1

⎞

⎟⎟

⎟⎟

⎟⎟

⎠ e^I

=

I∈T^(s+1)k

⎛

⎜⎜

⎜⎜

⎜⎜

⎝

I⊂I I∈T^k

⎛

⎝sgn(I,[I\I])(

j∈I

sgn(j, I_j)Ξ_j^Ij)

s![I\I]

s sgn(˜j₁,I˜¹, . . . ,˜j_s,I˜^s)(adj_sΞ)^I_˜j^˜1...I˜s

1...˜js)

⎞

⎟⎟

⎟⎟

⎟⎟

⎠ e^I,

where the last line is just a rewriting of the penultimate one. Indeed on expanding the sums the map, sending j_l toj;I^m toI_j;I¹, . . . ,I^m, . . . , I^s+1 to ˜I¹, . . . ,I˜^srespectively andj₁, . . . ,j_l, . . . , j_s+1 to ˜j₁, . . . ,˜j_srespectively is a bijection between the terms on the two sides of the last equality.

So, we have, by induction hypothesis, (s+ 1)!

I∈T^(s+1)k

I

s+1sgn(j₁, I¹, . . . , j_s+1, I^s+1)(adj_s+1Ξ)^I_j1¹_...j^...I_s+1^s+1

e^I

=

I∈T^(s+1)k

⎛

⎜⎜

⎜⎜

⎜⎜

⎝

I⊂I I∈T^k

sgn(I,[I\I])

⎛

⎝coefficient of e^I in π(Ξ)

⎞

⎠

×

⎛

⎝ coefficient of e^[I\I] in [π(Ξ)]^s

⎞

⎠

⎞

⎟⎟

⎟⎟

⎟⎟

⎠ e^I

=

I∈T^(s+1)k

coefficient of e^I in [π(Ξ)]^s+1

e^I = [π(Ξ)]^s+1.

This completes the induction proving the desired result.

Since we have seen that [π(Ξ)]^sdepends only on adj_sΞ, we are now in a position to define a linear projection for every value ofs. These maps will be useful later.

Definition 4.3. For every 2≤s≤min

n, _n

k−1

, we define the linear projection mapsπ_s:R ⁿ

k−1

s

×_n

s

→ Λ^ks(Rⁿ) by the condition,

π_s(adj_s(Ξ)) = [π(Ξ)]^s for allΞ∈R _n

k−1

×n.

Remark 4.4. It is clear that this condition uniquely defines the projection maps. For the sake of consistency, we define, π₁=πandπ₀ is defined to be the identity map fromRto R.

5. An important lemma

Lemma 5.1. Let 2≤k≤nandN = _n

k−1

. Consider the function

g(X, d) =f(π(X))−

min{N,n}

s=1

ds,adj_sX

whered= (d₁, . . . , d_min{N,n}),d_s∈R_N

s

×_n

s

for all1≤s≤min{N, n} andX ∈R^N×n. If for a given vector d, the function X → g(X, d) achieves a minimum over R^N×n, then for all 1 ≤ s ≤min{N, n}, there exists D_s∈Λ^ks such that,

ds,adj_sY=Ds, π_s(adj_sY) for all Y ∈R^N×n.

The lemma is technical and quite heavy in terms of notations. So before proceeding to prove the lemma as stated, it might be helpful to spell out the idea of the proof. The plan is always the same. In short, if the conclusion of the lemma does not hold, we can always choose a matrix X such that g(X, d) can be made to be smaller than any given real number, contradicting the hypothesis that the map X → g(X, d) assumes a minimum.

Proof. Let us fix a vector d and assume that for this d, the function X → g(X, d) achieves a minimum over R^N×n.

We will first show that all adjugates with a common index between subscripts and superscripts must have zero coefficients. More precisely, we claim that,

Claim 5.2. For any 2 ≤ k ≤n and for every 1 ≤ s ≤min{N, n}, for every J ∈ T^s, I =

I¹. . . I^s where I¹, . . . , I^s∈ T^k−1,we have,

(d_s)^I_J= 0 wheneverI∩J =∅.

We prove Claim5.2, using induction overs. To start the induction, we first show the case s= 1. Letj ∈I, whereI ∈ T^k−1. We chooseX =λe^j⊗e^I, then clearly π(X) = 0.Also,g(X, d) =f(0)−λ(d₁)^I_j. By lettingλ to +∞and −∞respectively, we deduce that (d₁)^I_j = 0, since otherwise we obtain a contradiction to the fact thatg achieves a finite minima.

Now we assume that Claim 5.2 holds for all 1 ≤ s ≤ p and prove the result for s = p+ 1. We consider (d_p+1)^I_j^1...Ip+1

1...jp+1 withj_l∈I^mfor some 1≤l, m≤p+ 1.

Now we first order the rest of the indices (other than the common index) in subscripts and the rest of the multiindices (other than the one with the common index) in superscripts. Let ˜I¹< . . . <I˜^p and ˜j₁ < . . . <˜j_p represent the multiindices and indices in the sets

I¹, . . . , I^p+1

\ {I^m}and {j₁, . . . , j_p+1} \ {j_l}respectively.

Now we choose,

X =λe^jl⊗e^Im+ p r=1

e^˜jr⊗e^I˜r.

Sincej_l∈I^m, we getπ(X) is independent ofλ. Also, all lower order non-constant adjugates ofX must contain the indexj_lboth in subscript and in superscript and hence their coefficients are 0 by the induction hypothesis.

Hence, the only non-constant adjugate ofX appearing in the expression forg(X, d) is, adj_p+1X_I¹_...I^p+1

j1...jp+1 = (−1)^αλ,

EXTERIOR CONVEXITY AND CLASSICAL CALCULUS OF VARIATIONS

whereαis a fixed integer. Now,

g(X, d) = (−1)^α+1λ(d_p+1)^I_j^1...Ip+1

1...jp+1 + constants.

Again as before, we let λ to +∞ and −∞ and we deduce, by the same argument, (d_p+1)^I_j1¹_...j^...I_p+1^p+1 = 0. This completes the induction and proves the claim.

At this point we split the proof in two cases, the case whenk is an even integer and the case whenk is an odd integer.

Case 1: k is even

Note that, unlessk = 2, it does not follow from above thatd_s = 0 for alls≥[ⁿ_k]. The possibility that two different blocks of multiindices in the superscript have some index in common has not been ruled out. Now we will show that the coefficients of two different adjugates having the same set of indices are related in the following way:

Claim 5.3. For everys≥1,

sgn(J;I) (d_s)^I_J= sgn( ˜J; ˜I) (d_s)^I_J^˜_˜, whenever J ∪I = ˜J ∪I,˜ with J,J˜ ∈ T^s , I =

I¹. . . I^s

=

I¹, . . . , I^s , I˜ =

I˜¹. . .I˜^s

= [ ˜I¹, . . . ,I˜^s], I¹, . . . , I^s,I˜¹, . . . ,I˜^s∈ T^k−1 andJ∩I=∅. In particular, given any U ∈ T^ks, there exists a constantD_U ∈R such that,

sgn(J;I) (d_s)^I_J =DU, (5.1)

for allJ∪I=U withJ ∈ T^s,I=

I¹. . . I^s

=

I¹, . . . , I^s

,I¹, . . . , I^s∈ T^k−1. We will prove the claim again by induction overs. We first prove it for the cases= 1.

For the cases= 1, we just need to prove, for any indexj, any multindexI∈ T^k−1 such that j∩I=∅, we have

sgn(j, I) (d₁)^I_j = sgn(˜j,I) (d˜ ₁)^I_˜j^˜, (5.2) where [j, I] = [˜j,I].˜ We chooseX =λsgn(j, I)e^j⊗e^I −λsgn(˜j,I)e˜ ^˜j⊗e^I˜.Clearly,π(X) = 0 and this gives,

g(X, d) =f(0) +λ

sgn(j, I) (d₁)^I_j−sgn(˜j,I) (d˜ ₁)^I_˜j^˜

,

where we have used Claim5.2to deduce that (d₂)^[I_[j˜j]^I]˜ = 0. Lettingλto +∞and−∞, we get (5.2).

Now we assume the result for all 1 ≤ s ≤ s₀ and show it for s = s₀+ 1. Let [I¹. . . I^s0+1j₁. . . j_s0+1] = [ ˜I¹. . .I˜^s0+1˜j₁. . .˜j_s0+1]. Note that the sets

I¹. . . I^s0+1j₁. . . j_s0+1 and

I˜¹. . .I˜^s0+1˜j₁. . .˜j_s0+1

are permuta- tions of each other, preserving an order relation given byj₁ < . . . < j_s0+1, ˜j₁< . . . <˜j_s0+1,I¹ < . . . < I^s0+1 and ˜I¹ < . . . <I˜^s0+1. Thus the aforementioned sets can be related by any permutation (of k(s₀+ 1) indices) that respects this order. Since any such permutation is a product ofk-flips, it is enough to prove the claim in case ofk-flips (cf.Def.7.2).

We now assume (J, I) and ( ˜J ,I) are related by a˜ k-flip interchanging the subscript j_lwith one index in the superscript block I^m and keep all the other indices unchanged. Also, we assume that after the interchange, the position of the multiindex containingj_l in the superscript is pand the new position of the index from the multiindexI^m in the subscript isq,i.e.,j_l∈I˜^p and ˜j_q ∈I^m.We also order the remaining indices and assume,

I˘= [ ˘I¹, . . . ,I˘^s0] ={I˘¹. . .I˘^s0}=

I¹. . .I^m. . . I^s0+1

, and

J˘= [˘j₁. . .˘j_s0] ={˘j₁. . .˘j_s0}=

j₁. . .j_l. . . j_s0+1

respectively. Now we choose,

X =λsgn(j_l, I^m)e^jl⊗e^Im−λsgn(˜j_q,I˜^p)e^˜jq⊗e^I˜p+

1≤r≤s0

e^˘jr⊗e^I˘r.

Note that π(X) is independent of λ. Also, all non-constant adjugates of X appearing with possibly non- zero coefficients in the expression for g(X, d) have, eitherj_l in subscript and I^min superscript or has ˜j_q as a subscript and ˜I^p as a superscript, but never both as then they have zero coefficients by Claim5.2. Also, these adjugates occur in pairs. More precisely, for every non-constant adjugate ofX appearing with possibly non-zero coefficients in the expression forg(X, d) havingj_l in subscript andI^m in superscript, there is one having ˜j_q in subscript and ˜I^p in superscript.

Let us show that, for any 1≤s≤s₀+ 1, any subset ¯J_s−1={¯j₁, . . . ,¯j_s−1} ⊂J˘ofsindices and any choice of ofs−1 multiindices ¯I¹, . . . ,I¯^s−1out ofs₀ multiindices ˘I¹, . . . ,I˘^s0, we have,

(adj_sX)^[I_[j^{m,I¯1,...,I¯s−1]}

lJ¯s−1]

sgn([j_lJ¯_s−1]; [I^m,I¯¹, . . . ,I¯^s−1]) =− (adj_sX)^{[ ˜}_[˜^I_j^{p,I¯1,...,I¯s−1]}

qJ¯s−1]

sgn([˜j_qJ¯_s−1]; [ ˜I^p,I¯¹, . . . ,I¯^s−1])· (5.3) Let a₁ be the position of j_l in

j_lJ¯_s−1

, a₂ be the position of ˜j_q in ˜j_qJ¯_s−1

, b₁ be the position of I^m in [I^m,I¯¹, . . . ,I¯^s−1] andb₂be the position of ˜I^p in

I˜^p,I¯¹, . . . ,I¯^s−1

. Sincekis even,

sgn([j_lJ¯_s−1]; [I^m,I¯¹, . . . ,I¯^s−1])

= (−1)^{{(a1−1)+(b1−1)}}sgn(j_l, I^m) sgn( ¯J_s−1;{I¯¹. . .I¯^s−1})

×sgn([j_l, I^m],[( ¯J_s−1;{I¯¹. . .I¯^s−1})]), and

sgn([˜j_qJ¯_s−1]; [ ˜I^p,I¯¹, . . . ,I¯^s−1])

= (−1)^{{(a2−1)+(b2−1)}}sgn(˜j_q,I˜^p) sgn( ¯J_s−1;{I¯¹. . .I¯^s−1})

×sgn([˜j_q,I˜^p],[( ¯J_s−1;{I¯¹. . .I¯^s−1})]).

We also have,

(adj_sX)^[I_[j^{m,I¯1,...,I¯s−1]}

lJ¯s−1] = (−1)^a1+b1sgn(j_l, I^m)λ

adj_s−1X_{[ ¯I}¹_,...,I¯s−1] [ ¯Js−1] , and

(adj_sX)^{[ ˜}_[˜^I_j^{p,I¯1,...,I¯s−1]}

qJ¯s−1] =−(−1)^a2+b2sgn(˜j_q,I˜^p)λ

adj_s−1X_{[ ¯I}¹_,...,I¯s−1] [ ¯Js−1] . Combining the four equations above, the result follows.

We now finish the proof of Claim5.3. Using (5.3), we have,

g(X, d) =λ

⎧⎪

⎪⎨

⎪⎪

⎩(−1)^α

sgn(J;I) (d_s0+1)^I_J−sgn( ˜J; ˜I) (d_s0+1)^I_J^˜_˜

+

s0

s=1

s

k_s,γ

⎛

⎜⎝

sgn([j_lJ¯_s−1]; [I^m,I¯¹, . . . ,I¯^s−1]) (d_s)^{[Im,I¯1,...,I¯s−1]} [jlJ¯s−1]

−sgn([˜j_qJ¯_s−1]; [ ˜I^p,I¯¹, . . . ,I¯^s−1]) (d_s)^{[ ˜Ip,I¯1,...,I¯s−1]} [^˜jqJ¯s−1]

⎞

⎟⎠

⎫⎪

⎪⎬

⎪⎪

⎭ + constants,