Invariants of multidimensional time series based on their iterated-integral signature

We introduce a novel class of features for multidimensional time series, that are invariant with respect to transformations of the ambient space. The general linear group, the group of rotations and the group of permutations of the axes are considered. The starting point for their construction is Chen's iterated-integral signature.


Introduction
The analysis of multidimensional time series is a standard problem in data science. Usually, as a first step, features of a time series must be extracted that are (in some sense) robust and that characterize the time series. In many applications the features should additionally be invariant to a particular group acting on the data. In character recognition on tablets for example, the angle from which the device is operated is usually not fixed. This leads to the requirement of rotation invariant features. In EEG analysis, invariants to the general linear group are benefical [EMZMN2012]. In some applications the labelling of coordinates is arbitrary, for example if we have observations of some system from a set of random or unlabelled sensors. Here permutation invariants are called for.
As any time series in discrete time can, via linear interpolation, be thought of as a multidimensional curve, one is naturally lead to the search of invariants of curves. Invariant features, of (mostly) two-dimensional curves have been treated using various approaches. Among the techniques are Fourier series (of closed curves) [Gra1972, ZR1972, KG1982], wavelets [CK1986], curvature based methods [MM1986, COSTH1998] and integral invariants [MCHYS2006]. Closest to our setting is probably the work [FKK2010].
The usefulness of iterated integrals in data analysis has recently been realized, see for example [LLN2013, Gra2013, KSHGL2017, YLNSJC2017] and the introduction in [CK2016]. Let us demonstrate the appearance of iterated integrals on a very simple example. Let X : [0, T ] → R 2 be a smooth curve. Say, we are looking for a feature describing this curve that is unchanged if one is handed a rotated version of X. Maybe the simplest one that one can come up with is the (squared) total displacement length |X T − X 0 | 2 . Now, where we applied the fundamental theorem of calculus twice and introduced the notation dX i r = X i r dr. We see that we have succeded in expressing this simple invariant in terms of iterated integrals of X; the collection of which is usually called its signature. The aim of this work can be summarized as describing all invariants that can be obtained in this way. It turns out, when formulated in the right way this search for invariants reduces to classical problems in invariant theory. In some sense this work can hence be seen as the application of classical results to the setting of invariant feature selection for time series. We also produce some mathematically new results. For example we exhibit, what seems for the first time, a basis for invariants with respect to the general linear group (Lemma 10), and state new geometric interpretations for certain terms of the signature (Section 3.4).
The paper is structured as follows. In the next section we introduce the signature of iterated integrals of a multidimensional curve, as well as some algebraic language to work with it. Based on this signature, we present in Section 3 and Section 4 invariants to the general linear group and the special orthogonal group. Both are based on classical results in invariants theory. In the case of the general linear group we are able to present a linear basis of the invariants. For completeness, we present in Section 5 the invariants to permutations, which have been constructed in [NRRZ2005]. In Section 6 we show how to use all these invariants if an additional (time) coordinate is introduced.
For readers who want to use these invariants without having to go into the technical results, we propose the following route. The required notation is presented in the next section. The invariants are presented in Proposition 11, Proposition 31 and Proposition 39. Examples are given in Section 3.1 (in particular Remark 14), Example 34 and Example 40. All these invariants are also implemented in the software package [D2018]. For a python package for calculating the iterated-integrals signature we propose using the package iisignature, as described in [Rei2017].

The signature of iterated integrals
By a multidimensional curve X we will denote a continuous mapping X : [0, T ] → R d of bounded variation 1 . The aim of this work is to find features (i.e. complex or real numbers) describing such a curve, that are invariant under the general linear group, the group of rotations and the group of permutations. Note, that in practical situation one is usually presented with a discrete sequence of data points in R d , a multidimensional time series. Such a time series can be easily transformed into a (piecewise) smooth curve by linear interpolation.
It was proven in [Che1957] (see [HL2010] for a recent generalization) that a curve X is almost completely characterized by the collection of its iterated integrals 2  The collection of all these integrals is called the signature of X. In a first step, we can hence reduce the goal Find functions Ψ : curves → R that are invariant under the action of a group G.
to the goal Find functions Ψ : signature of curves → R that are invariant under the action of a group G.
By the Shuffle identity (Lemma 1), any polynomial function on the signature can be re-written as a linear function on the signature. Assuming that arbitrary functions are well-approximated by polynomial functions, we are lead to the final simplification, which is the goal of this paper 1 The reader might prefer to just think of a (piecewise) smooth curve. The standard example in applications will be the piecewise linear interpolation of d-dimensional data measured at discrete time points. 2 Since X is of bounded variation the integrals are well-defined using classical Riemann-Stieltjes integration (see for example Chapter 6 in [Rud1964]). This can be pushed much further though. In fact the following considerations are purely algebraic and hence hold for any curve for which a sensible integration theory exists. A relevant example is Brownian motion which, although beeing almost surely nowhere differentiable, nonetheless admits a stochastic (Strantonovich) integral.
Find linear functions Ψ : signature of curves → R that are invariant under the action of a group G.

Algebraic underpinning
Let us introduce some algebraic notation in order to work with the collection of iterated integrals. Denote by T ((R d )) the space of formal power series in d non-commuting variables x 1 , x 2 , . . . , x d . We can conveniently store all the iterated integrals of the curve X in T ((R d )), by defining the signature of X to be Here the sum is taken over all n ≥ 0 and all i 1 , . . . , i n ∈ {1, 2}. For n = 0 the summand is, for algebraic reasons, taken to be the constant 1.
The algebraic dual of T ((R d )) is T (R d ), the space of polyonomials 3 in x 1 , x 2 , . . . , x d . The dual pairing, denoted by ·, · is defined by declaring all monomials to be orthonormal, so for example Here we write the element of T ((R d )) on the left and the element of T (R d ) on the right. We can "pick out" iterated integrals from the signature as follows The space T ((R d )) becomse an algebra by extending the usual product of monomials, denoted ·, to the whole space by bilinearity. Note that · is non-commutative.
On T (R d ) we usually use the shuffle product ¡ which, on monomials, interleaves them in all order-preserving ways, so for example Monomials, and hence homogeneous polynomials, have the usual concept of order or homogeneity. For n ≥ 0 we denote the projection on polynomials of order n by π n , so for example See [Reu1993] for more background on these spaces.
As mentioned above, every polynomial expression in terms of the signature can be re-written as a linear expression in (different) terms of the signature. This is the content of the following lemme, which is proven in [Ree1958].
Lemma 1 (Shuffle identity). Let X : [0, T ] → R d be a continuous curve of bounded variation, then for every a, b ∈ T (R d ) Remark 2. We have used this fact already in the introduction, where we confirmed by hand that We will use the following fact repeatedly, which also explains the commonly used name tensor algebra for T (R d ).
Lemma 3. The space of all multilinear maps on R d × . . . × R d (n-times) is in a one-to-one correspondence with homogeneous polynomials of order n in the non-commuting variables x 1 , . . . , x d by the following bijection with e i the i-th canonical basis vector of R d .

General linear group
Let GL(R d ) = {A ∈ R d×d : det(A) = 0}, be the general linear group of R d .
Definition 5. Define a linear action of GL(R d ) on T ((R d )) and T (R d ), by specifying on monomials Lemma 6. For all A ∈ R d×d , X a curve, Proof. It is enough to verify this on monomials φ = x 1 ..x m . Then By Lemma 43 the set {S(X) 0,T : X curve } spans T ((R d )). Hence φ is a GL invariant of weight w in the sense of Definition 4 if and only if for all A ∈ GL(R d ) Since the action respects homogeneity, we immediatly obtain that projection of invariants are invariants (take B = (det A) −w A in the following lemma): Bπ n φ = π n φ, for all n ≥ 1.
Proof. By definition, the action of GL on T (R d ) commutes with π n .
In order to apply classical results in invariant theory, we use the bijection poly between multilinear functions and noncommuting polynomials, given in Lemma 3.
The simplest multilinear function Ψ : (R d ) ×n → R, satisfying Ψ(Av 1 , .., Av n ) = det(A)Ψ(v 1 , .., v n ) that one can maybe think of, is the determinant itself. That is, n = d and where v 1 v 2 ..v n is the d × d matrix with columns v i . Up to a scalar this is in fact the only one, and invariants of higher weight are built only using determinants as a building block.
The following result is classical. There seem to be at least three ways to prove it. Weyl [Wey1946] uses the "Capelli idenities" and Dieudonne [DC1970, Section 2.5] uses the Young theory of the symmetric group. See [Gar1975] for a simple modern proof.
for all A ∈ GL(V ) and v 1 , . . . , v n ∈ V if and only if n = wd and ψ is the linear combination of functions of the form where σ can be any permutation of the numbers 1, . . . , wd.
To state the following theorem, we introduce the notion of Young diagrams, which play an important role in the representation theory of the symmetric group.
We associate to it a Young diagram, which is a collection of n boxes, with left-justified rows. There are r rows, with λ i boxes in the i-th row.
For example, the partition (4, 2, 1) of 7 gives the Young diagram A Young tableau is obtained by filling these boxes with the numbers 1, .., n. Continuing the example, the following is a Young tableau 2 3 7 1 5 4 6 A Young tableau is standard if the values in every row are increasing (from left to right) and are increasing in every column (from top to bottom). The previous tableau was not standard; the following is.
To our surprise the following statement on a linear basis for the invariants of Theorem 9 cannot be found in the literature.
Lemma 10. A linear basis for the invariants Theorem 9 for n = wd is given by where C i are the columns of Σ, and Σ ranges over all standard Young tableaus corresponding to the partition λ = (w, w, .., w) d times of n.

Here, for a sequence
Proof. We will show that the invariants form an irreducible representation of S n , for which an explicit basis can be given.
We first sketch how the irreducible representations for S n are constructed. Let us recall that a tabloid is an equivalence class of Young tableaux modulo permutations leaving the set of entries in each row invariant [Sag2013, Chapter 2]. For t a Young tableau denote {t} its tabloid, so for example The symmetric group S n acts on Young tableaux as We now show that the invariants are an irreducible representation isomorphic to Irrep (w,..,w) . First, S n acts on n-multilinear functions by permutation of the arguments This means, for tensors Ψ = w * 1 ⊗ .. ⊗ w * n (note the inverse) Denote the space of GL invariants of weight w given in Theorem 9 by invariants. It is a subvectorspace of n-multilinear functions, n = wd. Clearly S n · invariants = invariants. It is in fact irreducible, since for any σ ∈ S n the orbit of under S n spans invariants.
For a tabloid of shape (w, w, .., w) define This is a homomorphism of S n representations. Indeed, On the other hand with p := r τ −1 ( ) and Hence ι(τ · {t}) = τ · ι{t}), and ι is a homomorphism of S n representations. Hence, restricted to the irreducible representation Irrep (w,..,w) ι is either 0 or a bijection to its image [Sag2013, Theorem 1.6.5 (Schur's lemma)]. We now show that its image is invariants and establish a basis for invariants at the same time.
Define the standard Young tableau of shape (w, w, .., w) Then for any (standard) Young tableau t there exists unique σ t ∈ S n such that Indeed, since ι is a homomorphism of S n representation, In general, every π ∈ S n that is column-preserving for t f irst can be written as the product π 1 · .. · π w , with π j ranging over the permutations of the entries of the j-th column t f irst . Then as desired.
Applying Lemma 3 to Lemma 10 we get the invariants in T (R d ).
Proposition 11. A linear basis for the space of GL invariants of order n = wd is given by where C i are the columns of Σ, and Σ ranges over all standard Young tableaus corresponding to the partition λ = (w, w, .., w) d times of n.
Remark 12. By Lemma 7, for any invariant φ ∈ T (R d ) and n ≥ 1 we have that π n φ is also invariant. Hence the previous theorem characterizes all invariants we are interested in (Definition 4), not just homogeneous ones.
Remark 13. Note that each of these invariants φ consists only of monomials that contain every variable x 1 , . . . , x d at least once. This implies that S(X) 0,T , φ consists only of iterated integrals that contain every component X 1 , . . . , X d of the curve at least once. Hence, if at least one of these components is constant, the whole expression will be zero.
Since φ is invariant, this implies that S(X) 0,T , φ = 0 as soon as there is some coordinate transformation under which one component is constant, that is whenever the curve X stays in a hyperplane of dimension strictly less then d.

Examples
We will use the following short notation: We present the invariants described in Section 2 for some special cases of d and w.
We give the linear basis for certain dimensions d and orders w, given in Lemma 10.

− 21
Remark 14. Let us make clear that from the perspective of data analysis, the "invariant" of interest is really the action of this element in T (R d ) on the signature of a curve.
In this example, the real number Remark 15. This is a linear basis of invariants in the fourth level. If one takes algebraic dependencies into consideration, the set of invariants becomes smaller. To be specific, assume that one already has knowledge of the invariant of level 2 (i.e. S(X) 0,T , 12 − 21 ). If, say in a machine learning application, the learning algorithm can deal sufficiently well with nonlinearities, one should not be required to provide additionally the square of this number. In other words | S(X) 0,T , 12 − 21 | 2 can also be assumed to be "known". But, by the shuffle identity (Lemma 1 in the Appendix), this can be written as A similar analysis can also be carried out for the following invariants, but we refrain from doing so, since it can be easily done with a computer algebra system. Whatever the dimension d of the curve's ambient space, the space of invariants of weight 1 has dimension 1 and is spanned by Here, for a matrix C of non-commuting variables, The following lemma tells us that Inv d , although a formerly on level d, can be written in terms of expressions on level 1 and 2.
where x j denotes the omission of that argument.
Proof. We can express the determinant in terms of minors, with respect to any row (since the x i are non-commuting, this does not work with columns!). So e.g. for d = 3 for r = 0, 1, 2. Here, for a monomial of order n ≥ r, inserts the variable x i after position r, e.g.
Summing up we see that In general, for any i = 0, .., d − 1, and summing over r gives 3.3 The case d = 2, w = 1

Geometric interpretation
The invariant for d = 2, w = 1, namely φ = x 1 x 2 − x 2 x 1 has a simple geometric interpretation: it picks out (two times) 4 the area (signed, and with multiplicity) between the curve X and the cord spanned between its starting and endpoint. This follows from Green's theorem [Rud1964, Figure 1: A curve X = (X 1 , X 2 ) is shown, with shaded area given by 1 2 S(X) 0,T , Theorem 10.33], compare Figure 1.

Connection to correlation
Assume that X is a continuous curve, piecewise linear between some time point t i , i = 0, . . . , n. 5 The area is then explicitly calculated as Here, for two vectors a, b of length n the lag-one cross-correlation, which is a commonly used feature in data analysis.
In particular, if the curve starts at 0, we have which is an antisymmetrized version of the lag-one cross-correlation.
Remark 17. Note that it is immediate that the antisymmetrized version of the lag τ crosscorrelation, τ ≥ 2 are also invariants of the curve. Where they can be found in the signature S(X) is unknown to us.

Geometric interpretation
In this case, we have a geometric interpretation of the single invariant. Given a curve X, let V be a sixth of the invariant we have identified. Then which is a special case of Lemma 16. We have a clear geometric interpretation in terms of areas and increments.
If the curve is closed, then X 1 0,T = X 2 0,T = X 3 0,T = 0, so V is clearly 0. If the curve lies in a plane, by Remark 13, V is also 0.

A tetrahedron
If a, b, c and d are points in R 3 , then we say the signed volume of the tetrahedron with corners a,b,c and d (in order) is Its absolute value is the volume of the tetrahedron.
Proposition 18. A linearly interpolated curve through four points has V equal to the signed volume of the tetrahedron on them, and thus |V | is equal to the volume of its convex hull.
Proof. Without loss of generality we set a = 0. If the points are coplanar, then both V and the volumes are 0. We are concerned with the case where the points are in general position.
Consider the curve to be made of three line segments, between the points 0, b, c and d. Then the convex hull is a tetrahedron, whose volume is a third of the volume of the parallelepiped whose sides are the three line segments. Without loss of generality, 0, b and c (which form the base triangle, say) lie in the x 1 x 2 plane. V and the volumes are unchanged under a shear which fixes the x 1 x 2 plane and the base triangle but sends d to the x 3 axis. Then Consider a tetrahedron whose first point is 0 and last point is (0, 0, 1) on the x 3 -axis. We know V of the curve through the points is its signed volume, and we know This gives a slightly non-obvious formula for the signed volume of a tetrahedron as follows. Look along the edge of the tetrahedron from the beginning to the end. The signed volume is 1 3 of the length of the edge times the signed area of the triangle that you see.

Two tetrahedra
Let p 1 . . . p 5 be five points in general position. Consider the linearly interpolated curve through the points in order, made of 4 line segments. Pick axes so that p 1 is at the origin and p 5 is on the x 3 axis.
Proposition 20. V is the sum of the signed volumes of the tetrahedra with corners p 1 , p 3 , p 4 , p 5 and p 1 , p 2 , p 3 , p 5 .
Proof. The two tetrahedra both have p 1 p 5 as the edge from beginning to end. The sum of the signed volumes of the two tetrahedra is, by Proposition 19, 1 3 times the distance p 1 p 5 times the sum of the signed areas, projected onto the x 1 x 2 plane, of the disjoint triangles p 1 p 2 p 3 and p 1 p 3 p 4 . That projected area is exactly the signed area of the curve p 1 p 2 p 3 p 4 in the x 1 x 2 plane. Thus the sum of the signed volumes is 1 6 S(X) 0,T , 12 − 21 S(X) 0,T , 3 , which is V .
Proposition 21. Assume that, projected onto the x 1 x 2 plane, the points p 1 . . . p 5 form a convex quadrilateral. Assume, further, that the points p 2 . . . p 5 form a convex spherical quadrilateral when enlarged onto the unit sphere centred at p 1 and same for the points p 1 . . . p 4 around p 5 . Then the convex hull of p 1 . . . p 5 is the union of the tetrahedron with corners p 1 , p 3 , p 4 , p 5 and the one with corners p 1 , p 2 , p 3 , p 5 . These two tetrahedra have the same sign of signed volume, and they have disjoint interiors.
Proof. Because of the convex spherical quadrilaterals, the following edges must be on the boundary of the convex hull: p 1 p 2 , p 1 p 3 , p 1 p 4 , p 1 p 5 , p 2 p 5 , p 3 p 5 , p 4 p 5 . The following edges must be on the boundary because of any of the convexity assumptions: p 2 p 3 , p 3 p 4 .
Thus every point is a vertex of the convex hull. The convex hull is therefore made of two tetrahedra which share a triangular face. All the edges of the two tetrahedra p 1 , p 3 , p 4 , p 5 and p 1 , p 2 , p 3 , p 5 are on the boundary of the convex hull, so they must be the two.
In the plane, the two triples p 1 p 2 p 3 and p 1 p 3 p 4 form two triangles making up a convex quadrilateral. They must either both be specified clockwise or both anticlockwise. It follows that the signs of the signed volumes of the tetrahedra agree.
Proposition 22. Under the assumptions of Proposition 21, the curve has the property that |V | is the volume of the convex hull.
Proof. By Proposition 20, V is the sum of the signed volumes of the two tetrahedra, which by Proposition 21 make up the convex hull. They have the same sign, so |V | is the hull's volume.

More tetrahedra
The same argument will apply to more than five points p 1 . . . p n . Let γ be the linearly interpolated curve through the points (which is defined up to reparametrisation). Let V γ denote the value of V corresponding to the curve γ.
Proposition 24. If the points are in general position, are a convex polygon when projected onto the x 1 x 2 plane and with p 2 . . . p n convex when viewed from p 1 and p 1 . . . p n−1 convex when viewed from p n , then the linearly interpolated curve through the points has the property that |V γ | is the volume of its convex hull.

A curve
Definition 25. Let γ : [0, T ] → R 3 be any curve. Define its signed volume to be the following limit, if it exists Here γ π is the piecewise linear curve obtained from γ and the partition π of [0, T ].
Theorem 26. Let γ : [0, T ] → R 3 a continuous curve of bounded variation. Then its signed volume exists and Proof. Fix some sequence {π n } n∈N , of partitions of [0, T ] with |π n |→ 0. and interpolate γ linearly along each π n to obtain a sequence of linearly interpolated curves γ n . Then Signed-Volume(γ n ) = V γ n .
By stability of the signature we get convergence and this is independent of the particular sequence π n chosen.
The previous theorem is almost a tautology, but one corollary that then does not use signed volume is Corollary 27. Let γ : [0, T ] → R 3 be of bounded variation such that when collapsed perpendicular to the line through its endpoints, it forms a strictly convex closed curve, and when all but each end is projected onto a sphere around that end it forms a strictly convex curve. Then the signed volume equals the volume of the convex hull and we have This follows because any discretization will satisfy Proposition 24. An example is a piece of helix of up to a single revolution.

Rotations
Let be the group of rotations of R d .
for all A ∈ SO(R d ) and all curves X.
Alternatively, as explained in Section 3, where the action on T (R d ) was given in Definition 5.
Since det(X) = 1, any GL invariant of weight w ≥ 1 (Section 3) is automatically an SO invariant. But there are SO invariants that are not GL invariants (of any weight), for example, for d = 2, where P is a partition of {1, .., n} into sets of size 2 and sets of size d. In particular n = a·2+b·d for some a, b ∈ N.
is a polynomial on V n , viz a i e i 1 , v 1 .. e in , v n a i := ψ(e i 1 , .., e in ). Hence for some polynomial Ψ. Now the fact that ψ is multilinear forces Ψ to be of the form claimed.

Remark 30. For any vectors
Indeed, both sides conincide for v (i) = w (i) = e i and both change sign under permutation of two v's or two w's. Hence they coincide for any v (i) , w (i) ∈ {e 1 , .., e d }. Both are multilinear, so the coincide for all inputs.
We can then reformulate the theorem: ψ is invariant if and only if ψ is the linear combination of functions of the form where P is a partition of {1, .., n} into sets of size 2 and at most one set of size d.
In the language of invariants for T (R d ) we have Proposition 31. The SO invariants of homogeneity n are spanned by where Ψ ranges over the invariants of the previous remark and poly is given in Lemma 3.
In the case d = 2, here is another way to arrive at a full set of invariants, even giving an explicit basis. Taking inspiration from [Flu2000], which concerns rotation invariants of images, we work in the complex vector space T (C 2 ).
Theorem 32. Define The space of SO invariants on level n in T (C 2 ) is spanned freely by z = z j 1 · .. · z jn with #{r : j r = 1} = #{r : j r = 2}.
The space of SO invariants on level n in T (R 2 ) is spanned freely by with #{r : j r = 1} = #{r : j r = 2} and z 1 = 1.
Remark 33. In particular for d = 2 and n even, the dimension of rotation invariants on level n in T (R 2 ) is equal to n n/2 .

They form a basis
Now x j 1 ..x jn : j ∈ {1, 2} is a basis of π n T (C 2 ) with respect to C. Hence z j 1 ..z jn is (the map (x 1 , x 2 ) → (z 1 , z 2 ) is invertible). By Step 1 we have hence exhibited a basis (with respect to C) for all invariants in π n T (C 2 ).

Real invariants
The space of SO invariants on level n in T (C 2 ) is spanned freely by the set of z j 1 · .. · z jn with #{r : j r = 1} = #{r : j r = 2}.
Because z 3−j 1 · .. · z 3−jn is the complex conjugate of z j 1 · .. · z jn , this means that the space of SO invariants on level n in T (C 2 ) is spanned freely by the set of Re(z j 1 · .. · z jn ) and Im(z j 1 · .. · z jn ) with #{r : j r = 1} = #{r : j r = 2} and j 1 = 1.
This is an expression for a basis of the SO invariants in terms of real combinations of basis elements of the tensor space. They thus form a basis for the SO invariants for the free real vector space on the same set, namely π n T (R 2 ).
Example 34. We give maximally linearily independent subsets of the invariants given in Proposition 31, for certain dimensions d and orders w.
Then M :

Regarding the last point
S d then acts on T ((R d )) and T (R d ) via Definition 5. Explicitly, Definition 36. We call φ ∈ T (R d ) a permutation invariant if S(M (σ)X) 0,T , φ = S(X) 0,T , φ for all σ ∈ S d and all curve X.
Alternatively, as explained in Section 3, for all σ ∈ S d . Equivalently, Remark 37. An SO invariant is a a permutation invariant, if we restrict to even permutations.
We follow [NRRZ2005, Section 3]. To a monomial we associate the following set partition of [n] := {1, .., n} Example 38. Let d = 3, then Note that for every permutation σ ∈ S d , . · x σ(in) ). Example 40. Consider d = 3 Order n = 1 1 + 2 + 3 Assume now that X = (X 0 , X 1 , .., X d ) : [0, T ] → R 1+d . Here X 0 plays a special role, in that we assume that it is not affected by the space transformations under consideration. Often X 0 (t) = t is just a component keeping track of time.
Consider GL invariants for the moment. Consider the GL(R 2 ) invariant of weight 1 x 1 x 2 − x 2 x 1 .
Since elements of GL(R 2 ) leave the variable x 0 unchanged, a straightforward way to produce GL invariants presents itself: insert x 0 at the same position in every monomial. For example x 1 x 0 x 2 − x 2 x 0 x 1 is a GL(R 2 ) invariant of weight 1. We now formalize this idea and show that we get every GL invariant this way.
Define the linear map Remove of "removing instances of x 0 " on monomials, as Define the linear map of restriction to U on polynomials of order m by defining on monomials x i | U := x i| U so for example For z = (z 1 , .., z m+1 ) ∈ N m+1 denote by Insert z the linear operator on polynomials of order m by defining it on monomials as follows. For a monomial x i 1 ..x im of order m, Insert z inserts z 1 occurences of x 0 before x i 1 , z 2 occurences of x 0 before x i 2 , .., z m occurences of x 0 before x im and z m+1 occurences of x 0 after x im . For example Insert (2,1,4) x 1 x 2 = x 0 x 0 x 1 x 0 x 1 x 0 x 0 x 0 x 0 .
Theorem 42. The space of GL invariant of weight w, homogeneous of degree m is spanned by the polynomials with 0 ≤ n ≤ m, ψ is an GL invariant of weight w and homogeneity n and z ∈ N n+1 such that z = m − n.
Proof. Let n, ψ, z be as in the statement, then Insert z ψ is GL invariant of weight w. Indeed: for A = diag(1, A) ∈ GL(R d ), with A ∈ GL(R d ), A Insert z ψ = Insert z Aψ = (det A) w Insert z ψ. Now, since φ is GL invariant of weight w and since GL leaves span{x i : i = 0, ∈ U ; i j = 0, j ∈ U } invariant, we get that φ U is GL invariant of weight w. Clearly, there is 0 ≤ n ≤ m and i ∈ N n+1 such that Lastly, Remove φ U is GL invariant, since for A = diag(1, A) ∈ GL(R d ), with A ∈ GL(R d ), Hence every invariant is in the span of the set given in the statement.
The corresponding statements for rotations and permutations are completely analogous, so we omit stating them.
Combining this with the fact that left hand side of (4) is a closed set we get that x in · . . . · x i 1 ∈ span{π n (S(X) 0,1 ) : X curve }.
These elements span π n T ((R d )), which finishes the proof.