Recall that $K \langle \mathtt{1}, \ldots, \mathtt{d} \rangle$ is isomorphic to the free commutative algebra over the Lyndon words and if we grade each word by its length, the algebra homomorphism $$\phi: K[\mathtt{w} \ | \ \mathtt{w} \ \mathrm{Lyndon}] \cong K \langle \mathtt{1}, \ldots, \mathtt{d} \rangle, \ \mathtt{w} \mapsto \mathtt{w}$$ is an isomorphism of graded vector spaces. The inverse $\psi$ of this isomorphism is computed by lyndonShuffle. Dually, we have the isomorphism of graded vector spaces $$\psi^*: K[\mathtt{w} \ | \ \mathtt{w} \ \mathrm{Lyndon}]^* \cong K \langle \mathtt{1}, \ldots, \mathtt{d} \rangle^*, \ \alpha \mapsto \alpha \circ \psi.$$
We view the two vector spaces $K\langle \mathtt{1}, \ldots, \mathtt{d}\rangle^*$ and $K[\mathtt{w} \ | \ \mathtt{w} \ \mathrm{Lyndon}]^*$ as infinite-dimensional affine spaces. We are interested in the subset $\mathcal U_d$ of those points of $K\langle \mathtt{1}, \ldots, \mathtt{d}\rangle^*$ that define shuffle algebra homomorphisms. This is a Zariski closed set. As $\psi$ is an algebra homomorphism, they correspond to the points of $K[\mathtt{w} \ | \ \mathtt{w} \ \mathrm{Lyndon}]^*$ under $\psi^*$ that define algebra homomorphisms; these are parametrized by the points of the vector space $K^{\mathcal L}$, where $\mathcal L$ is the set of Lyndon words, via the map $$\eta: K^{\mathcal L} \to K[\mathtt{w} \ | \ \mathtt{w} \ \mathrm{Lyndon}]^*, \ x \mapsto ev_x.$$ In particular, $\mathcal U_d$ is parametrized by $\psi^* \circ \eta, \ x \mapsto (\mathtt{w} \mapsto \psi(\mathtt{w})(x))$.
Projecting $\mathcal U_d$ to the degree $k$ component, we obtain a subvariety of $((K^d)^{\otimes k})^* \cong (K^d)^{\otimes k}$, called the universal variety, $\mathcal U_{d,k}$.
As the map $\psi$ is compatible with the projection, $\psi^* \circ \eta$ restricts to a parametrization of $\mathcal U_{d,k}$: it is the image of the induced morphism $$K^{\mathcal L_k} \to (K\langle \mathtt{1}, \ldots, \mathtt{d}\rangle_k)^* \cong (K^d)^{\otimes k}, \ x \mapsto (\psi(\mathtt{w})(x) \cdot \mathtt{w})$$ where $\mathcal L_k$ is the set of Lyndon words of length at most $k$.
As usual, we can compute the ideal that cuts out the image variety as the kernel of the corresponding ring map $$K[x_{\mathtt w} \ | \ \mathtt{w} \text{ of length } k] \to K[y_{\mathtt w} \ | \ \mathtt{w} \text{ Lyndon of length } \leq k].$$ This map can be computed via lyndonShuffle. Let us do this in the example $d=3, k=3$.
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Let us compute the kernel.
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This agrees with the result in Table 2 of [1].
As $\phi$ is an isomorphism of graded vector spaces, we see that the variety $\mathcal U_{d,k}$ is parametrized by the vector of degree $k$ monomials in Lyndon words after a linear coordinate change on $(K^d)^{\otimes k}$. We can use this to simplify the computation of the universal variety.
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Let us compute the matrix of the coordinate change for $d=2, k=4$.
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Note that the coordinate change differs from the one described in Example 21 of [2] as our toric parametrization does not arise from the exponential map on the Lie algebra. We can easily construct this coordinate change as well, by using lieBasis and tensorExp:
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This is the matrix from Example 21 in [2]. Note that while we obtained the coordinate change by inverting the map that sends a word to its coefficient in the exponential (which is a linear combination of Lyndon word monomials), in [2] the coordinate change is obtained directly without computing the exponential. Both strategies yield the same result by Lemma 18 in loc. cit..
Améndola, C., Friz, P., & Sturmfels, B. (2019, January). Varieties of signature tensors. In Forum of Mathematics, Sigma (Vol. 7, p. e10). Cambridge University Press.
Galuppi, F. (2019). The rough Veronese variety. Linear algebra and its applications, 583, 282-299.
The source of this document is in PathSignatures.m2:337:0.