fixed bug in tensorExp, small changes to universal variety example

Author: felixlotter

PathSignatures.m2

@@ -271,7 +271,7 @@ Node
         Text
             This agrees with the result in Table 2 of @HREF("#ref1","[1]")@.
         Text
-            As $\phi$ is an isomorphism of graded vector spaces, we see that the variety $\mc U_{d,k}$ 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.
+            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.
         Example
             mons = flatten entries basis(3,Q);
             S = QQ[z_1..z_(length mons)];
@@ -304,7 +304,7 @@ Node
             M = sub(matrix apply(lpols, i -> (flatten entries (coefficients(i, Monomials => mons))#1) ),QQ);
             M^(-1)
         Text
-            This is the matrix from Example 21 in @HREF("#ref2","[2]")@ up to scalars.
+            This is the matrix from Example 21 in @HREF("#ref2","[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 @HREF("#ref2","[2]")@ the coordinate change is obtained directly without computing the exponential. Both strategies yield the same result by Lemma 18 in loc. cit..
         -- Text
         --     Let us compute a path variety after toric coordinate change.
         -- Example

PathSignatures/algebra.m2

@@ -279,21 +279,10 @@ lieBasis(Array, NCPolynomialRing) := (w,R) -> (
 
 lieBasis(List, NCPolynomialRing) := (l, R) -> lieBasis (new Array from l, R);
 
--- auxiliary functions for tensorExp
-expTermCoef = (t) -> (
-    m := max t;
-    counts := new MutableList from (m : 0);
-    for i from 0 to length(t)-1 do(
-       if(t#i > 0) then counts#(t#i - 1) = counts#(t#i - 1) + 1;
-    );
-    counts = toList(counts);
-    facs := apply(counts, i -> i!);
-    binom := product(facs);
-    return(1/binom);
-);
+
 
 expTerm = (tl,l) -> (
-    expTermCoef(l)*product(l,i->(tl_i))
+    return(1/(length(l))!)*product(l,i->(tl_i))
 )
 
 -- Given a tensor p with constant term 0, tensorExp(p,k) returns the k-th level component of exp(p)