to $F_{2}$ and vice versa. This justifies move (7) of 4.2. The situation where
$H$ contains a triangular face, which is move (8), is similar.

After all the reduction steps are performed, by 3.4, we are left with a graph
$H$ whose edges have multiplicity $1$ and $H$ is the medial graph of some
$3$-noncrossing matching $\xi$. This is move (1) of 4.2. The integer
$\alpha(H)_{\xi}$ is the number of ways that we end up in this situation. And
the integer $a_{\xi,(\sigma,\sigma^{\prime})}$ is the number of ways to do the
partition into two groves as desired, by Lemma 4.7. Summing over
$\xi\in\mathcal{TC}_{n}$, we obtain the coefficient on the right hand side of
Eq. 1, as desired. ∎

######  Proposition 4.9.

For each $\xi\in\mathcal{TC}_{n}$, we can lift $B_{\xi}$ to $G_{n,2}$.
Moreover, $\\{B_{\xi}\>|\>\xi\in\mathcal{TC}_{n}\\}$ is a basis of $G_{n,2}$.

###### Proof.

Fixing $n$, we first show that $B_{\xi}$’s are linearly independent. Recall
that for each $\xi\in\mathcal{TC}_{n}$, there is a corresponding cactus
network $\Gamma(\xi)$ whose medial pairing is $\xi$, and that the number of
edges of $\Gamma(\xi)$ equals $\\#\mathrm{cr}(\xi)$, the number of crossings
of $\xi$. Consider the computation of $\alpha(H)$ (4.2) for some $H\subset
2\Gamma(\xi)$. Each reduction step in 4.2 does not increase the number of
edges. Moreover, if $H$ contains any double edges, then those edges can be
removed by step (3) of 4.2. As a result, unless $H=\Gamma(\xi)$,
$\alpha(H)\in\mathbb{Z}\mathcal{TC}_{n}$ is a linear combination of
3-noncrossing matchings with strictly fewer crossings than $\xi$. And when
$H=\Gamma(\xi)$, we have $\alpha(H)=\xi$. This means that
$B_{\mu}(\Gamma(\xi))=0$ if $\\#\mathrm{cr}(\xi)\leq\\#\mathrm{cr}(\mu)$ and
$B_{\xi}(\Gamma(\xi))=\operatorname{wt}(\Gamma(\xi))\neq 0$.

If we have a nontrivial relation $\sum c_{\xi}B_{\xi}=0$, by choosing the
$\xi$ with the smallest number of crossings with $c_{\xi}\neq 0$ and
evaluating this relation $\sum c_{\xi}B_{\xi}=0$ on the cactus network
$\Gamma(\xi)$, we obtain a contradiction. Thus,
$\\{B_{\xi}\>|\>\xi\in\mathcal{TC}_{n}\\}$ is linearly independent. In light
of Theorem 4.8, this means that the matrix
$M=\\{a_{\xi,(\sigma,\sigma^{\prime})}\\}$ whose rows are indexed by
$\xi\in\mathcal{TC}_{n}$ and columns indexed by
$(\sigma,\sigma^{\prime})\in\mathcal{NP}_{n}\times\mathcal{NP}_{n}$ (which has
way more columns than rows) has full rank. We can pick any representative for
$B_{\xi}\in G_{n,2}$ as a linear combination of $L_{\sigma}L_{\sigma}$’s by
choosing a maximal invertible submatrix of $M$ and invert it.

As elements in $G_{n,2}$, $\\{B_{\xi}\>|\>\xi\in\mathcal{TC}_{n}\\}$ is a
basis since they are linearly independent and they span $G_{n,2}$ (Theorem
4.8). ∎

##  5\. Connections to the Temperley–Lieb immanant

The _Temperley-Lieb immanant_ $F_{\tau,T}$ is defined by the second author
[Lam15] as a function on a planar bipartite graph, or equivalently a function
on the affine cone over the Grassmannian $\operatorname{Gr}(k,n)$. However, we
take an orthogonal and self-contained approach and define $F_{\tau,T}\in
G_{n,2}$ as a linear combination of the Bush basis
$\\{B_{\xi}\>|\>\xi\in\mathcal{TC}_{n}\\}$.

###  5.1. Temperley-Lieb immanants in terms of the Bush basis

######  Definition 5.1.

A $(k,m)$-_partial noncrossing matching_ is a pair $(\tau,T)$, such that
$\tau$ is a matching of a subset $S(\tau)=S\subset\\{1,2,\ldots,m\\}$ of even
size that is noncrossing, i.e. if $[m]$ are arranged on a circle in order,
then edges formed by $\tau$ won’t cross, and that $T\subset[m]\setminus S$
with $|S|+2|T|=2k$.

Denote the set of $(k,m)$-partial noncrossing matchings as
$\mathcal{A}_{k,m}$. In the following, we take the parameters $k=n-1$ and
$m=2n$.

The framework of the next construction is similar to Section 5 of [Lam18]. For
each $3$-noncrossing matching $\xi\in\mathcal{TC}_{n}$, we define
$\beta(\xi)\in\mathbb{Z}\mathcal{A}_{n-1,2n}$ as follows.

Draw $\xi$ on a disk as a medial graph with $n$ strands on
$t_{1},\ldots,t_{2n}$. We recover the information of cactus networks
$\Gamma=\Gamma(\xi)$ and $\Gamma^{\vee}$ together in a single cactus, with a
procedure similar to the process described in Section 2.3. Label the boundary
vertices as $1,2,\ldots,2n$ such that $i$ is between $t_{i}$ and $t_{i+1}$ in
clockwise order. These boundary vertices are colored black. The strands in
$\xi$ divide the disk into regions. For each region, if there are boundary
vertices in this region, we identify them together; and if there are no
boundary vertices in this region, we add a black vertex for this interior
region. These identifications result in a cactus $S_{\zeta}$. Make the
intersections of strands in $\xi$ into white vertices and connect each white
vertex to the four black vertices that represent the four faces incident to
it. Denote this cactus graph as $N(\xi)$. See Fig. 15 for an example.

$1$$7$$4$$3$$2$$6$$5$$9$$10$$8$ Figure 15. For
$\xi=\\{(1,9),(2,4),(3,6),(5,7),(8,10)\\}$, the bipartite graph $N(\xi)$ with
$\xi$ draw in dashed lines, $\Gamma$ in black and $\Gamma^{\vee}$ in red

Note that the graph $N(\xi)$ can be viewed as $\Gamma=\Gamma(\xi)$ and
$\Gamma^{\vee}$ lying on top of each other, with intersections being the white
vertices in $N(\xi)$. Each edge of $\Gamma$ and $\Gamma^{\vee}$ consists of
two edges in $N(\xi)$ with endpoints having different colors, called “half-
edges” in $\Gamma$ and $\Gamma^{\vee}$

Let $Z_{1}\sqcup\cdots\sqcup Z_{r}$ be the corresponding noncrossing partition
of $[2n]$ so that we have $r$ distinct vertices on the boundary of the cactus
$S_{\zeta}$. Consider all the possible choices of $A$ that consists of the
following data:

  1. (1)

For every edge in $\Gamma$ and $\Gamma^{\vee}$, choose one of the two half-
edges so that every interior vertex has degree $2$ and every new boundary
vertex $Z_{i}$ has degree at most $2$.

  2. (2)

The chosen edges can be partitioned into vertex-disjoint cycles and paths,
whose endpoints are on the boundary. For each new boundary vertex $Z_{i}$ of
degree $0$, choose one vertex $j\in Z_{i}$ and put the rest
$Z_{i}\setminus\\{j\\}$ in $T(A)$; for each $Z_{i}$ and $Z_{i^{\prime}}$ of
degree $1$ that are the endpoints of a path, choose $j\in Z_{i}$ and
$j^{\prime}\in Z_{i^{\prime}}$, put this pair $(j,j^{\prime})$ in $\tau(A)$,
and put the rest $Z_{i}\setminus\\{j\\}$ and
$Z_{i^{\prime}}\setminus\\{j^{\prime}\\}$ in $T(A)$; for each $Z_{i}$ of
degree $2$, put $Z_{i}$ in $T(A)$. Let $\operatorname{cyc}(A)$ be the number
of cycles in $A$.

######  Lemma 5.2.

For a choice $A$ as above, $|\tau(A)|+2|T(A)|=2n-2$.

###### Proof.

Keep the notations as above. The cactus $S_{\zeta}$ contains $1+2n-r$ circles,
split into $(1+2n-r)+n+\\#\mathrm{cr}(\xi)$ regions by the $n$ strands in
$\xi$. Among these regions, there are $2n$ of them coming from the original
boundary vertices, so we have $1+n+\\#\mathrm{cr}(\xi)-r$ interior regions,
giving rise to $1+n+\\#\mathrm{cr}(\xi)-r$ interior black vertices in
$N(\xi)$. There are $\\#\mathrm{cr}(\xi)$ interior white vertices, so we have
$1+n+2\\#\mathrm{cr}(\xi)-r$ interior vertices and $r$ boundary vertices in
$N(\xi)$ in total. From $A$, we selected $2\\#\mathrm{cr}(\xi)$ edges,
contributing $4\\#\mathrm{cr}(\xi)$ to the total degree of all vertices. All
interior vertices have degree $2$, so the sum of degrees of boundary vertices
is $2r-2n-2$. Assume that $2s$ of them have degree $1$, that $r-n-s-1$ of them
have degree $2$, and that $n-s+1$ of them have degree $0$. We then have
$|\tau(A)|=2s$. Also by construction, we put every boundary vertex in $T(A)$
except one vertex of our choice from every new boundary vertex $Z_{i}$ with
degree $\leq 1$. Thus, $T(A)=(2n)-(n+s+1)=n-s-1$. This means
$|\tau(A)|+2|T(A)|=2n-2$ as desired. ∎

Then we define

$\beta(\xi):=\sum_{A}2^{\operatorname{cyc}(A)}(\tau(A),T(A))\in\mathbb{Z}\mathcal{A}_{n-1,2n}$

where the sum is over all selections $A$ as above.

######  Example 5.3.

First consider an extreme example of $\xi=\\{(1,2),(3,4),(5,6)\\}$ with $n=3$.
The cactus $S_{\zeta}$ identifies $2,4,6$ together. Since $\xi$ is
noncrossing, we do not need to select any half-edges, and the only choice that
we make is choosing a $2$ element subset of $\\{2,4,6\\}$ to be $T(A)$. As a
result,

$\beta(\\{(1,2),(3,4),(5,6)\\})=(\emptyset,\\{2,4\\})+(\emptyset,\\{2,6\\})+(\emptyset,\\{4,6\\}).$

######  Example 5.4.

Consider $\xi=\\{(1,3),(2,5),(4,6)\\}$ with $2$ crossings, with $N(\xi)$ shown
here

$4$$3$$2$$1$$6$$5$$t_{5}$$t_{4}$$t_{3}$$t_{2}$$t_{1}$$t_{6}$

All the $2^{4}=16$ selection of the half-edges are valid and we pick a few of
them here:

$4$$3$$2$$1$$6$$5$$4$$3$$2$$1$$6$$5$$4$$3$$2$$1$$6$$5$$4$$3$$2$$1$$6$$5$

with contributions

$\beta(\xi)=(\\{1,5\\},\\{6\\})+(\\{1,2,3,6\\},\emptyset)+2(\emptyset,\\{3,6\\})+(\\{2,6\\},\\{3\\})+\cdots$

where the third term has a coefficient $2$ because there is a cycle.

######  Definition 5.5.

Let $(\tau,T)\in\mathcal{A}_{n-1,2n}$. Define

$F_{\tau,T}=\sum_{\xi\in\mathcal{TC}_{n}}\beta(\xi)_{\tau,T}B_{\xi}$

where $\beta(\xi)_{\tau,T}\in\mathbb{Z}_{\geq 0}$ is the coefficient of
$(\tau,T)$ in $\beta(\xi)$.

###  5.2. Concordance

We use the notion of concordance from Section 5 of [Lam18], to be explained
further in Section 6.1. In 5.8, we will give a new derivation of the
“electrical” Plücker relations (Proposition 5.35 of [Lam18]).

We say that an $(n-1)$-element subset $I\in\binom{[2n]}{n-1}$ is _concordant_
with a noncrossing partition $\sigma$ if each part of $\sigma$ and each part
of the dual partition $\tilde{\sigma}$ contains exactly one element not in
$I$, viewing $\sigma$ as a noncrossing partition on $1,3,\ldots,2n-1$ and
$\tilde{\sigma}$ on $2,4,\ldots,2n$. In general, we also say $\sigma$ is
concordant with $I$, or $(\sigma,\tilde{\sigma})$ is concordant with $I$.
Given $I\in\binom{[2n]}{n-1}$, we use $\mathcal{E}(I)\subset\mathcal{NP}_{n}$
to denote the set of noncrossing partitions concordant with $I$.

######  Example 5.6.

Let $\sigma=(1,2,5\mid 3,4\mid 6)$,
$\tilde{\sigma}=(\tilde{1}\mid\tilde{2},\tilde{4}\mid\tilde{3}\mid\tilde{4},\tilde{5})$.
Then $\sigma$ is concordant with the subset $\\{3,4,7,9,12\\}$, but not
concordant with the subset $\\{1,3,7,9,12\\}$. See Fig. 16. We also draw the
Dyck path interpretation for later purposes (Section 6).

765432112111098Concordant

765432112111098NOT Concordant

$\bullet$123456789101112Concordant

$\bullet$123456789101112NOT Concordant

Figure 16. Example of concordant and not concordant subsets

For $I\in{2n\choose n-1}$, define

(2) $\Delta_{I}=\sum_{\sigma\in\mathcal{E}(I)}L_{\sigma}\in G_{n,1}.$

We say that a $(k,m)$-partial noncrossing matching $(\tau,T)$ is _compatible_
with $I,J\in{m\choose k}$ if

  1. (1)

$\tau$ is a matching on $(I\setminus J)\cup(J\setminus I)$ such that each
edges contains one vertex in $I\setminus J$ and the other vertex in
$J\setminus I$;

  2. (2)

$T=I\cap J$.

######  Theorem 5.7.

For $I,J\in{2n\choose n-1}$,

(3) $\Delta_{I}\Delta_{J}=\sum_{\tau,T}F_{\tau,T}$

where the sum is over all $(\tau,T)\in\mathcal{A}_{n-1,2n}$ compatible with
$I,J$.

###### Proof.

Both sides are in $G_{n,2}$. By 4.9, it suffices to compare the coefficient of
$B_{\xi}$ on both sides, for $\xi\in\mathcal{TC}_{n}$. We fix
$I,J\in{2n\choose n-1}$ and $\xi\in\mathcal{TC}_{n}$.

By Theorem 4.8,

$\Delta_{I}\Delta_{J}=\sum_{\sigma\in\mathcal{E}(I),\sigma^{\prime}\in\mathcal{E}(J)}L_{\sigma}L_{\sigma^{\prime}}=\sum_{\xi\in\mathcal{TC}_{n}}\sum_{\sigma\in\mathcal{E}(I),\sigma^{\prime}\in\mathcal{E}(J)}a_{\xi,(\sigma,\sigma^{\prime})}B_{\xi}.$

The coefficient
$a_{\xi,(I,J)}:=\sum_{\sigma\in\mathcal{E}(I),\sigma^{\prime}\in\mathcal{E}(J)}a_{\xi,(\sigma,\sigma^{\prime})}$,
by Lemma 4.7, equals the number of ways of splitting $\Gamma:=\Gamma(\xi)$
into two groves $F_{I}\sqcup F_{J}$ such that the boundary partition
$\sigma(F_{I})$ is concordant with $I$ and $\sigma(F_{J})$ is concordant with
$J$. Note that a splitting of $\Gamma$ corresponds to a splitting
$\tilde{F}_{I}\sqcup\tilde{F}_{J}$ of $\Gamma^{\vee}$ such that if $e\in
F_{I}$, then its corresponding edge $e^{\vee}$ in $\Gamma^{\vee}$ belongs to
$F_{J}$. The boundary partitions of $\tilde{F}_{I}$ and $\tilde{F}_{J}$ is
exactly the dual noncrossing partition $\tilde{\sigma}(F_{I})$ and
$\tilde{\sigma}(F_{J})$ respectively, by definition. We will think about
$F_{I}\sqcup F_{J}$ and $\tilde{F}_{I}\sqcup\tilde{F}_{J}$ simultaneously.

For the right hand side of Eq. 3, summing over $(\tau,T)$ compatible with
$I,J$,

$\sum_{\tau,T}F_{\tau,T}=\sum_{\xi\in\mathcal{TC}_{n}}\sum_{\tau,T}\beta(\xi)_{\tau,T}B_{\xi}=\sum_{\xi\in\mathcal{TC}_{n}}\sum_{A}2^{\operatorname{cyc}(A)}B_{\xi}$

where the final sum is over choices of $A$, with each choice $A$ consists of a
selection of half-edges of $\Gamma$ and $\Gamma^{\vee}$ and a certain
selection of vertices in each group of boundary vertices $Z_{i}$ identified
together (see the beginning of this section), such that $\tau(A),T(A)$ is
compatible with $I,J$.

To establish equality bijectively, we map each partition of groves
$F_{I}\sqcup F_{J}$ to a choice of $A$ described above, then we count the
cardinality of the preimage of each $A$. This map $\psi$ works as follows. The
connected components of $F_{I}\cup\tilde{F}_{I}$ give a partition $[2n]$ into
$n+1$ parts (Lemma 2.2 of [Lam18]) and since $\sigma(F)$ is concordant with
$I$, there is exactly one vertex for each part that does not lie in $I$,
called the _root_. We orient each tree in $F_{I}\cup\tilde{F}_{I}$ towards the
root (called a _rooting_) and for each edge $e\in F_{I}\cup\tilde{F}_{I}$,
choose the half-edge in $N(\xi)$ containing the source. Now, the selection $A$
consists of these half-edges $H$, together with $T(A)=I\cap J$ and $\tau(A)$
pairs up $I\setminus J$ and $J\setminus I$ based on the paths formed by these
half-edges. See Fig. 17.

$1$$12$$11$$10$$9$$8$$2$$3$$7$$4$$6$$5$

$1$$12$$11$$10$$9$$8$$2$$3$$7$$4$$6$$5$

$1$$12$$11$$10$$9$$8$$2$$3$$7$$4$$6$$5$

Figure 17. The map $\psi$ in the proof of Theorem 5.7 with
$\xi=(1,10|2,9|3,12|4,6|5,7|8,11)$, $I=\\{1,5,7,8,11\\}$,
$J=\\{3,4,7,9,12\\}.$ Top left: $F_{I}\cup\tilde{F}_{I}$; top right:
$F_{J}\cup\tilde{F}_{J}$; bottom: the selection $A$ in $N(\xi)$ with edges in
$\Gamma$ colored in black and edges in $\Gamma^{\vee}$ colored in red.

We first justify that this map $\psi$ is well-defined, i.e. the selection $A$
obtained satisfies that every internal vertex has degree $2$ in $H$, every
boundary vertex has degree at most $2$ in $H$, and that the pair
$(\tau(A),T(A))$ is compatible with $I,J$. Recall that $H\subset N(\xi)$ is
the set of half-edges that we selected. To begin with, each white interior
vertex has degree $2$. Each black interior vertex is not a a root in its
connected component in $F_{I}\cup\tilde{F}_{I}$ so it has outdegree $1$ in the
rooting, gaining one degree in $H$; similarly, it gains one degree in $H$ from
$F_{J}\cup\tilde{F}_{J}$ so it has degree $2$ in total. Likewise, each
boundary black vertex is either a root in $F_{I}\cup\tilde{F}_{I}$, in which
case it gains no degree in $H$, or not a root in $F_{I}\cup\tilde{F}_{I}$, in
which case it gains one degree in $H$. So considering $F_{J}\cup\tilde{F}_{J}$
as well, each boundary vertex has degree $\leq 2$.

The selection $A$ also consists of the data of $\tau(A)$ and $T(A)$ and we
need to show the following: First, from $\psi$, $\tau(A)$ can be chosen so
that $I\setminus J$ is paired with $J\setminus I$ given by the paths formed by
the half-edges described above; and second, $T(A)=I\cap J$ consists of
$|Z_{k}|-1$ vertices from $Z_{k}$ if $Z_{k}$ is of degree $0$ or $1$ in $H$
and consists of all vertices from $Z_{k}$ if $Z_{k}$ is of degree $2$ in $H$,
where $Z_{1},\ldots,Z_{m}$ is the set of boundary vertices in the cactus. The
claim concerning $T(A)$ is clear by counting: $v\in[2n]$ contributes degree
$x$ to $Z_{k}\ni v$ if $v$ appears $x$ times in $I\cup J$ as a multiset. As
for $\tau(A)$, it suffices to show that we cannot form a path with the two
endpoints both in $I\setminus J$. This is done by a parity argument. Notice
that as we traverse half-edges consecutively in a path or a cycle of $H$, we
alternate between black vertices and white vertices, and alternate between
edges in $F_{I}\cup\tilde{F}_{I}$ and edges in $F_{J}\cup\tilde{F}_{J}$. Thus,
if a path in $H$ starts with a vertex $v$ and an edge in
$F_{I}\cup\tilde{F}_{I}$, ends with a vertex $w$ and an edge in
$F_{J}\cup\tilde{F}_{J}$, then we know that $v\notin I$ and $w\notin J$,
meaning that $v\in J$ and $w\in I$ as they need to have degree $1$. This shows
that $(\tau(A),T(A))$ is compatible with $I,J$.

Now fix a selection $A$ such that $(\tau(A),T(A))$ is compatible with $I,J$.
We count the number of partitions of $\Gamma$ into groves $F_{I}\sqcup F_{J}$
such that $\sigma(F_{I}),\sigma(F_{J})$ are concordant with $I,J$
respectively. For each path $P$ in $A$, we can without loss of generality
assume that it starts with $v\in I\setminus J$ and ends with $w\in J\setminus
I$, as $\tau(A)$ is a matching that pairs $I\setminus J$ with $J\setminus I$.
As we traverse the half-edges in $P$, the first edge incident to $v$ needs to
be assigned to $F_{J}\cup\tilde{F}_{J}$ for a rooting to exist. Then the half-
edges must alternate between $F_{I}\cup\tilde{F}_{I}$ and
$F_{J}\cup\tilde{F}_{J}$ since for a rooting to exist, each internal black
vertex has outdegree $1$ in both $F_{I}\cup\tilde{F}_{I}$ and
$F_{J}\cup\tilde{F}_{J}$. This alternating property allows us to assign all
edges along this path $P$. Likewise, every cycle $C$ in $A$ has two possible
assignments to $F_{I}\cup\tilde{F}_{I}$ and $F_{J}\cup\tilde{F}_{J}$ via the
alternating property (see Fig. 18). Thus, every edge in $\Gamma$ and
$\Gamma^{\vee}$ is now assigned to either $F_{I}\cup\tilde{F}_{I}$ or
$F_{J}\cup\tilde{F}_{J}$. Notice that every directed tree whose internal
vertices have outdegree $1$ has a leaf that is the unique source, i.e. a root.
So the concordant property with $I$ and $J$ is exactly the same as the
existence of a rooting. All $2^{\operatorname{cyc}(A)}$ choices satisfy the
requirement. This finishes the comparison of coefficients of $B_{\xi}$ on both
sides of Eq. 3 as desired.

$4$$3$$2$$1$$6$$5$

$4$$3$$2$$1$$6$$5$

$4$$3$$2$$1$$6$$5$

$4$$3$$2$$1$$6$$5$

$4$$3$$2$$1$$6$$5$

$4$$3$$2$$1$$6$$5$

Figure 18. An example of two partitions achieving the same cycle in $N(\xi)$
with $\xi=(13|25|46)$, $I,J=\\{3,6\\}$. Left: $F_{I}\cup\tilde{F}_{I}$;
middle: $F_{J}\cup\tilde{F}_{J}$; right: the same selection $A$ of $N(\xi)$

∎

We have now established the following commutative diagram of various
expansions:

(4) $\leavevmode\hbox to135.59pt{\vbox
to55.24pt{\pgfpicture\makeatletter\hbox{\hskip 51.9981pt\lower-26.27036pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-51.9981pt}{-26.17052pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
15.91139pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.60585pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\Delta_{I}\Delta_{J}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
15.91139pt\hfil&\hfil\hskip 60.08675pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.78111pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${F_{\tau,T}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
12.08665pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
14.75102pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${L_{\sigma}L_{\sigma^{\prime}}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
14.75102pt\hfil&\hfil\hskip 57.57404pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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The existence of $F_{\tau,T}$’s satisfying Theorem 5.7 is equivalent to the
Plücker relations of the Plücker ring (see Theorem 3.10 and Theorem 3.1 of
[Lam15]). Using Eq. 2, we deduce the “electrical” Plücker relations.

######  Proposition 5.8 (Proposition 5.35 of [Lam18]).

For each $1\leq k<n-1$ and each
$I=\\{i_{1}<i_{2}<\dots<i_{n-1}\\},J=\\{j_{1}<j_{2}<\dots<j_{n-1}\\}$, we have

$\sum_{\sigma\in\mathcal{E}(I),\kappa\in\mathcal{E}(J)}L_{\sigma}L_{\kappa}=\sum_{I^{\prime},J^{\prime}}(-1)^{a(I^{\prime},J^{\prime})}\sum_{\sigma\in\mathcal{E}(I^{\prime}),\kappa\in\mathcal{E}(J^{\prime})}L_{\sigma}L_{\kappa}$

where:

  1. (1)

the summation is over $I^{\prime},J^{\prime}$ obtained from swapping the last
$k$ indices in $J$ with any $k$ indices in $I$, keeping the order in both;

  2. (2)

$a(I^{\prime},J^{\prime})$ is the total number of swaps needed to put both
subsets $I^{\prime}$ and $J^{\prime}$ in order.

##  6\. Tableaux basis of the Grove algebra

In this section, we provide a basis of $G_{n}$, building on results in
[BGKT21, CGS21]. We will select a set of relations $R$ from 5.8 and show that
$R$ is a Gröbner basis of the ideal $\mathfrak{I}_{n}$, which is defined to be
the ideal of polynomials in the variables
$\\{L_{\sigma}\\}_{\sigma\in\mathcal{NP}_{n}}$ that vanish on all electrical
networks $\Gamma$.

We work over ${\mathbb{C}}$ in this section, and $G_{n}$ denotes
$G_{n}\otimes{\mathbb{C}}$.

###  6.1. Embedding into the Grassmannian

Let $R(n-1,2n)$ denote the Plücker ring, the homogeneous coordinate ring of
the Grassmannian $\operatorname{Gr}(n-1,2n)$.

######  Theorem 6.1.

The formula

$\Delta_{I}=\sum_{\sigma\in\mathcal{E}(I)}L_{\sigma}\in G_{n,1}$

induces a graded ring homomorphism $\iota^{*}:R(n-1,2n)\to G_{n}$.

###### Proof.

Let $\pi:{\mathbb{C}}[\Delta_{I}\mid I\in\binom{[2n]}{n-1}]\to G_{n}$ be the
ring homomorphism given by the formula Eq. 2. The ring $R(n-1,2n)$ is the
quotient of ${\mathbb{C}}[\Delta_{I}]$ by the Plücker ideal generated by the
Plücker relations. We need to check that the Plücker ideal belongs to the
kernel of $\pi$.

It is shown in Theorem 3.10 of [Lam15] that the existence of elements
$F_{\tau,T}$ satisfying our Theorem 5.7 implies the Plücker relations, at
least set-theoretically. Since the ring $G_{n}$ has no nilpotents, it follows
that the entire Plücker ideal belongs to the kernel of $\pi$. ∎

The ring homomorphism $\iota^{*}:R(n-1,2n)\to G_{n}$ induces a morphism
between projective varieties

${\rm Proj}(G_{n})\to{\rm Proj}(R(n-1,2n)).$

Indeed, this is exactly the embedding $\iota$ of electrical networks into the
Grassmannian $\operatorname{Gr}(n-1,2n)$ constructed in [Lam18], and our
results give a new proof of this embedding. Note that in [Lam18], the formula
Eq. 2 is a theorem, whereas for us it is a definition.

The embedding is constructed in [Lam18] as follows. Recall that $C_{n}$
denotes the $n$-th Catalan number. Let $M=(M_{I\sigma})$ be the
$\binom{2n}{n-1}\times C_{n}$ matrix of $1$-s and $0$-s, with $1$-s in the
entries corresponding to pairs
$(I,\sigma)\in\binom{[2n]}{n-1}\times\mathcal{NP}_{n}$ that are concordant.
Let $\mathcal{X}_{n}=H\cap\operatorname{Gr}(n-1,2n)$ be the linear slice of
the Grassmannian, where $H\subset{\mathbb{P}}^{\binom{2n}{n-1}-1}$ is the
projective subspace given by the image of the matrix $M$. It is shown in
[Lam18] that $\mathcal{X}_{n,\geq
0}:=\mathcal{X}_{n}\cap\operatorname{Gr}(n-1,2n)_{\geq 0}$ is isomorphic to
the compactified space of electrical networks. Since $\mathcal{X}_{n,\geq 0}$
is Zariski-dense in $\mathcal{X}$, we have the following result.

######  Proposition 6.2.

The homogeneous coordinate ring of $\mathcal{X}$ is naturally isomorphic to
the grove algebra $G_{n}$. The ring homorphism $\pi:R(n-1,2n)\to G_{n}$ in
Theorem 6.1 induces the embedding
$\iota:\mathcal{X}_{n}\hookrightarrow\operatorname{Gr}(n-1,2n)$ of [Lam18].

###  6.2. Electrical networks and the Lagrangian Grassmannian

The subvariety $\mathcal{X}_{n}$ was further studied in the works of Chepuri,
George, and Speyer [CGS21] and by Bychkov, Gorbounov, Kazakov, and Talalaev
[BGKT21], who established the following result.

######  Theorem 6.3.

The subvariety $\mathcal{X}_{n}$ is isomorphic to the Lagrangian Grassmannian
$\operatorname{LG}(n-1,2n-2)$.

The appearance of the symplectic group in the study of electrical networks
earlier appeared in [LP15]. Combining Theorem 6.3 and 6.2, we have the
following result.

######  Proposition 6.4.

The dimension the graded piece $G_{n,d}$ of $G_{n}$ is $\dim
G_{n,d}=\\#\mathcal{C}_{n}^{(k)}$.

###### Proof.

By Borel-Weil theory, the embedding $\mathcal{X}_{n}$ of
$\operatorname{LG}(n-1,2n-2)$ into the Grassmannian
$\operatorname{Gr}(n-1,2n)$, and thus the projective space
${\mathbb{P}}^{\binom{2n}{n-1}-1}$ corresponds to a choice of line bundle on
$\operatorname{LG}(n-1,2n-2)$, or equivalently, to the choice of an
irreducible representation $V_{\omega}$ of the symplectic group
$\operatorname{Sp}(2n-2)$. Indeed, it is shown in [BGKT21] (see also [CGS21])
that $\omega=\omega_{n}$, where $n$ indexes the long simple root in the root
system of type $C_{n}$. Indeed, $V_{\omega}$ can be identified with the kernel
of the natural map
$\varphi:\bigwedge^{n-1}{\mathbb{C}}^{2n-2}\to\bigwedge^{n-3}{\mathbb{C}}^{2n-2}$;
see Theorem 17.5 of [FH91].

Each graded piece $G_{n,d}$ is itself an irreducible representation of
$\operatorname{Sp}(2n-2)$. Namely, we have $G_{n,d}\cong V_{d\omega}$. The
dimension of $V_{d\omega}$ is given by the Weyl character formula. Explicitly,
taking $\lambda=(d^{n-1})$ in Lemma 4.1 of [CS12], we get that this dimension
is equal to

(4) $\frac{\prod_{i=1}^{n-1}a_{i}\prod_{1\leq i<j\leq
n-1}(a_{i}-a_{j})(a_{i}+a_{j})}{\prod_{i=1}^{n-1}(2i-1)!}$

where $(a_{1},a_{2},\ldots,a_{n-1})=(d+n-1,d+n-2,\ldots,d+1)$. We manipulate
Eq. 4:

$\displaystyle=\prod_{1\leq i\leq n-1}(i+d)\prod_{1\leq i<j\leq
n-1}(i+j+2d)\left(\frac{\prod_{1\leq i<j\leq
n-1}(j-i)}{\prod_{i=1}^{n-1}(2i-1)!}\right)$ $\displaystyle=\prod_{1\leq i\leq
n-1}(i+d)\prod_{1\leq i<j\leq n-1}(i+j+2d)\frac{2^{n-1}}{\prod_{1\leq i\leq
n-1}(i+i)(i+i+1)\cdots(i+n-1)}$ $\displaystyle=\prod_{1\leq i\leq j\leq
n-1}\frac{i+j+2d}{i+j}.$

The result follows from 6.5.

∎

The following result follows from standard techniques in the enumeration of
noncrossing lattice paths; see [Ard15, Section 3.1.6].

######  Proposition 6.5.

We have

$\\#\mathcal{C}_{n}^{(k)}=\prod_{1\leq i\leq j\leq n-1}\frac{i+j+2k}{i+j}.$

Recall that in Section 2.4, we defined a partial order $\leq$ and a total
order $\preccurlyeq$ on Catalan objects. In this section, we use Dyck paths,
and for notations, we write $L_{P}$ where $P\in\mathcal{C}_{n}$, instead of
$L_{\sigma}$ where $\sigma\in\mathcal{NP}_{n}$, with the bijection between
Dyck paths and noncrossing partitions shown in Section 2.4. We label the nodes
of each Dyck path by $0,1,\ldots,2n$ and typically ignore the $0$.

We define a total ordering on monomials in $G_{n}$ as follows. For monomials
$L_{\mathbf{P}}=L_{P_{1}}\cdots L_{P_{d}}$ with
$P_{1}\preccurlyeq\cdots\preccurlyeq P_{d}$ and
$L_{\mathbf{Q}}=L_{Q_{1}}\cdots L_{Q_{d}}$ with
$Q_{1}\preccurlyeq\cdots\preccurlyeq Q_{d}$, we define $L_{\mathbf{P}}\prec
L_{\mathbf{Q}}$ if for some $1\leq k\leq d$, $P_{i}=Q_{i}$ for all $k+1\leq
i\leq d$ and $P_{k}\prec Q_{k}$. This is a lexicographic order where we
compare the larger sides first. Also, a monomial of lower degree always
precedes a monomial of higher degree. For a polynomial $f$ in $L_{\sigma}$’s,
its _leading term_ is the monomial with nonzero coefficient that is smallest
in the total order $\preccurlyeq$.

We say that a monomial $L_{\mathbf{P}}=L_{P_{1}}\cdots L_{P_{d}}$ is
_standard_ if $P_{1}\leq P_{2}\leq\cdots\leq P_{d}$, where we use the partial
order here. Thus, standard monomials of degree $d$ are indexed by
$\mathcal{C}_{n}^{(d)}$. We will soon see that standard monomials form a basis
of $G_{n}$.

######  Definition 6.6.

For a Dyck path $P$, its corresponding _Catalan subset_ $I(P)$ is an
$(n-1)$-element subset of $[2n]$ such that $\\{1\\}\cup\\{a+1\>|\>a\in
I(P)\\}$ is the positions of the upsteps in $P$. A subset $I\subset{2n\choose
n-1}$ is called a _Catalan subset_ if $I$ equals the Catalan subset of a Dyck
path $P$.

For a Catalan subset $I$, let $P(I)$ be the corresponding Dyck path. It is
also straightforward to see that $P\leq Q$ if and only if the $k$-th largest
element in $I(P)$ is greater than or equal to the $k$-th largest element in
$I(Q)$, for all $k=1,\ldots,n-1$. The following lemma is immediate from the
definition of Dyck paths.

######  Lemma 6.7.

The subset $I=\\{a_{1}<\cdots<a_{n-1}\\}$ is a Catalan subset if and only if
$a_{i}\leq 2i$ for all $1\leq i\leq n-1$.

######  Remark.

We see that $P\preccurlyeq Q$ if and only if $I(P)$ is lexicographically
larger than $I(Q)$. So for the remainder of the section, we will be extra
careful regarding whether a path is smaller than another path, or a subset is
smaller than another subset.

Observe that $I(P)$ is concordant with the noncrossing partition $\sigma(P)$.
See Fig. 19 for an example of $I=\\{1,3,4,5,10\\}$.

$\bullet$123456789101112 Figure 19. Concordant subset obtained from taking the
upsteps.

######  Lemma 6.8.

For a Catalan subset $I$, $P(I)$ is the minimum in the total order
$\preccurlyeq$ on $\mathcal{E}(I)$.

###### Proof.

Fix a Catalan subset $I$ and its Dyck path $P=P(I)$, and consider another Dyck
path $Q\preccurlyeq P$ which differs from $P$ at step $a+1$ where $Q$ goes
down and $P$ goes up and agrees with $P$. This means $a\in I$ by construction.
Consider the node right before step $a+1$, which is labeled by $a$ in $Q$ and
look horizontally to the left. Since $Q$ goes down at step $a+1$, we have that
$a$ is the maximum element in this connected component $Z$. Since $Q$ agrees
with $P$ before step $a$, every node in $Z$ is followed by an upstep, meaning
that $(Z\setminus\\{a\\})\subset I$. As $a\in I$, $Z\subset I$, and $I$ cannot
be concordant with $Q$. ∎

######  Definition 6.9.

For a pair of noncomparable (under $\leq$) Dyck path $P\succcurlyeq Q$, let
$I=I(P)=\\{a_{1}<\cdots<a_{n-1}\\}$ and $J=I(Q)=\\{b_{1}<\cdots<b_{n-1}\\}$ so
that $I(P)$ is lexicographically smaller than $I(Q)$. As $P\succcurlyeq Q$ and
$P,Q$ are noncomparable in $\leq$, let $k$ be the smallest index such that
$a_{k}>b_{k}$. Then define the Plücker relation of the form

$r_{P,Q}=\Delta_{I}\Delta_{J}-\sum_{I^{\prime},J^{\prime}}(-1)^{a(I^{\prime},J^{\prime})}\Delta_{I^{\prime}}\Delta_{J^{\prime}}$

where the sum is over $I^{\prime},J^{\prime}$ obtained from $I,J$ by swapping
the last $n-k$ entries $\\{a_{n-k},\ldots,a_{n-1}\\}$ of $I$ with all possible
choices of $(n-k)$-subsets of $J$, with
$\Delta_{I}=\sum_{K\in\mathcal{E}(I)}L_{K}$ and $a(I^{\prime},J^{\prime})$ is
the total number of swaps needed to put both $I^{\prime}$ and $J^{\prime}$ in
order, defined above.

Let $R=\\{r_{P,Q}\>|\>P\succcurlyeq Q\text{ noncomparable}\\}$ be the set of
all such relations.

######  Example 6.10.

Let $P$ be the blue Dyck path and $Q$ be the black Dyck path in Fig. 20 with
$I=I(P)=\\{1,2,5\\}$ and $J=I(Q)=\\{1,3,4\\}$. Then the first index $k$ such
that $a_{k}>b_{k}$ is $k=3$ with $a_{k}=5>b_{k}=4$. We have the relation

$r_{P,Q}=\sum_{\begin{subarray}{c}\sigma\in\mathcal{E}(\\{1,2,5\\})\\\
\kappa\in\mathcal{E}(\\{1,3,4\\})\end{subarray}}L_{\sigma}L_{\kappa}+\sum_{\begin{subarray}{c}\sigma\in\mathcal{E}(\\{1,2,3\\})\\\
\kappa\in\mathcal{E}(\\{1,4,5\\})\end{subarray}}L_{\sigma}L_{\kappa}-\sum_{\begin{subarray}{c}\sigma\in\mathcal{E}(\\{1,2,4\\})\\\
\kappa\in\mathcal{E}(\\{1,3,5\\})\end{subarray}}L_{\sigma}L_{\kappa}.$

$\bullet$1234567812345678 Figure 20. A pair of noncomparable Dyck paths

The following key lemma is our “straightening algorithm”.

######  Lemma 6.11.

For $P\succcurlyeq Q$, the leading term of $r_{P,Q}$ is $L_{Q}L_{P}$.

###### Proof.

Recall that the leading term is the smallest (in terms of Dyck paths)
monomial, compared from large to small. By Lemma 6.8, the leading term of
$\Delta_{I}$ is $L_{P(I)}$, if $I$ is a Catalan subset. And for a pair of
noncomparable Dyck paths $P\succcurlyeq Q$ with $I=I(P)$ and $J=I(Q)$, the
leading term of $\Delta_{I}\Delta_{J}$ is $L_{Q}L_{P}$.

Keep the notations as in this section. Let’s examine the set

$I^{\prime}=\\{a_{1},\ldots,a_{k-1},b_{m_{1}},\ldots,b_{m_{n-k}}\\}$

obtained from $I$ in the definition of $r_{P,Q}$ (6.9), where
$m_{1}<\cdots<m_{n-k}$. Without loss of generality, assume all elements in
$I^{\prime}$ are different, as we have $\Delta_{I^{\prime}}=0$ if otherwise.
For each $i=1,2,\ldots,n-1$, if $i\leq k-1$, we have $a_{1},\ldots,a_{i}\leq
2i$; and if $i\geq k$, we have $a_{1},\ldots,a_{k-1}\leq 2i$, and at least
$i-k+1$ number of elements from $\\{b_{m_{1}},\ldots,b_{m_{n-k}}\\}$ that are
less than or equal to $2i$ since $J=\\{b_{1}<\cdots<b_{n-1}\\}$ is a Catalan
subset. This means that for each $i$, there are at least $i$ elements from
$I^{\prime}$ that do not exceed $2i$, and by Lemma 6.7, $I^{\prime}$ is a
Catalan subset. Moreover, by definition of $k$, the smallest number among
$\\{b_{m_{1}},\ldots,b_{m_{n-k}}\\}$ is at most $b_{k}<a_{k}$. Thus, the
$i$-th smallest number of $I^{\prime}$ is smaller than or equal to the $i$-th
smallest number of $I$ for $i\leq k$; and the $k$-th smallest number of
$I^{\prime}$ is strictly smaller than the $k$-th smallest number of $I$. This
is saying $P(I^{\prime})\succcurlyeq P(I)=P$ and $P(I^{\prime})\neq P$.

For each term $L_{P^{\prime}}L_{Q^{\prime}}$ in
$\Delta_{I^{\prime}}\Delta_{J^{\prime}}$ in the relation $r_{P,Q}$, by Lemma
6.8 and the argument above, $P^{\prime}\succcurlyeq P(I^{\prime})\succcurlyeq
P\succcurlyeq Q$ and $P^{\prime}\neq P$. So the monomial
$L_{P^{\prime}}L_{Q^{\prime}}$ (no matter which one of $P^{\prime},Q^{\prime}$
is larger) is strictly larger than $L_{P}L_{Q}$ in our term order. As a
result, $L_{P}L_{Q}$ is the leading term of $r_{P,Q}$ as desired, with
coefficient $1$ from $\Delta_{I}\Delta_{J}$ that cannot be canceled. ∎

######  Theorem 6.12.

The set of relations $R$ is a Gröbner basis of the ideal $\mathfrak{I}_{n}$.
The graded piece $G_{n,d}$ of the Grove algebra has a linear basis given by
$\\{L_{\mathbf{P}}\>|\>\mathbf{P}\in\mathcal{C}_{n}^{(d)}\\}$.

###### Proof.

We have $R\subset\mathfrak{I}_{n}$ by construction. To show that
$\\{L_{\mathbf{P}}\>|\>\mathbf{P}\in\mathcal{C}_{n}^{(d)}\\}$ spans $G_{n,d}$,
we use induction on the total order on the monomials. If a monomial
$L_{\mathbf{Q}}$ is not standard, where
$\mathbf{Q}=(Q_{1}\preccurlyeq\cdots\preccurlyeq Q_{d})$, then there exists
some $k$ such that $Q_{k}\nleq Q_{k+1}$. We now have a noncomparable pair of
Dyck paths. Subtracting $L_{\mathbf{Q}}$ by the relation
$r_{Q_{k+1},Q_{k}}\cdot L_{Q_{1}}\cdots L_{Q_{k-1}}L_{Q_{k+2}}\cdots
L_{Q_{d}}$, we obtain a polynomial whose leading term is greater than
$L_{\mathbf{Q}}$. By the induction hypothesis, this difference can be written
as a linear combination of
$\\{L_{\mathbf{P}}\>|\>\mathbf{P}\in\mathcal{C}_{n}^{(d)}\\}$, so
$L_{\mathbf{Q}}$ can also be written as a linear combination of
$\\{L_{\mathbf{P}}\>|\>\mathbf{P}\in\mathcal{C}_{n}^{(d)}\\}$ in $G_{n}$.
Thus, standard monomials span. By 6.4 on the dimensions, we conclude that the
standard monomials form a basis of $G_{n}$. Moreover, this means that $R$
generates the ideal $\mathfrak{I}_{n}$, since any additional relation reduces
the dimension of some graded component $G_{n,d}$.

To see that $R$ is in fact a Gröbner basis of $\mathfrak{I}_{n}$, we need to
show that the leading terms of polynomials in $R$, which are
$\\{L_{P}L_{Q}\>|\>P,Q\text{ are noncomparable}\\}$, generate the initial
ideal $\mathrm{init}_{\preccurlyeq}(\mathfrak{I}_{n})$. In other words, it
suffices to show that for any $f\in\mathfrak{I}_{n}$, the leading term of $f$
is not standard. Assume the opposite that the leading term of $f$ is
$L_{\mathbf{Q}}$ with coefficient $1$, where $\mathbf{Q}=(Q_{1}\leq\cdots\leq
Q_{d})$. As explained in the previous paragraph, for any other monomial
$L_{\mathbf{Q}^{\prime}}$ in $f$, we can write it as a linear combination of
standard monomials which are weakly greater than $\mathbf{Q}^{\prime}$, thus
strictly greater than $\mathbf{Q}$. Applying this procedure, we obtain a
nonzero relation on the standard monomials, contradicting that standard
monomials form a basis. ∎

## References

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[1]organization=Pprime Institute CNRS, ENSMA, University of Poitiers,
addressline=1 Av. Clément Ader, postcode=86360, city=Chasseneuil-du-Poitou,
country=France [2]organization=Laboratory LIAS - ENSIP, University of
Poitiers, addressline=2 rue Pierre Brousse, postcode=86073, city=Poitiers,
country=France [3]organization=STELLANTIS, Advanced Innovation,
addressline=212 Bd Pelletier, city=Carrières-sous-Poissy, postcode=78955,
country=France

# Large scale response of a vehicle wake to on-road perturbations

Agostino Cembalo<EMAIL_ADDRESS>Jacques Borée
<EMAIL_ADDRESS>Patrick Coirault<EMAIL_ADDRESS>Clément Dumand<EMAIL_ADDRESS>
(April 2024)

###### Abstract

The aim of this research work is to analyse the large scale response of a
vehicle wake to on-road perturbations by using an instrumented vehicle and a
combination of scale one wind tunnel tests, track trials and on road
experiments. More precisely, in all these tests, we focus on the analysis of
the asymmetry of the pressure distribution at the base. Proper Orthogonal
Decomposition (POD) is used. For all cases considered, POD analysis reveals
two dominant modes, respectively associated with vertical and horizontal wake
large scale reorganisation. More than 50% of the total energy is carried by
these two modes and this value increases significantly for on-road tests.
Noteworthy, the low-frequency energy content of the temporal coefficients of
these modes is significantly higher on-road. Low frequencies (even very low
ones) then play a major role, corresponding to a quasi-static perturbation
domain of the velocity seen by the vehicle. We show that a quasi-steady
exploration of the on-road yaw angle statistical distribution during a wind
tunnel test captures phenomena similar to those observed on the road and is
therefore interesting to evaluate on-road aerodynamic performances. This also
opens perspectives for developing closed loop control strategies aiming to
maintain a prescribed wake balance in order to reduce drag experienced on the
road.

###### keywords:

Automotive aerodynamics, Drag Reduction, Wind Tunnel tests, On-Road tests

##  1 Introduction

The European Environment Agency has assessed the impact of transports on
greenhouse gases and state that ” _Transport is responsible for a quarter of
the EU’s greenhouse gas emissions, with road transport representing the
greatest share (72% in 2019)._ ”
111https://www.eea.europa.eu//publications/transport-and-environment-
report-2021 Moreover, it is estimated that passenger cars are still the
dominant transportation mean with a share of approximately 80%
222https://www.eea.europa.eu/publications/transport-and-environment-
report-2022/transport-and-environment-report/view. These studies highlight the
importance of reducing the greenhouse gases emissions of road transportation
and in this regards vehicle aerodynamics plays a crucial role. In fact, at
highway speeds, approximately 70% of the energy losses can be attributed to
aerodynamic forces (Kadijk and Ligterink [1], Hucho and Sovran [2]). In that
respect, vehicle manufacturers put a lot of effort in optimizing vehicle’s
aerodynamics both with numerical simulations and wind tunnel testing. However,
the vehicle’s surroundings are constantly changing in real-life scenarios and
the drag coefficient, usually optimised at zero yaw condition, varies
continuously.

As yaw angle is concerned, several studies have assessed an on-road average
yaw angle distribution (Carlino et al. [3], Garcia de la Cruz et al. [4],
Yamashita et al. [5], Stoll and Wiedemann [6]), showing that, on average, this
distribution corresponds to a quasi-normal distribution centered around 0°
with a standard deviation varying upon the external perturbations of the
velocity seen by the vehicle due not only to wind turbulence but also to
varying road surroundings such as guardrails, vehicles, bushes and trees among
others. Hence, only considering the zero yaw angle drag coefficient translates
into an under-prediction of the actual averaged drag coefficient. Howell et
al. [7] have addressed this problem by defining a wind averaged drag
coefficient, on a WLTP (Worldwide Harmonised Light vehicles Test Procedure)
road cycle, using quasi-steady approaches :

$C_{DWC}=0.530{C_{D}}_{0}+0.345{C_{D}}_{5}+0.130{C_{D}}_{10}+0.007{C_{D}}_{15},$

${C_{D}}_{i}$ ($i=0,5,10,15^{\circ}$) being the drag coefficient at a given
yaw angle and $C_{DWC}$ being the averaged drag coefficient based on a typical
wind distribution. The authors of this study also specifically highlight the
importance of reducing the sensitivity of the aerodynamic loads to the on-road
perturbations.

Cooper and Watkins [8] and Watkins and Cooper [9] successfully measured the
unsteady flow characteristics seen by a road vehicle. They showed that the
mean turbulence intensity is of the order of 5% with integral length scales of
the order of tens of meters. Moreover, turbulence seen by the vehicle has a
rich spectra over a wide range of frequencies and it has been found that the
wind gusts feed the spectrum in the low frequencies. Similar results have been
found by Wordley and Saunders [10], Wordley and Saunders [11] and Schröck et
al. [12]. The first two studies focus on the effect of four different terrain
variations on the turbulence scales and intensities. It is shown that ” _the
four different terrain overlap in the region of lower turbulence intensity (
$<5\%$) and smaller turbulent length scale ($<5\;m$)_”. The study of (Schröck
et al. [12]) emphasizes that, when the wind intensity grows, the energy
content of the turbulence spectrum seen by the vehicle shifts to the low
frequency region, resulting in about 75% of the energy content at frequencies
below $2\;Hz$.

All these studies show the importance of on-road turbulence on vehicles
aerodynamics. As an important follow-up on all these contributions, our
objective in this paper is to provide data and to analyze the spatio-temporal
distribution of the static pressure at the base of a vehicle submitted to such
on road perturbations. In particular, it is interesting to determine if the
external perturbations seen by the vehicle result in vertical or horizontal
large scale asymmetries of the wake that we expect to contribute significantly
to the mean drag experienced by the vehicle. To achieve this objective, we
equipped a vehicle with a Prandtl antenna and a Conrad probe at the front as
well as 49 pressure taps at the base. The vehicle was tested in a wind tunnel,
on a track and a ”RDE-like” route (RDE: Real Driving Emissions). Through the
collected data, we analyzed the pressure asymmetry of the wake and compared
results between wind tunnel experiments and real-world road testing.

The paper is organised as follows: Section 2 gives information on the
measurement systems installed in the car and Section 3 describes the different
experimental setup. In Section 4 we focus on the wind tunnel experiments at
zero yaw angle. Section 5 will highlight the major results obtained on-road
and some comparison with the wind tunnel results will be made. Section 6
provides a statistical comparison between the on-road tests and a test with a
dynamically varying yaw angle in the wind tunnel. Concluding remarks are
proposed in Section 7.

##  2 Measurement systems

A Citroën C4 Cactus, which is shown in Figure 1, has been used with some added
equipment. The vehicle has been equipped with a Prandtl antenna, a Conrad
probe, a GPS system and 49 pressure taps at the base. All the equipment is
shown in Figures 2 and 3.

Figure 1: Front-side and back view of the Citroën C4 Cactus, respectively on
the left and right-hand side. Figure 2: Measuring equipment installed on the
vehicle. (a) Prandtl antenna schematisation. (b) Conrad probe schematisation.
(c) GPS system mounted inside the vehicle. (d) Pressure scanner mounted in the
trunk of the car. Figure 3: Pressure taps location on the vehicle’s base. (a)
Physical location of the pressure taps. (b) Position schematisation of the
pressure taps.

Both Prandtl (Figure 2 a) and Conrad (Figure 2 b) probes have been constructed
internally in a Stellantis’ workshop, mainly following the instructions
detailed in Chue [13]. They have been calibrated and tested by Thomann [14].
The Prandtl antenna grants an angular tolerance of $\pm 10^{\circ}$ while the
Conrad probe has a maximum sensitivity of $60^{\circ}$ with a linear trend for
$-15^{\circ}\leq\beta\leq 15^{\circ}$. The probes are respectively linked with
a _Texxis DPS 50 - 0.5_ and a _Texxis DPS 50 - 2.5_ differential pressure
scanners. In Figure 2 (c) we can see the GPS system mounted inside the car.
Highlighted in green the _VBOX Racelogic_ which is the GPS system itself while
highlighted in red there is the _CAN_ box which gather the signals from the
two probes and send them to the _VBOX_. In Figure 2 (d), the differential
pressure scanner _Scanivalve MPS4264_ , mounted in the trunk of the car, is
shown. It counts 64 total pressure entries and works within a range of 8”
$H_{2}O$. Since it is a differential pressure scanner, this means that the
pressure on each of the 64 taps is given by $p_{i}={p_{a}}_{i}-p_{r}$, where
${p_{a}}_{i}$ is the absolute pressure on the $i_{th}$ tap, $p_{r}$ is the
reference pressure, and $p_{i}$ is the measured value on the $i_{th}$ tap. In
our specific case, the reference pressure is obtained using a pierced aluminum
bottle mounted inside the trunk of the car. This allow to always have the
static pressure in the trunk of the car while minimising the pressure
fluctuations by acting like a low-pass filter. In Figure 3 (a) we can see the
physical location of the pressure taps at the base of the car whereas in
Figure 3 (b) a schematisation of the pressure tap position is given. In the
latter, we can also visualize the location of the point considered as the
origin as well as the width $W=1150\;mm$ and height $H=900\;mm$ of the base.
The length of the vinyl tubes connected to the scanner varies considerably. At
the base, this length can range from 0.7 meters to 2.6 meters, depending on
the pressure tap considered. For each measurement channel and therefore each
tube length, a phase and amplitude correction was carried out by convolution
with a specifically determined transfer function (see Gravier [15]). To find
out the transfer function, the method proposed by Tijdeman and Bergh [16] was
used taking into account the geometric properties of the circuit for each
channel. In order to properly validate some sensitive information from the
physical model derived by these authors, such as for example the radius of the
vinyl tubes, a calibration was carried out on a specific test bench for
several lengths. The calibration bench also made it possible to verify that,
over the frequency range considered here (f $<\ 20Hz$) and after correction,
the phase and module of the signals are accurately corrected.

The acquired pressure data is then post-processed to calculate the pressure
coefficient:

$C_{p_{i}}=\frac{p_{i}-p_{r}}{Q},$ (1)

where $p_{i}$ represents the pressure measured at the $i_{th}$ pressure tap,
$p_{r}$ is the reference pressure in the vehicle trunk, and
$Q=\frac{1}{2}\rho_{\infty}U_{\infty}^{2}$ is the dynamic pressure, with
$\rho_{\infty}$ as the air density and $U_{\infty}$ as the free-stream
velocity. During on road tests, $U_{\infty}$ is the velocity deduced by the
GPS system at the time of the measurement.

The following notation will be used for spatial ($\langle A\rangle$) and
temporal ($\overline{A}$) averaging, respectively:

$\langle A\rangle(t_{m})=\frac{1}{N}\sum_{n=1}^{N}A(\textbf{x}_{n},\,t_{m}),$

$\overline{A}(x_{n})=\frac{1}{M}\sum_{m=1}^{M}A(\textbf{x}_{n},\,t_{m}),$

where $N$ and $M$ are the number of pressure taps at the base and the number
of time steps, respectively.

However, it is necessary to note that measuring the reference pressure on-road
remains a very complicated task because the slightest variation in the vehicle
results in a variation in the pressure. Comparisons between the speed data
obtained using the Prandtl antenna and those obtained using tachymetric
measurements have shown that the reference pressure is not constant. The
instantaneous difference between the pressure coefficient of each sensor and
the spatial average of the pressure coefficient at the base at the same
instant is defined as :

$C_{p}^{*}(\textbf{x}_{n},\,t_{m})=C_{p}(\textbf{x}_{n},\,t_{m})-\langle
C_{p}(t_{m})\rangle.$ (2)

Hereafter, we will always use $C_{p}^{*}(x_{n},\,t_{m})$ in order to eliminate
any temporal variation in the reference pressure during a test. By using
$C_{p}^{*}(x_{n},\,t_{m})$, we therefore focus on the asymmetry of the wake
corresponding to the deviation from the spatial average of the instantaneous
pressure distribution at the base.

##  3 Experimental setup

For this study, we were able to make use of the Stellantis wind tunnel located
in Orbassano (10043, Italy) (Stellato and Betti [17]), the Stellantis track of
La Ferté-Vidame (28340, France) and a RDE-like route (Yvelines, 78000,
France), which are shown in the Figures 4 and 5 respectively.

### Stellantis Wind Tunnel

Figure 4: Schematisation of the Stellantis Wind tunnel in Orbassano (10043,
Italy).

From the control room (1), operators can activate the wind tunnel. The test
section (2) is 3/4 open, with a variable nozzle section ranging from
$22\,\text{m}^{2}$ to $30.5\,\text{m}^{2}$ in a plenum measuring 10.5 meters
long, 12.2 meters wide, and 10.8 meters high. The turbulence rate at the
center of the section is less than 0.5%, and the boundary layer, at
$X=2.5\,\text{m}$ from the exit of the convergent, has a thickness of
$5\,\text{mm}$ with the boundary layer suction function activated. The maximum
test speed is $U_{\infty}=215\,\text{km/h}$ with a system of 5 moving belts,
one under the car and the remaining four under the wheels to allow their
rotation. The test speed for our tests is constant at
$U_{\infty}=110\,\text{km/h}$. The car is positioned on a turntable (3)
measuring 7 meters in diameter. Beneath the turntable, a six-component balance
(4) is installed to measure aerodynamic forces. The vehicle is attached to the
balance using four masts, with which we can also adjust the vehicle’s ground
clearance, set to the same value as measured with two people inside the
vehicle for road tests. Through the heat exchanger (5), we control the
temperature to remain constant at $T=22^{\circ}\text{C}$ throughout the tests.
The wind is set in motion with a motor-driven fan (6).

### On-road routes

Figure 5: On-road routes visualisation. (a) RDE-like route in the Yvelines
(78000, France). (b) Stellantis track of La Ferté-Vidame (28340, France).

The RDE-like route (part (a) of the Figure 5) is a route that partially
fullfils to the kilometer distribution in accordance with the new regulatory
guidelines for the vehicle homologation process (WLTP). It measures
approximately 68 kilometers and lasts about 1 hour. The whole route has, most
of the time, allowed vehicle speed above $70\;km/h$. This criterion was chosen
because an internal study at Stellantis was conducted to determine from which
speed aerodynamic efficiency becomes predominant compared to the rolling
resistance of the vehicle (Tuan [18]). This results in the following kilometer
distribution: $\sim 43\%$ highway (in green), $\sim 46\%$ rural (in red), and
$\sim 11\%$ urban (in blue). Atmospheric pressure information as well as wind
speed and orientation were obtained from averages of several weather stations
in the area.

Simultaneously, tests were conducted at the Stellantis site in La Ferté-Vidame
(28340, France) (part (b) of the Figure 5). This site consists of three
tracks: the Outer Track (in green), the Bel Air Track (in red), and the Park
Track (in blue). The colors correspond to the permitted speeds on the tracks,
so the Outer Track corresponds to a highway route, the Bel Air Track
corresponds to a rural route, and the Park Track corresponds to an urban
route. We mainly conducted tests on the Outer Track and the Bel Air Track. The
Outer Track measures approximately $\sim 6.5\;km$ and includes four main
turns. This track allows for a speed constantly over $90\;km/h$ throughout the
test. The Bel Air Track measures approximately $\sim 3.3\;km$ and includes
several short-radius turns, allowing for a speed of $70\leq V\leq 90$. These
tracks are connected by connecting ramps (sky-blue dashed lines). The La
Ferté-Vidame site also has a weather station capable of measuring wind speed
and direction, as well as temperature and atmospheric pressure. All tests
conducted on the tracks start at point 0 (in sky-blue), which corresponds to
the main entrance of the tracks. The direction of travel is indicated by red
triangles. However, each day, the direction of travel is reversed to be able
to conduct tests in both directions and ensure that the cars and the track
degrade uniformly and not biased by the direction of travel. In our tests, the
number of people in the vehicle is always two. There is a driver and the
person responsible for verifying the correct progress of the acquisition. The
passenger sits in the rear seat where a PC mounting system has been installed
to ensure safety requirements. A zero calibration is performed before each
test to compensate for any pressure signal drift that may be present. This
calibration is performed in a closed building to prevent small disturbances
from affecting the process. Acquisition is initiated just before starting the
test and stopped once the vehicle is stationary again.

##  4 Wind tunnel testing

The main objective of the wind tunnel sessions is to characterize the wake of
the car in terms of global motions and to analyse the frequency of these
motions. Subsequently, the same analysis will be carried out on the road to
allow for a comparison of the results.

Proper Orthogonal Decomposition (POD) will be used in that respect. This
analysis method was first introduced by Lumley [19] with the aim of
decomposing the flow’s turbulence into deterministic functions, each of which
can capture a portion of the flow’s turbulent kinetic energy. Several works
have summarized this method and explained how to use it (Chatterjee [20],
Cordier and Bergmann [21], Taira et al. [22], Weiss [23]).

In our specific case, the principle of POD, calculated here by the direct
method after substracting the time average of
$\overline{C^{*}_{p}(\textbf{x}_{i},t)}$, consists of decomposing the vector
$C^{*}_{p}(\textbf{x}_{i},t)$ into a set of deterministic functions
$\Phi^{(k)}(\textbf{x})$ that are modulated by N temporal coefficients
$a^{(k)}(t)$ as follows:

$C_{p}^{*}(\textbf{x}_{i},t)=\overline{C^{*}_{p}(\textbf{x}_{i},t)}+\sum_{k=1}^{N_{modes}}a^{(k)}(t)\Phi^{(k)}(\textbf{x}_{i}).$
(3)

It is important to recall that the functions $\Phi^{(k)}$ are orthogonal.

The decomposition is performed using singular value decomposition, that
provides the functions $\Phi^{(k)}$, which we will call modes hereafter, as
well as the eigenvector $\bm{\lambda}$, which contains the contribution of
each mode, in decreasing order, to the total variance. In our study, the
number of modes corresponds to the number of pressure sensors at the base,
which is 49 modes in total. This means that we have a signal decomposition in
the form:

$C_{p}^{*}(\textbf{x}_{i},t)=\overline{C^{*}_{p}(\textbf{x}_{i},t)}+\sum_{k=1}^{49}a^{(k)}(t)\Phi^{(k)}(\textbf{x}_{i}),\\\
$

with the total energy defined as follows:

$\xi=\sum_{i=1}^{N_{taps}}\overline{{C_{p_{i}}^{*}}^{2}}=\sum_{k=1}^{N_{modes}}\lambda_{k}.$

Additionally, thanks to the temporal coefficients of the modes ($a^{(k)}(t)$),
it is possible to analyze the frequency characteristics of the projection of
the velocity field onto each mode.

In what follows, we will concentrate on the two first POD modes because they
contain a significant contribution of the total variance. We remind that, by
using $C_{p}^{*}(\textbf{x},\,t)$, we focus on the asymmetry of the wake.

### Static yaw angle analysis

The first analysis concerns a case with zero yaw angle ($\beta=0^{\circ}$).
The results are presented in Figure 6.

Figure 6: Results of the wake analysis for a fixed zero yaw angle in the wind
tunnel. Presentation of the distribution of $\beta$, the first two modes, the
percentage of their contribution to the total energy, as well as spectral
analysis of the temporal coefficients associated with the two modes.

The distribution of $\beta$ shown in the Figure (top-left panel) has a mean
$\mu_{\beta}=-0.3^{\circ}$ and a standard deviation
$\sigma_{\beta}=0.13^{\circ}$. It is not centered at $\beta=0^{\circ}$ due to
some slight human errors, such as alignment to the symmetry plane of the wind
tunnel and alignment of the Conrad probe to the vehicle’s symmetry plane. The
latter error remains constant throughout our tests. This distribution of
$\beta$ corresponds to typical cases analyzed by automotive manufacturers,
i.e., a nearly constant yaw angle throughout the test. Under these conditions,
we performed the POD analysis. The energy contribution of the 49 modes is
presented in the bottom-left panel of the figure. We can observe that the
first two modes contribute significantly to the total wake energy
(approximately $45\%$), so we will analyze these two modes, which are
presented in the top-right panel of the figure. It is interesting to note that
these modes have well-defined footprints. For the first mode, we can associate
this footprint with a vertical tilting of the wake. In contrast, the footprint
of the second mode reveals a horizontal tilting. For this configuration, it is
relevant to analyze the characteristic frequencies of these two asymmetries.
To do this, we calculated the spectra of the two corresponding temporal
coefficients with a frequency resolution $\Delta f=5.10^{-2}\;Hz$. This
frequency resolution corresponds to a division of the signal into 53 blocks,
which will be the minimum number of blocks used subsequently. In the bottom-
right panel of the figure, a pre-multiplied spectrum is presented showing the
spectra of the two temporal coefficients. Regarding the frequencies, we chose
to represent dimensionless frequencies to compare the different tests
performed. For this purpose, we used the Strouhal number ($St=f.H/V$), which
is the product of the frequency and the advective time scale ($H/V$). Note
that the frequency range $f<20\;Hz$ discussed in section 2 corresponds here to
a Strouhal number of $St<\ 0.6$, which means that phases and amplitudes are
accurately corrected for all taps in the relevant frequency range. Moreover,
from measurements performed with reduced scale models in a wind tunnel
(Haffner et al. [24]) we do not expect significant higher frequency
contributions for large scale events. We observe a large band distribution of
energy for both $a^{(1)}$ and $a^{(2)}$, with a more defined peak at
$St\approx 10^{-1}$ for $a^{(2)}$. As a measure of the important role of low
frequencies we can observe that 62% and 63% of the energy of the mode for the
first and second coefficients, respectively, are contained for Strouhal values
less than $10^{-1}$.

##  5 On-road results

The response of the wake of the vehicle to on-road perturbations of the
velocity seen by the vehicle is analysed in this section. We conducted
numerous tests and a subset is shown in Table 1. The two on-road tests
presented hereafter are the two test cases highlighted in bold in the Table.
These two tests provide a comprehensive overview of the wake behavior under
different conditions, allowing us to draw meaningful conclusions about the
vehicle’s wake behaviour when compared to wind tunnel tests at zero yaw angle.
The robustness of these results is discussed in detail in Cembalo [25]. As
shown in Table 1, the conclusions discussed below apply to all on-road tests
with only minor differences on energy content in the POD decomposition or
spectral energy density of the temporal coefficients.

Routes | $V_{wind}[km/h]$ | $O_{wind}$ | $P_{atm}[hPa]$ | $\mu_{\beta}\,[^{\circ}]$ | $\sigma_{\beta}\,[^{\circ}]$ |  % ($\Phi_{1}+\Phi_{2}$) |  % $a_{1}$/$a_{2}$ ($St\leq 10^{-1}$) | $\xi/\xi_{0}$
---|---|---|---|---|---|---|---|---
Track | 22 | North-West | 975 | -0.45 | 1.74 | 62 | 79 and 79 | 3.01
Track | 25 | South-West | 975 | -0.19 | 1.68 | 62 | 79 and 79 | 5.52
Track | 13 | South | 974 | -0.44 | 1.52 | 63 | 78 and 79 | 3.57
Outer-Track | 14 | South | 979 | 0.02 | 1.37 | 56 | 76 and 81 | 2.71
Track | 16 | South-West | 978 | -0.08 | 1.54 | 63 | 77 and 79 | 2.79
Track | 8 | South | 987 | -0.31 | 1.04 | 69 | 74 and 69 | 3.01
Outer-Track | 3 | South-Est | 988 | -0.34 | 1.24 | 56 | 75 and 78 | 2.3
Outer-Track | 6 | Est | 987 | 0.01 | 1.15 | 58 | 77 and 77 | 2.43
Outer-Track | 6 | North-Est | 987 | 0 | 1.10 | 55 | 75 and 77 | 2.24
RDE-like | 8 | South-West | 1021 | -0.02 | 1.23 | 55 | 79 and 70 | 3.26
RDE-like | 9 | South-Est | 1021 | -0.02 | 1.07 | 63 | 81 and 70 | 3.66
RDE-like | 12 | North-Est | 1015 | -0.54 | 1.56 | 57 | 76 and 67 | 2.84
Highway | 12 | North-Est | 1015 | -0.18 | 1.63 | 53 | 72 and 72 | 1.98
RDE-like | 21 | North | 1009 | -0.18 | 1.65 | 55 | 76 and 72 | 3.66
RDE-like | 25 | North-Est | 1009 | 0.1 | 1.4 | 55 | 76 and 70 | 3.03
RDE-like | 3 | South-West | 1022 | 0.02 | 1.41 | 63 | 72 and 65 | 6.22
RDE-like | 6 | West | 1021 | 0.1 | 1.54 | 60 | 69 and 69 | 2.85
Table 1: Some statistics to show the robustness of the on-road tests.
Hereafter, the two tests in bold will be analysed in detail.

In Figure 7 (a), the results are from a test on the La Ferté-Vidame track,
while in part (b), it corresponds to a test on the RDE-like route. Atmospheric
pressure information as well as wind speed and orientation for the two tests
are detailed in Table 1.

Figure 7: Results of wake analysis on track and on the RDE-like route.
Presentation of the distribution of $\beta$, the first two modes, and the
percentage contribution to the total energy as well as spectral analysis of
the temporal coefficients associated with the two modes. (a) Results for test
08 on August 6, 2021, at the La Ferté-Vidame track. (b) Results for test 04 on
July 22, 2021, on the RDE-like route.

Regarding the test conducted at La Ferté-Vidame, the direction of travel was
counterclockwise, as indicated by the red triangles in Figure 5. The test
route includes approximately 4 laps on the Outer track and 2 laps on the Bel
Air track. There is very little traffic on this road closed to public. When
compared to RDE test conditions, the surroundings of the road on the
Stellantis track is repeated at each lap while it displays more diversity on
public roads.

It is interesting to note that the distributions of $\beta$ are very different
to that observed in Figure 6. On the track (part (a) of the Figure 7), the
distribution has a mean $\mu_{\beta}=-0.45^{\circ}$ and a standard deviation
$\sigma_{\beta}=1.74^{\circ}$, while on the RDE-like route (part (b) of the
Figure 7), the distribution has a mean $\mu_{\beta}=-0.02^{\circ}$ and a
standard deviation $\sigma_{\beta}=1.07^{\circ}$. On the track, the
distribution is not centered at $\beta=0^{\circ}$, which is consistent with
the counterclockwise direction of the laps. Additionally, the wind speed was
higher during the track test, which may explain the difference observed in the
standard deviation.

POD modes 1 and 2 have a very clear large scale signature corresponding to
respectively vertical and horizontal balance of the wake. They are very
similar for both on-road tests. They are also qualitatively similar to the
main POD modes obtained in the wind tunnel at zero yaw angle. However, some
details differ, for example the signature of mode 2 at the bottom part of the
base. The contribution of these two modes to the total energy is now 63%. This
is a very significant increase when compared to the wind tunnel tests (45%
contribution for these two modes).

On the RDE-like route, a slight increase in the energy contained in mode 1
(about 44%) and a slight decrease in mode 2 (about 19%) is observed. However,
the sum of the energy contained in the two modes remains almost unchanged.
Looking now at the premultiplied spectra of the temporal coefficients
associated to modes 1 and 2, an increase of the energy in the low frequency
domain is observed for the first mode. For all these tests, 70% to 80% of the
energy content is measured for Strouhal numbers lower than 0.1. Note that, for
on-road situations, the vehicle velocity $V$ used in the computation of the
Strouhal number in figure 7 is the average vehicle velocity during the test.

A higher frequency resolution ($\Delta f=5.10^{-3}\;Hz$) was obtained by using
longer blocks and therefore a lower number of blocks for averaging the spectra
($N_{blocks}\leq 13$). The results are presented in Figure 8. The presence of
a large band contribution for Strouhal numbers lower than $10^{-2}$ is very
clear, especially for the first mode corresponding to the vertical wake
balance (Figure 7). We observe that very low frequencies (down to $St\approx
10^{-3}$) contribute to the power spectrum. At the average speed maintained
during the track test, the distance traveled by the car during a period
$T=1/f$ can be estimated as $d\approx V.T=H/St$ which is very large for
Strouhal numbers lower than $10^{-2}$. We stress that a quasi steady response
of the wake is expected for these low Strouhal numbers. This is an important
aspect because these low frequencies do not only come from wind turbulence.
They probably also originate from the variation in the flow seen by the car
due to a variable environment having a wide range of longitudinal scales along
the road. Indeed, driving along road portions having similar characteristics
(freely exposed to side winds, protected in a forest, exposed to urban area
features, …) will induce low frequency components corresponding to the time
taken to travel along these portions. Of course, sharp changes in wind
directions occur while entering to - or emerging from - a sheltered portion.
However, careful experimental study by Haffner et al. [24] using a model
geometry at reduced scale shows that natural or provoked wake switching
between right/left asymmetry is a slow process taking place in approximately
30 convective time scales. This corresponds to $St\approx 1/30$. The slow
reorganisation of the large scale wake therefore acts as a low-pass filter
that cuts high frequency disturbances.

Figure 8: Results of the spectral analysis of the temporal coefficients with a
frequency resolution of $\Delta f=5.10^{-3}[Hz]$ for the track and RDE-like
route tests. (a) Coefficients $a^{(1)}$ and $a^{(2)}$ on the track. (b)
Coefficients $a^{(1)}$ and $a^{(2)}$ on the RDE-like route.

##  6 Statistical analysis at varying yaw angle in the wind tunnel

Going back to the wind tunnel, we can try to reproduce the yaw angle
distribution observed on the road. A subset of yaw angles of the vehicle is
first chosen. One single continuous wind tunnel test of approximately 20
minutes - the total number of sample is 120687 at a sampling rate of 100 Hz -
is used for the test shown in Figure 9. During the test, our choice is to
impose a variable measurement time for each vehicle yaw angle of the subset in
order to have a data set representative of the statistical distribution of yaw
angles observed on the road.

Figure 9: Results of the wake analysis for a varying yaw angle in the wind
tunnel. Presentation of the distribution of $\beta$, the first two modes and
the percentage of their contribution to the total energy. The time averaged
value $\overline{C^{*}_{p}(\textbf{x},t)}$ at the base for the full test is
shown on bottom right plot

Modes 1 and 2 are very similar to on road modes. We observe that nearly 63% of
the turbulent kinetic energy is contained in the first two modes. This is also
very similar to the energy distribution for on-road tests. Moreover, the
energy contained in the first mode has almost doubled compared to the static
yaw angle test, which seems to be due to the variation of the yaw angle along
the tests, similarly to what we could observe on the road. This may be a good
indication of the interest for making these quasi steady tests during vehicle
development to test the robustness of a vehicle design to yaw angle
perturbations.

Figure 10: Comparison of total variance $\xi$ for the different analyzed
tests. In sky blue, the wind tunnel case with fixed $\beta$. In dark blue, the
wind tunnel case with varying $\beta$. In yellow and red, the cases on track
and on the RDE-like route, respectively on the left and on the right.

The total energy $\xi$ measured in the different tests is compared in Figure
10 to that measured in the wind tunnel for a fixed yaw angle. This ratio
increases significantly for the test with dynamic variation of $\beta$ in the
wind tunnel, reaching more than double the reference value
($\xi/\xi_{0}\approx 2.5$). The total energy further increases on the road. On
the track, the difference is similar to the cases with dynamically varying
$\beta$ in the wind tunnel ($\xi/\xi_{0}\approx 3.01$), while on the RDE-like
route, the total variance increases even more ($\xi/\xi_{0}\approx 3.66$)
compared to the value observed in the wind tunnel for fixed yaw, which is a
very significant difference.

The imposed quasi static variations of the yaw angle are interesting in order
to widen the statistical distribution of the projection of instantaneous
pressure footprints on the main POD modes that represent the wake vertical and
horizontal asymmetries. These modes are very similar to on-road main modes and
this is an interesting property for wind tunnel tests to be confirmed on other
vehicle geometries.

Figure 11: Distribution of the temporal coefficients $a^{(1)}$ and $a^{(2)}$
as function of the yaw angle $\beta$ in the varying yaw angle test. The red
dot is the conditional average of the temporal coefficient for each
orientation of the vehicle.

Figure 11 shows the distribution of the first and second temporal coefficients
as function of the yaw angle. Each group corresponds to a prescribed
orientation of the vehicle during the dynamic tests. The red dot is the
conditional average of the temporal coefficient for each orientation of the
vehicle. We see that the POD decomposition is able to describe a large span of
wake states at the base. It is well known that drag increases with yaw angle.
For the largest positive or negative yaw angles considered here, the change of
sign of $a^{(2)}$ corresponds to a change in left/right asymmetry. As
$a^{(1)}$ is concerned, we see that positive values correspond to large yaw
angles while small yaw angles - and therefore minimum drag - are associated
statistically to negative values of $a^{(1)}$. The time averaged value
$\overline{C^{*}_{p}(\textbf{x},t)}$ at the base for the full test is also
shown in Figure 9. This mean distribution displays approximately a left/right
symmetry and a positive pressure gradient (pressure increases when moving
upward). Using the POD decomposition a low order representation of
$C^{*}_{p}(\textbf{x},t)$ can be obtained using the first two modes only. From
the signature of the mean distribution, from the spatial distribution of the
two dominant modes shown in Figure 9 and from the conditional values of the
temporal coefficients discussed above, it is easy to figure out that the wake
is evolving from a positive vertical asymmetry at about zero yaw to a
horizontal asymmetry at larger yaw. Indeed, at larger yaw, a positive value of
$a^{(1)}$ reduces the vertical gradient while a non zero $a^{(2)}$ corresponds
to horizontal pressure gradients. Considering the similarities with on-road
results, we therefore conclude that a low frequency exploration of such wake
large scale states is performed during on-road travel. Therefore, even a slow
closed loop control able to maintain a prescribed wake balance would be
interesting for drag reduction. This approach is proposed for a reduced scale
model using actuated flaps located at the back of a model vehicle in Cembalo
[25] and Cembalo et al. [26].

##  7 Key findings and Conclusions

The aim of this research work was to analyse the effect of on-road
perturbations on the asymmetry of a vehicle wake. We described the vehicle
used for testing in a full-scale wind tunnel, on a track and on a RDE-like
route. Additionally, we detailed the measurement systems and experimental
setup. To analyze the wake characteristics in the wind tunnel and on the road,
Proper Orthogonal Decomposition (POD) of the pressure distribution on the base
of the vehicle was employed. The main results are summarized below:

  * •

Wind tunnel tests with fixed yaw substantially deviate from road conditions,
showing lower modal energy content and different wake modal footprints, yet
still corresponding to similar large scale vertical or horizontal wake
asymmetries. The low-frequency energy content of the temporal coefficients of
the mode is significantly different from that observed on-road.

  * •

Road tests exhibit a nearly normal distribution of yaw angle, with values
consistently between $\pm 10^{\circ}$ and $-5^{\circ}\leq\beta\leq 5^{\circ}$
for over 95% of the time. POD analysis also reveals two primary modes,
associated with vertical and horizontal wake asymmetries, respectively. More
than 60% of the total energy is carried by these two modes. Low frequencies
(even very low ones down to $St\approx 10^{-3}$) play a major role,
corresponding to a quasi-static perturbation domain. Indeed, over 70% of the
energy of the two main modes is contained within frequencies $St\leq 10^{-1}$;

  * •

A wind tunnel run can be designed by selecting a subset of vehicle yaw and a
variable measurement time for each yaw. The large data set obtained is then
representative of the statistical distribution of yaw angles observed on the
road. This single run captures phenomena similar to those observed on the
road. The footprint of the two main modes are very similar. The energy content
in the first two modes is higher and closer to that measured on road. These
tests also show that multiple large scale wake states contribute to the first
two modes. Considering the similarities with on-road results, we conclude that
a low frequency large span exploration of such wake states is performed during
on-road travel.

Further studies should complete this analysis. It would be interesting to
analyse the effect of pitch angle variations on these large scale modes.
Moreover, the tests have been conducted on a single vehicle. It would be
interesting to see the robustness of these results on other types of vehicle.
We also stress that a quasi steady response of the wake is expected for the
energy containing low Strouhal numbers observed on the road. Therefore, the
use of steady CFD computations for a subset of yaw angles is interesting to
evaluate a vehicle on-road characteristics. Moreover, even a slow closed loop
control able to maintain a prescribed wake balance would be promising for drag
reduction strategies. This is an ongoing work on 2/5 reduced scale models
using recursive subspace-based predictive control as proposed in Cembalo [25]
and Cembalo et al. [26].

## Acknowledgements

The authors would like to warmly thank Jean Charles Boueilh for invaluable
support during the experiments, as well as Yann Goraguer for moral and
technical assistance.

## Funding

This research project was funded by STELLANTIS and Ministry for Higher
Education and Research. Agostino Cembalo wishes to acknowledge support from
ANRT scholarship.

## Competing interest

The authors report no conflict of interest.

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