2.1 Graphs: simple and Cayley

This subsection establishes the basic definitions, results and notation from graph theory needed below. A graph is a tuple $\Gamma = (V,E)$ where V is the set of vertices and $E \subset V\times V$ is the set of edges. Graphs here are understood to be undirected and simple (i.e., finite, loopless and without multiple edges). We will always take $\Gamma $ to be connected and may remind the reader of this assumption where particularly important. Write $E(\Gamma )$ for the edge set of a graph $\Gamma $ and $V(\Gamma )$ for the vertex set. Inclusion $v \in \Gamma $ means $v \in V(\Gamma )$ .

If the vertex set V has some natural linear order (in particular, when $V = [n]$ ), then an edge between vertices $i < j$ will always be written with the lower vertex first $(i,j)$ , unless explicitly stated otherwise. Note that these edges are undirected, so an edge $(i,j)$ is the same as an edge $(j,i)$ .

We denote graphs pictorially with circles as vertices and lines as edges between them; for example, we would display a particular graph $\Gamma $ on $9$ vertices as

The induced subgraph of $\Gamma $ with vertex set $V \setminus A$ is , where $E' = E \cap \left ( V \setminus A \times V \setminus A\right )$ . We write $\Gamma - v$ for $\Gamma \setminus \{v\}$ . When collapsing a subgraph in drawing, we reference the subgraph in a square to distinguish that there are multiple vertices being referenced, and double lines connecting to acknowledge the possibility of multiple edges. For example, we may display $\Gamma $ above as

if the structure within $\Gamma \setminus \left \{v_1 ,v_2\right \}$ is not needed.

A path in $\Gamma $ from vertex $v_0$ to vertex $v_\ell $ of length $\ell $ is a sequence of vertices $(v_0,v_1,...,v_\ell )$ , where $(v_k,v_{k+1}) \in E(\Gamma )$ for $k=0,...,\ell -1$ . Define the distance $d(v,w)$ between v and w as the minimum length over all paths from v to w, and let if no such path exists.

In other words, a graph is k-connected if there exists no cut set A where $\left | A \right | <k$ . The following is an equivalent characterization used in $\S $ 4.

An R-labeled graph is a tuple $(V,E,L)$ , where $(V,E)$ is a graph and L is a function $L \colon E \to R$ for some set R. A Cayley graph of a group G and a set of generators S is the graph $\left (G,\{(g,h) \mid g^{-1}h \in S\}\right )$ . Cayley graphs are usually directed graphs, but all generators considered here will be involutions, and so the Cayley graphs will be undirected simple graphs. Note that $g^{-1}h \in S$ if and only if $h=gs$ for $s \in S$ , so edges in a Cayley graph correspond to right multiplication by generators.

This paper concerns graphs $\Gamma $ on vertex set $[n]$ and labeled Cayley graphs of the symmetric group with generators being some set of transpositions. The edge labels are principle ideals in $\mathbb {C}[t_\bullet ]$ .

Note $w^{-1}v$ is conjugate to $wv^{-1}$ , so if $w = v(i,j)$ , then $\mathcal {L}(w,v) = \langle t_{w(i)}-t_{w(j)}\rangle = \langle t_{v(i)}-t_{v(j)}\rangle .$ Note also that $\mathcal {L}$ is defined whenever $wv^{-1}$ is a transposition.

Example 2.5. Let $\Gamma = \left ([3],\{(1,2),(2,3)\}\right )$ . Then $\mathcal {G}_\Gamma $ has vertex set $S_3$ , edges $\{(w,v) \mid w^{-1}v \in \{(1,2),(2,3)\}$ , and labels of the form $\langle t_i-t_j\rangle $ , where $i,j \in [3]$ . Below is $\mathcal {G}_\Gamma $ , with labeling ideals denoted by generators.

Consider the edge $(132,312)$ . These permutations have their first and second positions swapped, corresponding to right multiplication by $(1,2) \in E(\Gamma )$ . The edge is labeled $\langle t_1-t_3\rangle $ because these permutations have the entries $1$ and $3$ swapped, corresponding to left multiplication by $(1,3)$ .

The $\Gamma $ -length of a permutation $w \in S_n$ is

(2.2)

This is also the value of $d(e,w)$ in $\mathcal {G}_\Gamma $ . When $\Gamma $ is the path graph, $\Gamma $ -length is the traditional length function on permutations.

2.2 Splines

This section introduces the ring of splines on a labeled Cayley graph. The lemmas in this subsection are well known and straightforward, but we include proofs for completeness.

To distinguish from polynomials, we always denote a spline with a bar.

Example 2.7. Again, consider $\Gamma = \left ([3],\{(1,2),(2,3)\}\right )$ . Drawn below (omitting edge-labels) are three examples of splines on $\mathcal {G}_\Gamma $ .

So $\bar {\rho }_1(w) = t_1$ for all $w \in S_3$ , $\bar {\rho }_2(w) = t_{w(1)}$ for all $w \in S_3$ , and ${ \bar {\rho _3}(w) = \begin {cases} t_1-t_2 & \text {if }w = 213 \\ t_3-t_2 & \text {if }w = 231 \\ 0 & \text {otherwise.} \end {cases}}$

The set of splines is closed under addition, as well as multiplication.

Proof. Let $(w,v) \in E(\mathcal {G}_\Gamma )$ . By assumption, $\bar {\rho }(w)-\bar {\rho }(v) \in \mathcal {L}(w,v)$ and $\bar {\sigma }(w)-\bar {\sigma }(v) \in \mathcal {L}(w,v)$ . We have

$$ \begin{align*} \bar{\rho}(w)\bar{\sigma}(w) - \bar{\rho}(v)\bar{\sigma}(v) &= \bar{\rho}(w)\bar{\sigma}(w) - \bar{\sigma}(v)\bar{\rho}(w) +\bar{\sigma}(v)\bar{\rho}(w) - \bar{\rho}(v)\bar{\sigma}(v) \\ &= \bar{\rho}(w)\left(\bar{\sigma}(w) - \bar{\sigma}(v)\right) +\bar{\sigma}(v)\left(\bar{\rho}(w) - \bar{\rho}(v)\right), \end{align*} $$

and the sum is clearly in $\mathcal {L}(w,v)$ .

Definition 2.9. The ring of splines on $\mathcal {G}_\Gamma $ is the subring

of ${ \prod _{w \in S_n} \mathbb {C}[t_\bullet ]}$ with pointwise addition and multiplication.

We now construct two sets of splines and the identity spline, each are elements of $\mathcal {M}_{\Gamma }$ for all $\Gamma $ . Let

The ring $\mathcal {M}_{\Gamma }$ is an infinite-dimensional $\mathbb {C}$ -vector space in the natural way and can also be viewed as a finitely generated graded $\mathbb {C}[t_\bullet ]$ -module in two ways via the following module actions:

(2.3) $$ \begin{align} f\left(t_1,\ldots,t_n\right).\bar{\rho} = f\left(\bar{t}_1,\ldots,\bar{t}_n\right)\bar{\rho} \end{align} $$

and

(2.4) $$ \begin{align} f\left(t_1,\ldots,t_n\right).\bar{\rho} = f\left(\bar{x}_1,\ldots,\bar{x}_n\right)\bar{\rho}, \end{align} $$

where the right-hand side of both (2.3) and (2.4) work by substituting splines for variables in to the polynomial f then multiplying as in the ring structure of $\mathcal {M}_{\Gamma }$ . For both actions, the constant $f(0,\ldots ,0)$ is naturally mapped to . Since $\mathcal {M}_{\Gamma }$ is a $\mathbb {C}[t_\bullet ]$ -submodule of $\prod _{w \in S_n} \mathbb {C}[t_\bullet ]$ for either module action, it is finitely generated. We call the module action (2.3) the left action and the module action (2.4) the right action of $\mathbb {C}[t_\bullet ]$ on $\mathcal {M}_{\Gamma }$ . Given any $\omega \in S_n$ , both actions may be twisted by sending $f \mapsto \omega f$ first in the polynomial ring. Both the left and right actions are naturally compatible with the grading on $\mathcal {M}_{\Gamma }$ .

Example 2.11. Let $\bar {\rho } \in \mathcal {M}_{\Gamma }$ and let $f(t_\bullet ) = t_1^3 + t_2^2 + t_3$ . Let $\omega =(1,2,3) \in S_n$ . The left action of f on $\bar {\rho }$ evaluated at any $v \in S_n$ is

$$\begin{align*}f(t_\bullet).\bar{\rho}(v) = \left[((\bar{t}_1)^3 + (\bar{t}_2)^2 + \bar{t}_3)\bar{\rho}\right](v) = (t_1^3 + t_2^2 + t_3)\bar{\rho}(v),\end{align*}$$

the right action of f on $\bar {\rho }$ evaluated at any $v \in S_n$ is

$$\begin{align*}f(t_\bullet).\bar{\rho}(v) =\left[((\bar{x}_1)^3 + (\bar{x}_2)^2 + \bar{x}_3)\bar{\rho}\right](v) = (t_{v(1)}^3 + t_{v(2)}^2 + t_{v(3)})\bar{\rho}(v),\end{align*}$$

the $\omega $ -twisted left action of f on $\bar {\rho }$ evaluated at any $v \in S_n$ is

$$\begin{align*}f(t_\bullet).\bar{\rho}(v) = \left[((\bar{t}_{\omega(1)})^3 + (\bar{t}_{\omega(2)})^2 + \bar{t}_{\omega(3)})\bar{\rho}\right](v) = (t_2^3 + t_3^2 + t_1)\bar{\rho}(v) ,\end{align*}$$

and the $\omega $ -twisted right action of f on $\bar {\rho }$ evaluated at any $v \in S_n$ is

$$\begin{align*}f(t_\bullet).\bar{\rho}(v) = \left[((\bar{x}_{\omega(1)})^3 + (\bar{x}_{\omega(2)})^2 + \bar{x}_{\omega(3)})\bar{\rho}\right](v)= (t_{v(2)}^3 + t_{v(3)}^2 + t_{v(1)})\bar{\rho}(v).\end{align*}$$

The ring of splines has a $S_n$ -module structure, originally defined for Hessenberg graphs in [Reference Tymoczko28, Reference Tymoczko29].

Definition 2.12. Let $\bar {\rho } \in \mathcal {M}_{\Gamma }$ . The dot action of $S_n$ on $\mathcal {M}_{\Gamma }$ is given by

for $w,v \in S_n$ . Any $\omega \in S_n$ may twist the dot action by first sending $v \to \omega v \omega ^{-1}$ (conjugating by $\omega $ ). Since conjugation is an inner automorphism of $S_n$ , the standard and $\omega $ -twisted $S_n$ -module structures on $\mathcal {M}_{\Gamma }$ are isomorphic.

Using our standard for visualizing splines, the dot action by w moves polynomials around $\mathcal {G}_\Gamma $ by sending the polynomial at v to $wv$ (for all $v \in S_n$ ) and then acts on every polynomial by w as in Equation (2.1).

Example 2.13. The dot action of the transposition $(1,2)$ on the spline $\bar {\rho }_3$ from Example 2.7 is computed below.

Computed below is the $\omega = (1,2,3)$ -twisted action of the transposition $(1,2)$ on the spline $\bar {\rho }_3$ from Example 2.7. Note this is the same as the untwisted action of $(1,2,3)(1,2)(1,2,3)^{-1} = (2,3)$ .

Remark 2.14. The dot action is well defined. We have for all $(v_1,v_2) \in E(\mathcal {G}_\Gamma )$ that

$$ \begin{align*} w \cdot \bar{\rho}(v_1) - w \cdot \bar{\rho}(v_2) &= w\bar{\rho}(w^{-1}v_1)- w\bar{\rho}(w^{-1}v_2) \\ &= w(\bar{\rho}(w^{-1}v_1)- \bar{\rho}(w^{-1}v_2)) \\ &\in w\mathcal{L}(w^{-1}v_1,w^{-1}v_2). \end{align*} $$

If $v_1v_2^{-1} = (i,j)$ , then $w^{-1}v_1v_2^{-1}w = (w^{-1}(i),w^{-1}(j))$ . So

$$\begin{align*}w\mathcal{L}(w^{-1}v_1,w^{-1}v_2) = \left\langle w(t_{w^{-1}(i)}-t_{w^{-1}(j)}) \right\rangle = \left\langle t_i-t_j \right\rangle = \mathcal{L}(v_1,v_2).\end{align*}$$

Thus, $w \cdot \bar {\rho }(v_1) - w \cdot \bar {\rho }(v_2) \in \mathcal {L}(v_1,v_2)$ , and $w \cdot \bar {\rho } \in \mathcal {M}_{\Gamma }$ .

Finally, consider the quotients

(2.5)

and

(2.6)

Call $\mathrm {L}_\Gamma $ and $\mathrm {R}_\Gamma $ the left and right quotients of $\mathcal {M}_{\Gamma }$ , respectively. As $\mathbb {C}[t_\bullet ]$ -modules for the left and right action, both quotients are , where I is the ‘irrelevant ideal’ $\left \langle t_1,...,t_n\right \rangle $ of $\mathbb {C}[t_1,\ldots ,t_{n}]$ . Thus, $\mathrm {L}_{\Gamma }$ and $\mathrm {R}_{\Gamma }$ each inherit the structure of a finite-dimensional graded $\mathbb {C}$ -vector space from the left- and right-module structure of $\mathcal {M}_{\Gamma }$ , respectively. Any homogeneous module-generating set over $\mathbb {C}[t_\bullet ]$ projects to a spanning set over $\mathbb {C}$ in the quotient.

The ideals $\left \langle \bar {t}_1,\ldots ,\bar {t}_n\right \rangle $ and $\left \langle \bar {x}_1,\ldots ,\bar {x}_n\right \rangle $ are homogeneous and $S_n$ -equivariant, and so the graded $S_n$ -module structure on $\mathcal {M}_{\Gamma }$ projects to graded $S_n$ -representations on both $\mathrm {L}_{\Gamma }$ and $\mathrm {R}_{\Gamma }$ . Symmetric functions are formal power series in $\{x_1,x_2,...\}$ invariant under permuting the variables. The Frobenius character map gives an isomorphism from the algebra of representations of symmetric groups to the algebra of symmetric functions. The two bases of symmetric functions we consider are Schur functions $\{s_\lambda \}$ , which correspond to irreducible representations, and homogeneous symmetric functions { $h_\lambda \}$ , which correspond to induced representations of trivial representations on Young subgroups to symmetric groups. Both Schur and homogeneous symmetric functions are indexed by integer partitions. Denote the Frobenius character of these (q-graded) $S_n$ -representations as $\mathbf {ch}\left (\mathrm {L}_{\Gamma }\right )$ and $\mathbf {ch}\left (\mathrm {R}_{\Gamma }\right )$ , respectively. Since both $\mathbf {ch}\left (\mathrm {L}_{\Gamma }\right )$ and $\mathbf {ch}\left (\mathrm {R}_{\Gamma }\right )$ correspond to graded representations, and all representations are sums of irreducible representations, both $\mathbf {ch}\left (\mathrm {L}_{\Gamma }\right )$ and $\mathbf {ch}\left (\mathrm {R}_{\Gamma }\right )$ are manifestly Schur-positive graded symmetric functions.

Example 2.15. Again, consider $\Gamma = \left ([3],\{(1,2),(2,3)\}\right )$ . Then

$$\begin{align*}\mathbf{ch}\left(\mathrm{L}_{\Gamma}\right) = s_3 + (s_{2,1} + 2s_3)q + s_3q^2=h_3 + (h_{2,1}+h_3)q + h_3q^2 \end{align*}$$

and

$$\begin{align*}\mathbf{ch}\left(\mathrm{R}_{\Gamma}\right) = s_3 + 2s_{2,1}q + s_3q^2. \end{align*}$$

The following Lemma 2.16 is useful for computer calculations.

Lemma 2.16. Let $\Gamma $ and $\Gamma '$ be two graphs on $[n]$ , and . Then

$$\begin{align*}\mathcal{M}_{\Gamma \cup \Gamma'} = \mathcal{M}_{\Gamma} \cap \mathcal{M}_{\Gamma'} \end{align*}$$

Proof. This easily follows from the set-theoretic definition

$$\begin{align*}\mathcal{M}_{\Gamma} = \left\{ \left.\bar{\rho} \in \prod_{w \in S_n} \mathbb{C}[t_\bullet]\; \right|\; \bar{\rho}(w) - \bar{\rho}(v) \in \mathcal{L}(w,v) \text{ for all } (w,v) \in E(\mathcal{G}_\Gamma)\right\}.\\[-48pt]\end{align*}$$

2.3 Isomorphisms

It is natural to expect that if two graphs $\Gamma $ and $\Gamma '$ on $[n]$ are isomorphic, that the resulting algebraic structures on $\mathcal {M}_{\Gamma }$ and $\mathcal {M}_{\Gamma '}$ should also have meaningful isomorphisms between them. This section shows that an isomorphism $\Gamma \to \Gamma '$ induces a labeled-graph isomorphism $\mathcal {G}_\Gamma \to \mathcal {G}_{\Gamma '}$ , a ring isomorphism $\mathcal {M}_{\Gamma } \to \mathcal {M}_{\Gamma '}$ , a collection of different $\mathbb {C}[t_\bullet ]$ -module isomorphisms $\mathcal {M}_{\Gamma } \to \mathcal {M}_{\Gamma }$ , and an $S_n$ -module isomorphism $\mathcal {M}_{\Gamma } \to \mathcal {M}_{\Gamma }$ that leads to equalities $\mathbf {ch}\left (\mathrm {L}_{\Gamma }\right ) = \mathbf {ch}\left (\mathrm {L}_{\Gamma '}\right )$ and $\mathbf {ch}\left (\mathrm {R}_{\Gamma }\right ) = \mathbf {ch}\left (\mathrm {R}_{\Gamma '}\right )$ .

Throughout this subsection, let $\Gamma $ and $\Gamma '$ be graphs on $[n]$ and say that $\omega \colon \Gamma \to \Gamma '$ is a graph isomorphism. Then $\omega $ is also naturally an element of $S_n$ , viewed as a bijection from $[n]$ to itself. Let $\omega $ denote both the graph isomorphism and associated permutation.

Our first construction is an isomorphism between the corresponding labeled Cayley graphs. The following Lemma 2.17 states that $\mathcal {G}_\Gamma $ and $\mathcal {G}_{\Gamma '}$ are related as graphs by conjugation, and the associated labels are related via the action on ideals induced by the action on polynomials in Equation (2.1).

Proof. Conjugation is a group automorphism of $S_n$ . Say $(v_1,v_2) \in E(\Gamma )$ and in particular that $v_1^{-1}v_2 = (i,j) \in E(\Gamma )$ . Then

$$ \begin{align*} \left(\omega v_1 \omega^{-1}\right)^{-1}\left(\omega v_2\omega^{-1}\right) &= \omega v_1^{-1} v_2\omega^{-1} \\ &= \omega(i,j)\omega^{-1} \\ &= \left(\omega(i),\omega(j)\right) \in E(\Gamma'). \end{align*} $$

Thus, conjugation by $\omega $ defines a graph isomorphism $\mathcal {G}_\Gamma \to \mathcal {G}_{\Gamma '}$ . For the labels on $\mathcal {G}_\Gamma $ and $\mathcal {G}_{\Gamma '}$ , the computation above also shows that if $v_1v_2^{-1} = (p,q)$ , then $\left (\omega v_1 \omega ^{-1}\right )\left (\omega v_2\omega ^{-1}\right )^{-1} = (\omega (p),\omega (q))$ . It follows that $\left (\omega v_1 \omega ^{-1},\omega v_2\omega ^{-1}\right ) \in E(\mathcal {G}_{\Gamma '})$ is labeled $\left \langle t_{\omega (p)} - t_{\omega (q)}\right \rangle = \omega \left \langle t_p-t_q\right \rangle $ . The claim follows.

Define $\Omega \colon \mathcal {M}_{\Gamma } \to \mathcal {M}_{\Gamma '}$ by . The following Proposition 2.18 proves $\Omega $ is a ring isomorphism and is actually a consequence of Lemma 2.17 and a more general Proposition of Gilbert, Tymoczko and Viel [Reference Gilbert, Tymoczko and Viel15, Prop 2.7]. We include the proof here for completeness.

Proof. Let $\bar {\rho } \in \mathcal {M}_{\Gamma }$ . First, we show that $\Omega (\bar {\rho }) \in \mathcal {M}_{\Gamma '}$ . Let $(v_1,v_2) \in E(\mathcal {G}_{\Gamma '})$ . By Lemma 2.17, there is an edge $(\omega ^{-1}v_1\omega ,\omega ^{-1}v_2\omega ) \in E(\mathcal {G}_\Gamma )$ , and so $\bar {\rho }(\omega ^{-1}v_1\omega )-\bar {\rho }(\omega ^{-1}v_2\omega ) \in \mathcal {L}(\omega ^{-1}v_1\omega ,\omega ^{-1}v_2\omega )$ . Now we have

$$ \begin{align*} \Omega(\bar{\rho})(v_1) - \Omega(\bar{\rho})(v_2) &= \omega\bar{\rho}\left(\omega^{-1} v_1 \omega\right) - \omega\bar{\rho}\left(\omega^{-1} v_2 \omega\right) \\ &= \omega\left(\bar{\rho}\left(\omega^{-1} v_1 \omega\right) - \bar{\rho}\left(\omega^{-1} v_2 \omega\right)\right) \\ &\in \omega \mathcal{L}\left(\omega^{-1} v_1 \omega,\omega^{-1} v_2 \omega\right) = \mathcal{L}'(v_1,v_2). \end{align*} $$

Thus, $\Omega (\bar {\rho }) \in \mathcal {M}_{\Gamma '}$ . It is easy to verify that this map is a ring homomorphism, and the inverse from $\mathcal {M}_{\Gamma '}$ to $\mathcal {M}_{\Gamma }$ is constructed in the same manner with the map $\omega ^{-1} \colon \Gamma ' \to \Gamma $ .

The following lemma gives three instances in which $\Omega $ is also a module isomorphism between $\mathcal {M}_{\Gamma }$ and $\mathcal {M}_{\Gamma '}$ .

Proof. Both $\mathbb {C}[t_\bullet ]$ -module statements follow from two straightforward computations,

$$\begin{align*}\Omega(\bar{t}_i)(v) =\bar{t}_{\omega(i)}(v)\; \text{ and }\; \Omega(\bar{x}_i)(v) = \bar{x}_{\omega(i)}(v). \end{align*}$$

Say for the left action, if $f \in \mathbb {C}[t_\bullet ]$ and $\bar {\rho } \in \mathcal {M}_{\Gamma }$ , then $\Omega (f(t_1,\ldots ,t_n).\bar {\rho }) = f(\bar {t}_{\omega (1)},\ldots ,\bar {t}_{\omega (n)})\Omega (\bar {\rho })$ , precisely the twisted action. The same holds for the right $\mathbb {C}[t_\bullet ]$ -action to the $\omega $ -twisted right $\mathbb {C}[t_\bullet ]$ -action. It is easy to show that ring isomorphism $\Omega ^{-1}$ is the inverse for $\Omega $ as a $\mathbb {C}[t_\bullet ]$ -module morphism for both pairs of actions, and so $\Omega $ is a $\mathbb {C}[t_\bullet ]$ -module isomorphism as in (1) and (2).

Given the dot action on $\mathcal {M}_{\Gamma }$ , the induced action of $u \in S_n$ on $\bar {\rho } \in \mathcal {M}_{\Gamma '}$ is

To check that the action is compatible with multiplication of elements $v,u \in S_n$ , compute

$$ \begin{align*} (v,(u,\bar{\rho})) &= \left(v,\Omega\left( u \cdot \Omega^{-1}(\bar{\rho})\right)\right) \\ &= \Omega\left( v \cdot \Omega^{-1}\Omega\left( u \cdot \Omega^{-1}(\bar{\rho})\right)\right) \\ &= \Omega\left( v \cdot \left( u \cdot \Omega^{-1}(\bar{\rho})\right)\right)\\ &= \Omega\left( vu \cdot \Omega^{-1}(\bar{\rho})\right)\\ &= (vu,\bar{\rho}). \end{align*} $$

Now compute for $u,v \in S_n$ that

$$ \begin{align*} (u,\bar{\rho})(v) &= \Omega\left( u \cdot \Omega^{-1}(\bar{\rho})\right) (v) \\ &= \omega \left( u\cdot \Omega^{-1}(\bar{\rho})\right)(\omega^{-1}v\omega) \\ &= \omega u \left(\Omega^{-1}(\bar{\rho})\right)(u^{-1}\omega^{-1}v\omega) \\ &= \omega u \omega^{-1}\bar{\rho}(\omega u^{-1}\omega^{-1}v)\\ &= \omega u \omega^{-1} \cdot \bar{\rho}(v). \end{align*} $$

This is precisely the $\omega $ -twisted dot action of u on $\mathcal {M}_{\Gamma '}$ . Again, by computing with $\Omega ^{-1}$ , it follows that $\Omega $ is an $S_n$ -module isomorphism.

Note that any generating set for the left or right $\mathbb {C}[t_\bullet ]$ -module structures on $\mathcal {M}_{\Gamma }$ must necessarily be generators for the $\omega $ -twisted versions as well. As such, when searching for generators, we may choose any graph isomorphic to $\Gamma $ for explicit calculations.

By Proposition 2.20, we may consider any graph isomorphic to $\Gamma $ when calculating $\mathbf {ch}\left (\mathrm {L}_{\Gamma }\right )$ and $\mathbf {ch}\left (\mathrm {R}_{\Gamma }\right )$ .

3.1 Implied conditions on splines

This subsection gives algebraic conditions that an element $\bar {\rho } \in \mathcal {M}_{\Gamma }$ must satisfy that are not explicitly in the definition. Specifically, given $w,v \in S_n$ , we want to infer conditions on $\bar {\rho }(w) - \bar {\rho }(v)$ when $(w,v)$ is not necessarily an edge in $\mathcal {G}_\Gamma $ . Let $w,v \in S_n$ and $(w=v_0,v_1,\ldots ,v_m=v)$ be a path from w to v in $\mathcal {G}_\Gamma $ . Say for each edge $(v_{k-1},v_k)$ that $v_kv_{k-1}^{-1} = (i_k,j_k)$ , so that $\mathcal {L}(v_{k-1},v_k) = \left \langle t_{i_k} - t_{j_k}\right \rangle $ for each $k=1,\ldots ,m$ . Then

(3.1) $$ \begin{align} \bar{\rho}(w)-\bar{\rho}(v) = \sum_{k=1}^{m} \left(\bar{\rho}(v_{k-1})-\bar{\rho}(v_k)\right) \in \left\langle t_{i_k} - t_{j_k} \mid k \in [m] \right\rangle. \end{align} $$

Define

(3.2) $$ \begin{align} I_B^w := \left\langle t_{w(i)}-t_{w(j)} \mid (i,j) \in B \subseteq E(T) \right\rangle. \end{align} $$

Lemma 3.3 below is particularly useful when $\bar {\rho } \in \mathcal {M}_{\Gamma }$ satisfies $\bar {\rho }(v) = 0$ for some $v \in S_n$ . Recall that we identify $B \subset E(\Gamma )$ with a subset of transpositions, and write $w\langle B \rangle $ for the left coset at w of the reflection subgroup generated by the transpositions in B.

Proof. Let $w^{-1}v = b_1\cdots b_m$ , where $b_1,\ldots ,b_m \in B$ . Let $(w = v_0, v_1,\ldots , v_m=v)$ be the path from w to v, where $v_{k}^{-1}v_{k-1} = b_k \in B$ for all $k \in [m]$ . Say that $v_{k}v_{k-1}^{-1} = (i_k,j_k)$ , so that $\mathcal {L}(v_{k-1},v_{k})= t_{i_k} - t_{j_k}$ . By Equation (3.1),

$$\begin{align*}\bar{\rho}(w)-\bar{\rho}(v) \in \left\langle t_{i_k} - t_{j_k} \mid k \in [m] \right\rangle. \end{align*}$$

For each $k \in [m]$ , since $v_kv_{k-1}^{-1}= (i_k,j_k)$ , we have that $b_k = v_k^{-1}v_{k-1} = \left (v_k^{-1}(i_k),v_k^{-1}(j_k)\right )$ . Each edge $b_k \in B$ , so the integers $v_k^{-1}(i_k) = (wb_1\cdots b_k)^{-1}(i_k)$ and $v_{k}^{-1}(j_k) = (wb_1\cdots b_k)^{-1}(j_k)$ must be in the same connected component of $([n],B)$ .

Since $(wb_1\cdots b_k)^{-1} = (b_k\cdots b_1)w^{-1}$ , it follows that $w^{-1}(i_k)$ and $w^{-1}(j_k)$ are vertices in the same connected component of $([n],B)$ for all $k \in [m]$ . If $(q_0,...,q_\ell )$ is a path in $([n],B)$ from $q_0 = w^{-1}(i_k)$ to $q_{\ell } = = w^{-1}(j_k)$ , then $t_{q_0} - t_{q_\ell } = \sum _{r=1}^\ell t_{q_{r-1}} - t_{q_r}$ , and thus, $t_{w^{-1}(i_k)}-t_{w^{-1}(j_k)} \in I_B^e$ . It follows that $t_{i_k} - t_{j_k} \in I_B^w$ for all $k \in [m]$ , and so $\bar {\rho }(w)-\bar {\rho }(v)\in I_B^w$ .

A monomial ideal in $\mathbb {C}[t_\bullet ]$ is an ideal I generated by monomials. Monomial ideals are particularly nice when computing intersections; if $I_1 = \langle m_1,\ldots ,m_k\rangle $ and $I_2 = \langle n_1,\ldots ,n_\ell \rangle $ are both monomial ideals, then $I_1 \cap I_2 = \langle \mathrm {lcm}(m_i,n_j) \mid i \in [k],\; j\in [\ell ]\rangle $ .

Let T be a spanning tree of $\Gamma $ , where $E(T) = \{(a_1,b_1),\ldots ,(a_{n-1},b_{n-1})\}$ . Ideals of the form $\langle t_{a_i}-t_{b_i} \mid (a_i,b_i) \in B \subset E(T) \rangle $ can be considered monomial ideals, via the graded automorphism

$$\begin{align*}\mathbb{C}[t_1,\ldots,t_n] \cong \mathbb{C}[t_{a_1}-t_{b_1},\ldots,t_{a_{n-1}}-t_{b_{n-1}},t_n] \end{align*}$$

defined by $t_i \mapsto \begin {cases}t_{a_i}-t_{b_i} &\text {if } i \in [n-1] \\ t_n & \text {if }i=n \end {cases}$ . Since $t_i \mapsto t_{w(i)}$ is also a graded automorphism of $\mathbb {C}[t_\bullet ]$ , the ideals $I_B^w$ from Equation 3.2 can also be considered as monomial ideals (taking care to fix T and $w \in S_n$ ). We will fix T and w and then treat ideals of the form $I_B^w$ as monomial ideals to compute intersections in the proofs of Theorem 3.7 and Lemma 4.1 in the following two sections.

3.2 Coset splines and trees

This subsection establishes a set of splines called coset splines (Definition 3.4) that generate $\mathcal {M}_{\Gamma }$ as a module over the polynomial ring when $\Gamma $ is a tree. This subsection also identifies a subset of those coset splines that generate $\mathcal {M}_{\Gamma }$ as a ring when $\Gamma $ is a tree.

Definition 3.4. Let $\Gamma $ be a tree, and $B \subseteq E$ . The coset spline at the identity $\bar {f}_e^B \colon S_n \to \mathbb {C}[t_\bullet ]$ is

$$\begin{align*}\bar{f}_e^B(w) := \begin{cases} { \prod\limits_{(i,j) \in E \setminus B} \left(t_{w(i)} - t_{w(j)}\right)} & w \in \langle B \rangle \\ 0 & \text{otherwise} \end{cases} \end{align*}$$

The coset spline at w is $\bar {f_w^B} := w \cdot \bar {f}_e^B$ . We adopt the conventions that a product over the empty set $\emptyset $ is $1$ (so ) and that the subgroup generated by the empty set is the identity (so $\langle \emptyset \rangle = \{e\}$ ).

Example 3.5. Again, consider $\Gamma = \left ([3],\{(1,2),(2,3)\}\right )$ . Drawn below are three examples of coset splines on $\mathcal {G}_\Gamma $ .

Proof. It suffices to show $\bar {f}_e^B$ is a spline. Let $w \in \langle B \rangle $ and $v \in S_n$ , where $vw^{-1} = (i,j) \in E$ .

If $(i,j) \in E \setminus B$ , then $v \notin \langle B \rangle $ , and so $\bar {f}_e^B(v) = 0$ . Thus, $\bar {f}_e^B(w) - \bar {f}_w^B(v) = t_{w(i)}-t_{w(j)} \in \mathcal {L}(w,v)$ , as desired.

If $(i,j) \in B$ , then

$$ \begin{align*} \bar{f}_e^B(w) - \bar{f}_e^B(v) &= \prod_{(r_1,s_1) \in E \setminus B} \left(t_{w(r_1)} - t_{w(s_1)}\right) - \prod_{(r_2,s_2) \in E \setminus B} \left(t_{w(i,j)(r_2)} - t_{w(i,j)(s_2)}\right) \\ &= w\left(\prod_{(r_1,s_1) \in E \setminus B} \left(t_{r_1} - t_{s_1}\right) - (i,j)\left(\prod_{(r_2,s_2) \in E \setminus B} \left(t_{r_2} - t_{s_2}\right)\right)\right)\\ &= w\left(\sum_{0\leq p,q} g_{pq}(t_\bullet) t_i^p t_j^q -(i,j)\left(\sum_{0\leq r,s} g_{rs}(t_\bullet) t_i^r t_j^s\right)\right)\\ &= w\left(\sum_{0\leq p,q} g_{pq}(t_\bullet) (t_i^p t_j^q- t_i^q t_j^p)\right). \end{align*} $$

As $t_i^p t_j^q- t_i^q t_j^p \in \left \langle t_i-t_j\right \rangle $ , it follows that $\bar {f}_e^B(w) - \bar {f}_e^B(v) \in \left \langle t_{w(i)} - t_{w(j)} \right \rangle = \mathcal {L}(w,v)$ . Thus, $\bar {f}_e^B$ is a spline.

Now we prove that coset splines are uniquely determined by the coset. For all $u \in \langle B \rangle $ , we have

$$ \begin{align*} u \cdot \bar{f}_e^B(w) &= \begin{cases}{ u\left(\prod\limits_{(i,j) \in E - B} \left(t_{u^{-1}w(i)} - t_{u^{-1}w(j)}\right)\right)} & u^{-1}w \in \langle B \rangle \\ 0 & \text{otherwise} \end{cases}\\ &= \begin{cases} { \prod\limits_{(i,j) \in E - B} \left(t_{w(i)} - t_{w(j)}\right)} & w \in \langle B \rangle \\ 0 & \text{otherwise} \\ \end{cases}\\ &= \bar{f}_e^B(w). \end{align*} $$

If $w\langle B \rangle = v\langle B \rangle $ , then $w=vu$ , for some $u \in \langle B \rangle $ , and so

$$\begin{align*}\bar{f}_w^B = w\cdot \bar{f}_e^B = (vu)\cdot \bar{f}_e^B = v\cdot (u\cdot \bar{f}_e^B) = v\cdot \bar{f}_e^B = \bar{f}_v^B.\\[-34pt] \end{align*}$$

Note that the following Theorem 3.7 is independent of the left or right module structure.

Proof. Let $\bar {\rho }\in \mathcal {M}_{\Gamma }$ , we will show that $\bar {\rho } \in \mathbb {C}[t_\bullet ]\{\bar {f}_B^w \mid w \in S_n, B \subseteq E(\Gamma )\}$ by induction on containment of the support $\mathrm {supp}(\bar {\rho })$ . If $\bar {\rho } \equiv 0$ , this is clearly in the span of the coset splines, and the base case $\mathrm {supp}(\bar {\rho }) = \emptyset $ is done. Otherwise, $\mathrm {supp}(\bar {\rho }) \neq \emptyset $ , and we assume all splines $\bar {\kappa }$ where $\mathrm {supp}(\bar {\kappa }) \subsetneq \mathrm {supp}(\bar {\rho })$ are in $\mathbb {C}[t_\bullet ]\{\bar {f}_B^w \mid w \in S_n, B \subseteq E(\Gamma )\}$ . Replacing $\bar {\rho }$ by if necessary, we assume $\bar {\rho }(e) = 0$ . This also handles the case where $\mathrm {supp}(\bar {\rho }) = S_n$ .

Fix $w \in S_n$ such that $\bar {\rho }(w) \neq 0$ and w is adjacent in $\mathcal {G}_\Gamma $ to some $w' \in S_n$ where $\bar {\rho }(w') = 0$ . Define

Since $\bar {\rho }(w') = 0$ , this set is nonempty. Each element in $\mathcal {B}_w$ is a generating set for a reflection subgroup whose left coset at w contains an element not in $\mathrm {supp}(\bar {\rho })$ . Note if $B \subset B'$ and $B \in \mathcal {B}_w$ , then $B' \in \mathcal {B}_w$ .

By Lemma 3.3,

(3.3)

Following the logic of Subsection 3.1 (i.e., treating $\{t_{w(i)} - t_{w(j)} \mid (i,j) \in E(\Gamma )\}$ as variables), $\mathcal {I}_\rho ^w$ is a monomial ideal generated by the monomials that are contained within every element of the intersection.

A monomial $\mathfrak {m} = { \prod\limits _{(i,j) \in E} \left (t_{w(i)}-t_{w(j)}\right )^{\alpha _{ij}}}$ is contained within the ideal $\mathcal {I}_\rho ^w$ if and only if for every $B \in \mathcal {B}_w$ , there is at least one $(i,j) \in B$ such that $\alpha _{ij}> 0$ . For generators of $\mathcal {I}_\rho ^w$ , it suffices to consider only those monomials such that $\alpha _{ij} \in \{0,1\}$ for all $(i,j) \in E(\Gamma )$ . Since $\alpha _{ij} \in \{0,1\}$ , the monomials that generate $\mathcal {I}_\rho ^w$ are a subset of $\{\bar {f}_w^B(w) \mid B \subseteq E(\Gamma )\}$ . In particular, we have the equality $\left \langle \bar {f}_w^D(w) \mid D \subseteq E(\Gamma ),\;\bar {f}_w^D(w) \in \mathcal {I}_\rho ^w \right \rangle = \mathcal {I}_\rho ^w$ .

Consider the coset splines $\{\bar {f}_w^D \mid \bar {f}_w^D(w) \in \mathcal {I}_\rho ^w\}$ . By definition, for any $D \subset E(\Gamma )$ ,

$$\begin{align*}\bar{f}_w^{D}(w) = \prod_{(i,j) \in E(\Gamma)\setminus D} t_{w(i)} - t_{w(j)}. \end{align*}$$

Now $\bar {f}_w^{D}(w)\in \mathcal {I}_\rho ^w$ if and only if $\left (E(\Gamma )\setminus D\right ) \cap B \neq \emptyset $ for all $B \in \mathcal {B}_w$ . Thus, $\bar {f}_w^{D}(w)\in \mathcal {I}_\rho ^w$ if and only if $B \not \subset D$ for all $B \in \mathcal {B}_w$ . Since $\mathcal {B}_w$ is closed under supersets, $\bar {f}_w^{D}(w)\in \mathcal {I}_\rho ^w$ if and only if $D \notin \mathcal {B}_w$ . Thus,

$$ \begin{align*} \bar{\rho}(w) \in \mathcal{I}_\rho^w = \left\langle \bar{f}_w^D(w) \mid \text{ for all } v \in w\langle D \rangle, \; \bar{\rho}(v) \neq 0 \right\rangle. \end{align*} $$

Let $\bar {f} \in \mathbb {C}[t_\bullet ]\{\bar {f}_w^D \mid \text { for all } v \in w\langle D \rangle ,\; \bar {\rho }(v) \neq 0 \}$ such that $\bar {\rho }(w) = \bar {f}(w)$ (a different $\bar {f}$ may be chosen for the left and right module structure, but either way, such a $\bar {f}$ exists since it is only required to agree with $\bar {\rho }$ at w). Since $\mathrm {supp}(\bar {f}) \subseteq \mathrm {supp}(\bar {\rho })$ and $\bar {f}(w) = \bar {\rho }(w) \neq 0$ , it follows that $\mathrm {supp}(\bar {\rho }) \supsetneq \mathrm {supp}(\bar {\rho }-\bar {f})$ . Thus, $\bar {\rho }-\bar {f} \in \mathbb {C}[t_\bullet ]\{\bar {f}_B^w \mid w \in S_n, B \subseteq E(\Gamma )\}$ . Since $\bar {f}$ is also a sum of coset splines, $\bar {\rho } \in \mathbb {C}[t_\bullet ]\{\bar {f}_B^w \mid w \in S_n, B \subseteq E(\Gamma )\}$ .

The collection of all coset splines is not a minimal generating set. One might significantly decrease the size of this set by fixing the linear order on $S_n$ in the proof of Lemma 3.2, and only considering the largest (by support) coset splines supported ‘above’ a permutation. There is no guarantee that these generators are minimal for all degrees, but it is easy to reason that this collection is minimal for the module $\mathcal {M}_{\Gamma }^{\leq 2}$ generated by the constant, linear and quadratic splines.

We also achieve a generating set for $\mathcal {M}_{\Gamma }$ as a ring in Corollary 3.8 below.

Proof. It follows immediately from the definition that

$$\begin{align*}f_w^B = \prod_{s \in E(\Gamma) \setminus B}f_w^{E(\Gamma) \setminus\{s\}}. \end{align*}$$

So every coset spline except is a product of linear coset splines, which generate $\mathcal {M}_{\Gamma }$ together with either $\{\bar {t}_i \mid i \in [n]\}$ or $\{\bar {x}_i \mid i \in [n]\}$ by Theorem 3.7.

We can leverage Theorem 3.7 to compute $\mathcal {M}_{\Gamma }$ for all graphs $\Gamma $ . Any graph $\Gamma $ can be expressed as the union of spanning trees $\Gamma = T_1 \cup \cdots \cup T_k$ . Lemma 2.16 says that ${ \mathcal {M}_{\Gamma } = \bigcap _{i=1}^k \mathcal {M}_{T_i}}$ , and Theorem 3.7 gives explicit generators for each $\mathcal {M}_{T_i}$ . This is most useful in computer calculations, where the task of constructing modules from generators and intersecting them can be completed by a computer algebra system.

5.1 Some relations

This set $\mathcal {F}_\Gamma $ is not a $\mathbb {C}$ -basis of $\mathcal {M}_{\Gamma }^1$ . Indeed, the following Lemmas 5.5, 5.6 and 5.7 give relations between elements of $\mathcal {F}_\Gamma $ .

The first set of relations in the Lemma 5.5 are relatively straightforward.

Proof. Relation $(1)$ is easy, as is relation $(2)$ once it is noted that the support of each $\bar {f}_A^{(i,j)}$ for a fixed $(i,j)$ is disjoint. Fix $w \in S_n$ .

For relation (3), compute that

$$ \begin{align*} \sum_{k=1}^n \bar{y}_{G,k}^j(w) &= \sum_{\substack{k \in [n]\\w^{-1}(k) \in G}} t_k - t_{w(j)} = \left(\sum_{\substack{k\in [n]\\w^{-1}(k) \in G}} t_k\right) - \left| G \right|t_{w(j)} \\ &= \left(\sum_{r \in G} t_{w(r)}\right) - \left| G \right|t_{w(j)} = \left(\left(\sum_{r\in G} \bar{x}_r\right) - \left| G \right|\bar{x}_j\right)(w). \end{align*} $$

For relation (4), note that either $w^{-1}(k) = j$ or $w^{-1}(k) \in G$ for one and only one connected component G of $\Gamma - j$ . As such, if $w^{-1}(k) = j$ , in which case w is not in the support of any of the $\bar {y}_{G,k}^j$ and likewise $\bar {t}_k(w)-\bar {x}_j(w) = 0$ . Otherwise, $w^{-1}(k) \in G$ for some particular connected component G and $\bar {y}_{G,k}^j(w) = t_k-t_{w(j)}$ . This is precisely $\bar {t}_k(w) - \bar {x}_j(w)$ , and we have (4).

Lemma 5.6 below shows that if j is a cut vertex and a connected component G of $\Gamma - j$ is connected to j by a cut edge $(i,j)$ , then the spline $\bar {y}_{G,k}^j$ can be written as a sum of other splines from $\mathcal {F}_\Gamma $ . In particular, it will allow us to remove from $\mathcal {F}_\Gamma $ the splines in $\mathcal {Y}_\Gamma $ that correspond to components connected by cut edges.

Lemma 5.6. Let $\Gamma $ be a graph, j a cut vertex and $(i,j)$ a cut edge, choosing $G_{(i,j)}$ to be the component containing the vertex i. Let C be the connected component of the forest with vertex set $V(G_{(i,j)}) \cup \{j\}$ and edge set $\{s \mid s\text { is a cut edge of }G_{(i,j)}\} \cup \{(i,j)\}$ that contains the vertex j. Then for all $k \in [n]$ ,

$$\begin{align*}\sum_{\substack{v \in C-j \\v \vdash \Gamma}} \sum_{\substack{ G\\ G \cap C = \emptyset}} \bar{y}_{G,k}^v + \sum_{a \in E(C)} \sum_{\substack{A \ni k\\ \left| A \right| = \left| G_a \right|}} \bar{f}_A^{a} = \bar{y}_{G_{(i,j)},k}^j, \end{align*}$$

where G in the first double-sum is a connected component of $\Gamma - v$ , where $G_a$ is the connected component that is a subset of $G_{(i,j)}$ , and where A in the second double-sum is a subset of $[n]$ .

Since Lemma 5.6 is rather technical, we will walk through an example before seeing the full proof. Let $\Gamma $ be the graph on $13$ vertices below:

where $(11,12)$ is cut edge and $G_{(11,12)}$ is the component that contains $11$ . The forest of cut edges with vertex set $V(G_{(11,12)}) \cup \{12\}$ has edge set $\{(1,4),(8,11),(9,11),(10,11),(12,11)\}$ . If we mark in $\Gamma $ the component of this forest that contains the vertex $11$ with double lines, we get the following graph:

In essence, Lemma 5.6 says that for all $k \in [13]$ , the spline $\bar {y}_{G_{(11,12)},k}^{12}$ can be written as a sum of some splines in $\mathcal {C}_\Gamma $ associated to the cut edges $(8,11)$ , $(9,11)$ and $(10,11)$ (since they are a part of that marked tree) and some splines in $\mathcal {Y}_\Gamma $ associated to cut vertex $8$ , since $\Gamma - 8$ has components that ‘hang off of’ that marked tree.

For each of the cut edges $a \in \{(8,11),(9,11),(10,11)\}$ , we must choose $G_a$ to be the component contained within $G_{(11,12)}$ , so $V(G_{(9,11)}) = \{1,...,11\}$ , $V(G_{(9,11)}) = \{9\}$ , and $V(G_{(10,11)}) = \{10\}$ .

More formally, Lemma 5.6 states that for all $k \in [13]$ , the following equality holds (we will denote the specific subgraph by their vertex set):

$$\begin{align*}\bar{y}_{[5],k}^8 + \bar{y}_{\{6,7\},k}^8 + \sum_{\substack{\left| A \right| = 8 \\ k \in A}} \bar{f}_A^{(8,11)}+ \sum_{\substack{\left| A \right| = 1 \\ k \in A}} \bar{f}_A^{(9,11)}+ \sum_{\substack{\left| A \right| = 1 \\ k \in A}} \bar{f}_A^{(10,11)}+ \sum_{\substack{\left| A \right| = 11 \\ k \in A}} \bar{f}_A^{(11,12)} = \bar{y}_{[11],k}^{12}.\end{align*}$$

The proof proceeds in cases by the value of $w^{-1}(k)$ . For example, say we wished to evaluate both sides of the above expression at a $w \in S_{13}$ such that $w^{-1}(k) = 3$ . This makes it easier to determine which splines in the sum above on the left are supported at w. In particular,

1. 1. ${ \bar {y}_{[5],k}^8(w) = t_k - t_{w(8)}}$ since $3 \in [5]$ ,

2. 2. ${ \bar {y}_{\{6,7\},k}^8(w) = 0}$ since $3 \notin \{6,7\}$ ,

3. 3. ${ \sum _{\substack {\left | A \right | = 8 \\ k \in A}} \bar {f}_A^{(8,11)}(w) = t_{w(8)} - t_{w(11)}}$ since the set $A = w(V(G_{(8,11)})$ contains $w(3) = k$ ,

4. 4. ${ \sum _{\substack {\left | A \right | = 1 \\ k \in A}} \bar {f}_A^{(9,11)}(w) = 0}$ and ${ \sum _{\substack {\left | A \right | = 1 \\ k \in A}} \bar {f}_A^{(10,11)}(w) = 0}$ since $k \notin w(\{9\})$ and $k\notin w(\{10\})$ , and finally,

5. 5. ${ \sum _{\substack {\left | A \right | = 11 \\ k \in A}} \bar {f}_A^{(11,12)}(w) = t_{w(11)} - t_{w(12)}}$ since the set $A = w(V(G_{(11,12)})$ contains $w(3) = k$ .

If we add these all up, the sum telescopes and the evaluation is

$$\begin{align*}(t_k - t_{w(8)}) + (t_{w(8)} - t_{w(11))} + (t_{w(11)} - t_{w(12)}) = t_k - t_{w(12)},\end{align*}$$

which is precisely ${ \bar {y}_{[11]}^{12}(w)}$ . The essence of the proof, which we will now provide, is that the splines in the sum with support at a particular $w \in S_n$ can be determined from any simple path from $w^{-1}(k)$ to j and that the sum always telescopes as it did in the example above.

Proof of Lemma 5.6.

Let $w \in S_n$ . We will show that both sides of the claimed equality are equal when evaluated at w. Let $k \in [n]$ , and we will proceed in cases based off of the value $w^{-1}(k)$ .

First, say $w^{-1}(k) \notin V(G_{(i,j)})$ . All paths that begin in $G_{(i,j)}$ and leave must contain the edge $(i,j)$ and thus visit the vertex j. In particular, for any $v \in C$ where $c \vdash \Gamma $ , the connected component of $\Gamma - v$ that contains $w^{-1}(k)$ also contains $j \in C$ . So $w^{-1}(k) \notin G$ for any G in the first double-sum, and so

$$\begin{align*}\sum_{\substack{v \in C-j \\v \vdash \Gamma}} \sum_{\substack{ G\\ G \cap C = \emptyset}} \bar{y}_{G,k}^v(w) = 0.\end{align*}$$

If $a \in E(C)$ , then $G_a$ was chosen to be contained within $G_{(i,j)}$ , so $w^{-1}(k) \notin G_a$ as well. In particular, if $k \in A$ , then $w^{-1}(A) \neq G_a$ , and so

$$\begin{align*}\sum_{a \in E(C)} \sum_{\substack{A \ni k\\\left| A \right| = \left| G_a \right|}} \bar{f}_A^{a}(w) = 0.\end{align*}$$

It is direct from the definition that $\bar {y}_{G_{(i,j)},k}^v(w) = 0$ , and so the claim holds when $w^{-1}(k) \notin G_{(i,j)}$ .

Now assume that $w^{-1}(k) \in G_{(i,j)}$ . Let $P = \left (p_0,\ldots ,p_\ell , p_{\ell +1} \right )$ be a simple path from to . Note that $p_\ell $ must be the vertex i. This path P may start outside of C, but must eventually enter the tree C. Say that $m \in \{0,...,\ell \}$ is the lowest index such that $p_m \in C$ . Since $i \in C$ and $p_\ell = i$ , this integer m does exist. Simple paths from vertex to vertex within trees are unique, so there is a unique simple path from $p_m$ to j in C. This path is .

First, we will determine the value of $\bar {f}_A^{a}(w)$ for $a \in E(C)$ . Say the edge $a \in C$ is not an edge in the path $P^C$ . Then the vertices $w^{-1}(k)$ and j are in the same connected component of the graph $([n],E(\Gamma )-a)$ . In particular, $w^{-1}(k) \notin G_a$ , so if $k \in A$ , then $w^{-1}(A) \neq V(G_a)$ . It follows that for all A such that $k \in A$ , if $a \notin P^C$ , then $\bar {f}_A^{a}(w) = 0$ .

However, if $a \in P^C$ , then we may let , and then $k \in A$ . So for each $a \in P^C$ , there exists a single spline $\bar {f}_A^{a}$ in the sum that is supported at w. In particular, if $a = (p,q)$ , then $\bar {f}_A^{q}(w) = t_{w(p)} - t_{w(q)}$ .

At this point, we have that

$$\begin{align*}\sum_{\substack{v \in C-j \\v \vdash \Gamma}} \sum_{\substack{ G\\ G \cap C = \emptyset}} \bar{y}_{G,k}^v(w) + \sum_{a \in E(C)} \sum_{\substack{A \ni k\\\left| A \right| = \left| G_a \right|}} \bar{f}_A^{a} = \sum_{\substack{v \in C-j \\v \vdash \Gamma}} \sum_{\substack{ G\\ G \cap C = \emptyset}} \bar{y}_{G,k}^v(w) + \sum_{(p,q) \in P^C} t_{w(p)} - t_{w(q)}. \end{align*}$$

Now we will have two cases: if $w^{-1}(k) \in C$ and if $w^{-1}(k) \notin C$ .

Case 1: $w^{-1}(k) \in C$ . So $m = 0$ . Then for all $v \in G_{(i,j)}$ , if a connected component G of $\Gamma - v$ contains $w^{-1}(k)$ , then so does $G \cap C$ . In particular,

$$\begin{align*}\sum_{\substack{v \in C-j \\v \vdash \Gamma}} \sum_{\substack{ G\\ G \cap C = \emptyset}} \bar{y}_{G,k}^v(w) = 0.\end{align*}$$

Now we compute

$$ \begin{align*} \sum_{(r,v) \in P^C} t_{w(v)} - t_{w(r)} &=\sum_{i=0}^\ell t_{w(p_i)} - t_{w(p_{i+1})} \\ &= t_{w(p_0)} - t_{w(p_{\ell+1})}\\ &= t_k - t_{w(j)}. \end{align*} $$

This is precisely $\bar {y}_i^j(w)$ , and so the equality holds if $w^{-1}(k) \in C$ .

Case 2: $w^{-1}(k) \notin C$ . Then $m \neq 0$ , and consider the vertex $p_{m-1}$ . Since $p_m \in C$ and $\Gamma - p_m$ separates $w^{-1}(k)$ from j, the vertex $p_m$ is a cut vertex of $\Gamma $ . If $v' \in C$ is any vertex other than $p_m$ , then $p_m$ and $w^{-1}(k)$ are in the same connected component of $\Gamma - v'$ (connected via the path $(p_0,...,p_m)$ ). In particular, any connected component of $\Gamma - v'$ that contains $w^{-1}(k)$ intersects nontrivially with C. So for $v=p_m \in C-j$ , there is precisely one component G of $\Gamma - v$ that contains $w^{-1}(k)$ . If this component G intersected nontrivially with C, then $(p_{m-1},p_m)$ would have to be an edge in C, but $(p_{m-1},p_m)$ was explicitly assumed not to be a cut edge. In particular, w is supported on one and only one spline in the first double-sum (the one where $v = p_m$ and $G \ni p_{m-1})$ ), and so

$$\begin{align*}\sum_{\substack{v \in C-j \\v \vdash \Gamma}} \sum_{\substack{ G\\ G \cap C = \emptyset}} \bar{y}_{G,k}^v(w) = t_k - t_{w(p_m)}. \end{align*}$$

It follows that

$$ \begin{align*} \sum_{\substack{v \in C-j \\v \vdash \Gamma}} \sum_{\substack{ G\\ G \cap C = \emptyset}} \bar{y}_{G,k}^v(w) + \sum_{(p,q) \in P^C} t_{w(p)} - t_{w(q)} &= t_k - t_{w(p_m)} + \sum_{r=m}^\ell t_{w(p_r)} - t_{w(p_{r+1})} \\ &= t_{k} - t_{w(p_m)} + t_{w(p_m)} - t_{w(p_{\ell+1})}\\ &= t_k - t_{w(j)}. \end{align*} $$

So in either case, the sum evaluates to $\bar {y}_{G_{(i,j)},k}^j(w)$ .

Now Lemma 5.6 two very important consequences. First, as mentioned, it will allow us to disregard those splines $\bar {y}_{G,k}^j$ where G is connected to j via a cut edge. Second, observe we only required j to be a cut vertex so that $\bar {y}_{G,k}^j$ is defined. We may, however, remove this restriction and ‘force through’ the argument as follows. If $(p,q)$ is a cut edge and q is not a cut vertex, then q is a leaf in $\Gamma $ . Let $G_{(p,q)}$ be the connected component containing p (and thereby all of $[n] \setminus \{q\}$ ). We might abuse notation and let for all $w \in S_n$ ,

$$ \begin{align*} \bar{y}_{G_{(p,q)},k}^q(w) &= \begin{cases} t_k - t_{w(q)} & \text{if }w^{-1}(k) \in [n] \setminus\{q\} \\ 0 &\text{otherwise} \end{cases} \\ &= \bar{t}_k(w) - \bar{x}_q(w), \end{align*} $$

and get another relation from Lemma 5.6. A consequence of this is Lemma 5.7 below.

Lemma 5.7. Let $(i,j)$ be a leaf edge in $\Gamma $ with j the leaf vertex, and $G_{(i,j)}$ the connected component of $([n],E(\Gamma ) \setminus \{s\})$ that contains i. Then for all $A \subset [n]$ such that $\left | A \right | = \left | G_{(i,j)} \right | = n-1$ , we have that

$$\begin{align*}\bar{f}_A^{(i,j)} \in \mathbb{C}\left\{\bar{\rho} \in \mathcal{F}_\Gamma\left| \bar{\rho} \neq \bar{f}_B^{(i,j)} \text{ for any }B \subset [n] \right. \right\}.\end{align*}$$

Proof. Let C be the connected component of the forest with vertex set $V(G_{(i,j)}) \cup \{j\} = [n]$ and edge set $\{s \mid s\text { is a cut edge of }\Gamma )\}$ that contains the vertex j. By Lemma 5.6 and the discussion above, for all $k \in [n]$ ,

$$\begin{align*}\sum_{\substack{v \in C-j \\v \vdash \Gamma}} \sum_{\substack{ G\\ G \cap C = \emptyset}} \bar{y}_{G,k}^v + \sum_{a \in E(C)} \sum_{\substack{A \ni k\\\left| A \right| = \left| G_a \right|}} \bar{f}_A^{a} = \bar{t}_k - \bar{x}_j, \end{align*}$$

where G in the first double-sum is a connected component of $\Gamma - v$ , $G_{a}$ is the connected component of $([n],E(\Gamma )\setminus \{(i,j)\})$ that is a subgraph of $G_{(i,j)}$ (i.e., does not contain j), and A in the second double-sum is a subset of $[n]$ . In particular,

$$\begin{align*}\sum_{\substack{\left| A \right| = n-1 \\ k \in A}} \bar{f}_A^{(i,j)}= \bar{t}_k - \bar{x}_n- \sum_{\substack{v \in C-j \\v \vdash \Gamma}} \sum_{\substack{ G\\ G \cap C = \emptyset}} \bar{y}_{G,k}^v - \sum_{\substack{a \in E(C) \\ a\neq (i,j)}} \sum_{\substack{A \ni k\\\left| A \right|=\left| G_a \right| }} \bar{f}_A^{a}. \end{align*}$$

Now the right-hand side is in ${ \mathbb {C}\left \{\bar {\rho } \in \mathcal {F}_\Gamma \left | \bar {\rho } \neq \bar {f}_B^{(i,j)} \text { for any }B \subset [n] \right. \right \}}$ . Let ${ \bar {\sigma }_k :=\sum _{\substack {\left | A \right | = n-1 \\ k \in A}} \bar {f}_A^{(i,j)},}$ so the above relation says ${ \bar {\sigma }_k \in \\C\left \{\bar {\rho } \in \mathcal {F}_\Gamma \left | \bar {\rho } \neq \bar {f}_B^{(i,j)} \text { for any }B \subset [n] \right. \right \}}$ . For each $p \in [n]$ , we have that

$$\begin{align*}\bar{f}_{[n] \setminus \{p\}}^{(i,j)} = \left(\frac{1}{n-1} \sum_{k \in [n]} \bar{\sigma}_k\right) - \bar{\sigma}_p. \end{align*}$$

Thus, ${ } \bar {f}_{[n] \setminus \{p\}}^{(i,j)} \in \mathbb {C}\left \{\bar {\rho } \in \mathcal {F}_\Gamma \left | \bar {\rho } \neq \bar {f}_B^{(i,j)} \text { for any }B \subset [n] \right. \right \}$ , and as all subsets of size $\left | A \right | = n-1$ take the form $A = [n] \setminus \{p\}$ , the claim follows.

Example 5.8. If $\Gamma $ is the graph below where $(12,13)$ is a leaf,

then Lemma 5.7 states that $\mathbb {C}\mathcal {F}_\Gamma $ is identical to ${ \mathbb {C}\left \{\bar {\rho } \in \mathcal {F}_\Gamma \left | \bar {\rho } \neq \bar {f}_B^{(12,13)} \text { for any }B \subset [n] \right. \right \}}$ .

Since $(1,4)$ is also a leaf, we have that $\mathbb {C}\mathcal {F}_\Gamma $ is equal to ${ \mathbb {C}\left \{\bar {\rho } \in \mathcal {F}_\Gamma \left | \bar {\rho } \neq \bar {f}_B^{(1,4)} \text { for any }B \subset [n] \right. \right \}}$ as well. Note we cannot remove the coset splines for $(1,4)$ and $(12,13)$ from $\mathcal {F}_\Gamma $ at the same time and maintain the $\mathbb {C}$ -span; we have to pick a particular leaf to remove and stick with it.