Notation 2.2. We denote by ${{\mathbf {Vect}}}$ the category of vector spaces and linear maps.

Notation 2.4. We denote by ${{{\mathbf {GrVect}}}}$ the category of graded vector spaces and grading preserving linear maps. Sometimes we will implicitly consider an ungraded vector space (such as ${\mathbb {Q}}$ ) as a graded vector space concentrated in degree $0$ .

Notation 2.5. For $k \in {\mathbb {Z}}$ , we denote by $\mathrm {s}^{k}(-)$ a k-fold degree shift. For V a graded vector space, the graded vector space $\mathrm {s}^{k} V$ is given by . When $k = 1$ , we just write .

Notation 2.6. Let $m \in {\mathbb {Z}}$ . A graded symplectic form of degree $-m$ on a graded vector spaces V is a nondegenerate bilinear form of degree $-m$

$$\begin{align*}\langle-, -\rangle \colon V \mathop{\otimes} V \longrightarrow {\mathbb{Q}} \end{align*}$$

such that $\langle v, w \rangle = (-1)^{ | v | | w | + 1} \langle w, v \rangle $ (i.e. it is graded anti-symmetric).

We denote by ${{\mathbf {Sp}}}^{m}$ the category of finite-dimensional graded vector spaces equipped with a graded symplectic form of degree $-m$ with morphisms those linear maps of degree $0$ that preserve the bilinear form.

Notation 2.9. Let $(C_{*,*}, d^1, d^2)$ be a double complex. We denote by $\operatorname {\mathrm {Tot}} C$ its total complex – that is, the chain complex with underlying graded vector space

and differential $d = d^1 + d^2$ .

Proof. The map f canonically lifts to a map $g \colon C_{*,*} \to \left ( {\mathrm {H}}_{0} (C_{\bullet ,*}, d^2), d_1 \right )$ of double complexes, where the target is equipped with the double complex structure concentrated in bidegrees $(p, 0)$ . The map g induces an isomorphism between the $E^1$ -pages of the spectral sequences associated to these double complexes whose zeroth differential is given by $d^2$ (see, for example, Weibel [Reference WeibelWei95, §5.6]). By the Comparison Theorem for spectral sequences, this implies that f is a quasi-isomorphism (see, for example, [Reference WeibelWei95, Theorem 5.2.12]).

Proof. The first claim follows directly from the definition of the Grothendieck construction. For the second claim, note that the G-module $\coprod _{s \in S} X_s$ is isomorphic to the left Kan extension of X along the canonical projection ${\mathop {\mathchoice {\textstyle }{}{}{} \int _{G}}} S \to G$ . Taking H-coinvariants of this corresponds to left Kan extending further along the quotient map $G \to {{G}/{H}}$ . Since Kan extensions compose, the result is isomorphic to the left Kan extension of X along the composite ${\mathop {\mathchoice {\textstyle }{}{}{} \int _{G}}} S \to {{G}/{H}}$ .

Proof. This follows easily from the definitions.

Proof. The left Kan extension in the statement is, at the unique object $*$ of H, isomorphic to ${\mathrm {colim}}_{F \mathbin {\downarrow } *}\ (X \circ {\mathrm {pr}})$ , where ${\mathrm {pr}} \colon F \mathbin {\downarrow } * \to \mathcal G$ denotes the projection. The action of an element $h \in H$ on this object is induced by the functor $F \mathbin {\downarrow } * \to F \mathbin {\downarrow } *$ given by $(G, h') \mapsto (G, h h')$ (note that this functor commutes with the projection). That this is isomorphic, as an H-module, to the description given in the statement follows from Lemma 2.17.

Notation 2.24. Let $\mathscr M$ a ${{{\mathbf {\Sigma }}}}$ -module. We define a functor ${\mathscr M}[{-}] \colon {{{\mathbf {GrVect}}}} \to {{{\mathbf {GrVect}}}}$ by

$$\begin{align*}V \longmapsto \bigoplus_{k \in {\mathbb{N}_0}} \left( {\mathscr M({\underline{k}})^{\mathrm{op}}} \mathop{\otimes}{}_{\!\!\Sigma{}_{{\underline{k}}}} V^{\otimes{\underline{k}}} \right) \end{align*}$$

and call it the Schur functor associated to $\mathscr M$ .

Proof. We define the isomorphism by sending the element of

represented by $S \in {{{\mathbf {\Sigma }}}}$ and $(a \colon S \to T) \in S \mathbin {\downarrow } {{{\mathbf {\Sigma }}}}$ as well as the elements

$$\begin{align*}p \mathop{\otimes} {\textstyle\bigotimes}_{t \in T} q_t \in \mathscr P(T) \mathop{\otimes} \mathscr Q(a) \qquad\mathrm{and}\qquad {\textstyle\bigotimes}_{s \in S} v_s \in V^{\otimes S} \end{align*}$$

to $\varepsilon \in \{\pm 1\}$ times the element of

represented by $T \in {{{\mathbf {\Sigma }}}}$ and as well as the elements

$$\begin{align*}p \in \mathscr P(T) \qquad\mathrm{and}\qquad q_t \mathop{\otimes} {\textstyle\bigotimes}_{s \in F_t} v_s \in \mathscr Q(F_t) \mathop{\otimes} V^{\otimes F_t} \end{align*}$$

where $\varepsilon $ is the sign incurred by permuting the two expressions

$$\begin{align*}p \mathop{\otimes} {\textstyle\bigotimes}_{t \in T} q_t \mathop{\otimes} {\textstyle\bigotimes}_{s \in S} v_s \quad\mathrm{and}\quad p \mathop{\otimes} {\textstyle\bigotimes}_{t \in T} \left( q_t \mathop{\otimes} {\textstyle\bigotimes}_{s \in F_t} v_s \right) \end{align*}$$

into each other.

Proof. The map lifts to an isomorphism $(\mathrm {s} {\mathbb {Q}})^{{{\otimes}} k} \mathop{\mathop{\otimes}} V^{{{\otimes}} k} \to (\mathrm {s} V)^{{{\otimes}} k}$ that is compatible with the ${\Sigma _{k}}$ -actions. Passing to quotients again, we obtain the desired statement.

Notation 2.30. For a cyclic operad $\mathscr C$ , we write ${\mathscr C \left (\mkern -4mu\left (n\right )\mkern -4mu\right )}$ for $\mathscr C(n-1)$ equipped with its right ${\Sigma _{n}}$ -action. We consider a cyclic operad as a ${{{\mathbf {P}}}}$ -module via (the opposite of) this extended right action. (Note that this is different from the ${{{\mathbf {P}}}}$ -module associated to the underlying operad.) In particular, the Schur functor associated to $\mathscr C$ is ${\mathscr C}[{V}] = \bigoplus _{n \ge 1} {\mathscr C \left (\mkern -4mu\left (n\right )\mkern -4mu\right )} \mathop{\mathop{\otimes}}{\hspace{-2pt}}_{{\Sigma _{n}}} V^{{{\otimes}} n}$ .

Proof. Checking antisymmetry and the Jacobi identity are straightforward, though tedious, computations. The only trick one has to use is that $v\ {}_i{\circ }_j\ w$ being nonzero implies that $ | {v_i} | + | {w_j} | = m$ . We omit the details. The claim about the functoriality is clear from the definitions.

In the case $\mathscr C = \operatorname {\mathscr {L}ie}$ , a different proof of this fact can be found in [Reference Berglund and MadsenBM20, equation (6.7)].

Proof. The composition

is a degree $-m$ isomorphism of the form

$$\begin{align*}\begin{pmatrix} 0 & f \\ f' & 0 \end{pmatrix} \end{align*}$$

since $\langle v, {v'}\rangle = 0 = \langle w, {w'}\rangle $ for all $v, v' \in V$ and $w, w' \in W$ by our assumption that $m \neq 2n$ (which also implies $m \neq 2(m - n)$ ). In particular, f and $f'$ are degree $-m$ isomorphisms. Noting that $f(w) = \langle w, - \rangle $ , this implies the first claim. The second follows from $f(w)(v) = \langle w, v\rangle = - (-1)^{n(m-n)} \langle v, w \rangle $ .

Proof. Let $f \in {\mathrm {GL}}(V)$ . Then the computation

$$ \begin{align*} &\; \langle {{\left( f(v), {({f^{-1}})^\vee}(\mathrm{s}^{m} \phi) \right)}}, {{\left( f(v'), {({f^{-1}})^\vee}(\mathrm{s}^{m} \phi') \right)\rangle}} \\ =&\; (\phi \circ {f^{-1}})(f(v')) - (-1)^{n(m-n)} (\phi' \circ {f^{-1}})(f(v)) \\ =&\; \phi(v') - (-1)^{n(m-n)} \phi'(v) \\ =&\; \langle {(v, \mathrm{s}^{m} \phi),} {(v', \mathrm{s}^{m} \phi')\rangle} \end{align*} $$

shows that the map in the statement is actually well defined. It is clearly injective. We will now show that it is also surjective. For this purpose, let $f \oplus g \in \Gamma $ , where $f \in {\mathrm {GL}}(V)$ and $g \in {\mathrm {GL}}(\mathrm {s}^{m} {V^\vee })$ . By using that ${f^{-1}} \oplus {g^{-1}} \in \Gamma $ as well, we obtain

$$\begin{align*}g(\phi)(v) = \langle{g(\mathrm{s}^{m} \phi),} v \rangle = \langle {\mathrm{s}^{m} \phi}, {{f^{-1}}(v)}\rangle = \phi({f^{-1}}(v)) = {({f^{-1}})^\vee}(\phi)(v) \end{align*}$$

and hence $g = {({f^{-1}})^\vee }$ . This shows surjectivity.

Proof. There is a canonical isomorphism

$$\begin{align*}H^{\otimes S} \mathop{\otimes} V^{\otimes I} \mathop{\otimes} W^{\otimes J} \cong \bigoplus_{(A,B)} (V, \mathrm{s}^{m} {V^\vee})^{\otimes (A, B)}, \end{align*}$$

where $(A,B)$ runs over all (ordered) partitions of $S \amalg I \amalg J$ into two subsets such that $I \subseteq A$ and $J \subseteq B$ . This is an isomorphism of $({\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}} \times \Gamma )$ -modules where a morphism $(f, g, h)$ of ${\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}}$ acts on both sides by permuting the factors (on the right-hand side this maps the $(A, B)$ summand into the $(c(A), c(B))$ summand, where ) and an element $g \in {\mathrm {GL}}(V) \cong \Gamma $ acts diagonally (acting on ${V^\vee }$ by ${({g^{-1}})^\vee }$ ).

Now note that the $\Gamma $ -coinvariants of $(V, \mathrm {s}^{m} {V^\vee })^{{{\otimes}} (A, B)}$ are trivial when $|A| \neq |B|$ since $2 \in {\mathrm {GL}}(V)$ acts via multiplication by

. Hence, the inclusion

becomes an isomorphism after taking $\Gamma $ -coinvariants.

Now let T be a finite linearly ordered set. Then there is, for any $(A, B) \in \operatorname {OM}(T)$ , a canonical isomorphism of $\Gamma $ -modules

given by the following permutation: we consider the tensor product on the right-hand side to be indexed over the set $A \times \{0, 1\}$ equipped with the lexicographical order and permute according to the bijection $\psi _{A, B} \colon T \to A \times \{0, 1\}$ that sends $a \in A$ to $(a, 1)$ and $b \in B$ to $({(\mu _{A, B})^{-1}}(b), 0)$ .

The isomorphisms $\operatorname {sgn}_{A, B}(\psi _{A, B}) \cdot \Psi _{A, B}$ assemble into an isomorphism of $({\Sigma _{T}} \times \Gamma )$ -modules

where an element $c \in {\Sigma _{T}}$ acts on the right-hand side by sending the factor corresponding to $(A, B)$ into the factor corresponding to $(c(A), c(B))$ via

$$\begin{align*}c_*^{A,B} \left( {\textstyle\bigotimes}_{a \in A} (\mathrm{s}^{m} \phi_a \mathop{\otimes} v_a) \right) = \operatorname{sgn}_{A, B}(c) \cdot \left( {\textstyle\bigotimes}_{a \in A} (\mathrm{s}^{m} \phi_{{\tau^{-1}}(a)} \mathop{\otimes} v_{{\sigma^{-1}}(a)}) \right), \end{align*}$$

where $\sigma $ and $\tau $ are as in the statement of the lemma. Said differently, we have

$$\begin{align*}c_*^{A,B} = \operatorname{sgn}_{A, B}(\psi_{A, B}) \operatorname{sgn}_{c(A), c(B)}(\psi_{c(A), c(B)}) \cdot ( \Psi_{c(A), c(B)} \circ c_* \circ {\Psi_{A, B}^{-1}} ), \end{align*}$$

which maybe makes it clearer why $\Psi $ is compatible with the ${\Sigma _{T}}$ -action.

Now consider the case that $T = S \amalg I \amalg J$ . Then it is clear that the isomorphism $\Psi $ restricts to an isomorphism

of $({\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}} \times \Gamma )$ -modules.

Lastly, note that there is a $\Gamma $ -module isomorphism

$$ \begin{align*} \Phi \colon \mathrm{s}^{m} {V^\vee} \mathop{\otimes} V &\longrightarrow \mathrm{s}^{m} {\mathrm{End}}(V) \\ \mathrm{s}^{m} \phi \mathop{\otimes} v &\longmapsto \mathrm{s}^{m} (v' \mapsto \phi(v') v), \end{align*} $$

where $\Gamma \cong {\mathrm {GL}}(V)$ acts on ${\mathrm {End}}(V)$ by conjugation. Together with the canonical isomorphism $(\mathrm {s}^{m} {\mathrm {End}}(V))^{{{\otimes}} A} \cong \mathrm {s}^{m |A|} ({\mathrm {End}}(V))^{{{\otimes}} A}$ these isomorphisms combine into an isomorphism

as desired. For the claims made about the induced $({\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}})$ -module structure on the left-hand side and the explicit description of $\Theta _1$ , note that, when is a basis of V, then $\Phi (\mathrm {s}^{m} {e_j^\vee } \mathop{\mathop{\otimes}} e_i) = \mathrm {s}^{m} \operatorname {E}_{i, j}$ , where $\operatorname {E}_{i, j}$ is with respect to e.

Proof. This can, for example, be found in [Reference LodayLod98, 9.2.5, 9.2.8, and 9.2.10].Footnote ²

Proof. On each summand, $\Theta _2$ is given by a shift of the isomorphism in Proposition 3.8. Now let $(A, B) \in \operatorname {OM}(S, I, J)$ and $\omega \in {\Sigma _{A}}$ , and fix a basis of V. We will also use the notation from Lemma 3.7. First note that by Proposition 3.8, the elements $[{\operatorname {E}_{1, {\omega (1)}} \mathop{\mathop{\otimes}} \dots \mathop{\mathop{\otimes}} \operatorname {E}_{a, {\omega (a)}}}]$ of ${\left ({\mathrm {End}}(V)^{{{\otimes}} A}\right )}_\Gamma $ , with $\omega $ running over ${\Sigma _{A}}$ , form a basis. Hence, it is enough to check the compatibility of $\Theta _2$ with the $({\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}})$ -action on those elements. To this end, we note that $\Theta _2 \left ( \mathrm {s}^{m |A|} [{\operatorname {E}_{1, {\omega (1)}} \mathop{\mathop{\otimes}} \dots \mathop{\mathop{\otimes}} \operatorname {E}_{a, {\omega (a)}}}] \right ) = \mathrm {s}^{m |A|} \omega $ and

$$ \begin{align*} &\; \Theta_2 \left( {\operatorname{sgn}_{A, B}(c)} \cdot \mathrm{s}^{m |A|} [{\operatorname{E}_{{\sigma^{-1}}(1), {\omega({\tau^{-1}}(1))}} \mathop{\otimes} \dots \mathop{\otimes} \operatorname{E}_{{\sigma^{-1}}(a), {\omega({\tau^{-1}}(a))}}}] \right) \\ &\quad = \mathrm{s}^{m |A|} (\operatorname{sgn}_{A, B}(c) \cdot \sigma \omega {\tau^{-1}}), \end{align*} $$

which is what we wanted to show.

Proof. We start by rewriting the expression yielded by Corollary 3.9 and Remark 3.10. We have isomorphisms of $({\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}})$ -modules

$$ \begin{align*} \bigoplus_{(A, B) \in \operatorname{OM}(S, I, J)} \mathrm{s}^{m |A|} {{\mathbb{Q}}[{\Sigma_{A}}]} &\cong \bigoplus_{(A, B) \in \operatorname{OM}(S, I, J)} \mathrm{s}^{m |A|} {{\mathbb{Q}} \langle {\mathop{Bij}}{(A,B)} \rangle} \\ &\cong \bigoplus_{(A, B, \mu) \in \operatorname{OM}_\mu(S, I, J)} \mathrm{s}^{m |A|} {{\mathbb{Q}} \langle \mu \rangle}, \end{align*} $$

where, for the first isomorphism, we identify ${\Sigma _{A}}$ with the set of (not necessarily order preserving) bijections ${\mathop {Bij}}{(A,B)}$ via the map $\omega \mapsto \mu _{A,B} \circ {\omega ^{-1}}$ . For the second isomorphism, we set . An element $(f, g, h) \in {\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}}$ acts on $\operatorname {OM}_\mu (S, I, J)$ via

where , and on ${{\mathbb {Q}} \langle \mu \rangle }$ by sending $\mu $ to $\operatorname {sgn}_{A,B}(c) \cdot (c \mathbin {.} \mu )$ .

Now we have isomorphisms of $({\Sigma _{I}} \times {\Sigma _{J}})$ -modules

$$ \begin{align*} {\left( (P^{\mathrm{op}} \mathop{\otimes}{}_{\!\!\Sigma_{S}} H^{\otimes S}) \mathop{\otimes} V^{\otimes I} \mathop{\otimes} W^{\otimes J} \right)}_\Gamma &= {\left( {(P \mathop{\otimes} H^{\otimes S})}_{{\Sigma_{S}}} \mathop{\otimes} V^{\otimes I} \mathop{\otimes} W^{\otimes J} \right)}_\Gamma \\ &\cong {\left( {(P \mathop{\otimes} H^{\otimes S} \mathop{\otimes} V^{\otimes I} \mathop{\otimes} W^{\otimes J})}_\Gamma \right)}_{{\Sigma_{S}}} \\ &\cong {\left( P \mathop{\otimes} {(H^{\otimes S} \mathop{\otimes} V^{\otimes I} \mathop{\otimes} W^{\otimes J})}_\Gamma \right)}_{{\Sigma_{S}}} \end{align*} $$

since the actions of ${\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}}$ and $\Gamma $ commute and since $\Gamma $ acts trivially on P. By Lemma 3.7, Corollary 3.9 and Remark 3.10, as well as the preceding discussion, the rightmost expression above is isomorphic to

(3.2) $$ \begin{align} {\Big( P \mathop{\otimes} \bigoplus_{(A, B) \in \operatorname{OM}(S, I, J)} \mathrm{s}^{m |A|} {{\mathbb{Q}}[{\Sigma_{A}}]} \Big)_{{\Sigma_{S}}}} \cong {\Big( \bigoplus_{(A, B, \mu) \in \operatorname{OM}_\mu(S, I, J)} \left( P \mathop{\otimes} \mathrm{s}^{m |A|} {{\mathbb{Q}} \langle \mu \rangle} \right) \Big)_{{\Sigma_{S}}}} \end{align} $$

as $({\Sigma _{I}} \times {\Sigma _{J}})$ -modules. To describe the action on the sum on the right-hand side, let $(f, g, h) \in {\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}}$ and set . Then it sends the summand corresponding to $(A, B, \mu )$ to the summand corresponding to $(c(A), c(B), c \mathbin {.} \mu )$ by acting on P via f and on ${{\mathbb {Q}} \langle \mu \rangle }$ as described above. The quotient is taken with respect to the action of ${\Sigma _{S}} \cong {\Sigma _{S}} \times \{{\mathrm {id}}_{I\}} \times \{{\mathrm {id}}_{J\}}$ .

Now we can apply Lemma 2.16. Noting that there is a canonical isomorphism

$$\begin{align*}{\mathop{\mathchoice{\textstyle}{}{}{} \int_{{\Sigma_{S}} \times {\Sigma_{I}} \times {\Sigma_{J}}}}} \operatorname{OM}_\mu(S, I, J) \cong {{\mathbf{Match}}}^{S}_{I,J} \end{align*}$$

of categories over ${\Sigma _{I}} \times {\Sigma _{J}}$ , we obtain an isomorphism of the right-hand side of (3.2) with

$$\begin{align*}\mathop{\operatorname{Lan}_{{{\mathbf{Match}}}^{S}_{I,J} \to {\Sigma_{I}} \times {\Sigma_{J}}}} (P \mathop{\otimes} \operatorname{sgn}_{n, m}) \end{align*}$$

as $({\Sigma _{I}} \times {\Sigma _{J}})$ -modules, where a morphism $(f, g, h)$ of ${{\mathbf {Match}}}^{S}_{I,J}$ acts on P by $f \in {\Sigma _{S}}$ . To this we can, in turn, apply Lemma 2.18 to obtain the desired description in terms of a colimit: given an object $(A, B, \mu ) \in {{\mathbf {Match}}}^{S}_{I,J}$ and an element $(g, h) \in {\Sigma _{I}} \times {\Sigma _{J}}$ , we pick the lift $({\mathrm {id}}_{S}, g, h) \colon (A, B, \mu ) \to (A, B, c \mathbin {.} \mu )$ , where .

The explicit description of the isomorphism is obtained by chasing through the explicit descriptions of its constituent parts.

Proof. We have

$$\begin{align*}{\left( {\mathscr P} [{H_i}] \mathop{\otimes} V_i^{\otimes I} \mathop{\otimes} W_i^{\otimes J} \right)}_{\Gamma_i} \cong \bigoplus_{k \in {\mathbb{N}_0}} {\left( ( {\mathscr P({\underline{k}})^{\mathrm{op}}} \mathop{\otimes}{}_{\!\!\Sigma_{{\underline{k}}}} {H_i}^{\otimes {\underline{k}}}) \mathop{\otimes} V_i^{\otimes I} \mathop{\otimes} W_i^{\otimes J} \right)}_{\Gamma_i}\end{align*}$$

by definition of ${\mathscr P}[{-}]$ . When $k \le \dim {H_i} - |I| - |J|$ , there is an isomorphism

(3.3)

provided by Proposition 3.13. It is compatible with the maps $H_i \to H_{i+1}$ (i.e., it is a natural isomorphism of sequential diagrams, where the left-hand side is considered as a constant functor). To see this, we chose a basis of $V_i$ (the degree n part of $H_i$ ), extend it to a basis of $V_{i+1}$ , and use the explicit description of the isomorphism (3.3). Hence we have, for each $k \in {\mathbb {N}_0}$ , an isomorphism

which, by setting , assemble into an isomorphism

since colimits commute with direct sums. Since the inclusion ${{{\mathbf {Match}}}^{{{\mathbf {P}}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}} \to {{\mathbf {Match}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ is an equivalence of categories, it induces a canonical isomorphism

which finishes the construction.

We will now prove the claimed explicit description of the composed isomorphism. To this end, consider the element represented by $(S, A, B, \mu ) \in {{\mathbf {Match}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ and $p \mathop{\mathop{\otimes}} \mathrm {s}^{m |A|} 1 \in \mathscr P(S) \mathop{\mathop{\otimes}} \operatorname {sgn}_{n, m}$ . It is equal, for any choice of bijection $f \colon S \to {\underline {|S|}}$ , to the element represented by $\left ( {\underline {|S|}}, c(A), c(B), {{c}|}_B \circ \mu \circ {({{c}|}_A)^{-1}} \right )$ and $\operatorname {sgn}_{A, B}(c) \cdot \left ( f_*(p) \mathop{\mathop{\otimes}} \mathrm {s}^{m |A|} 1 \right )$ , where . On such an element, the isomorphism is, by construction, given by the isomorphism of Proposition 3.13. The representative one obtains using the explicit description of that isomorphism represents the same element as the representative we provided in the statement of this proposition (by translating it back to S via ${f^{-1}}$ ). In the same way, one proves that the action of ${\Sigma _{I}} \times {\Sigma _{J}}$ is as claimed.

Proof. Let $p, q \colon {{\mathbf {Match}}}_{I,J} \to {{\mathbf {Match}}}_{\varnothing ,\varnothing }$ be the canonical functors given, on objects, by

$$ \begin{align*} p (S, A, B, \mu) &= (S \amalg I \amalg J, A, B, \mu) \\ q (S, A, B, \mu) &= (T, A, B, \mu), \end{align*} $$

where T is equal to $S \amalg I \amalg J$ as a set but is equipped with the linear order uniquely determined by the following two properties:

• • The element $a \in A$ is the predecessor of $\mu (a) \in B$ .

• • Its restriction to A agrees with the linear order on A obtained by restricting the one of $S \amalg I \amalg J$ .

The identity of the underlying sets determines a natural isomorphism $p \to q$ . The map in the statement is obtained by applying $\operatorname {sgn}_{n, m}$ to this natural transformation.

Proof. We have isomorphisms

by Proposition 3.15 and Lemmas 3.18 and 2.26. Moreover, by definition,

so that the leftmost colimit above is isomorphic to

and hence, it is enough to show that the Grothendieck construction of the covariant functor $U \colon {{\mathbf {Match}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}} \to {{{\mathbf {Cat}}}}$ given by $(F, S, T, \mu ) \mapsto F \mathbin {\downarrow } {{{\mathbf {\Sigma }}}}$ (where a map acts by precomposition with its inverse) is canonically isomorphic to ${{\mathbf {DirGraph}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ . For this, we define a functor ${\mathop {\mathchoice {\textstyle }{}{}{} \int }} U \to {{\mathbf {DirGraph}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ , given, on objects respectively morphisms, by

$$ \begin{align*} \left( (F, S, T, \mu), a \colon F \to N \right) &\longmapsto \left( F, N, S, T, \mu, a \right) \\ \left( (f \colon F \to F', {\mathrm{id}}_{I}, {\mathrm{id}}_{J}), k \colon a \circ {f^{-1}} \to a' \right) &\longmapsto \left( f, {\mathrm{pr}}(k) \right), \end{align*} $$

where ${\mathrm {pr}} \colon F \mathbin {\downarrow } {{{\mathbf {\Sigma }}}} \to {{{\mathbf {\Sigma }}}}$ is the projection. This is clearly an isomorphism.

The explicit description of the $({\Sigma _{I}} \times {\Sigma _{J}})$ -action and the explicit description of the isomorphism follow from the explicit descriptions in Proposition 3.15 and Lemmas 3.18 and 2.26.

Proof of Theorem 3.29.

Lemma 2.29 and Proposition 3.19 provide us with an isomorphism $\Phi ^{\prime }_{I,J}$ on the underlying $({\Sigma _{I}} \times {\Sigma _{J}})$ -modules (in graded vector spaces). Let $G = (F, N, S, T, \mu , a) \in {{\mathbf {DirGraph}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ be a directed $(I,J)$ -graph and

(3.5)

an element. Denoting by $[x]$ the element represented by G and x, we set

(the sign $\gamma (G)$ serves to cancel out a sign encountered below). We have $\Phi _{I,J}([x]) = [{\Phi ^G_{I,J}(x)}]$ , where we set with y given by

(3.6)

(where $s \colon {\underline {|S|}} \to S$ is the unique order-preserving bijection) and the sign $\varepsilon (x) \in \{\pm 1\}$ is the sign incurred by permuting

$$\begin{align*}({\mathrm{s}} 1)^{\otimes N} \mathop{\otimes} \bigotimes_{v \in N} {\mathrm{s}}^{-m} \xi_v \mathop{\otimes} \bigotimes_{s \in S} (h_s \mathop{\otimes} h_{\mu(s)}) \quad \mathrm{into} \quad \bigotimes_{v \in N} {\mathrm{s}} \bigg( {\mathrm{s}}^{-m} \xi_v \mathop{\otimes} \bigotimes_{f \in {a^{-1}}(v)} h_f \bigg) \mathop{\otimes} \bigotimes_{f \in I \amalg J} h_f \end{align*}$$

according to the Koszul sign rule.

We will now identify the result of pulling the Chevalley–Eilenberg differential $d = d_0 + d_1$ (cf. Definition 2.28) back to ${\mathfrak {DG}}^{m}{(\mathscr C)}_{I,J}$ along the isomorphism $\Phi _{I,J}$ . To this end, we first evaluate it on $\tilde y$ . We have $d_0 = 0$ (since the differential of $ {({\mathrm {s}}^{-m} \mathscr C)}[{H_i}]$ is trivial) and

(3.7)

where . In the following, we will, for ease of notation, identify each ${a^{-1}}(v)$ with $\{1, \dots , | v | \}$ via the unique order-preserving bijection and set , where $f(v,i)$ is the i-th element of ${a^{-1}}(v)$ . We recall

and note that, by definition of $h_f$ , the expression $\langle {h_{v';j}}, {h_{v;i}}\rangle $ is zero unless $\{i, j\} = \{s, \mu (s)\}$ for some $s \in S \cap F$ . Hence, the same is true for the summand in (3.7) corresponding to $(i, j)$ . We thus obtain the following description:

where $\delta _s \in \{\pm 1\}$ is some sign. We note that is equal to some sign $\delta ^{\prime }_s \in \{\pm 1\}$ times the image under $\Phi _{I,J}$ of the element $[z]$ represented by the directed $(I,J)$ -graph $\operatorname {ctr}_{s}(G)$ decorated by $\xi _{a(s)}\ {}_s{\circ }_{\mu (s)}\ \xi _{a(\mu (s))}$ at the collapsed vertex ${\mathrm {pr}}(a(s)) = {\mathrm {pr}}(a(\mu (s)))$ and by $\xi _v$ at all other vertices v. (Note that the family of basis elements $h_\bullet $ obtained for $\operatorname {ctr}_{s}(G)$ does not agree with the family obtained for the noncontracted flags of G. However, they are equivalent under the $\Gamma _i$ -action.)

We will now determine the sign $\delta ^{\prime }_s$ in a special case. To this end, let $s \in S \cap F$ and such that $a(s) \neq a(t)$ . Assume that and are, in this order, the first two elements of N. Moreover, assume that $a \colon F \to N$ is order-preserving and that s is the last element of ${a^{-1}}(v)$ and that t is the first element of ${a^{-1}}(v')$ . (Note that any directed $(I,J)$ -graph is isomorphic to one of this form.) We have

by definition. We first note that $ \langle {h_{v';t}}, {h_{v;s}}\rangle = 1$ , that $\gamma (G) \gamma (\operatorname {ctr}_{s}(G)) = (-1)^{m |N| + n}$ , and that

$$ \begin{align*} c &= | {{\mathrm{s}} y_v} | + \left( | {h_{v;s}} | - m \right) \left( | {h_{v'}} | - m \right) + | {h_{v';t}} | | {h_{v';t+1,\deg{v'}}} | + 1 \\ &= \left( 1 - m + \operatorname{w}(v) \right) + (n - m) \left( {\operatorname{w}(v')} - m \right) + (m - n) \left( {\operatorname{w}(v')} - (m - n) \right) + 1, \end{align*} $$

which reduces the question to determining $\varepsilon (x) \varepsilon (z)$ . To this end, we factor the permutation defining $\varepsilon $ into three parts: firstly permuting the $h_f$ among each other, secondly interleaving the ${\mathrm {s}}^{-m} \xi _v$ and $h_v$ and thirdly interleaving the ${\mathrm {s}} 1$ and $y_v$ . The deviation between $\varepsilon (x)$ and $\varepsilon (z)$ for the first part is given by $-1$ to the power of

$$\begin{align*}m \left( { | {v} | ^{\mathrm{out}}} - 1 \right) + m \left( {\operatorname{w}(v)} - n \right), \end{align*}$$

for the second part, it is $-1$ to the power of

$$\begin{align*}\left( | {h_s} | + | {h_t} | \right) \left( |N| - 2 \right) (-m) + | {h_v} | (-m) \equiv |N| m + \operatorname{w}(v) m \pmod 2, \end{align*}$$

and for the the third part, it is $-1$ to the power of

$$\begin{align*}\left( {- m} + | {h_s} | + | {h_t} | \right) |N| + | {y_v} | = -m + \operatorname{w}(v), \end{align*}$$

so that, putting everything together, we obtain $\delta ^{\prime }_s = (-1)^{m ({ | {v} | ^{\mathrm {out}}} - 1)}$ .

Lastly, we show that, for any isomorphism of directed $(I,J)$ -graphs $\chi \colon G \to G'$ , we have

$$\begin{align*}[{ d^G_s(\Phi^G_{I,J}(x)) }] = [{ d^{G'}_{\chi(s)}(\Phi^{G'}_{I,J}(\chi \mathbin{.} x)) }], \end{align*}$$

which will finish the identification of the differential. The statement is clear up to sign, so it is enough to show that those agree as well. We write $\chi \mathbin {.} x = \alpha x'$ , where $\alpha \in \{\pm 1\}$ and $x'$ is the element represented by $G'$ and the labels $(\chi \mathbin {.} \xi _{{\chi ^{-1}}(w)})_{w \in N'}$ . Lastly, we define $y'$ analogously to (3.6), so that $\Phi ^{G'}_{I,J}(x') = \gamma (G') \varepsilon (x') y'$ .

By definition, we have $[{\gamma (G) \varepsilon (x) y}] = \Phi _{I,J}([x]) = \Phi _{I,J}({[\chi \mathbin {.} x]}) = [{\alpha \gamma (G') \varepsilon (x') y'}]$ . Noting that $\gamma (G) = \gamma (G')$ , it follows from the definitions that $\alpha \gamma (G) \varepsilon (x) \gamma (G') \varepsilon (x') = \alpha \varepsilon (x) \varepsilon (x')$ is the sign induced by permuting

$$\begin{align*}\bigotimes_{v \in N} {\mathrm{s}} \bigg( {\mathrm{s}}^{-m} \xi_v \mathop{\otimes} \bigotimes_{f \in {a^{-1}}(v)} h_f \bigg) \quad \mathrm{into} \quad \bigotimes_{w \in N'} {\mathrm{s}} \bigg( {\mathrm{s}}^{-m} \xi_{{\chi^{-1}}(w)} \mathop{\otimes} \bigotimes_{g \in {a^{\prime-1}}(w)} h_{{\chi^{-1}}(g)} \bigg) \end{align*}$$

(here, we use that the families $(h_f)_{f \in F}$ and $(h^{\prime }_{\chi (f)})_{f \in F}$ are equivalent under the action of $\Gamma _i$ for large enough i; hence, we can assume that they are equal, which we will do from now on). This sign is the product of the sign $\alpha '$ incurred by permuting $\bigotimes _{v \in N} {\mathrm {s}} y_v$ into $\bigotimes _{w \in N'} {\mathrm {s}} y_{{\chi ^{-1}}(w)}$ and the signs $\alpha _v$ incurred by permuting $\bigotimes _{f \in {a^{-1}}(v)} h_f$ into $\bigotimes _{g \in {a^{\prime -1}}(\chi (v))} h_{{\chi ^{-1}}(g)}$ .

It remains to evaluate the sign difference between $d^G_s(y)$ and $d^{G'}_{\chi (s)}(y')$ . For this, write , and such that $v < v'$ . Also let such that $w < w'$ . Then the relevant summands of $d_1(\tilde y)$ and $d_1(\tilde y')$ are the ones indexed by $(v, v')$ and $(w, w')$ . Note that $-1$ to the power of $b_{v,v'} + b^{\prime }_{w,w'}$ , the sign difference between $\textstyle \bigwedge _{\tilde v \in V \setminus \{v, v'\}} {\mathrm {s}} y_{\tilde v}$ and $\textstyle \bigwedge _{\tilde w \in V' \setminus \{w, w'\}} {\mathrm {s}} y^{\prime }_{\tilde w}$ , and $\alpha '$ and $\alpha _{\tilde v}$ for $\tilde v \in V \setminus \{v, v'\}$ cancel each other out, except for, if $\chi $ swaps the order of v and $v'$ , a sign of $-1$ to the power of $1 + | {y^{\prime }_w} | | {y^{\prime }_{w'}} | $ . Now note that, in the case that $\chi $ swaps the order of v and $v'$ , we have ${[y^{\prime }_{w'}, y^{\prime }_w]} = (-1)^{1 + | {y^{\prime }_w} | | {y^{\prime }_{w'}} | } {[y^{\prime }_w, y^{\prime }_{w'}]}$ in the summand of $d_1(\tilde y')$ . Hence, the only sign that has not yet canceled is $\alpha _v \alpha _{v'}$ .

Thus, it remains to show that the summand of ${[y_v, y_{v'}]}$ corresponding to $(s, t)$ equals the summand of ${[y^{\prime }_{\chi (v)}, y^{\prime }_{\chi (v')}]}$ corresponding to $(\chi (s), \chi (t))$ , up to a sign of $\alpha _v \alpha _{v'}$ . (For ease of notation, we assume from now on that $a(s) = v$ , and we set , , , and , possibly changing notation from before.) These two summands agree in $ {({\mathrm {s}}^{-m} \mathscr C)}[{H_i}]$ up to sign. The difference in signs is given by $-1$ to the power of $a_{v,v'}^{s,t} + a_{w,w'}^{s',t'}$ times the sign incurred by permuting

into $h^{\prime }_w\ {}_{s'}{\tilde {\circ }}_{t'}\ h^{\prime }_{w'}$ . Now we note that $-1$ to the power of $a_{v,v'}^{s,t} + 1 + m + m ( | {h_{v;s}} | + | {h_{v'}} | )$ is precisely the sign incurred by permuting $h_v \mathop{\mathop{\otimes}} h_{v'}$ into $(h_v\ {}_s{\tilde {\circ }}_t\ h_{v'}) \mathop{\mathop{\otimes}} h_{v';t} \mathop{\mathop{\otimes}} h_{v;s}$ and analogously for $a_{w,w'}^{s',t'}$ . Noting that $ | {h_{v;s}} | = | {h^{\prime }_{w;s'}} | $ and $ | {h_{v'}} | = | {h^{\prime }_{w'}} | $ , this implies the claim since $\alpha _v \alpha _{v'}$ is the sign incurred by permuting $h_v \mathop{\mathop{\otimes}} h_{v'}$ into $h^{\prime }_w \mathop{\mathop{\otimes}} h^{\prime }_{w'}$ .

Now we prove that $\Phi _{\varnothing ,\varnothing }$ is a map of graded coalgebras and that $\Phi _{I,J}$ is compatible with the comodule structures over these coalgebras. It is clear that $\Phi _{\varnothing ,\varnothing }$ is compatible with the counits. We will now identify the comodule structure map $\rho $ of ${\mathfrak {DG}}^{m}{(\mathscr C)}_{I,J}$ obtained by pulling back the one of the Chevalley–Eilenberg chains. When $I = J = \varnothing $ , this also deals with the comultiplication $\Delta $ of the coalgebra structure since, in this case, we have $\Delta = \rho $ on both sides (see Remark 3.23). We continue to use the notation of (3.5) and (3.6). We have

(3.8)

where the sum runs over all subsets $N'$ of N and $\delta (N') \in \{\pm 1\}$ is the sign incurred by permuting ${\textstyle \bigotimes }_{v \in N} {\mathrm {s}} y_v$ into ${\textstyle \bigotimes }_{v \in N'} {\mathrm {s}} y_v \mathop{\mathop{\otimes}} {\textstyle \bigotimes }_{w \in N \setminus N'} {\mathrm {s}} y_w$ . Note that (the equivalence class of) $\textstyle \bigwedge _{v \in N'} {\mathrm {s}} y_v$ is trivial except when $\mu (S \cap {a^{-1}}(N')) = T \cap {a^{-1}}(N')$ . Hence, we can also consider the sum in (3.8) to run only over all neighbor-closed vertex sets $N'$ of G. For such an $N'$ , we set

where . We denote the element of ${\mathfrak {DG}}^{m}{(\mathscr C)}$ represented by ${\mathrm {ind}}_{G}(N')$ and $x_{N'}$ by $[{x_{N'}}]$ , and the element of ${\mathfrak {DG}}^{m}{(\mathscr C)}_{I,J}$ represented by ${\mathrm {coind}}_{G}(N')$ and $x^{\prime }_{N'}$ by $[{x^{\prime }_{N'}}]$ . We have

$$ \begin{align*} \Phi_{\varnothing,\varnothing}([{x_{N'}}]) &= \gamma \left( {{\mathrm{ind}}_{G}(N')} \right) \varepsilon(x_{N'}) \cdot \textstyle\bigwedge_{v \in {N'}} {\mathrm{s}} y_v \\ \Phi_{I,J}([{x^{\prime}_{N'}}]) &= \gamma \left( {{\mathrm{coind}}_{G}(N')} \right) \varepsilon(x^{\prime}_{N'}) \cdot \left( { \textstyle\bigwedge_{v \in {N \setminus N'}} {\mathrm{s}} y_v } \mathop{\otimes} {\textstyle\bigotimes}_{f \in I \amalg J} h_f \right) \end{align*} $$

so that it only remains to see that

$$\begin{align*}\gamma(G) \varepsilon(x) \delta(N') \gamma \left( {{\mathrm{ind}}_{G}(N')} \right) \varepsilon(x_{N'}) \gamma \left( {{\mathrm{coind}}_{G}(N')} \right) \varepsilon(x^{\prime}_{N'}) \end{align*}$$

is equal to the sign in the definition of the comodule structure of ${\mathfrak {DG}}^{m}{(\mathscr C)}_{I,J}$ . Noting that

$$\begin{align*}\gamma(G) \gamma \left( {{\mathrm{ind}}_{G}(N')} \right) \gamma \left( {{\mathrm{coind}}_{G}(N')} \right) = (-1)^{m |N'| |{N \setminus N'}|}, \end{align*}$$

this is clear from the definitions.

Proof. The condition $2n < m < 3n$ implies that

$$\begin{align*}{ {({\mathrm{s}}^{-m} \mathscr C)}[{H_i}]}^+ = \left( {\mathrm{s}}^{-m} {\mathscr C \left(\mkern-4mu\left(2 \right)\mkern-4mu\right)} \mathop{\otimes}{}_{\!\!\Sigma_{2}} (W_i \mathop{\otimes} W_i) \right) \oplus \bigoplus_{k \ge 3} {\mathrm{s}}^{-m} {\mathscr C \left(\mkern-4mu\left(k \right)\mkern-4mu\right)} \mathop{\otimes}{}_{\!\!\Sigma_{k}} H_i^{\otimes k} \subseteq {({\mathrm{s}}^{-m} \mathscr C)}[{H_i}], \end{align*}$$

where $W_i$ is the degree $m - n$ part of $H_i$ . Then the explicit description of the isomorphism of Theorem 3.29, together with Remark 3.30, implies that there is an isomorphism from the differential graded $({\Sigma _{I}} \times {\Sigma _{J}})$ -submodule $D^{\prime }_*$ of ${\mathfrak {DG}_{\mathrm {trunc}}^{m}}{(\mathscr C)}_{I,J}$ spanned by the decorated graphs with at most $\dim H_i - |I| - |J|$ internal flags to the differential graded $({\Sigma _{I}} \times {\Sigma _{J}})$ -submodule $C^{\prime }_*$ of the right-hand side of (3.9) spanned by the elementary tensors involving at most $\dim H_i$ elements of $H_i$ (including $V_i$ and $W_i$ ).

We will now prove that $C^{\prime }_p$ is equal to degree p of the right-hand side of (3.9) when $p \le \alpha \dim H_i$ . To this end, we take elements

and note that $ | x | = \sum _{v \in N} \left ( 1 - m + | {\xi _v} | + \operatorname {w}(v) \right )$ , where . Due to the truncation, we have that

$$\begin{align*}\operatorname{w}(v) \begin{cases} = 2 (m - n), & \mathit{if}\ \ k_v = 2 \\ \ge k_v n, & \mathit{if}\ \ k_v \ge 3 \end{cases} \end{align*}$$

and that $k_v \ge 2$ . Setting and similarly for $N_{\ge 3}$ , we thus obtain

$$\begin{align*}| x | \ge |{N_2}| (1 - m + 2 (m-n)) + \sum_{v \in N_{\ge 3}} (1 - m + k_v n) \end{align*}$$

since $ | {\xi _v} | \ge 0$ by assumption. We note $1 - m + 2(m - n) \ge 2$ and, when $k_v \ge 3$ , that

$$\begin{align*}\frac{3}{1 - m + 3n} (1 - m + k_v n) \ge k_v \end{align*}$$

since $\frac {3n}{1 - m + 3n} \ge 1$ as $-3n < 1 - m < 0$ . Setting

we thus obtain

$$\begin{align*}\beta | x | \ge |{N_2}| 2 + \sum_{v \in N_{\ge 3}} k_v = \sum_{v \in N} k_v \end{align*}$$

and hence that $\beta | x | + |I| + |J|$ is larger or equal to the number of elements of $H_i$ involved in y. Furthermore,

$$\begin{align*}\beta | x | + |I| + |J| = \beta | y | + (1 - \beta n) |I| + (1 - \beta (m-n)) |J| \le \beta |y| \end{align*}$$

since $ | v | = n |I|$ and $ | w | = (m - n) |J|$ . Hence, $\beta | y | = {\alpha ^{-1}} | y | \le \dim H_i$ implies $[y] \in C^{\prime }_*$ .

Using the same argument, we can show that $D^{\prime }_p = {\mathfrak {DG}_{\mathrm {trunc}}^{m}}{(\mathscr C)}_p$ for $p \le \alpha \dim H_i$ . (A combinatorial argument using graphs would produce a better bound here; however, we will not need this.)