1 Introduction
The starting point of this article is the following manifold with boundary
(i.e., the gfold connected sum of ${\mathrm {S}^{k}} \times {\mathrm {S}^{l}}$ with an open disk removed). Given this, a natural object to consider is the classifying space
of the topological monoid of homotopy automorphisms of $M^{k, l}_{g, 1}$ that fix the boundary pointwise. It classifies fibrations with fiber $M^{k, l}_{g, 1}$ under the trivial fibration with fiber its boundary (see, for example, [Reference Hepworth and LahtinenHL15, Appendix B]). In particular, the cohomology of ${\mathrm {B}} {{\mathrm {aut}}_{\partial }}(M^{k, l}_{g, 1})$ consists of the characteristic classes for such fibrations.
This cohomology appears to be hard to understand completely. However, at least rationally, it is possible to say something about the behavior of the stabilization maps
obtained by gluing on, along one of its boundary components, a copy of ${\mathrm {S}^{k}} \times {\mathrm {S}^{l}}$ with two disjoint open disks removed. Namely, in the case $2 \le k = l$ , Berglund–Madsen [Reference Berglund and MadsenBM20] have shown this map to be an isomorphism in a certain stable range of degrees,Footnote ^{1} and this was extended by Grey [Reference GreyGre19] to the cases $3 \le k \le l \le 2k  2$ .
Moreover, [Reference Berglund and MadsenBM20] contains, in the case $3 \le k = l$ , a combinatorial description (in terms of graph complex homology) of the stable cohomology (i.e., the limit of the maps (1.1) as g goes to infinity). In this paper, we provide a similar description when $k \neq l$ . The following is our main result.
Theorem 1.1 (see Corollary 5.9).
Let $3 \le k < l \le 2k  2$ and $2 \le g$ be integers. Then there is, in cohomological degrees $\le g  2$ , an isomorphism of graded algebras
compatible with the stabilization maps on the lefthand side. Here we set , denote by ${\mathfrak {UG}}^{m} {\operatorname {(\mathscr {L}ie)}}$ the mtwisted graph complex associated to the cyclic Lie operad (with its canonical coalgebra structure), and write ${()^\vee }$ for linear dualization.
In particular, after stabilizing, we obtain an isomorphism
of graded algebras.
Remark. We also prove a version of Theorem 1.1 for cohomology with certain local coefficients; see Theorem 5.7. In this generality, the description we give is in terms of a directed graph complex (see below). It should be possible to relate this to an undirected graph complex, just as in the case of trivial coefficients above, but we do not carry this out in this paper.
Remark. The proof of Theorem 1.1 does not rely on the stability result of Grey [Reference GreyGre19] (though we do reuse some of its ingredients). In particular, we obtain a new proof of his result (for the manifolds $M^{k, l}_{g, 1}$ ), while also roughly doubling the stable range. This is made possible by a result of Li–Sun [Reference Li and SunLS19], which yields a slope $1$ vanishing of the rational cohomology of ${\mathrm {GL}}_{n}({\mathbb {Z}})$ with certain coefficients. That a result of this form would lead to an improvement of the stable range had been suggested by Krannich [Reference KrannichKra20, p. 1070].
Surprisingly, the graph complex appearing in Theorem 1.1 only depends on the dimension $k + l$ and not on k and l themselves. Moreover, when $k + l$ is even, it is the same graph complex that appears in [Reference Berglund and MadsenBM20] – that is, in the case $k = l$ (when $k + l$ is odd, it is defined similarly to the even case but appears to have quite different homology; see further below). For both of these observations, there seems to be no a priori reason to expect them to be true. The second one is particularly interesting when considering our proof, where first a different graph complex appears, of which we then show that it actually has the same homology as ${\mathfrak {UG}}^{k + l  2} {\operatorname {(\mathscr {L}ie)}}$ .
We will now describe the terms appearing on the righthand sides of the isomorphisms of Theorem 1.1 in more detail. First, note that
by a classical result of Borel [Reference BorelBor74]. The graph complex ${\mathfrak {UG}}^{m} {(\mathscr C)}$ , for $\mathscr C$ a cyclic operad, is also a wellknown object. Versions of it were first described by Kontsevich [Reference Kontsevich, Corwin, Gelfand and LepowskyKon93, Reference Kontsevich and JosephKon94], but they have, since then, appeared in many different places; see, for example, [Reference Getzler and KapranovGK98, Reference Conant and VogtmannCV03, Reference Lazarev and VoronovLV08, Reference Berglund and MadsenBM20].
We now sketch the definition of this graph complex. It is generated by graphs (potentially with loops and parallel edges) equipped with an orientation of each edge and a labeling of each vertex with an element of $\mathscr C$ of cyclic arity the valence of the vertex (which we assume to be at least three). We consider a vertex to have homological degree $1  m$ and an edge to have homological degree m. The resulting graded vector space is quotiented by the actions of the automorphism groups of the graphs, taking into account the homological degrees of the vertices and edges and yielding an extra sign $(1)^{m+1}$ whenever the orientation of an edge is flipped. The differential is defined to be the sum over all possible contractions of nonloop edges. When an edge is contracted, the label of the new vertex is obtained via the cyclic operad composition of the labels of the two contracted vertices, as in the following picture.
The graph complex has the structure of a cocommutative differential graded coalgebra with comultiplication given by the sum over all possible ways to partition the connected components of a graph into two sets.
Note that, up to regrading, the graph complex and, in particular, its homology only depend on the parity of m. Even though this homology could theoretically be computed directly from its definition, this has only be feasible in low degrees so far. Its structure in higher degrees remains largely mysterious. By a result of Kontsevich [Reference Kontsevich, Corwin, Gelfand and LepowskyKon93] (for m even; see also Conant–Vogtmann [Reference Conant and VogtmannCV03]) and Lazarev–Voronov [Reference Lazarev and VoronovLV08] (for m odd), Lie graph complex homology is closely related to the rational homology (with twisted coefficients if m is odd) of the group $\mathrm {Out}(F_g)$ of outer automorphisms of the free group on g generators. See Remark 4.30 for a summary of what is known about this. In particular, computations of Brun–Willwacher [Reference Brun and WillwacherBW23] imply that ${\mathrm {H}}_{p} ({\mathfrak {UG}}^{m} \operatorname {(\mathscr {L}ie)})$ is trivial for $0 < p \le 3m + 2$ when $m \ge 3$ is odd (see Remark 4.30). Using this, we obtain the following consequence of our main theorem.
Corollary 1.2. Let $3 \le k < l \le 2k  2$ and $2 \le g$ be integers such that $k + l$ is odd. Then there is, in cohomological degrees $\le {\min } \left ( g  2, 3(k + l)  4 \right )$ , an isomorphism of graded algebras
compatible with the stabilization maps on the lefthand side.
Sketch of the proof of Theorem 1.1
From now on, we fix some $3 \le k < l \le 2k  2$ and drop them from the notation. The starting point of our proof are methods recently developed by Berglund–Zeman [Reference Berglund and ZemanBZ22] which in particular yield an isomorphism
for any $\Gamma ^{\mathbb {Z}}_g$ representation P (the details of applying their methods to relative selfequivalences are worked out in the upcoming paper [Reference Berglund and StollBS] joint with Berglund). Here, the differential graded Lie algebra $\mathfrak g_g$ is the Quillen model of the 1connected cover of ${\mathrm {B}} {{\mathrm {aut}}_{\partial }}(M_{g, 1})$ . We write ${\mathrm {H}}_{\mathrm {CE}}^{*} (\mathfrak g_g)$ for its Chevalley–Eilenberg cohomology, and we set
where R is a commutative ring and ${\langle {},{}\rangle _\cap }$ is the intersection pairing. The Lie algebra $\mathfrak g_g$ has an explicit description which was shown by Berglund–Madsen [Reference Berglund and MadsenBM20] to be isomorphic to
equipped with the trivial differential. Here, $\operatorname {\mathscr {L}ie}$ is (the symmetric sequence underlying) the cyclic Lie operad, $\mathrm {s}$ denotes a degree shift, ${\mathscr A} [{}]$ is the Schur functor associated to $\mathscr A$ , and ${()}^+$ denotes positive truncation.
The next step is to simplify the righthand side of (1.2). To this end, we prove, using the previously mentioned work of Li–Sun [Reference Li and SunLS19], that the canonical map
is an isomorphism in a stable range when P is an algebraic representation of $\Gamma ^{\mathbb {Q}}_g$ . (See also work of Krannich [Reference KrannichKra20], where an analogous argument is made to obtain a similar stable range in the case $k = l$ .)
This shows that it is enough to identify the (stable) cohomology of the invariants $\left ( {\operatorname {C}_{\mathrm {CE}}^{*}}(\mathfrak g_g) \mathop{\mathop{\otimes}} P \right )^{\Gamma ^{\mathbb {Q}}_g}$ . For sake of exposition, we will now focus on the case $P = {\mathbb {Q}}$ . In the paper, we do carry out the first part of the argument for a certain class of nontrivial representations as well, though. Similar situations, arising from ${\mathrm {Sp}}_{2g}$ or $O_{g,g}$ instead of ${\mathrm {GL}}_{g}$ , have been considered by Kontsevich [Reference Kontsevich, Corwin, Gelfand and LepowskyKon93, Reference Kontsevich and JosephKon94] (see also Conant–Vogtmann [Reference Conant and VogtmannCV03]) in the symplectic case and by Berglund–Madsen [Reference Berglund and MadsenBM20] in both cases. In our situation, we use coinvariant theory methods similar to those of [Reference Berglund and MadsenBM20] to obtain a description of the stable cohomology in terms of a directed version of the Lie graph complex, which we denote by ${\mathfrak {DG}}^{m}{{\operatorname {(\mathscr {L}ie)}}}$ . It is defined in the same way as ${\mathfrak {UG}}^{m}{\operatorname {(\mathscr {L}ie)}}$ , except that every edge is equipped with a direction, which (unlike an orientation) must be respected by automorphisms of the graph.
Theorem 1.3 (see Corollary 3.33).
There is an isomorphism
of differential graded coalgebras, compatible with the stabilization maps on the lefthand side. Here, ${\mathfrak {DG}_{\mathrm {trunc}}^{m}}{(\operatorname {\mathscr {L}ie})}$ is the subcomplex of ${\mathfrak {DG}}^{m} {\operatorname {(\mathscr {L}ie)}}$ spanned by truncated directed graphs, which are those that do not have vertices of valence zero or one and whose vertices of valence two have only incoming edges.
Remark. At the heart of this theorem actually lies an unstable identification of a certain subcomplex of ${{\operatorname {C}^{\mathrm {CE}}_{*}} (\mathfrak g_g)} _{\Gamma _g}$ . Moreover, we prove a more general statement, which also applies to a certain class of nontrivial representations P, as well as to all cyclic operads and not just $\operatorname {\mathscr {L}ie}$ .
Although less ubiquitous than undirected graph complexes, directed versions have appeared in the literature before; see, for example, [Reference WillwacherWil14, Reference ŽivkovićŽiv20, Reference Dolgushev and RogersDR19]. Surprisingly, it turns out that the homology of the truncated directed graph complex ${\mathfrak {DG}_{\mathrm {trunc}}^{m}}{(\operatorname {\mathscr {L}ie})}$ is actually isomorphic to the one of ${\mathfrak {UG}}^{m} {\operatorname {(\mathscr {L}ie)}}$ . We will now explain the idea of this argument. Restricting the differential of ${\mathfrak {DG}_{\mathrm {trunc}}^{m}}{(\operatorname {\mathscr {L}ie})}$ to the two edges incident to a vertex of valence two (which must be labeled by the identity operation) yields the following picture.
Thus, intuitively, taking homology should kill the difference between the two orientations of an edge. This can be formalized using a spectral sequence argument to prove the following theorem. Together with the preceding results, this implies Theorem 1.1.
Theorem 1.4 (see Theorem 4.28).
Let $\mathscr C$ be a cyclic operad such that ${\mathscr C \left (\mkern 4mu\left (2 \right )\mkern 4mu\right )} \cong {{\mathbb {Q}} \langle {\mathrm {id}} \rangle }$ . Then there is, for any $m \in {\mathbb {N}_0}$ , an isomorphism
of graded coalgebras.
Remark. We only carry out this argument in the situation arising from trivial coefficients $P = {\mathbb {Q}}$ . There should exist a more general statement of this form, though.
Remark. For the nontruncated directed graph complex, an argument similar to this has been sketched by Willwacher [Reference WillwacherWil14, Appendix K].
Related work and further research
Building on their work for homotopy automorphisms, Berglund–Madsen [Reference Berglund and MadsenBM20] also computed, for $k \ge 3$ , the stable cohomology of the classifying spaces ${\mathrm {B}} \widetilde {\mathrm {Diff}}_\partial \left ( M^{k,k}_{g,1} \right )$ of block diffeomorphisms. Combining their methods with the methods of our paper (as well as an upgrade of the methods of [Reference Berglund and ZemanBZ22] to automorphisms of bundles, which is planned to be contained in the upcoming paper [Reference Berglund and StollBS] joint with Berglund or a followup) should yield a computation of the stable cohomology of ${\mathrm {B}} \widetilde {\mathrm {Diff}}_\partial \left ( M^{k,l}_{g,1} \right )$ for $3 \le k < l \le 2k  2$ . This is currently work in progress. It should significantly simplify and extend a result of Ebert–Reinhold [Reference Ebert and ReinholdER22], who compute, in a handson way, the stable cohomology of ${\mathrm {B}} \widetilde {\mathrm {Diff}}_\partial \left ( M^{k,k+1}_{g,1} \right )$ up to degree roughly k. From this, they deduce, using classical methods as well as a recent result of Krannich [Reference KrannichKra22], the stable cohomology of ${\mathrm {B}} \mathrm {Diff}_\partial \left ( M^{k,k+1}_{g,1} \right )$ in the same range. A complete calculation of the stable cohomology of block diffeomorphisms would, in the same way, allow to extend this result to degree roughly $2k$ . Moreover, one might hope to extend this calculation even further, for example by using methods recently developed by Krannich–RandalWilliams [Reference Krannich and RandalWilliamsKR21]. This is the goal of current work in progress joint with Krannich.
One question one might hope to attack with this approach is to determine the images of the cohomology classes coming from the graph complex under the morphism
induced by the forgetful map. When $k = l$ , the stable cohomology of the righthand side has been computed completely in celebrated work of Galatius–RandalWilliams [Reference Galatius and RandalWilliamsGR14], but even in that case, it is mostly unknown what the graph cohomology classes map to. Results in this direction have the potential to yield very interesting information about both the cohomology of the diffeomorphism groups as well as the homology of the graph complex and thus the homology of $\mathrm {Out}(F_g)$ .
Structure of this paper
In Section 2, we fix some conventions, recall various definitions and prove basic lemmas needed throughout the rest of the paper. In Section 3, we use coinvariant theory to prove Theorem 1.3. In Section 4, we prove Theorem 1.4 using a spectral sequence argument. Finally, Section 5 contains the topological part, in which we combine everything to obtain Theorem 1.1.
2 Preliminaries
In this section, we collect a number of basic conventions, notations, definitions and lemmas which we will use throughout the rest of this paper.
2.1 Graded vector spaces
Convention 2.1. The base field is ${\mathbb {Q}}$ .
Notation 2.2. We denote by ${{\mathbf {Vect}}}$ the category of vector spaces and linear maps.
Convention 2.3. All gradings are by ${\mathbb {Z}}$ .
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 kfold 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 $m$
such that $\langle v, w \rangle = (1)^{  v   w  + 1} \langle w, v \rangle $ (i.e. it is graded antisymmetric).
We denote by ${{\mathbf {Sp}}}^{m}$ the category of finitedimensional 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.
Remark 2.7. Note that a morphism of ${{\mathbf {Sp}}}^{m}$ is automatically injective.
2.2 Double complexes
Convention 2.8. A double complex is a bigraded vector space $C_{*,*}$ equipped with a differential $d^1$ of bidegree $(1, 0)$ and a differential $d^2$ of bidegree $(0, 1)$ such that $d^1 \circ d^2 =  d^2 \circ d^1$ .
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 $d = d^1 + d^2$ .
Lemma 2.10. Let $(C_{*,*}, d^1, d^2)$ be a double complex concentrated in nonnegative degrees such that ${\mathrm {H}}_{*} (C_{p,*}, d^2)$ is concentrated in degree $0$ for all p. Denote by
the map of chain complexes given by the canonical projections $C_{p, 0} \to {\mathrm {H}}_{0} (C_{p,*}, d^2)$ and the trivial map on $C_{p, q}$ for $q> 0$ . Then f is a quasiisomorphism.
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 quasiisomorphism (see, for example, [Reference WeibelWei95, Theorem 5.2.12]).
2.3 Group actions
Definition 2.11. Let $\mathcal G$ be a groupoid and $\mathcal C$ a category. A left $\mathcal G$ module in $\mathcal C$ is a functor $M \colon \mathcal G \to \mathcal C$ , and a right $\mathcal G$ module is a functor $M \colon {{\mathcal G}^{\mathrm {op}}} \to \mathcal C$ .
Remark 2.12. Identifying a group G with its associated oneobject groupoid, a left Gmodule in $\mathcal C$ is an object of $\mathcal C$ together with a left Gaction, and a right Gmodule in $\mathcal C$ is an object of $\mathcal C$ together with a right Gaction.
Convention 2.13. Unless otherwise stated, ‘module in $\mathcal C$ ’ will mean ‘left module in $\mathcal C$ ’. In particular, group actions will be from the left. If we omit the category $\mathcal C$ , we mean a (left or right) module in the category ${{{\mathbf {GrVect}}}}$ .
Definition 2.14. For $\mathcal G$ a groupoid, $\mathcal C$ a category and M a (left) $\mathcal G$ module in $\mathcal C$ , we will denote by $M^{\mathrm {op}}$ the right $\mathcal G$ module in $\mathcal C$ given by the composition
where the first map is the isomorphism given by the identity on objects and by taking the inverses of morphisms.
Remark 2.15. If $\mathcal G = G$ is a group, this means that $M^{\mathrm {op}}$ is the right Gmodule with the opposite action (i.e., ).
2.4 Kan extensions
The following lemmas will be useful later.
Lemma 2.16. Let G be a group and S a Gset (i.e., a functor $S \colon G \to {{{\mathbf {Set}}}}$ ). Then a functor X from the Grothendieck construction ${\mathop {\mathchoice {\textstyle }{}{}{} \int _{G}}} S$ to some category $\mathcal C$ can be equivalently described as a family $(X_s)_{s \in S}$ of objects of $\mathcal C$ equipped with morphisms $g_s \colon X_s \to X_{g \mathbin {.} s}$ for every $g \in G$ and $s \in S$ such that they are functorial in the sense that $h_{g \mathbin {.} s} \circ g_s = (h g)_s$ .
Moreover, let $H \trianglelefteq G$ be a normal subgroup. If $\mathcal C$ is cocomplete, then there is an isomorphism
of $({{G}/{H}})$ modules in $\mathcal C$ . On the lefthand side $g \in G$ , acts on $\coprod _{s \in S} X_s$ by mapping $X_s$ to $X_{g \mathbin {.} s}$ via $g_s$ ; the action of ${{G}/{H}}$ is the induced action on the coinvariants. On the righthand side, we have the left Kan extension of X along the composite ${\mathop {\mathchoice {\textstyle }{}{}{} \int _{G}}} S \to G \to {{G}/{H}}$ of the canonical projection and the quotient map.
Proof. The first claim follows directly from the definition of the Grothendieck construction. For the second claim, note that the Gmodule $\coprod _{s \in S} X_s$ is isomorphic to the left Kan extension of X along the canonical projection ${\mathop {\mathchoice {\textstyle }{}{}{} \int _{G}}} S \to G$ . Taking Hcoinvariants 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 ${\mathop {\mathchoice {\textstyle }{}{}{} \int _{G}}} S \to {{G}/{H}}$ .
We will now provide an explicit identification of the left Kan extension in the lemma above. This takes the form of the following (more general) lemmas.
Lemma 2.17. Let $\mathcal G$ be a groupoid, H a group and $F \colon \mathcal G \to H$ a functor. Denote by $\mathcal G^F \subseteq \mathcal G$ the wide subcategory of those morphisms that F maps to the neutral element $e_H \in H$ . Also denote by $*$ the unique object of H considered as a groupoid, and by $F \mathbin {\downarrow } *$ the corresponding comma category. Then there is a fully faithful functor
given by sending an object $G \in \mathcal G^F$ to $(G, e_H \colon F(G) = * \to *)$ and a morphism $g \colon G \to G'$ to itself.
Moreover, assume that, for all $G \in \mathcal G$ and $h \in H$ , there exists a morphism $g_{G,h} \colon G \to K_{G,h}$ in $\mathcal G$ such that $F(g_{G,h}) = h$ . Then $\Phi _F$ is an equivalence of categories with an inverse (up to natural isomorphism) given by the functor
given on objects by $(G, h) \mapsto K_{G,h}$ and on morphisms by $(g \colon (G, h) \to (G', h')) \mapsto g_{G',h'} g {g_{G,h}^{1}}$ . (Note that $\Psi _F$ depends on the choice of $g_{G,h}$ for all $G \in \mathcal G$ and $h \in H$ .) The natural isomorphism ${\mathrm {id}}_{F \mathbin {\downarrow } *} \to \Phi _F \circ \Psi _F$ is, at an object $(G, h)$ , given by the map $g_{G,h} \colon G \to K_{G,h}$ .
Proof. This follows easily from the definitions.
Lemma 2.18. In the situation of the second part of Lemma 2.17, let $\mathcal C$ be a cocomplete category and $X \colon \mathcal G \to \mathcal C$ a functor. Furthermore, for an element $h \in H$ , let $A_h$ denote the functor $\mathcal G^F \to \mathcal G^F$ that is given by $G \mapsto K_{G,h}$ on objects and that sends a morphism $g \colon G \to G'$ to $g_{G',h} g {g_{G,h}^{1}}$ . Then there is an isomorphism
of Hmodules in $\mathcal C$ . Here an element $h \in H$ acts on the righthand side via the functor $A_h$ and the natural transformation ${\mathrm {inc}}_{\mathcal G^F} \to {{\mathrm {inc}}_{\mathcal G^F}} \circ A_h$ given, at an object $G \in \mathcal G^F$ , by $g_{G,h} \colon G \to K_{G,h}$ .
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 Hmodule, to the description given in the statement follows from Lemma 2.17.
2.5 Symmetric sequences
Definition 2.19. We denote by ${{{\mathbf {LinSet}}}}$ the category with objects finite sets equipped with a linear order and morphisms the maps of sets (not necessarily respecting the order), by its maximal subgroupoid, and by ${{{\mathbf {P}}}}$ the skeleton of ${{{\mathbf {\Sigma }}}}$ spanned by the objects for $n \in {\mathbb {N}_0}$ .
Remark 2.20. There exist canonical equivalences of categories ${{{\mathbf {LinSet}}}} \to {{{\mathbf {FinSet}}}}$ , ${{{\mathbf {\Sigma }}}} \to {\operatorname {Core}({{\mathbf {FinSet}}})}$ , and ${{{\mathbf {P}}}} \to {\operatorname {Core}({{\mathbf {FinSet}}})}$ , though neither of them admit a canonical inverse equivalence. The inclusion ${{{\mathbf {P}}}} \to {{{\mathbf {\Sigma }}}}$ , however, is an equivalence that does admit a canonical inverse equivalence – namely, by identifying an object $S \in {{{\mathbf {\Sigma }}}}$ with an object in ${{{\mathbf {P}}}}$ via the unique map that respects the linear order.
Definition 2.21. Let $\mathcal V$ be a symmetric monoidal category and $V \in \mathcal V$ . Then there is a functor $V^{{{\otimes}} } \colon {{{\mathbf {P}}}} \to \mathcal V$ given on objects by ${\underline {n}} \mapsto V^{{{\otimes}} n}$ and defined on morphisms via the symmetrizer isomorphism. Pulling this back via the canonical equivalence ${{{\mathbf {\Sigma }}}} \to {{{\mathbf {P}}}}$ , we obtain a functor ${{{\mathbf {\Sigma }}}} \to \mathcal V$ which we also denote by $V^{{{\otimes}} }$ .
Similarly, for $S \in {{{\mathbf {\Sigma }}}}$ and $(V_s)_{s \in S}$ a family of objects of $\mathcal V$ , we write $\bigotimes _{s \in S} V_s$ for $V_{s(1)} \mathop{\mathop{\otimes}} \dots \mathop{\mathop{\otimes}} V_{s(S)}$ , where $s \colon {\underline {S}} \to S$ is the unique orderpreserving bijection. We will also use the notation $\bigotimes _{s \in S} v_s$ to denote elementary tensors of this tensor product.
Definition 2.22. For $\mathscr M$ a ${{{\mathbf {\Sigma }}}}$ module and $a \colon S \to T$ a morphism of ${{{\mathbf {LinSet}}}}$ , we write
where ${a^{1}}(t) \subseteq S$ has the linear order induced from the one of S. This construction is functorial in $a \in {{{\mathbf {\Sigma }}}} \mathbin {\downarrow } {{{\mathbf {\Sigma }}}}$ , with the comma category being taken over ${{{\mathbf {LinSet}}}}$ .
Definition 2.23. Let $\mathscr P$ and $\mathscr Q$ be two ${{{\mathbf {\Sigma }}}}$ modules. Then we can form their composition product – that is, the ${{{\mathbf {\Sigma }}}}$ module $\mathscr P \circ \mathscr Q$ given on objects by
where the slice category $S \mathbin {\downarrow } {{{\mathbf {\Sigma }}}}$ is taken over ${{{\mathbf {LinSet}}}}$ . A morphism $f \colon S \to S'$ of ${{{\mathbf {\Sigma }}}}$ acts via the map
induced by the functor ${f^{1}} \mathbin {\downarrow } {{{\mathbf {\Sigma }}}} \colon S \mathbin {\downarrow } {{{\mathbf {\Sigma }}}} \to S' \mathbin {\downarrow } {{{\mathbf {\Sigma }}}}$ as well as the natural transformation $\mathscr P(T) \mathop{\otimes} \mathscr Q(a) \to \mathscr P(T) \mathop{\otimes} \mathscr Q(a \circ {f^{1}})$ given by the identity of $\mathscr P(T)$ and the map $\mathscr Q(a) \to \mathscr Q(a \circ {f^{1}})$ induced by the morphism $(f, {\mathrm {id}}_{T}) \colon a \to a \circ {f^{1}}$ of ${{{\mathbf {\Sigma }}}} \mathbin {\downarrow } {{{\mathbf {\Sigma }}}}$ .
2.6 Schur functors
Notation 2.24. Let $\mathscr M$ a ${{{\mathbf {\Sigma }}}}$ module. We define a functor ${\mathscr M}[{}] \colon {{{\mathbf {GrVect}}}} \to {{{\mathbf {GrVect}}}}$ by
and call it the Schur functor associated to $\mathscr M$ .
Remark 2.25. There is a canonical, and natural in V, isomorphism
where ${{{\mathbf {P}}}}$ acts diagonally on the tensor product. We can also replace ${{{\mathbf {P}}}}$ by ${{{\mathbf {\Sigma }}}}$ since they are canonically equivalent.
Lemma 2.26. Let $\mathscr P$ and $\mathscr Q$ be ${{{\mathbf {\Sigma }}}}$ modules. Then there is a natural isomorphism
with an explicit description as given in the proof.
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
to $\varepsilon \in \{\pm 1\}$ times the element of
represented by $T \in {{{\mathbf {\Sigma }}}}$ and as well as the elements
where $\varepsilon $ is the sign incurred by permuting the two expressions
into each other.
2.7 Differential graded Lie algebras
Definition 2.27. Let L be a differential graded Lie algebra. Its positive truncation is the differential graded Lie subalgebra ${L}^+ \subseteq L$ given by
where $\delta $ is the differential of L.
Definition 2.28. Let L be a differential graded Lie algebra. We denote by ${\operatorname {C}^{\mathrm {CE}}_{*}} (L)$ its Chevalley–Eilenberg complex (i.e., the underlying graded vector space of the free graded commutative algebra ${\mathrm {\Lambda }}(\mathrm {s} L)$ ), equipped with the differential $d = d_0 + d_1$ , where
where $\delta $ is the differential of L.
The Chevalley–Eilenberg complex comes equipped with the structure of a cocommutative differential graded coalgebra. Its counit $\varepsilon \colon {\operatorname {C}^{\mathrm {CE}}_{*}} (L) \to {\mathbb {Q}}$ is given by $1$ on the empty wedge and $0$ on all higher wedges. The comultiplication $\Delta \colon {\operatorname {C}^{\mathrm {CE}}_{*}} (L) \to {\operatorname {C}^{\mathrm {CE}}_{*}} (L) \mathop{\mathop{\otimes}} {\operatorname {C}^{\mathrm {CE}}_{*}} (L)$ is given by
where the sum runs over all subsets A of the set ${\underline {k}} = \{1, \dots , k\}$ . Here, $\varepsilon (A) \in \{\pm 1\}$ is the sign incurred by permuting $\mathrm {s} x_1 \mathop{\mathop{\otimes}} \dots \mathop{\mathop{\otimes}} \mathrm {s} x_k$ into ${\textstyle \bigotimes }_{a \in A} \mathrm {s} x_a \mathop{\mathop{\otimes}} {\textstyle \bigotimes }_{b \in {\underline {k}} \setminus A} \mathrm {s} x_b$ .
Moreover, we write ${\mathrm {H}}_{\mathrm {CE}}^{*} (L)$ for the Chevalley–Eilenberg cohomology of L – that is, the cohomology of the linear dual of ${\operatorname {C}^{\mathrm {CE}}_{*}} (L)$ , equipped with its graded commutative algebra structure.
Lemma 2.29. Let ${\mathrm {\Lambda }} \mathrm {s}$ denote the ${{{\mathbf {\Sigma }}}}$ module given by ${\mathrm {\Lambda }} \mathrm {s}(S) = (\mathrm {s} {\mathbb {Q}})^{{{\otimes}} S}$ . Then there is a canonical isomorphism of graded vector spaces, natural in $V \in {{{\mathbf {GrVect}}}}$ ,
where .
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.
2.8 The convolution Lie algebra
Recall that a cyclic operad is an operad $\mathscr C$ such that the right action of ${\Sigma _{n}}$ on $\mathscr C(n)$ extends to a right action of ${\Sigma _{n+1}}$ in a way compatible with the composition operations (see Getzler–Kapranov [Reference Getzler and KapranovGK95, §2]). Unless explicitly stated otherwise, we will always work in the category of graded vector spaces.
Notation 2.30. For a cyclic operad $\mathscr C$ , we write ${\mathscr C \left (\mkern 4mu\left (n\right )\mkern 4mu\right )}$ for $\mathscr C(n1)$ 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 ${\mathscr C}[{V}] = \bigoplus _{n \ge 1} {\mathscr C \left (\mkern 4mu\left (n\right )\mkern 4mu\right )} \mathop{\mathop{\otimes}}{\hspace{2pt}}_{{\Sigma _{n}}} V^{{{\otimes}} n}$ .
Definition 2.31. For a cyclic operad $\mathscr C$ , natural numbers $k, l \in {\mathbb {N}_{\ge 1}}$ , operations $p \in {\mathscr C \left (\mkern 4mu\left ({k} \right )\mkern 4mu\right )}$ and $q \in {\mathscr C \left (\mkern 4mu\left ({l} \right )\mkern 4mu\right )}$ , and $1 \le i \le k$ and $1 \le j \le l$ , we set
where is the cyclic permutation and ${\circ }_i$ denotes the ith partial composition of the underlying operad of $\mathscr C$ . In the case that $i = k$ , we set .
Remark 2.32. Via the canonical equivalence ${{{\mathbf {\Sigma }}}} \to {{{\mathbf {P}}}}$ , the operations ${}_i{\circ }_j$ of Definition 2.31 generalize to operations
where $S, T \in {{{\mathbf {\Sigma }}}}$ , and $s \in S$ and $t \in T$ . Here,
equipped with the linear order such that
with the linear order on each block inherited from S respectively T.
Definition 2.33. Let $V \in {{\mathbf {Sp}}}^{m}$ , $k, l \in {\mathbb {N}_0}$ , $1 \le i \le k$ and $1 \le j \le l$ . We denote by
the map given by
where, for $a \le b$ , we set and analogously for $w_{a,b}$ .
Lemma 2.34. Let $\mathscr C$ be a cyclic operad in graded vector spaces that is concentrated in even degrees and let $V \in {{\mathbf {Sp}}}^{m}$ . Then $ {(\mathrm {s}^{m} \mathscr C)}[{V}]$ becomes a graded Lie algebra by setting
for $\xi \in {\mathscr C \left (\mkern 4mu\left ({k} \right )\mkern 4mu\right )}$ , $v \in V^{{{\otimes}} k}$ , $\zeta \in {\mathscr C \left (\mkern 4mu\left ({l} \right )\mkern 4mu\right )}$ , and $w \in V^{{{\otimes}} l}$ . This is functorial – that is, the restricted functor ${{ {(\mathrm {s}^{m} \mathscr C)}[{}]}}_{{{\mathbf {Sp}}}^{m}} \colon {{\mathbf {Sp}}}^{m} \to {{{\mathbf {GrVect}}}}$ lifts to a functor to graded Lie algebras, which we also denote by ${(\mathrm {s}^{m} \mathscr C)}[{}]$ .
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)].
Remark 2.35. The condition that $\mathscr C$ is concentrated in even degrees is purely for convenience, as it simplifies the signs occurring.
3 The directed graph complex
In this section, our goal is to prove a version of Theorem 1.3 with coefficients. That is, we want to identify (in a stable range)
where $\Gamma _g^{\mathbb {Q}}$ is the group of automorphisms of the graded vector space ${\widetilde {\operatorname {H}}_{*}}(M_{g,1}; {\mathbb {Q}})$ that respect the intersection pairing, Q is some $\Gamma ^{\mathbb {Q}}_g$ module, and the graded Lie algebra structure on the Schur functor is obtained from Lemma 2.34.
To this end, we begin, in the first two subsections, with studying ${( {\mathscr P} [{H}] \mathop{\mathop{\otimes}} Q)}_{{\mathrm {Aut}}(H)}$ , where $\mathscr P$ is a ${{{\mathbf {\Sigma }}}}$ module, H is an object of ${{\mathbf {Sp}}}^{m}$ that is concentrated in two degrees $\neq \frac {m}{2}$ , and Q is an ${\mathrm {Aut}}(H)$ representation. From the results we obtain for this situation, we will, via Lemmas 2.26 and 2.29, be able to deduce a description as desired. The arguments we will use are similar to those of [Reference Berglund and MadsenBM20, §9] for the case where H is concentrated in degree $\frac m 2$ . We will restrict ourselves to the case that $Q = V^{{{\otimes}} I} \mathop{\mathop{\otimes}} W^{{{\otimes}} J}$ for some finite linearly ordered sets I and J, where V and W are the two nontrivial homogeneous pieces of H. Identifying the result in this case with its induced $({\Sigma _{I}} \times {\Sigma _{J}})$ action allows one to deduce the result for more general representations via Schur–Weyl duality (see e.g. [Reference Fulton and HarrisFH04, §6.1] or [Reference EtingofEti+11, §5.19]).
Throughout this section, we fix two integers n and m such that $m \neq 2n$ . (They correspond to $k  1$ and $k + l  2$ above.)
3.1 Coinvariants of monomial functors
In this subsection, we let $H = V \oplus W$ be a finitedimensional graded vector space such that V is concentrated in degree n and W is concentrated in degree $m  n \neq n$ . Moreover, we assume that H is equipped with a graded symplectic form $\langle , \rangle $ of degree $m$ (see Notation 2.6). We write for the group of degree $0$ symplectic automorphisms of H.
Let S, I and J be finite linearly ordered sets and P a ${\Sigma _{S}}$ module. Our goal for this subsection is to identify the coinvariants
as a $({\Sigma _{I}} \times {\Sigma _{J}})$ module only in terms of P, without reference to H. We begin with some basic observations about the structures of H and $\Gamma $ .
Lemma 3.1. The map $W \to \mathrm {s}^{m} {V^\vee }$ given by $w \mapsto \mathrm {s}^{m} \langle w,  \rangle $ is a degree $0$ isomorphism. After identifying W with $\mathrm {s}^{m} {V^\vee }$ via this isomorphism, the inner product on $H = V \oplus \mathrm {s}^{m} {V^\vee }$ is given by $\langle {(v, \mathrm {s}^{m} \phi )}, {(v', \mathrm {s}^{m} \phi ')\rangle } = \phi (v')  (1)^{n(mn)} \phi '(v)$ .
Proof. The composition
is a degree $m$ isomorphism of the form
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(mn)} \langle v, w \rangle $ .
Using the preceding lemma we will, in the following, implicitly identify H with $V \oplus \mathrm {s}^{m} {V^\vee }$ . We now give a description of $\Gamma $ in these terms.
Lemma 3.2. The map ${\mathrm {GL}}(V) \to {\mathrm {Aut}}(V \oplus \mathrm {s}^{m} {V^\vee }, \langle , \rangle ) = \Gamma $ given by $f \mapsto f \oplus {({f^{1}})^\vee }$ is an isomorphism of groups. (Here, we implicitly identify automorphisms of $\mathrm {s}^{m} {V^\vee }$ with automorphisms of ${V^\vee }$ .)
Proof. Let $f \in {\mathrm {GL}}(V)$ . Then the computation
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
and hence $g = {({f^{1}})^\vee }$ . This shows surjectivity.
By basic linear algebra, there is a canonical isomorphism of graded vector spaces $V \mathop{\mathop{\otimes}} \mathrm {s}^{m} {V^\vee } \cong \mathrm {s}^{m} {\mathrm {End}}{(V)}$ (here, ${\mathrm {End}}(V)$ is concentrated in degree $0$ since V is concentrated in a single degree). We will now use this to identify ${(H^{{{\otimes}} S} \mathop{\mathop{\otimes}} V^{{{\otimes}} I} \mathop{\mathop{\otimes}} W^{{{\otimes}} J})}_\Gamma $ in terms of ${\mathrm {End}}{(V)}$ . To make this precise, we need to introduce some notation.
Definition 3.3. Let S be a finite linearly ordered set, A and B two disjoint subsets of S such that (i.e., an ordered partition of S into two subsets), and $V_1$ and $V_2$ two graded vector spaces. Then we denote by $(V_1, V_2)^{{{\otimes}} (A, B)}$ the tensor product $\bigotimes _{s \in S} V_s$ , where if $s \in A$ and if $s \in B$ .
A morphism $f \colon S \to S'$ of ${{{\mathbf {\Sigma }}}}$ induces a map
by permuting the factors. When $V_1$ is concentrated in degree n and $V_2$ in degree $mn$ , we denote by $\operatorname {sgn}_{A,B}(f) \in \{\pm 1\}$ the sign incurred by this permutation – that is, it is chosen such that
holds.
Definition 3.4. Let S be a finite linearly ordered set. We denote by $\operatorname {OM}(S)$ the set of ordered matchings of S (i.e., (ordered) pairs $(A, B)$ such that A and B are disjoint subsets of S with $A = B$ and ). (In particular, $\operatorname {OM}(S)$ is empty if $S$ is odd.) This assembles into a functor $\operatorname {OM} \colon {{{\mathbf {\Sigma }}}} \to {{{\mathbf {Set}}}}$ by letting a morphism $f \colon S \to S'$ act via $(A, B) \mapsto (f(A), f(B))$ .
Also note that the linear order of S restricts to a linear order on A and a linear order on B. In particular, there is a unique orderpreserving bijection $A \to B$ . We will denote this bijection by $\mu _{A,B}$ .
Definition 3.5. Let S, I and J be finite linearly ordered sets. We denote by $\operatorname {OM}(S, I, J) \subseteq \operatorname {OM}(S \amalg I \amalg J)$ the subset of those ordered matchings $(A, B)$ such that $I \subseteq A$ and $J \subseteq B$ . This assembles into a functor $\operatorname {OM} \colon {{{\mathbf {\Sigma }}}} \times {{{\mathbf {\Sigma }}}} \times {{{\mathbf {\Sigma }}}} \to {{{\mathbf {Set}}}}$ by letting a morphism $(f, g, h)$ act via $f \amalg g \amalg h$ .
Remark 3.6. Note that there is a canonical bijection $\operatorname {OM}(S, I, J) \cong \operatorname {OM}(S)$ , which is natural in ${{{\mathbf {\Sigma }}}} \times {{{\mathbf {\Sigma }}}} \times {{{\mathbf {\Sigma }}}}$ .
Lemma 3.7. Let S, I and J be finite linearly ordered sets. Then there is an isomorphism of $({\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}})$ modules
where $g \in {\mathrm {GL}}(V) \cong \Gamma $ acts on $f \in {\mathrm {End}}(V)$ via the conjugation $g \mathbin {.} f = g f {g^{1}}$ , and diagonally on the tensor products. The $({\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}})$ module structure on the lefthand side is given as follows: a morphism $(f, g, h)$ of ${\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}}$ maps the factor corresponding to $(A, B) \in \operatorname {OM}(S, I, J)$ into the factor corresponding to $(c(A), c(B))$ via the map given by
where , , and , and the $\operatorname {E}_{i, j}$ are singleentry matrices with respect to some fixed basis of V (the resulting map is independent of this choice).
Moreover, $\Theta _1$ can be chosen such that, for any basis e of V, it fulfills, on the factor corresponding to $(A, B) \in \operatorname {OM}(S, I, J)$ ,
where the $E_{i,j}$ are with respect to e.
Proof. There is a canonical isomorphism
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 righthand 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 righthand 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 righthand side by sending the factor corresponding to $(A, B)$ into the factor corresponding to $(c(A), c(B))$ via
where $\sigma $ and $\tau $ are as in the statement of the lemma. Said differently, we have
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
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 lefthand 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.
We have now reduced finding a description of ${\left (H^{{{\otimes}} S} \mathop{\mathop{\otimes}} V^{{{\otimes}} I} \mathop{\mathop{\otimes}} W^{{{\otimes}} J}\right )}_\Gamma $ independent of H to finding a description of ${\left ({{\mathrm {End}}(V)}^{{{\otimes}} A}\right )}_\Gamma $ that is independent of V. Such a description is provided by the classical coinvariant theory for the general linear group.
Proposition 3.8 (Fundamental theorem of coinvariant theory).
Let $k \in {\mathbb {N}_0}$ such that $k \le \dim V$ . Then there is an isomorphism of vector spaces
such that $[\operatorname {E}_{\omega (1), {\sigma (1)}} \mathop{\mathop{\otimes}} \dots \mathop{\mathop{\otimes}} \operatorname {E}_{\omega (k), {\sigma (k)}}] \mapsto {\omega ^{1}} \sigma $ for all $\omega , \sigma \in {\Sigma _{k}}$ and any basis of V.
Proof. This can, for example, be found in [Reference LodayLod98, 9.2.5, 9.2.8, and 9.2.10].Footnote ^{2}
Applying this to our situation, we obtain the following corollary.
Corollary 3.9. Let S, I and J be finite linearly ordered sets such that $S + I + J \le 2 (\dim V)$ . Then there is an isomorphism of $({\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}})$ modules
where ${\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}}$ acts on the lefthand side as in Lemma 3.7 and on the righthand side as follows: an element $(f, g, h)$ maps the factor corresponding to $(A, B) \in \operatorname {OM}(S, I, J)$ into the factor corresponding to $(c(A), c(B))$ via the map
where , and $\sigma $ and $\tau $ are as in Lemma 3.7.
Moreover, $\Theta _2$ can be chosen such that, for any basis of V and $\omega \in {\Sigma _{A}}$ , it fulfills, on the factor corresponding to $(A, B) \in \operatorname {OM}(S, I, J)$ , that
where we identify A with via the unique orderpreserving bijection.
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
which is what we wanted to show.
Remark 3.10. Setting $T = S \amalg I \amalg J$ , we note that $\Theta _2$ restricts to an isomorphism of $({\Sigma _{S}} \times {\Sigma _{I}} \times {\Sigma _{J}})$ modules on the summands indexed by $\operatorname {OM}(S, I, J) \subseteq \operatorname {OM}(T)$ .
We have now almost achieved what we set out do to in this subsection, as we have obtained, when $S + I + J \le 2 (\dim V)$ , an isomorphism
with the righthand side independent of H as desired. To generalize this to expressions of the form ${((P^{\mathrm {op}} \mathop{\mathop{\otimes}}{\hspace{2pt}}_{\Sigma _{S}}\, H^{{{\otimes}} S}) \mathop{\mathop{\otimes}} V^{{{\otimes}} I} \mathop{\mathop{\otimes}} W^{{{\otimes}} J})}_\Gamma $ , we will use the abstract lemmas of Section 2.4. To state the result in its simplest form, we need to introduce a bit of notation.
Definition 3.11. Let I and J be finite linearly ordered sets. We denote by ${{\mathbf {Match}}}_{I,J}$ the following groupoid:

• Objects are tuples $(S, A, B, \mu )$ with S a finite linearly ordered set, $(A, B) \in \operatorname {OM}(S, I, J)$ an ordered matching, and $\mu \colon A \to B$ a bijection.

• A morphism $(S, A, B, \mu ) \to (S', A', B', \mu ')$ is a tuple $(f, g, h)$ of bijections $f \colon S \to S'$ , $g \in {\Sigma _{I}}$ , and $h \in {\Sigma _{J}}$ , such that the map fulfills $c(A) = A'$ , $c(B) = B'$ , and ${{c}}_B \circ \mu = \mu ' \circ {{c}}_A$ .
Moreover, for $S \in {{{\mathbf {\Sigma }}}}$ , we denote by ${{\mathbf {Match}}}^{S}_{I,J} \subseteq {{\mathbf {Match}}}_{I,J}$ the full subgroupoid spanned by objects of the form $(S', A', B', \mu ')$ with $S' = S$ . Also note that there is a canonical functor ${\mathrm {pr}}_{I,J} \colon {{\mathbf {Match}}}_{I,J} \to {\Sigma _{I}} \times {\Sigma _{J}}$ given on morphisms by sending $(f, g, h)$ to $(g, h)$ . We denote by ${{\mathbf {Match}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}} \subseteq {{\mathbf {Match}}}_{I,J}$ the wide subgroupoid of those morphisms that are sent to $({\mathrm {id}}_{I}, {\mathrm {id}}_{J})$ by ${\mathrm {pr}}_{I,J}$ , and analogously for ${{\mathbf {Match}}}^{S}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}} \subseteq {{\mathbf {Match}}}^{S}_{I,J}$ .
Definition 3.12. We write $\operatorname {sgn}_{n, m}$ for the ${{\mathbf {Match}}}_{I,J}$ module given by
(see Definition 3.3). Said differently, it is $\mathrm {s}^{m A} {\mathbb {Q}}$ on which a morphism $(f, g, h)$ acts by $\operatorname {sgn}_{A, B}(c)$ , where .
With only a little bit more work, the lemmas of Section 2.4 now specialize to the following. (See [Reference Berglund and MadsenBM20, Corollary 9.8] for a similar statement in the case where H is concentrated in a single degree and $I = J = \varnothing $ .)
Proposition 3.13. Let S, I and J be finite linearly ordered sets such that $S + I + J \le 2 (\dim V) = \dim H$ , and let P be a ${\Sigma _{S}}$ module. Then there is an isomorphism of $({\Sigma _{I}} \times {\Sigma _{J}})$ modules
where a morphism $(f, {\mathrm {id}}_{I}, {\mathrm {id}}_{J})$ of ${{\mathbf {Match}}}^{S}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ acts on P by $f \in {\Sigma _{S}}$ , and $\Gamma $ acts trivially on P and diagonally on the tensor product. The action of an element $(g, h) \in {\Sigma _{I}} \times {\Sigma _{J}}$ on the lefthand side is given by sending an element represented by $(A, B, \mu ) \in {{\mathbf {Match}}}^{S}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ and $x \in P \mathop{\mathop{\otimes}} \operatorname {sgn}_{n, m}$ to the element represented by $(A, B, {{c}}_B \circ \mu \circ {({{c}}_A)^{1}})$ and $\operatorname {sgn}_{A, B}(c) x$ , where .
Moreover, the isomorphism above can be chosen such that, for any basis e of V, it maps the element of the colimit represented by $(A, B, \mu ) \in {{\mathbf {Match}}}^{S}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ and $p \mathop{\mathop{\otimes}} \mathrm {s}^{m A} 1 \in P \mathop{\mathop{\otimes}} \operatorname {sgn}_{n, m}$ to $[{p \mathop{\mathop{\otimes}} {\textstyle \bigotimes }_{t \in S \amalg I \amalg J} h_t}]$ where
(here $a \colon {\underline {A}} \to A$ is the unique orderpreserving bijection).
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
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
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
as $({\Sigma _{I}} \times {\Sigma _{J}})$ modules. To describe the action on the sum on the righthand 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
of categories over ${\Sigma _{I}} \times {\Sigma _{J}}$ , we obtain an isomorphism of the righthand side of (3.2) with
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.
3.2 Coinvariants of Schur functors
In this subsection, we want to compute
as a $({\Sigma _{I}} \times {\Sigma _{J}})$ module, where $\mathscr P$ is some ${{{\mathbf {\Sigma }}}}$ module, and H, V, W and $\Gamma $ are as in the preceding subsection. Noting that $ {\mathscr P} [{H}]$ is a sum of monomial functors in H, Proposition 3.13 yields an identification as desired in a range depending on the dimension of H. To obtain a description of the whole thing, we thus need to let the dimension of H go to infinity.
More precisely, we fix the following notation for the rest of this section:
is a sequence of maps of ${{\mathbf {Sp}}}^{m}$ (see Notation 2.6) such that $H_i$ is concentrated in dimensions n and $mn$ and such that the sequence $\dim {H_i}$ diverges. Moreover, we write and denote by $V_i$ and $W_i$ the degree n and $m  n$ parts of $H_i$ , respectively.
Note that there are maps $\Gamma _i \to \Gamma _{i+1}$ given by extending an automorphism of $H_i$ by the identity on the symplectic complement of $H_i$ in $H_{i+1}$ . Moreover, the maps $H_i \to H_{i+1}$ become $\Gamma _i$ equivariant when $H_{i+1}$ is equipped with the restricted action. In particular, for any functor F from ${{\mathbf {Sp}}}^{m}$ to some cocomplete category $\mathcal C$ , we obtain a sequence ${F(H_1)}_{\Gamma _1} \to {F(H_2)}_{\Gamma _2} \to \dots $ and thus can make sense of the expression ${\mathrm {colim}}_{i \in {\mathbb {N}}}\ {F(H_i)}_{\Gamma _i}$ .
Definition 3.14. A compatible basis of the $V_i$ is a sequence $e_1, e_2, \dots $ of elements of ${\mathrm {colim}}_{i}\ V_i$ such that, for each $i \ge 1$ , each of the elements $e_1, \dots , e_{\dim {V_i}}$ is represented by an (automatically unique) element of $V_i$ , and the set of these representatives forms a basis of $V_i$ .
Proposition 3.15. Let I and J be finite linearly ordered sets and $\mathscr P$ a ${{{\mathbf {\Sigma }}}}$ module. Then there is an isomorphism of $({\Sigma _{I}} \times {\Sigma _{J}})$ modules
where a morphism $(f, {\mathrm {id}}_{I}, {\mathrm {id}}_{J})$ of ${{\mathbf {Match}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ acts on P by $f \in {\Sigma _{S}}$ . The action of an element $(g, h) \in {\Sigma _{I}} \times {\Sigma _{J}}$ on the lefthand side is given by sending an element represented by $(S, A, B, \mu ) \in {{\mathbf {Match}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ and $x \in \mathscr P(S) \mathop{\mathop{\otimes}} \operatorname {sgn}_{n, m}$ to the element represented by $(S, A, B, {{c}}_B \circ \mu \circ {({{c}}_A)^{1}})$ and $\operatorname {sgn}_{A, B}(c) x$ , where .
Moreover, this isomorphism can be chosen such that, for any compatible basis $(e_j)_{j \in {{\mathbb {N}_{\ge 1}}}}$ of the $V_i$ , it maps an element represented by $(S, A, B, \mu ) \in {{\mathbf {Match}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ and $p \mathop{\mathop{\otimes}} \mathrm {s}^{m A} 1 \in \mathscr P(S) \mathop{\mathop{\otimes}} \operatorname {sgn}_{n, m}$ to the element $ [{p \mathop{\mathop{\otimes}} {\textstyle \bigotimes }_{t \in S \amalg I \amalg J} h_t}]$ of
where $h_t$ is as in Proposition 3.13 (and we use the definition of $ {\mathscr P} [{H_i}]$ of Remark 2.25).
Proof. We have
by definition of ${\mathscr P}[{}]$ . When $k \le \dim {H_i}  I  J$ , there is an isomorphism
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 lefthand 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.
3.3 Coinvariants of double Schur functors
In this subsection, our goal is to compute
as a $({\Sigma _{I}} \times {\Sigma _{J}})$ module, where $\mathscr P$ and $\mathscr Q$ are ${{{\mathbf {\Sigma }}}}$ modules, and $H_i$ , $V_i$ , $W_i$ and $\Gamma _i$ are as in the preceding subsection. By Lemma 2.26, we have an isomorphism $ {\mathscr P} { [{\mathscr Q} [{H_i}]]} \cong {(\mathscr P \circ \mathscr Q)}[{H_i}]$ . Hence, we can use Proposition 3.15 to obtain a description as desired. We now introduce some notation to state the result in a nice way.
Definition 3.16. Let I and J be finite linearly ordered sets. We denote by ${{\mathbf {DirGraph}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ the following groupoid:

• Objects are directed $(I,J)$ graphs – that is, tuples $(F, N, S, T, \mu , a)$ with F and N finite linearly ordered sets, $(S, T) \in \operatorname {OM}(F, I, J)$ an ordered matching, $\mu \colon S \to T$ a bijection, and $a \colon F \to N$ a map of sets.

• A morphism from $(F, N, S, T, \mu , a)$ to $(F', N', S', T', \mu ', a')$ is a tuple of bijections $(f \colon F \to F'$ , $k \colon N \to N')$ such that $c(S) = S'$ , $c(T) = T'$ , ${{c}}_T \circ \mu = \mu ' \circ {{c}}_S$ , and $k \circ a = a' \circ f$ , where we set .
When $I = J = \varnothing $ , we just write ${{\mathbf {DirGraph}}}$ and call its objects directed graphs.
Remark 3.17. We think of a directed $(I,J)$ graph as a directed graph with I inputs and J outputs. Under this viewpoint, we have the following: N is the set of vertices (or nodes), F is the set of internal flags (or halfedges), a describes to which vertex a flag is attached, S is the set of those flags which are sources (i.e., outgoing) which includes the inputs, similarly T is the set of those flags which are targets (i.e., incoming) which includes the outputs, and $\mu $ describes the (directed) edges (i.e., $s \in S$ and $\mu (s) \in T$ are connected and form an edge).
Lemma 3.18. There is an isomorphism from $\operatorname {sgn}_{n, m}$ to the ${{\mathbf {Match}}}_{I,J}$ module given by
where a morphism $(f, g, h) \colon (S, A, B, \mu ) \to (S', A', B', \mu ')$ of ${{\mathbf {Match}}}_{I,J}$ acts by permuting the factors according to ${{c}}_A \colon A \to A'$ , where . This isomorphism maps $\mathrm {s}^{m A} 1 \in \operatorname {sgn}_{n, m}$ to $\varepsilon \cdot \mathrm {s}^{m A} 1 \in (\mathrm {s}^{m} {\mathbb {Q}})^{{{\otimes}} A}$ , where $\varepsilon $ is the sign incurred by permuting
where if $t \in A$ and if $t \in B$ .
Proof. Let $p, q \colon {{\mathbf {Match}}}_{I,J} \to {{\mathbf {Match}}}_{\varnothing ,\varnothing }$ be the canonical functors given, on objects, by
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.
The following proposition contains the desired description of the $({\Sigma _{I}} \times {\Sigma _{J}})$ module ${\mathrm {colim}}_{i \in {\mathbb {N}}}\ {( {\mathscr P} {[{\mathscr Q} [{H_i}]]} \mathop{\mathop{\otimes}} V_i^{{{\otimes}} I} \mathop{\mathop{\otimes}} W_i^{{{\otimes}} J} )}_{\Gamma _i}$ . (See [Reference Berglund and MadsenBM20, Theorem 9.12] for a similar statement in the case where H is concentrated in a single degree and $I = J = \varnothing $ .)
Proposition 3.19. Let I and J be finite linearly ordered sets and let $\mathscr P$ and $\mathscr Q$ be ${{{\mathbf {\Sigma }}}}$ modules. Then there is an isomorphism of $({\Sigma _{I}} \times {\Sigma _{J}})$ modules
where a morphism $(f, k)$ of ${{\mathbf {DirGraph}}}_{{\mathrm {id}}_{I},{\mathrm {id}}_{J}}$ acts on