Self-duality of the lattice of transfer systems via weak factorization systems
Evan E. Franchere, Kyle Ormsby, Angélica M Osorno, Weihang Qin, Riley Waugh
aa r X i v : . [ m a t h . A T ] F e b SELF-DUALITY OF THE LATTICE OF TRANSFER SYSTEMSVIA WEAK FACTORIZATION SYSTEMS
EVAN E. FRANCHERE, KYLE ORMSBY, ANG´ELICA M. OSORNO, WEIHANG QIN,AND RILEY WAUGH
Abstract.
For a finite group G , G -transfer systems are combinatorial objects whichencode the homotopy category of G - N ∞ operads, whose algebras in G -spectra are E ∞ G -spectra with a specified collection of multiplicative norms. For G finite Abelian,we demonstrate a correspondence between G -transfer systems and weak factorizationsystems on the poset category of subgroups of G . This induces a self-duality on thelattice of G -transfer systems. Introduction
In stable homotopy theory, commutative ring spectra are algebras over an E ∞ operad.The equivariant story is more subtle. In [BH15], A. Blumberg and M. Hill introduced N ∞ operads to capture the varying classes of multiplicative norm maps supported by equivari-ant ring spectra. By work of Blumberg and Hill [BH15], P. Bonventre and L. Pereira [BP],J. Guti´errez and D. White [GW18], and J. Rubin [Rub], we know that the homotopy cate-gory of N ∞ operads may be identified with the lattice of indexing systems for the group ofequivariance. The indexing system associated to an operad records which norm maps arebeing encoded by the operad. Further work of Rubin [Rub20] and S. Balchin, D. Barnes,and C. Roitzheim [BBR], identified transfer systems as the essential combinatorial data ofindexing systems, thus proving that the homotopy category of N ∞ operads is equivalentto the lattice of transfer systems. The combinatorics of this lattice thus plays a centralrole in the study of equivariant ring spectra.Fix a finite group G . We recall the basics of G - N ∞ operads and G -transfer systems inSection 3. For the purposes of this introduction, note that a G -transfer system is a relation R on Sub( G ), the subgroup lattice of G , that refines inclusion and satisfies the followingconditions: • (reflexivity) H R H for all H ≤ G , • (transitivity) K R H and L R K implies L R H , • (closed under conjugation) K R H implies that ( gKg − ) R ( gHg − ) for all g ∈ G , • (closed under restriction) K R H and M ≤ H implies ( K ∩ M ) R M .In other words, a transfer system is a sub-poset of Sub( G ) closed under conjugation andrestriction. These objects form a bounded lattice Tr( G ) ordered under refinement (seeProposition 3.7).In this paper, we establish that Tr( G ) is self-dual whenever G is Abelian. Our proofproceeds via a surprising connection with weak factorization systems. A weak factorizationsystem on a category C is a pair of classes of morphisms ( L , R ) satisfying the factorization nd lifting axioms specified in Definition 4.5. Considering Sub( G ) as a poset category, weshow in Theorem 4.12 that there is a bijective correspondence between transfer systemsand weak factorization systems given by R ←→ ( (cid:27) R , R )where (cid:27) R denotes the morphisms in Sub( G ) having the left-lifting property with respectto R .When G is Abelian, the subgroup lattice Sub( G ) carries a self-duality ∇ . In Theorem 4.20,we prove that φ : Tr( G ) −→ Tr( G ) R 7−→ (( (cid:27) R ) op ) ∇ is a self-duality on Tr( G ). This generalizes the self-duality on transfer systems for cyclicgroups of squarefree order observed in [BBPR]; see Theorem 4.25. We anticipate that theself-duality of Tr( G ) will prove useful in future enumerative work on transfer systems and N ∞ operads. Organization.
Sections 2 and 3 cover necessary background material on partially orderedsets and transfer systems, respectively.The real work is contained in Section 4, which is organized into five subsections. Subsection4.1 abstracts the notion of a transfer system for an Abelian group to the context ofarbitrary posets. Subsection 4.2 introduces weak factorization systems and proves thatthey are in bijection with transfer systems on a poset (Theorem 4.12). In subsection4.3, we prove Theorem 4.20 on self-duality of the transfer system lattice. Subsection4.4 compares our self-duality (which is defined for every finite Abelian group) to theself-duality of [BBPR] on Tr( G ) for G cyclic of squarefree order. In Subsection 4.5, weillustrate a numerical symmetry of the duality, namely the number of “slats” in a transfersystem on a cyclic group of order p n q for p, q distinct primes.Finally, Section 5 produces a direct bijection between transfer systems for a cyclic groupof order p n , p prime, and noncrossing partitions of { , , . . . , n } . This gives a novel proofof the Catalan enumeration of such transfer systems originally found in [BBR], and wededuce a new corollary linking minimal generation of transfer systems to the Narayananumbers. This section is independent of the rest of the paper. Notation.
We use the following notation throughout. • G — a finite group, eventually Abelian. • Sub( G ) — the subgroup lattice of G . • Tr( G ) — the lattice of transfer systems on G under refinement. • P — a poset, considered either as a set with a relation or as a category in whichthere is at most one morphism between each object. • [ n ] — the poset { < < · · · < n } . • B n — the Boolean poset of subsets of { , , . . . , n } under inclusion. • D n — the divisor poset of n under divisibility. • C n — the cyclic group of order n . • P op — the dual of a poset P . • R — a transfer system, considered either as a relation or a collection of morphisms. (cid:27) M and M (cid:27) — morphisms with the left (resp. right) lifting property with respectto a class of morphisms M . • L (cid:27) R — the property L ⊆ (cid:27) R (equivalently, R ⊆ L (cid:27) ). • ∇ — self-duality on a poset. Acknowledgements.
The authors thank Jonathan Rubin, who suggested this topic, andConstanze Roitzheim, who explained the contents of [BBR, BBPR], and suggested avenuesof exploration. We also thank Hugh Robinson for helpful correspondence and for authoringOEIS entry A092450. This research was supported by NSF grant DMS-1709302.2.
Preliminaries on posets
In this section, we briefly collect some well-known facts and examples from the theoryof partially ordered sets and lattices. We refer the reader to [Sta12, Chapter 3] for acomprehensive reference.Recall that a partially ordered set or poset ( P , ≤ ) consists of a set P , together with abinary relation that is reflexive, antisymmetric and transitive. We say that x < y is a cover relation in P if there is no z ∈ P such that x < z < y .We represent a finite poset P with a Hasse diagram . This is a graph whose vertices arethe elements of P , edges are cover relations, and such that if x < y , then x is drawn below y . Example . Let n be a natural number.(1) We denote by [ n ] the set { , , . . . , n } with its usual order structure.(2) The Boolean poset B n is the poset of subsets of { , , . . . , n } under inclusion. Itis isomorphic to the product of n copies of [1] with itself. The Hasse diagramcorresponds to the edges of an n -dimensional cube.(3) The set of positive divisors of n forms a poset D n under divisibility. If n = p a . . . p a k k is the prime factorization of n , then D n is isomorphic to [ a ] × · · · × [ a k ]by identifying the exponents of the primes. Its Hasse diagram is a k -dimensionalgrid.(4) For a group G , we denote by Sub( G ) the poset of subgroups of G under inclusion.Note that Sub( C n ) is isomorphic to D n , where C n denotes the cyclic group of order n .The Hasse diagrams for these posets are illustrated in Fig. 1.For x and y in a poset P , their least upper bound, if it exists, is denoted by x ∨ y and iscalled the join . Similarly, their greatest lower bound is denoted by x ∧ y and is called the meet . A lattice is a poset for which every pair of elements has both a join and a meet. Aposet is bounded if it has a least and a greatest element.All of the posets of Example 2.1 are bounded lattices. In the case of Sub( G ), the meet isgiven by intersection, while the join is the subgroup generated by the union. Computer calculations of transfer systems that agreed with A092450 were our first hint at the linkbetween transfer and weak factorization systems. ∅
31 213 2312123(1) (2) (3)
Figure 1.
Hasse diagrams for (1) [5], (2) B , and (3) [3] × [2] ∼ = D p q ∼ =Sub( C p q ) for p, q distinct primes. Remark . As noted in [Sta12, Proposition 3.3.1], if a finite poset P has all meets andhas a greatest element, then it is a lattice. Dually, if P has all joins and has a leastelement, then it is a lattice.Given a poset P , we consider it as a category whose objects are the elements of P andwhose morphisms are given by the relation ≤ . In other words, there is a unique morphismfrom x to y whenever x ≤ y , and no morphisms otherwise. Note that extant diagram in P commutes.The dual of P , denoted by P op , is the poset with the same underlying set but with relationreversed. Note that as categories, P op is precisely the opposite category of P .3. Transfer systems and N ∞ operads Transfer systems, as originally and independently defined by J. Rubin [Rub20, Definition3.4] and S. Balchin, D. Barnes, and C. Roitzheim [BBR], are meant to isolate the essentialdata necessary to record all the norms/transfer maps encoded by N ∞ operads. Here werecall the necessary definitions, make the preceding statement precise, and record somebasic facts about transfer systems.Let G be a finite group and for n ≥ S n denote the symmetric group on n letters. Definition 3.1. A G - N ∞ operad is a symmetric operad O on G -spaces satisfying thefollowing three properties: • for all n ≥ O ( n ) the G × S n -space is S n -free, • for every Γ ≤ G × S n , the Γ-fixed point space O ( n ) Γ is empty or contractible, and • for all n ≥ O ( n ) G is nonempty.A map of G - N ∞ operads ϕ : O → O is a morphism of operads in G -spaces, and as such,the map at level n is G × S n -equivariant. The associated category of G - N ∞ operads isdenoted N ∞ - Op G .A map ϕ : O → O of G - N ∞ operads is a weak equivalence if ϕ : O ( n ) Γ → O ( n ) Γ is a weak homotopy equivalence of topological spaces for all n ≥ ≤ G × S n . he associated homotopy category (formed by inverting weak equivalences) is denotedHo( N ∞ - Op G ). Remark . Every G - N ∞ operad O is a naive E ∞ operad, and thus parametrizes anoperation that is associative and commutative up to coherent higher homotopies. Inaddition, O admits norms for the finite H -sets T for which O ( | T | ) Γ( T ) is nonempty, where H is a subgroup of G and Γ( T ) ≤ G × S | T | is the graph of a permutation representationof T . This is the sense in which G - N ∞ operads parametrize admissible norms.We now turn to transfer systems, which we will eventually relate back to N ∞ operads.Recall that Sub( G ) denotes the poset of subgroups of G under inclusion. For g ∈ G and H ∈ Sub( G ), let g H = gHg − ∈ Sub( G ) denote the g -conjugate of H . Definition 3.3.
Let G be a finite group. A G -transfer system is a relation R on Sub( G )that refines the inclusion relation and satisfies the following properties: • (reflexivity) H R H for all H ≤ G , • (transitivity) K R H and L R K implies L R H , • (closed under conjugation) K R H implies that g K R g H for all g ∈ G , • (closed under restriction) K R H and M ≤ H implies ( K ∩ M ) R M .A G -transfer system R can alternatively be described as a partial order on Sub( G ) thatrefines ≤ and is closed under conjugation and under restriction.We represent transfer systems by drawing the corresponding directed graph, ignoring thetrivial edges ( i.e. , self loops). Note that this is not the Hasse diagram for the correspondingposet, as it will include non-covering relations. Remark . If G is a Dedekind group (so all subgroups are normal) conjugation is triviallysatisfied. We will later concentrate on Abelian groups, the most common class of Dedekindgroups. Definition 3.5.
Let Tr( G ) denote the poset of all G -transfer systems ordered underrefinement. Thus, R ≤ R if and only if for all K, H ∈ Sub( G ), if K R H then K R H .Note that if we consider a binary relation on a set S as a subset of S × S , refinement isjust set inclusion.The following construction, based on the work of [BH15, GW18, BP, Rub20, Rub, BBR],links G - N ∞ operads and G -transfer systems. Given O ∈ N ∞ - Op G , define R O by the rule K R O H ⇐⇒ K ≤ H and O ([ H : K ]) Γ( H/K ) = ∅ where Γ( H/K ) is the graph of some permutation representation H → S [ H : K ] of H/K . Theorem 3.6.
The assignment N ∞ - Op G −→ Tr( G ) O O induces an equivalence Ho( N ∞ - Op G ) ≃ Tr( G ) (considering the poset Tr( G ) as a category). e conclude this section by recalling a few more facts about transfer systems. Proposition 3.7.
The poset (Tr( G ) , ≤ ) is a bounded lattice.Proof. The least element in Tr( G ) is given by the equality relation in Sub( G ), while thegreatest element is given by the inclusion relation, showing Tr( G ) is bounded. The inter-section of two transfer systems is a transfer system, thus giving the meet. By Remark 2.2,we get the desired result. (cid:3) The join of two transfer systems can be explicitly described as the transfer system gener-ated by their union [Rub20, Theorem A.2].Previous work has revealed the cardinality and structure of transfer systems on the fol-lowing groups: C p n [BBR], C pq and C pqr [BBPR], C p × C p , Q , S , D p [Rub20]. Weexpand on the case of C p n , for which the collection of transfer systems is enumerated bythe Catalan numbers. Proposition 3.8 ([BBR, Theorems 1 and 2]) . | Tr( C p n ) | = Cat( n + 1) , where Cat( n + 1) is the ( n + 1) th Catalan number. Moreover, the lattice structure on Tr( C p n ) corresponds to the Tamari lattice. Here the Tamari lattice is the poset of binary trees with n + 1 leaves ordered by treerotation, first explored by D. Tamari [Tam62]. It forms the skeleton of K n +2 , the n -dimensional associahedron [Sta63].The original enumeration of Tr( C p n ) in [BBR] proceeds by checking that | Tr( C p n ) | satisfiesthe recurrence formula for the Catalan numbers. In Section 5, we present an alternateproof based on noncrossing partitions.4. A categorical approach to transfer systems
In this section we define transfer systems for arbitrary finite posets, generalizing the def-inition for Dedekind groups. We show that for a finite lattice P , there is a one-to-onecorrespondence between transfer systems on P and weak factorization systems on P . Us-ing this, we prove that Tr( P ) is self-dual whenever P is a finite self-dual lattice. Wecompare this result with the involution defined for G = C p ...p n in [BBPR].4.1. Transfer systems on posets.
We now generalize the definition of a transfer systemto an arbitrary poset and characterize them in categorical terms.
Definition 4.1.
Let P = ( P , ≤ ) be a poset. A transfer system on P consists of a partialorder R on P that refines ≤ and such that for all x, y, z ∈ P , if x R y , z ≤ y , and x ∧ z exists, then ( x ∧ z ) R z .As noted above, for a Dedekind group G , a G -transfer system is the same as a transfersystem on Sub( G ). Proposition 4.2.
Let P be a poset considered as a category. A collection of morphisms R is a transfer system if and only if R is a subcategory that contains all objects and isclosed under pullbacks. roof. Reflexivity translates to containing the identity morphism for all objects, whiletransitivity translates to being closed under composition. Note that for a diagram xz y in P , the pullback, if it exists, is given by x ∧ z . Thus, being closed under restrictiontranslates precisely to being closed under pullbacks. (cid:3) Weak factorization systems.
We now explore a surprising connection betweentransfer systems and weak factorization systems. A standard reference for weak factor-ization systems is [MP12, § Definition 4.3.
Let C be a category and let i : A → B and p : X → Y be morphisms in C . If for every f and g that make the square A XB Y i g pf ∃ λ commute, there exists a lift λ such that the two triangles above commute, we say i hasthe left lifting property with respect to p , or equivalently, p has the right lifting property with respect to i . Definition 4.4.
Let M and N be a classes of morphisms in C . We define (cid:27) M = { i | i has the left lifting property with respect to all p ∈ M} and M (cid:27) = { p | has the right lifting property with respect to all i ∈ M} . Note that
M ⊂ (cid:27) N if and only if N ⊂ M (cid:27) ; we write M (cid:27) N when this holds. Definition 4.5. A weak factorization system in a category C consists of a pair ( L , R ) ofclasses of morphisms in C such that(1) every morphism f in C can be factored as f = pi with i ∈ L and p ∈ R , and(2) L = (cid:27) R and R = L (cid:27) . The collection of weak factorization systems on a category C forms a poset under inclusionof the right set R , or equivalently, under reverse inclusion of the left set L . Remark . Recall that if C is a category, the opposite category C op is constructed byreversing the direction of morphisms in C . Note that ( L , R ) is a weak factorization systemon C if and only if ( R op , L op ) is a weak factorization system on C op .In order to relate weak factorization systems with transfer systems, we will need a helpfulproperty of weak factorization systems as well as a re-characterization of weak factorizationsystems. Proofs of these results can be found, for example, in [MP12, § roposition 4.7. Let ( L , R ) be a weak factorization system on a category C . Then R contains all isomorphisms in C , and is closed under composition, pullbacks, and retracts,and dually for L .Remark . In a poset P , the only isomorphisms are the identity maps, and the onlyretract of a morphism is itself. Thus, Propositions 4.2 and 4.7 imply that if ( L , R ) is aweak factorization system in a lattice P , then R is a transfer system on P . Proposition 4.9.
Let L and R be a pair of classes of morphisms in C . Then ( L , R ) is aweak factorization system on C if and only if (1) every morphism f in C can be factored as f = pi with i ∈ L and p ∈ R , (2) L (cid:27) R , and (3) L and R are closed under retracts. We use this result to construct weak factorization systems from transfer systems on aposet.
Proposition 4.10.
Let P be a finite lattice and let R be a transfer system on P . Thenthere is a unique weak factorization system ( L , R ) on P .Proof. For a weak factorization system we need L = (cid:27) R , so L is uniquely determinedby R . Since P is a poset, the only retract of a morphism x → y is itself. Thus, byProposition 4.9, it suffices to show that every morphism can be factored, since L = (cid:27) R implies L (cid:27) R .Let x → y be a morphism in P . If x → y is in L we are done because we can factorwith the identity. Suppose then that x → y is not in L . This means that there exists acommutative diagram x zy w with z → w ∈ R that does not admit a lift, meaning that y z , and hence y ∧ z < y .Then, since y ≤ w and R is closed under restriction, we get that y ∧ z → y is in R . If x → y ∧ z is in L , we found a factorization of x → y . If not, we can repeat this step, andsince P is finite, this process must terminate eventually. (cid:3) Remark . The proof makes it clear that one may reformulate Proposition 4.10 so thatit applies to infinite lattices if we add the condition that R is closed under transfinitecomposition.Remark 4.8 and Proposition 4.10 combine to give the following result. Theorem 4.12.
Let P be a finite lattice. Then R ←→ ( (cid:27) R , R ) gives an isomorphism between the poset of transfer systems on P and the poset of weakfactorization systems on P . e conclude this section with an explicit description of the collection (cid:27) R for a transfersystem R on P . Definition 4.13.
Let P be a poset. For a transfer system R ∈
Tr( P ), define the downwardextension of R to be E ( R ) = { z → y | there exists x ∈ P such that z ≤ x < y and x → y ∈ R} . Proposition 4.14.
Let R be a transfer system on a finite lattice P . Then (cid:27) R = E ( R ) c , where ( − ) c denotes the complement of the collection of morphisms.Proof. We will begin by showing that E ( R ) c ⊆ (cid:27) R . Let a → b ∈ E ( R ) c . Then we need toshow that given a commutative diagram a xb y with x → y ∈ R , there exists a lift. Since we are working with a category coming froma poset, to satisfy the lifting property it will suffice to show that b ≤ x in P . Note that b ≤ y , so by restriction we have that x ∧ b → b ∈ R . Moreover, a ≤ x ∧ b , so the assumptionthat a → b / ∈ E ( R ) implies that x ∧ b = b , or equivalently, that b ≤ x as desired.For the other inclusion, we will proceed by proving the contrapositive. To that end,suppose that a → b ∈ E ( R ). Then there exists x ∈ P such that x = b , a ≤ x and x → b ∈ R . Consider the commutative diagram a xb b in P . Given that x < b , there exists no lift, and hence a → b is not in (cid:27) R , as wanted. (cid:3) Self-duality.
In this section we prove the main result of this paper, namely, that if P is self-dual, so is its lattice of transfer systems Tr( P ). Definition 4.15.
Let P be a poset. We say P is self-dual if there exists a bijection ∇ : P → P such that x ≤ y if and only if y ∇ ≤ x ∇ . We call ∇ a duality for P , and write x ∇ insteadof ∇ ( x ).Note that we can consider ∇ as an isomorphism of categories P op → P . There is nocondition on ∇ being an involution, although it will be in the examples of interest to us. Moreover, there are examples of self-dual posets for which there is no order-reversing involution, see[Sta12, Chapter 3, Exercise 3]. xample . For a natural number n , the poset [ n ] is self-dual by mapping i to n − i .This duality extends to the product poset [ n ] × · · · × [ n k ]. Example . Recall the Boolean lattice B n of Example 2.1 (2). The function that sends asubset to its complement is an order-reversing involution, and hence a duality for B n . Notethat under the isomorphism B n ∼ = [1] n , this duality matches with the one in Example 4.16. Example . The poset D n of positive divisors of n is self-dual by mapping k to n/k .Recall from Example 2.1 (3) that if n = p a . . . p a k j is the prime decomposition of n , then D n is isomorphic to [ a ] × · · · × [ a k ]. Under this isomorphism, the duality of D n coincideswith the one of Example 4.16. Example . As noted in [Sch94, Theorem 8.1.4], the lattice of subgroups Sub( G ) isself-dual for every finite Abelian group G . The bijection ∇ is constructed using a (non-canonical) isomorphism between G and G ∗ = Hom( G, C × ). In the case that G = C n ,there is an explicit order-reversing involution given by C k C n/k , which coincides withthe one in Example 4.18 via the identification of Sub( C n ) with D n . Theorem 4.20. If ( P , ∇ ) is a self-dual lattice, then Tr( P ) is self-dual, with duality φ : Tr( P ) → Tr( P ) given by φ ( R ) = (( (cid:27) R ) op ) ∇ . Moreover, if ∇ is an involution so is φ .Proof. Given a transfer system R on P , we consider the weak factorization system ( (cid:27) R , R )on P . By Remark 4.6, ( R op , ( (cid:27) R ) op ) is a weak factorization system on P op . The isomor-phism ∇ takes this to a weak factorization system (( R op ) ∇ , (( (cid:27) R ) op ) ∇ ) on P . Thus, byRemark 4.8 the collection (( (cid:27) R ) op ) ∇ is a transfer system on P .If R ′ is another transfer system on P , we have that R ⊆ R ′ if and only if (cid:27) R ′ ⊆ (cid:27) R .This shows that R ⊆ R ′ if and only if φ ( R ′ ) ⊆ φ ( R ), since ( − ) op and ( − ) ∇ preserve andreflect the containment relation, respectively.Let ∆ denote the inverse of ∇ , considered again as an isomorphism P op → P . Standardmanipulations using the fact that ( (cid:27) R ) (cid:27) = R (see Proposition 4.10) and ( (cid:27) M ) op =( M op ) (cid:27) prove that φ is a bijection with inverse φ − ( R ) = (( (cid:27) R ) op ) ∆ . In particular, if ∇ is an involution, so is φ . (cid:3) As a consequence of Proposition 4.14, we obtain an explicit description of the involution φ . Corollary 4.21.
Let ( P , ∇ ) be a self-dual lattice. Then the involution φ satisfies that φ ( R ) = (( E ( R ) op ) ∇ ) c . .4. Cyclic groups of squarefree order.
In [BBPR], Balchin and collaborators definean order-reversing involution on the lattice of transfer systems for the cyclic group oforder p . . . p n , where p , . . . , p n are distinct primes. In this section we prove that theirinvolution coincides with the one defined above. Remark . Let G = C p ...p n . As mentioned in Example 2.1, the lattice Sub( G ) isisomorphic to the Boolean lattice B n , and the involutions of Examples 4.17 and 4.19coincide via this isomorphism.The Hasse diagram of B n consists of the edges of an n -dimensional cube, and thus we canconsider its 2 n facets. Borrowing notation from [BBPR], for all i = 1 , . . . n , we denoteby B i and T i the bottom and top facets, respectively. These correspond to the vertices a ∈ { , } n with a i = 0 for the bottom and a i = 1 for the top. Given a transfer system R on B n , we can restrict it to a facet to obtain a transfer system therein.We now recall the involution Φ n : Tr( B n ) → Tr( B n ) of [BBPR, § Construction 4.23.
For n ≥
1, the involution Φ n on Tr( B n ) is defined inductively asfollows: • If n = 1, Φ exchanges the trivial transfer system with the full transfer system. • Suppose Φ n is defined for some n ≥
1, and let
R ∈
Tr( B n +1 ). Then Φ n +1 ( R )is obtained by applying Φ n to R restricted to each facet, and placing the resultin the opposite facet. Lastly, we add the long diagonal edge ~ → ~ R did notcontain any nontrivial edges with target ~ et al. prove that this function is well defined and is indeed an order-reversinginvolution on Tr( B n ) (see [BBPR, Theorem 4.5, Proposition 4.6]). Example . We will show how to compute Φ ( R ) where R is the following transfersystem on B . We first perform Φ to R restricted to each of the six faces and place the result on theopposite face. For example, to obtain the restriction of Φ ( R ) to T , we perform Φ restricted to B .
00 100010 110000 100010 110 001 101011 111 Φ R| B Φ ( R| B ) Φ ( R ) | T After this is done to each facet, we assemble the results. At the end, we decide whetheror not to include the long diagonal. In this case, since R contains the edge from (0 , , , , ( R ) is shown below. Φ R Φ ( R ) Theorem 4.25.
Consider the Boolean lattice B n with duality ∇ given by swapping and . Then the involution φ of Theorem 4.20 is equal to Φ n .Proof. We proceed by induction, using the explicit description of φ from Corollary 4.21,which uses the downward extension of Definition 4.13. A quick calculation shows that theresult holds when n = 1.Assume the result holds for the n -dimensional Boolean lattice, and let R be a transfersystem on B n +1 . First notice that the long diagonal ~ → ~ φ ( R ) if and only if it is notin E ( R ). By the definition of E ( R ), this happens exactly when R contains no nontrivialedges with target ~
1, as needed.Let F be a facet of B n +1 . By the inductive hypothesis and Corollary 4.21, we have thatΦ n ( R| F ) = φ ( R| F ) = ((( E F ( R| F )) op ) ∇ F ) c F , where E F and ∇ F denote the downward extension and the duality within F , and c F takesthe complement within F as well.Recall from Construction 4.23 that, ignoring the long diagonal, Φ n +1 ( R ) is obtained bytaking Φ n ( R| F ) and placing it in the opposite face, and taking the union over all faces.If F is either B i or T i , the map ∇ F swaps between 0 and 1 in all coordinates except forthe i th one, while the process of taking a subset of F and placing it in the opposite faceis done by swapping 0 and 1 in the i th coordinate and keeping all other coordinates thesame. Thus, the two processes combined amount to applying ∇ . Hence it suffices to provethat(4.26) (cid:0) E ( R ) c (cid:1) | ∂ = [ F ( E F ( R| F )) c F . Restricting to the boundary ∂ = S F on the left-hand side has the effect of ignoring thelong diagonal. Figure 2.
The involution φ acting on a [3] × [1] transfer system. Notethat the involution exchanges the 3-slat transfer system on the left withthe 1-slat transfer system on the right.Since F is an interval in B n +1 (it contains all the vertices between its bottom and topvertices), we have that E F ( R| F ) = E ( R ) | F . Thus, ( E F ( R| F )) c F = ( E ( R ) | F ) c F = ( E ( R ) c ) | F , from which (4.26) follows. (cid:3) Slats — numerical symmetry of the duality.
We expect the existence of theinvolution to aid in proving enumeration results, as was done in [BBPR] to count thenumber of transfer systems for [1] × [1] × [1]. In fact, we first suspected the existence ofan involution when attempting to enumerate the transfer systems for [ n ] × [1] and noticedthere was a symmetry in the results when we restricted to counting transfer systems witha given number of “slats”, as we now explain. Throughout this section we fix n ≥ Definition 4.27.
Let 0 ≤ k ≤ n . The k th slat in the poset [ n ] × [1] is ( k, ≤ ( k, R is a transfer system on [ n ] × [1], by the restriction property, if the k th slat is in R , sois the i th slat for all 0 ≤ i < k . For 0 ≤ k ≤ n , we let S k denote the set of all transfersystems for which the k th slat is the top slat. We let S − be the set of transfer systemswith no slats.Note that we can alternatively describe S k as the collection of transfer systems with k + 1slats.Recall the involution ∇ on [ n ] × [1] of Example 4.16. It sends ( i, j ) to ( n − i, − j ). Proposition 4.28.
The involution φ of Theorem 4.20 exchanges the sets S k and S n − k − .Proof. We use the explicit description of φ from Corollary 4.21. Since slats are coveringrelations in [ n ] × [1], a given slat is in R if and only if it is in E ( R ). Note furthermorethat ∇ sends the k th slat to the ( n − k )th slat. It follows that R contains the k th slat ifand only if φ ( R ) does not contain the ( n − k )th slat, thus proving the result. (cid:3) . Transfer systems and noncrossing partitions
While it is already known that C p n -transfer systems can be counted with Catalan numbers,we re-prove this result by presenting a natural bijection between noncrossing partitions ofthe set { , . . . , n } and C p n -transfer systems. Additionally, we will use this bijection andthe structure of noncrossing partitions in order to relate Narayana numbers and transfersystems. We first recall the definition of a noncrossing partition.Recall that Sub( C p n ) is isomorphic to the poset [ n ] = { < < · · · < n } of Example 2.1(1).Thus for ease of notation we work with transfer systems on [ n ]. Definition 5.1.
A partition of the set { , . . . , n } is noncrossing if for all 0 ≤ a < b Let R ∈ Tr([ n ]). We say i → j ∈ R is a maximal edge in R if i = j andthere exists no k ≥ j such that i → k ∈ R . We denote the set of all maximal edges in R by M ( R ). Figure 3. Example of maximal edges in a transfer system in the lattice[6]. The maximal edges are marked in red.Note that there is at most one maximal edge starting at a given vertex. We now constructthe bijection between Tr([ n ]) and NC n +1 . Definition 5.3. Let ( P , ≤ ) be a poset, and let F be a binary relation on P that refines ≤ . We denote by hF i the minimal transfer system that contains F as given in [Rub20,Construction A.1]. We say F is a generating set for hF i .For a transfer system R on P , we define the minimal generating number of R to be theminimal cardinality of a generating set for R . Any generating set with minimal cardinalityis called a minimally generating set .The following lemma tells us that the maximal edges of R form a minimally generatingset for R . emma 5.4. Let R ∈ Tr([ n ]) . Then R = h M ( R ) i , and moreover, M ( R ) is a minimallygenerating set for R .Proof. The equality follows from the fact that any nontrivial edge in R is the restrictionof an edge in M ( R ).Consider the set S of vertices in [ n ] that are the sources of nontrivial edges in R . If i ∈ S then i → ( i + 1) ∈ R by restriction. Since edges of this form represent covering relations,we know these edges cannot be obtained by closing under transitivity. Thus the onlyedges that could generate the edge i → ( i + 1) are precisely edges of the form i → j where j ≥ i + 1. From this it is then clear that in order to generate R , a generating set musthave a size of at least | S | . Since | M ( R ) | = | S | , the claim follows. (cid:3) Definition 5.5. Let π be a noncrossing partition of [ n ]. Let J ( π ) be the set of nontrivialedges in [ n ] connecting each element of a class with the largest element in that class.0 1 2 3 4 5 0 1 2 3 4 5 π J ( π ) Figure 4. An example of a noncrossing partition π and edge set J ( π ) Lemma 5.6. Let π be a noncrossing partition of [ n ] . Then the transfer system h J ( π ) i consists of the closure under restriction of J ( π ) .Proof. As noted in [Rub20, Construction A.1], h J ( π ) i is constructed by first closing underrestriction and then under transitivity. Thus, it is enough to prove that the closure underrestriction of J ( π ) is already closed under transitivity. Suppose i → j and j → k areobtained by restriction from J ( π ). This means that there exist j ′ ≥ j and k ′ ≥ k suchthat i and j ′ are in the same class and j ′ is the largest element therein, and similarly, j and k ′ are in the same class and k ′ is the largest element therein. Since π is noncrossing,we must have that k ′ ≤ j ′ . Then i → k is the restriction of i → k ′ , which is in J ( π ). (cid:3) Theorem 5.7. Let R ∈ Tr([ n ]) . Let ψ ( R ) be the partition of [ n ] associated to the equiv-alence relation generated by M ( R ) . Then (1) ψ ( R ) is a noncrossing partition, and (2) the map ψ : Tr([ n ]) → NC n +1 is a bijection with inverse χ ( π ) = h J ( π ) i .Proof. We denote by ∼ R the equivalence relation generated by M ( R ). Using the transi-tivity of a transfer system, one can check that for i < j ∈ [ n ], i ∼ R j if and only if either i → j ∈ M ( R ) or there exists k > j such that i → k ∈ M ( R ) and j → k ∈ M ( R ). Notethat in either case, there exists k ≥ j such that i → k ∈ M ( R ) and j → k ∈ R .To prove that ψ ( R ) is a noncrossing partition, let a < b < c < d in [ n ] such that a ∼ R c and b ∼ R d . Thus, there exist e ≥ c and f ≥ d such that a → e and b → f are in M ( R ),and c → e and d → f are in R . By restriction, we get that a → b and b → c are in R , and hus a → f and b → e are also in R by transitivity. Since a → e and b → f are maximal,that implies that e = f , and hence a , b , c , and d are all in the same equivalence class, asrequired.We now prove that ψ is a bijection. Note that Lemma 5.6 implies that for a noncrossingpartition π , J ( π ) = M ( χ ( π )) . Since the partition generated by J ( π ) is precisely π , it follows that ψ ( χ ( π )) = π .Similarly, the description above of the equivalence relation generated by M ( R ) impliesthat M ( R ) = J ( ψ ( R )) . That combined with Lemma 5.4 proves that χ ( ψ ( R )) = R , thus showing that ψ and χ are indeed inverses of each other. (cid:3) With ψ established as a bijection, we can now use ψ to directly enumerate transfer systemsminimally generated by edge sets of a certain size. Note that refinement of partitions givesan order relation on the collection of noncrossing partitions. Proposition 5.8 ([Kre72], see also [Sim00]) . The poset of NC n under refinement formsa lattice called the Kreweras lattice . It is graded by the function rank( π ) = n − bk( π ) , where bk( π ) is the number of blocks of π .Remark . Beware that the Tamari lattice is a strict extension of the Kreweras lattice. Proposition 5.10 ([Kre72], see also [Sim00]) . Let NC n ( k ) denote the number of partitionsin NC n with rank k . Then NC n ( k ) is given by the Narayana number NC n ( k ) = 1 n (cid:18) nk (cid:19)(cid:18) nk − (cid:19) . With this rank property established in the context of noncrossing partitions, we can thenuse χ to see what this rank looks like for transfer systems. Proposition 5.11. Let π ∈ NC n +1 . Then the rank of π in the Kreweras lattice is equalto the minimal generating number for χ ( π ) .Proof. By Lemma 5.4, the minimal generating number for χ ( π ) is equal to the cardinalityof M ( χ ( π )). As noted in the proof of Theorem 5.7, M ( χ ( π )) = J ( π ). Note that J ( π )consists precisely of the elements of [ n ] that are not maximal within their class in π , thus,the cardinality of J ( π ) is n + 1 − bk( π ), as desired. (cid:3) Corollary 5.12. Let Tr k ([ n ]) be the set of transfer system on [ n ] minimally generated by k edges. Then | Tr k ([ n ]) | = NC n +1 ( k ) = 1 n + 1 (cid:18) n + 1 k (cid:19)(cid:18) n + 1 k − (cid:19) . eferences [BBPR] Scott Balchin, Daniel Bearup, Clelia Pech, and Constanze Roitzheim, Equivariant homotopycommutativity for G = C pqr , arXiv e-prints (2020), arXiv:2001.05815.[BBR] Scott Balchin, David Barnes, and Constanze Roitzheim, N ∞ -operads and associahedra , arXive-prints (2019), arXiv:1905.03797v2.[Bec48] H. W. Becker, Rooks and Rhymes , Math. Mag. (1948), no. 1, 23–26.[BH15] Andrew J. Blumberg and Michael A. Hill, Operadic multiplications in equivariant spectra, norms,and transfers , Adv. Math. (2015), 658–708.[BP] Peter Bonventre and Luis A. 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Department of Mathematics, Reed College, Portland, OR 97202, USA Email address : [email protected] Department of Mathematics, Reed College, Portland, OR 97202, USA Email address : [email protected] Department of Mathematics, Reed College, Portland, OR 97202, USA Email address : [email protected] Department of Mathematics, Reed College, Portland, OR 97202, USA Email address : [email protected] Department of Mathematics, Reed College, Portland, OR 97202, USA Email address : [email protected]@reed.edu