Hall monoidal categories and categorical modules Tashi Walde Abstract We construct so called Hall monoidal categories (and Hall modules there- over) and exhibit them as a categorification of classical Hall and Hecke algebras (and certain modules thereover). The input of the (functorial!) construction are simplicial groupoids satisfying the 2-Segal conditions (as introduced by Dy- ckerhoff and Kapranov [DKa]), the main examples come from Waldhausen’s S-construction. To treat the case of modules, we introduce a relative version of the 2-Segal conditions. Furthermore, we generalize a classical result about the representation the- ory of symmetric groups to the case of wreath product groups: We construct a monoidal equivalence between the category of complex G ≀ Sn-representations (for a fixed finite group G and varying n ∈ N) and the category of “G- equivariant” polynomial functors; we use this equivalence to prove a version of Schur-Weyl duality for wreath products. This paper is, up to minor modifications, the author’s Master’s thesis as submitted to the University of Bonn on July 22, 2016. arXiv:1611.08241v2 [math.CT] 16 Feb 2017 i Contents 1 Introduction 1 1.1 Auniversalperspective. .. .. .. 1 1.2 Hallmonoidalcategories . 2 1.3 Can we also construct “Hall modules”? . 4 1.4 A Schur-Weyl duality for wreath products via “G-equivariant” poly- nomialfunctors .............................. 5 1.5 Acknowledgements ............................ 7 1.6 Notationsandconventions . 7 2 The classical Hall algebra 10 2.1 Thesimplicialgroupoidofflags . 11 2.2 Functions on isomorphism classes . 12 2.3 The classical Hall algebra, revised . ... 13 2.4 Example: vector spaces and free group representations over F1 .... 14 3 Preliminaries on (hypo-)simplicial objects 17 3.1 NotationandConventions . 17 3.2 Hypo-simplicialobjects. 17 3.2.1 Relativesimplicialobjects . 18 3.3 The 1-Segalcondition........................... 20 3.3.1 Weak monoid objects and tensor products . 21 3.3.2 TheMacLanepentagon . 23 3.3.3 Modulesovermonoidobjects . 24 3.3.4 Monoidal categories and monoidal modules . 25 3.4 Laxandoplaxmorphisms . 25 3.4.1 (Lax)monoidhomomorphisms. 26 3.5 The 2-Segalcondition........................... 27 3.5.1 The 2-Segal condition for simplicial groupoids . 28 3.5.2 The relative 2-Segalcondition . 31 4 The generalized Hall construction 34 4.1 Monoidalleftderivatorsofgroupoids . ... 34 4.1.1 Adhocdefinitions. .. .. .. 34 4.1.2 Multi-valued tensor products . 37 4.1.3 Naturality of multi-valued tensor products . ... 38 4.1.4 Componentwise compatibility with homotopy colimits ..... 39 4.2 The generalized Hall construction . .. 42 4.2.1 Constructthedataonobjects . 42 4.2.2 Checktheproperties . 43 ii iii Contents 4.2.3 Functoriality ........................... 45 5 Hall monoidal categories and categorical modules 48 5.1 Examples ................................. 49 5.2 The finitary case and decategorification . ... 50 5.2.1 The fat Hall algebra and the dimension map . 52 5.2.2 Sectionsofthedimensionmap. 55 5.3 Idempotent decomposition of the Hall monoidal category ....... 57 5.4 Functoriality................................ 60 5.5 Hall (categorical) modules via bounded flags . .... 61 5.5.1 Quotientdata........................... 61 5.5.2 The quotient datum of a group of automorphisms . 62 5.5.3 The 2-Segal relative simplicial groupoid of bounded flags . 63 5.6 Duality pairing on Hall (categorical) modules . ..... 66 5.6.1 Bi-relative simplicial groupoids . 67 5.6.2 Theabstractcategoricalpairing . 68 5.6.3 Morphisms induce bi-relative simplicial groupoids . ...... 69 5.6.4 Thecategoricalpairing . 70 5.6.5 The finitary case and decategorification . 71 5.7 Example: vector spaces and free group representations over F1 .... 72 5.7.1 TheHallmonoidalcategory . 72 5.7.2 ParabolicHallmodules. 73 5.8 Convolutionalgebras . .. .. .. 74 5.8.1 Thefibersimplicialgroupoid. 75 5.8.2 Thecaseofgroupactions . 76 5.8.3 The classical Hecke algebra, revisited . .. 77 5.8.4 The Hall-Hecke monoidal category . 77 5.8.5 Convolutionmodules . 78 6 Equivariant polynomial functors 80 6.0.1 Motivation............................. 80 6.0.2 Overview ............................. 81 6.1 Polynomial functors and linearization . .... 83 6.1.1 Decomposition into homogeneous pieces . 84 6.1.2 Lifting the multilinear component to the wreath product ... 86 6.1.3 Linearization gives an equivalence of categories . ..... 87 6.2 Multi-representablefunctors . .. 90 6.2.1 Decomposing multi-representable functors . ... 90 6.2.2 Extending to the semidirect product . 92 6.3 Inrepresentationtheory... 92 iv 6.3.1 ...offinitealgebras... 92 6.3.2 ...over a field of characteristic zero . .. 94 6.4 The Hall monoidal category of F1[G]−free ............... 95 6.5 Thecharacteristicmap . 95 6.5.1 Symmetric polynomial class functions . 96 6.5.2 An intrinsic description of the characteristic map . ..... 96 7 Schur-Weyl duality for wreath products 99 7.1 Overview.................................. 99 7.2 Matrixcoefficients ............................100 7.3 Polynomial representations of matrix groups . .100 7.4 The n-standardrepresentation . .101 7.5 The ring of polynomial functions on Ad(G) ...............102 7.6 Decomposing the n-standardrepresentation . 103 7.7 Schur-Weyl duality for wreath products . .104 Appendix A Abstract Nonsense 107 A.1 The Grothendieck construction . 107 A.2 Semidirect products of categories . 109 A.2.1 Universalproperty . .110 A.2.2 Functoriality . .. .. ..111 A.2.3 Enriched semidirect product . 111 A.2.4 Wreathproducts . .113 A.3 Calculusofcanonicalmates . .113 A.4 Someenrichedcategorytheory. 115 References 117 1 Introduction 1. Introduction The theory of Hall algebras began more than a century ago when Steinitz ob- served [Ste01] that (for each fixed prime p) there is an associative (!) product A · B := t(C, A, B) · C C X on (formal sums of) isomorphism classes {A, B, C, . } of finite abelian (p-)groups, where t(C, A, B) counts the number of subgroups A′ ⊆ C such that A′ ∼= A and C / A′ ∼= B. This product yields an associative algebra with a basis consisting of λi partitions by identifying each finite abelian p-group λ Z / (p ) with its type λ. The topic remained forgotten for more than fifty yearsL until Hall rediscovered what is nowadays known as Hall’s algebra of partitions [Hal59]. This most classical of Hall algebras was and is of great interest due to its close relationship to several fundamental objects in mathematics such as symmetric functions [Mac95] and flag varieties. Hall algebras came back into the spotlight in the early 90s due to Ringel’s ground- breaking discovery [Rin90] that the positive part of the quantized enveloping algebra Uv(g) (of a simple complex Lie algebra g) can be realized by applying Hall’s con- −→ struction to the category Q−repFq of finite dimensional representations over Fq of the corresponding Dynkin quiver. Many variants of Hall algebras have since been introduced and studied; see Schiff- mann’s lecture notes [Sch09b] [Sch09a] for a detailed overview. 1.1. A universal perspective Dyckerhoff and Kapranov [DKa] propose the following perspective on Hall algebras: The various collections A {X[n] := {0= A0 ֒→ A1 ֒→ . ֒→ An of flags of length n ∈ N in a certain suitable category1)A can be naturally organized into a simplicial object2) X A : ∆op −→ C which is known in algebraic K-theory as Waldhausen’s S-construction [Wal85]. We are supposed to view X A as the “universal” Hall algebra; all other Hall algebras 1)In the classical case of Steinitz and Hall, for instance, A would be the category of finite abelian p-groups. For us A will always be a proto-abelian category [Dyc15, Definition 1.2]. 2)For us X A will always be a simplicial groupoid, i.e. C := Grpd. Depending on the precise nature of the category A other types of simplicial objects (e.g. simplicial spaces or simplicial stacks) can be used [Dyc15]. 2 are created out of X A by applying appropriate specializations C ? (called transfer theories [DKa, Section 8.1]). Example 1.1.1 (see Section 2). The classical Ringel-Hall algebra hall(A) (of a proto- abelian category A) can be recovered [Dyc15, Proposition 2.19] by taking C := Grpd to be the category of groupoids and by passing to (finitely supported) functions on isomorphism classes. ♦ Associativity and unitality of the various Hall algebras are encoded universally in the simplicial object X A as the so called 2-Segal conditions [DKa, Section 2.3]. The 2-Segal conditions can be formulated for any simplicial object; it is therefore tempting to consider other simplicial objects X : ∆op → C which do not necessarily arise through Waldhausen’s S-construction. Example 1.1.2 (see Section 5.8). The classical Hecke algebra H(G,H) associated to an inclusion H ⊆ G of finite groups can be obtained [DKa, Example 8.2.11] by taking the 2-Segal simplicial groupoid (called the Hecke-Waldhausen construction) (2) (2) G,H X :[n] 7−→ B H ×B G ··· ×B G B H (with n +1 many factors) and passing to (finitely supported) functions on isomor- phism classes. ♦ The Ringel-Hall algebra and the Hecke algebra can thus be put into the same context: both algebras can be obtained by applying the transfer theory V : Grpd → Z−Mod, V : A 7→ Z(π0A) (of finitely supported functions on isomorphism classes) to an appropriate 2-Segal simplicial groupoid X ; in both cases we simply speak of the Hall algebra hall(X ) of the corresponding simplicial groupoid X . 1.2. Hall monoidal categories When reading about functions on isomorphism classes of a groupoid the following objection should immediately jump to mind: Objection 1.2.1. [Dyc15] If we are to consider “maps” out of a groupoid A then surely we should not be using functions π0A → Z on isomorphism classes but rather functors A →V for some suitably chosen category V. ♠ This idea leads to a categorification of the Hall algebra. 3 Introduction Theorem 1.2.2 (see Section 4.2).