Generators for the representation rings of certain wreath products
aa r X i v : . [ m a t h . R T ] F e b Generators for the representation rings ofcertain wreath products
Nate HarmanJuly 27, 2018
Abstract
Working in the setting of Deligne categories, we generalize a resultof Marin that hooks generate the representation ring of symmetricgroups to wreath products of symmetric groups with a fixed finitegroup or Hopf algebra. In particular, when we take the finite groupto be cyclic order 2 we recover a conjecture of Marin about Coxetergroups in type B.
In [3], Deligne defined the categories
Rep ( S t ) for t an arbitrary complexnumber. In the context of the Church-Farb framework of representationstability [1] we may think of these Deligne categories at generic values of t as models for stable categories of representations of the symmetric group. Inparticular they satisfy the following “stable” properties: • For generic t , Rep ( S t ) is semisimple with irreducible objects ˜ V ( λ ) in-dexed by partitions. These interpolate the irreducible representations V ( λ ( n )) of S n with n >>
0, where we add a sufficiently long first rowto λ to make it the right size. In particular these representations areknown to have polynomial growth of dimension Dim ( V ( λ ( n ))) = p λ ( n ),and in the Deligne category we have Dim ( ˜ V ( λ )) = p λ ( t ). • If k ∈ Z + we have induction functors Rep ( S k ) ⊠ Rep ( S t ) → Rep ( S t + k ),where Rep ( S k ) denotes the usual category of complex representations1f S k . The multiplicity ˜ c νλ,µ of ˜ V ( ν ) in Ind( V ( λ ) ⊠ ˜ V ( µ )) is equal tothe stable limit of Littlewood-Richardson coefficients c ν ( n + k ) λ,µ ( n ) . Similarstatements hold for restriction, with an appropriate version of Frobe-nius reciprocity. • The structure constants for the tensor product are the so called reducedKronecker coefficients which are the stable limits of the Kronecker co-efficients.In [8] Marin proves that hooks, i.e. partitions of the form ( n − k, k )generate the representation ring of Rep( S n ). While he doesn’t use the lan-guage, his proof mostly takes place in the stable setting and his argumentshows that hooks freely generate the stable representation ring. This thenimplies they must generate in the classical setting (although not freely). Sowe may think of this result as an application of stable representation theoryto classical representation theory.In the Deligne category setting Marin’s result appears in Deligne’s originalpaper [3], saying that the Grothendieck ring of the Deligne category is freelygenerated by objects corresponding to hooks. The result for the classical casefollows by projecting from the Deligne category onto Rep( S n ), and lookingat the induced map of Grothendieck rings.The proof is done by defining a filtration on the Deligne category suchthat the associated graded Grothendieck ring is isomorphic in a natural wayto the ring ( L n K (Rep( S n )) , · ) with multiplication coming from inducingrepresentations from S n × S m to S n + m . This ring is well known to be iso-morphic to the ring of symmetric functions, and the elementary symmetricfunctions correspond to hooks.Deligne categories for wreath products with a finite group or Hopf alge-bra were defined by Knop [7]. In [9] Mori defined wreath product Delignecategories associated to an arbitrary k -linear category C , which is a tensorcategory whenever C is. As for the symmetric group these may be thoughtof as stable versions of the more classical wreath product categories in waysanalogous to those listed above. See [5] for a more detailed overview of repre-sentation theory in complex rank, including discussions of the constructionsmentioned here.Motivated by a conjecture of Marin about generalizing his result to Cox-eter groups in type B (which are wreath products), the goal of this paper isto prove similar results about the Grothendieck rings of these Deligne cate-2ories. By projecting from the Deligne categories to classical representationcategories, we obtain systems of generators for the representation rings ofwreath products with finite groups, answering the conjecture of Marin in thecase when the finite group is cyclic of order 2. Acknowledgements
The author is grateful to P. Etingof for suggesting the question, and to V.Ostrik for his many helpful comments on the preliminary version of thispaper. This material is based upon work supported by the National ScienceFoundation Graduate Research Fellowship. W n ( C ) and S t ( C ) First we will review Mori’s construction of the wreath product Deligne cat-egories S t ( C ), and the aspects of the theory useful for our purposes. Thissection is largely expository, and all the proofs will be left to the references.Fix k an algebraically closed field of characteristic zero, and let C be a k -linear semisimple tensor category with a unit object satisfying End C ( ) = k . Definition 2.1 ( Mori [9] Definition 3.13 ) . Consider the action of thesymmetric group S n on the category C ⊠ n by permuting the factors. Wedenote the equivariantization of this action by W n ( C ) and refer to it as awreath product category.In the case that C is the category of representations of a finite group G ,the category W n ( C ) equivalent to the representation category of the wreathproduct S n ( G ) (also denoted by G ≀ S n or G n ⋊ S n ) of G with a symmetricgroup. Many of the standard facts about the representation theory of thesegroups hold in the categorical setting as well, we will highlight some of theseproperties that are important for our purposes. Proposition 2.2 ( Mori [9] sections 3.5 and 5.3 ) . Since we will be mostly working at the level of the Grothendieck group, the semisimplecondition can be relaxed to an artinian condition and the main result will still hold butwe will assume it for simplicity. . If I ( C ) is an indexing set for equivalence classes of irreducible objectsin C , then the irreducible objects of W n ( C ) are indexed by the set: P C n = { λ : I ( C ) → P such that | λ | := X U ∈ I ( C ) | λ ( U ) | = n } where P denotes the set of partitions. We denote the irreducible objectcorresponding to λ ∈ P C n by V ( λ ) .2. The natural inclusions S n × S m ֒ → S n + m give rise to induction functors Ind W n + m ( C ) W n ( C ) ⊠ W m ( C ) : W n ( C ) ⊠ W m ( C ) → W n + m ( C ) admitting two sided adjoints Res W n + m ( C ) W n ( C ) ⊠ W m ( C ) referred to as restrictionfunctors.3. If we take the Grothendieck ring of W ∗ ( C ) := L n W n ( C ) with a tensorstructure corresponding to induction, then the map [ V ( λ )] → O U ∈ I ( C ) s λ ( U ) is a graded isomorphism with the ring N U ∈ I ( C ) Λ where Λ is the ringof symmetric functions, and s λ is a Schur function. We now want to construct categories S t ( C ) which interpolate W n ( C ) tonon-integer values of t analogously to the construction of the Deligne category Rep ( S t ) as explained by Comes and Ostrik in [2]. To do this we will definecertain well behaved standard objects in W n ( C ) for different values of n .Let I = { i , i , . . . , i k } be a finite set and U I = ( U i ) i ∈ I a collection ofobjects in C . If n ≥ k we can consider U i ⊠ U i ⊠ ... ⊠ U i k ⊠ ⊠ n − k C as anobject of C ⊠ k ⊠ W n − k ( C ). We will let [ U I ] n ∈ W n ( C ) denote the image of thisunder the appropriate induction functor, when n < k it will be convenientto define [ U I ] n to be the zero object. We have the following lemma: Lemma 2.3 ( Mori [9] definition 4.5 and lemma 4.9 ) .
1. For finite sets I and J and collections of objects U I , V J as abovethere exists an explicitly described vector space H ( U I , V J ) such that Hom W n ( C ) ([ U I ] n , [ V J ] n ) ∼ = H ( U I , V J ) for all n sufficiently large. . There exists a map Φ : H ( V I , W K ) ⊗ H ( U I , V J ) → H ( U I , W K ) ⊗ k [ x ] such that for n sufficiently large the composition of Φ with the evalua-tion at n map ev n : H ( U I , W K ) ⊗ k [ x ] → H ( U I , W K ) corresponds to thecomposition map Hom W n ( C ) ([ V J ] n , [ W K ] n ) ⊗ Hom W n ( C ) ([ U I ] n , [ V J ] n ) → Hom W n ( C ) ([ U I ] n , [ W k ] n ) . Using this lemma we are able to define an auxiliary category S t ( C ) foran arbitrary number t ∈ k as follows: Objects are symbols h U I i t where U I is a finite collection of objects of C . Morphisms from h U I i t to h V J i t are given by the space H ( U I , V J ) from thefirst part of the previous lemma. Composition is given by ev t ◦ Φ where Φ is the map from the second partof the previous lemma, and ev t is the evaluation at t map. Definition 2.4 ( Mori [9] Definitions 4.10 and 4.16 ) . The wreath prod-uct Deligne category S t ( C ) is defined as the pseudo-abelian or Karoubianenvelope of S t ( C ) , it comes equipped a natural tensor structure coming fromthe tensor structure of C .Now we want to explicitly describe these Deligne categories S t ( C ) andhow they interpolate the wreath product categories W n ( C ). First define theset P C by: P C = ∪ n P C n = { λ : I ( C ) → P| λ ( U ) = ∅ for all but finitely many U } and for λ ∈ P C and n sufficiently large define λ n ∈ P C n by: λ n ( U ) = (cid:26) λ ( U ) : U = C ( n − | λ | , λ ( U )) : U = C (2.1)That is λ n is obtained from λ by adding a long first row to the partitioncorresponding to the unit object of C , and leaving the rest the same. If n isnot sufficiently large, i.e. if ( n − | λ | , λ ( )) is not a valid partition, then it willbe convenient to define λ n = 0. We have the following description of S t ( C ): Theorem 2.5 ( Mori [9] Theorems 4.13 and 5.6 ) .
1. There exists a bijection between P C and indecomposable objects of S t ( C ) .We denote these corresponding indecomposable objects by ˜ V ( λ ) .2. When t / ∈ Z ≥ the category S t ( C ) is semisimple. . For n ∈ Z ≥ there exists a full, essentially surjective tensor functor S n ( C ) → W n ( C ) which we will refer to as a projection functor.4. For all λ ∈ P C there exists N such that the image of ˜ V ( λ ) under theprojection from S n ( C ) to W n ( C ) is isomorphic to V ( λ n ) for all n ≥ N . In order to pass results from S t ( C ) at generic values of t to the moreclassical wreath product categories W n ( C ) we will need a more detailed de-scription of what happens at integer values of t . For our purposes howeverit will be convenient to do this after we have defined a filtration in the nextsection. Remark . These sequences of irreducible representations V ( λ n ) of S n ( G )are exactly those that show up in finitely generated F I G modules, as definedby Gan and Li in [6] and developed further by Sam and Snowden in [10].This is consistent with the philosophy that these Deligne categories can bethought of as models for a stable representation category. | λ | filtration An immediate consequence of Mori’s construction in terms of the objects[ U I ] is the existence of induction functors W n ( C ) ⊠ S t ( C ) → S t + n ( C ) whichinterpolate the induction functors W n ( C ) ⊠ W m ( C ) → W n + m ( C ) where we fix n and let m grow to infinity. This defines a categorical action of the tensorcategory W ∗ ( C ) on S ∗ ( C ) := L t ∈ k S t ( C ).Similarly, we get restriction functors the other direction which are twosided adjoints to the induction functors and interpolate the correspondingrestriction functors between the genuine wreath product categories.Mori’s construction ensures that objects of S t ( C ) occur as summands inobjects of the form: Ind S t ( C ) W k ( C ) ⊠ S t − k ( C ) ( W ⊠ S t − k ( C ) ) (2.2)for some natural number k and W ∈ W k ( C ). In particular we could describe˜ V ( λ ) as the unique indecomposable summand of: M ( λ ) := Ind S t ( C ) W n ( C ) ⊠ S t − n ( C ) ( V ( λ ) ⊠ S t − n ( C ) )not occurring as a summand in an object of the form (2.2) for any k < n .This suggests we consider the following filtrations:6 efinition 2.7 (The | λ | -filtration) . Define a filtration on S t ( C ) by puttingan object M in the k th level of the filtration if M occurs as a summand inan object of the form of (2.2). Analogously define such filtrations on thecategories W n ( C ).Immediately we see that ˜ V ( λ ) is minimally contained in the | λ | th levelof the filtration on S t ( C ). Similarly V ( λ ) is minimally contained in the ( n − µ ( ) )th level of the filtration on W n ( C ).By the nature of its definition, this filtration is well behaved with re-spect to the induction functors. If we let c γλ,µ denote the structure constantsfor ( L n K ( W n ( C )) , · ), which are just products of the usual Littlewood-Richardson coefficients by Proposition 2.2 part 3. Then by looking at theLittlewood-Richardson rule with one partition having a long first row, we getthe following relation:Ind S t ( C ) W n ( C ) ⊠ S t − n ( C ) ( V ( λ ) ⊠ ˜ V ( µ )) = ( M c γλ,µ ˜ V ( γ )) ⊕{ terms lower in the filtration } (2.3)Frobenius reciprocity and similar analysis of the Littlewood-Richardsonrule gives us the “lead term” for restrictions of indecomposables:Res S t ( C ) W n ( C ) ⊠ S t − n ( C ) ( ˜ V ( µ )) = ( ⊠ ˜ V ( µ )) ⊕ { terms M ⊠ ˜ V ( ν ) with | ν | < | µ |} (2.4)In the case of C = V ect k this S t ( C ) is equivalent to Rep ( S t ), and ourfiltration agrees with the filtration defined by Deligne in [3] and by Marin in[8]. In that case the filtration was also well behaved with respect to the in-ternal tensor structure, and we had that the associated graded Grothendieckring was isomorphic to the ring of symmetric functions. In our setting wehave the following generalization: Lemma 2.8.
The filtration defined above gives the Grothendieck ring K ( S t ( C )) the structure of a filtered ring. Moreover, the associated graded ring is thenisomorphic to ( L n K ( W n ( C )) , · ) , where the isomorphism sends the image of [ ˜ V ( λ )] to [ V ( λ )] .Proof. Let M ( λ ) = Ind S t ( C ) W n ( C ) ⊠ S t − n ( C ) ( V ( λ ) ⊠ S t − n ( C ) ). At the level of theGrothendieck ring we have that [ M ( λ )] = [ ˜ V ( λ )]+ { terms lower in the filtration } .So inductively we can conclude that [ ˜ V ( µ ) ⊗ ˜ V ( λ )] and [ ˜ V ( µ ) ⊗ M ( λ )] have7he same highest order terms with respect to this filtration. To simplify thenotation we will now begin omitting the subscripts and superscripts from theinduction and restriction functors, they all will go between W n ( C ) ⊠ S t − n ( C )and S t ( C ); ˜ V ( µ ) ⊗ M ( λ ) = ˜ V ( µ ) ⊗ Ind( V ( λ ) ⊠ )= Ind(Res( ˜ V ( µ )) ⊗ ( V ( λ ) ⊠ ))By (2.4) this becomes:˜ V ( µ ) ⊗ M ( λ ) = Ind((( ⊠ ˜ V ( µ ) ⊕{ terms M ⊠ ˜ V ( ν ) with | ν | < | µ |} ) ⊗ ( V ( λ ) ⊠ ))= Ind(( V ( λ ) ⊠ ˜ V ( µ )) ⊕ { terms M ⊠ ˜ V ( ν ) with | ν | < | µ |} )Which by (2.3) becomes:˜ V ( µ ) ⊗ M ( λ ) = M c γλ,µ ˜ V ( γ ) ⊕ { summands lower in the filtration } Since the coefficients c γλ,µ were the structure constants for the inductionproduct, we see that indeed the associated graded Grothendieck ring of S t ( C )is isomorphic to the induction ring ( L n K ( W n ( C )) , · ).This combined with proposition 2.2 part 3 immediately gives us the fol-lowing useful corollary: Corollary 2.9.
The associated graded Grothendieck ring of S t ( C ) at generic t with respect to the | λ | -filtration is isomorphic to N U ∈ I ( C ) Λ . The map sendsirreducible objects to products of Schur functions. t Our goal is to obtain a collection of generators for the representation ringsof wreath products S n ( G ), or more generally the Grothendieck rings of the8ategorical wreath products W n ( C ). To do this we want to take a nice systemof generators for these relatively well behaved Deligne categories and passthem down to these more classical categories.In general some care needs to be taken in the Deligne category settingwhen taking t to be a positive integer. What happens in this case is handledin depth in the case of symmetric groups by Comes and Ostrik in [2], andtheir results were extended to the wreath product setting by Mori [9]. Wewill outline the parts of the theory that are relevant for our purposes.The Deligne category S t = n ( C ) fails to be semisimple or even abelian, butit has a tensor structure and we can still consider the split Grothendieck ringspanned by indecomposable objects. We want to think of these as being alink between the Grothendieck rings of W n ( C ), and the Grothendieck ringsof S t ( C ) at generic t . We can make this precise via projection and lifting. Theorem 3.1 ( Description of projection, Mori [9] theorem 5.6 ) . Theprojection functor of theorem 2.5 induces a surjective homomorphism fromthis split Grothendieck ring to the Grothendieck ring of W n ( C ) . Explicitly, itsends ˜ V t = n ( λ ) to either V ( λ n ) if n − | λ | ≥ λ ( C ) or the zero object otherwise. In particular we have the following immediate corollary:
Corollary 3.2.
A collection of generators of the split Grothendieck ring of S t = n ( C ) consisting of elements of the form [ ˜ V t = n ( λ )] projects to a collectionof generators of the Grothendieck ring of W n ( C ) consisting of elements of theform [ V ( λ n )] . Next we want to relate the split Grothendieck rings at positive integer t to the Grothendieck ring at generic t . We need to be a bit careful because theindecomposable objects ˜ V t = n ( λ ) of S t = n ( C ) are not always flat deformationsof the irreducible objects ˜ V ( λ ) in the Deligne categories for generic t . In par-ticular we do not expect the map [ ˜ V t = n ( λ )] [ ˜ V ( λ )] to be a homomorphismof between these rings. We can relate these by the process of lifting, whichhas the following (simplified) description: Theorem 3.3 ( Description of lifting, after [2] lemma 5.20 ) . There exists a map Lift t from objects of S t = n ( C ) to objects of S T ( C ) where T is a formal variable, satisfying:1. Lift t descends to an isomorphism from the split Grothendieck ring S t = n ( C ) to the Grothendieck ring of S T ( C ) . . This isomorphism sends [ ˜ V t = n ( λ )] to either [ ˜ V ( λ )] or [ ˜ V ( λ )] + [ ˜ V ( λ ′ )] for an explicitly described λ ′ depending on the combinatorics of n and λ satisfying | λ ′ | < | λ | .Proof. The existence of a lift follows from standard facts about lifting ofidempotents, and the proof is identical to that for
Rep ( S t ). The combina-torics follows from the case of Rep ( S t ) via an explicit identification of theblocks of S t ( C ) with blocks of Rep ( S t ′ ) given by Mori ([9], Prop 5.4).In particular, the additional “error” term picked up when lifting is lowerin the filtration. So while the map [ ˜ V t = n ( λ )] [ ˜ V ( λ )] is not an isomorphism(or even a homomorphism) between the split Grothendieck ring at integer t and the Grothendieck ring at generic t , it is an isomorphism at the level ofassociated graded rings. We now have the following key lemma: Lemma 3.4. If { [ ˜ V ( λ )] | λ ∈ J } is a collection of generators for the as-sociated graded Grothendieck ring of S t ( C ) at generic values of t for someindexing set J ⊂ P C , then { [ V ( λ n )] | λ ∈ J } is a collection of generators forthe Grothendieck ring of W n ( C ) .Proof. The previous remark allows us to transfer this to a system of gener-ators for the associated graded Grothendieck ring at integer t . A standardupper triangularity argument tells us that any lift of this system of genera-tors will also generate the split Grothendieck ring. Corollary 3.2 allows usto transfer this down to W n ( C ). Remark . Understanding these non-semisimple Deligne categories andtheir split Grothendieck rings in better detail is of independent interest asthey seem to lie somewhere between classical representation theory and sta-ble representation theory. Recently Inna Entova-Aizenbud [4] used the non-semisimple Deligne categories for symmetric groups to find new identitiesinvolving the reduced and non-reduced Kronecker coefficients.
Our goal is to write down explicit systems of generators for the Grothendieckrings of the wreath product categories W n ( C ) consisting of irreducible objects.10y lemma 3.4 it suffices to find an explicit system of generators for theassociated graded Grothendieck ring of S t ( C ) at generic t . By corollary 2.9this is isomorphic to the tensor product of one copy of the ring Λ of symmetricfunctions for each irreducible object of C .Once translated into the language of symmetric functions, for each irre-ducible object of C we need to choose a collection of Schur functions whichgenerate the ring of symmetric functions. Two well known such collectionsare the elementary and complete homogeneous symmetric functions, whichare Schur functions corresponding to λ = (1 k ) or λ = ( k ) respectively. Allthat is left is to describe which objects these correspond to.First let’s handle the case when the irreducible object of C we are lookingat is the unit object. The elementary and complete homogeneous symmet-ric functions just correspond to the usual hook and length two partitionrepresentations of the natural copy of Rep ( S t ) inside S t ( C ). If C is the rep-resentation category of a finite group G and t = n is a natural number thenunder the projection to the category of representations of S n ( G ) these arejust the corresponding representations of S n with trivial action of G .If V ∈ I ( C ) is a nontrivial irreducible then let V k,ε := V ⊠ k ⊗ ε ∈ W k ( C )where ε is either the sign or trivial representation for the copy of Rep ( S k ) in W k ( C ). Under our correspondence the elementary (resp. complete homoge-nous) symmetric functions correspond to the objects Ind S t ( C ) W k ( C ) ⊠ S t − k ( C ) ( V k,ε ⊠ )for ε the sign representation (resp. trivial).Putting it all together in the case where C is the representation categoryof a finite group G with m equivalence classes of irreducible representations,we get 2 m easy to describe systems of generators for the representation ring ofthe wreath product S n ( G ). The previous remarks imply the following result: Theorem 4.1. (Systems of generators for representation rings) If G is a finite group, then the representation ring of the wreath product S n ( G ) is generated by either the hook or length two partition representa-tions of S n along with induced representations Ind S n ( G ) S k ( G ) × S n − k ( G ) (( V ⊗ k ⊗ ε V ) ⊠ S n − k ( G ) ) for each irreducible representation V of G and k ≤ n , where ε V is aconsistent (not depending on k ) choice of either the trivial or sign characterof the symmetric group S k . .1 The abelian group case and Marin’s conjecture Of particular interest is the case when G is an abelian group, this includesCoxeter groups in type B as well as the complex reflection groups G ( m, , n ).Here we can get an alternate description of the objects corresponding to theelementary symmetric functions.The sign-twisted representations Ind S n ( G ) S k ( G ) × S n − k ( G ) (( V ⊗ k ⊗ ε ) ⊠ S n − k ( G ) )described in the previous section appear as summands in exterior powers ofthe relatively easy to describe representations Ind S n ( G ) G × S n − ( G ) ( V ⊠ S n − ( G ) ),along with other more complicated terms coming from the exterior powers of V itself. However if we start with a nontrivial character χ of G these otherterms all vanish and these exterior powers coincide with our sign-twistedrepresentations.These exterior powers are perhaps closer to a direct generalization ofhooks for these wreath products, and can be taken to be in our generatingsets by choosing the elementary symmetric functions over the complete ho-mogeneous symmetric functions for each character of G . In particular if G isabelian then all of its irreducible representations are characters and Theorem4.1 becomes: Theorem 4.2. (Hook-like generating sets) If G is a finite abelian group, then the representation ring of the wreathproduct S n ( G ) is generated by the reflection representation of S n , the n -dimensional representations Ind S n ( G ) G × S n − ( G ) ( χ ⊠ S n − ( G ) ) for nontrivial char-acters χ of G , and exterior powers thereof.Remark . While this paper was being written the author was informedthat this result for wreath products with Abelian groups has been recentlyproven by Schlank and Stapleton using different methods.Now let’s specialize to the case when G is cyclic of order 2. The wreathproducts S n ( G ) are Coxeter groups W of type B n . If χ is the nontrivialrepresentation of G , then the reflection representation of W is given by V :=Ind S n ( G ) G × S n − ( G ) ( χ ⊠ S n − ( G ) ). Next we let U be the reflection representation of S n , upgraded to a S n ( G ) representation by letting G act trivially. Translatedinto this language our theorem becomes: Theorem 4.4. (Marin’s conjecture 6.2 for type B n ) For W a Coxeter group of type B n the representation ring R ( W ) is gen-erated by Λ k U, Λ k V, k ≥ . eferences [1] T. Church and B. Farb, Representation theory and homological stability ,arXiv:1008.1368v2[2] J. Comes, V. Ostrik,
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