Ideals in Deligne's tensor category Rep(GL_δ)

1.3 MB·English

by  Jonathan Comes

IDEALS IN DELIGNE’S TENSOR CATEGORY Rep(GL )
δ
JONATHANCOMES
2 Abstract. WegiveaclassificationofidealsinRep(GL )forarbitraryδ.
δ
1
0
2

n 1. Introduction
a
J 1.1. LetKdenoteafieldofcharacteristiczero,V ad-dimensionalvectorspaceover
6 K, and GLd = GL(V) the corresponding general linear group. The usual tensor
2 product of vector spaces gives Rep(GL ), the category of finite dimensional rep-
d
resentations of GLd, the structure of a K-linear symmetric tensor category. More
]
T generally, given integers m,n ≥ 0 the standard tensor product of super vector
R spaces gives Rep(GL(m|n)), the category of representations of the general lin-
ear supergroup GL(m|n), the structure of a K-linear symmetric tensor category
.
h and the natural representation has categorical dimension (i.e. super dimension)
at m−n. Given an arbitrary δ ∈ K, Deligne has defined a K-linear symmetric ten-
m sor category Rep(GL ) containing a “natural” object of categorical dimension δ,
δ
[ whichisuniversalamongallsuchcategories[Del2,Proposition10.3]. Inparticular,

Rep(GL(m|n)) along with the natural representation of GL(m|n) prescribe a ten-
1
sor functor F : Rep(GL ) → Rep(GL(m|n)) whenever δ = m−n. Let I(m|n)
v m|n δ
9 denote the collection of objects sent to zero by Fm|n. I(m|n) is an example of an
6 “ideal” in Rep(GL ). More generally, we have
δ
6
Definition 1.1.1. AcollectionofobjectsI inatensorcategoryT iscalledanideal
5
. of T if the following conditions are satisfied:
1
(i) X⊗Y ∈I whenever X ∈T and Y ∈I.
0
2 (ii) If X ∈ T, Y ∈ I and there exist α : X → Y, β : Y → X such that
1 β◦α=id , then X ∈I.
X
:
v We will call an ideal in T nontrivial if it contains a nonzero object, but does not
i containallobjectsinT. Themainresultofthispaperisaclassificationofidealsof
X
Rep(GL ). More precisely, we will show the set {I(m|n) | m,n∈Z ,m−n=δ}
r δ ≥0
a isacompletesetofpairwisedistinctnontrivialidealsinRep(GL )(Theorem4.2.1).
δ
1.2. Arelatednotionisthatofatensor ideal, i.e.acollectionofmorphismswhich
is closed under arbitrary compositions and tensor products (see, for instance [CK,
§2.4]). A classification of tensor ideals in Rep(GL ) would be a nice result, and
δ
the main result of this paper can be viewed as a step towards that classification.
For more details on the relationship between ideals and tensor ideals we refer the
reader to [CK, §2.5].
On the other hand, ideals in the sense of Definition 1.1.1 are interesting in their
own right. For instance, nontrivial ideals in tensor categories will sometimes admit
modifiedtraceanddimensionfunctionsinthesenseof[GKP]. In[CK]itwasshown
Date:January30,2012.
1
2 JONATHANCOMES
thatifδ ∈Z thentheuniquenontrivialidealinDeligne’sRep(S )alwaysadmits
≥0 δ
a modified trace. It would be interesting to determine which nontrivial ideals in
Rep(GL ) admit modified traces.
δ
1.3. Theproofofthemainresultofthispaperreliesheavilyonresultsfrom[CW].
The following items concerning the category Rep(GL ) can be found in loc. cit.:
δ
I. An explicit construction of Rep(GL ) using walled Brauer diagrams.
δ
II. A classification of indecomposable objects by bipartitions.
III. A procedure for decomposing tensor products of indecomposable objects.
IV. A complete description of the indecomposable objects in the ideal I(m|n).
Although we do not need the explicit construction of Rep(GL ) in this paper,
δ
items II, III, and IV above play a crucial role in our proof of the classification of
ideals. WefocusonaspecialclassofindecomposableobjectsinRep(GL )whichare
δ
labelled by so-called almost (m|n)-cross bipartitions (see §2.4). Roughly speaking,
to prove the main result we use item III above to show that every nontrivial ideal
in Rep(GL ) is, in some sense, generated by an indecomposable object labelled by
δ
an almost (m|n)-cross bipartition. Then we use item IV above to conclude that all
nontrivial ideals must be of the form I(m|n).
The decomposition of tensor products in Rep(GL ) is governed by the combina-
δ
torialpropertiesofBrundanandStroppel’sweightdiagramsintroducedin[BS1-4].
Accordingly, these weight diagrams also play a significant role in this paper.
1.4. Thepaperisorganizedasfollows. In§2weintroducebipartitionsandweight
diagrams,thecombinatorialobjectsinthispaper. Weproveseveralbasicfactscon-
cerning these combinatorial objects, paying special attention to the almost (m|n)-
cross bipartitions. In §3 we recall the classification of indecomposable objects in
Rep(GL ), as well as some useful formulae for decomposing tensor products in
δ
Rep(GL ) found in [CW]. The remainder of §3 is devoted to proving some techni-
δ
calresultsondecomposingtensorproducts. Finally,in§4weproveourclassification
of ideals in Rep(GL ).
δ
2. Bipartitions and weight diagrams
In this section we lay out the combinatorial tools used in this paper, namely
bipartitionsandtheweightdiagramsofBrundanandStroppel. Sinceittakeslittle
space to do so, we provide proofs to most of the properties we will need concerning
thesecombinatorialobjects. However,wenotethatmanytheseobservationscanbe
found (at least implicitly) in [BS1-5] or even in [CW]. We will follow the notation
in [CW].
2.1. Bipartitions. Apartition isantupleofnonnegativeintegersα=(α ,α ,...)
1 2
such that α ≥ α for all i > 0, and α = 0 for all but finitely many i. Let P
i i+1 i
denote the set of all partitions and let ∅=(0,0,...)∈P. We write |α|=(cid:80) α
i>0 i
for the size of α. Given α,β ∈ P, we say α is contained in β and write α ⊂ β
if α ≤ β for all i > 0. We identify a partition α with its Young diagram which
i i
consists of left-aligned rows of boxes, with α boxes in the ith row (reading from
i
top to bottom). For example,
.
IDEALS IN DELIGNE’S TENSOR CATEGORY Rep(GLδ) 3
Elements of P ×P are called bipartitions. Given a bipartition λ, we let λ• and
λ◦ denote the partitions such that λ=(λ•,λ◦). We will write |λ|=|λ•|+|λ◦| for
the size1 of λ. Given bipartitions λ and µ, we say λ is contained in µ and write
λ⊂µ if λ• ⊂µ• and λ◦ ⊂µ◦. Given a bipartition λ=(λ•,λ◦), let Add•(λ) (resp.
Add◦(λ)) denote the set of bipartitions obtained from λ by adding a box to λ•
(resp. λ◦). Similarly, let Rem•(λ) (resp. Rem◦(λ)) denote the set of bipartitions
obtained from λ by removing a box from λ• (resp. λ◦).
2.2. Weight diagrams. Given δ ∈K and a bipartition λ, set
I (λ) ={λ•,λ•−1,λ•−2,...},
∧ 1 2 3
I (λ,δ) ={1−δ−λ◦,2−δ−λ◦,3−δ−λ◦,...}.
∨ 1 2 3
The weight diagram of λ with parameter δ, denoted x (δ), is the diagram obtained
λ
by labeling the integer vertices on the number line according to the following rule:
label the the ith vertex by

(cid:13) if i(cid:54)∈I (λ)∪I (λ,δ),
 ∧ if i∈I∧(λ)\I∨(λ,δ),
∧ ∨
∨ if i∈I (λ,δ)\I (λ),
 × if i∈I∨(λ)∩I (∧λ,δ).
∧ ∨
For example,
,
and
(1)
It will be important for us to understand how adding/removing a box to a
bipartition affects its weight diagram. We record this in the following proposition
leaving the proof as a straightforward exercise for the reader.
Proposition 2.2.1. Suppose λ and µ are bipartitions and δ ∈Z.
(i) µ ∈ Rem•(λ) (equivalently λ ∈ Add•(µ)) if and only if the labels on x (δ)
λ
and x (δ) are all the same except an adjacent pair of vertices which are labelled as
µ
in one of the following four cases:
x (δ) ×(cid:13) ∧(cid:13) ×∨ ∧∨
µ
x (δ) ∨∧ (cid:13)∧ ∨× (cid:13)×
λ
1Notethatthisdiffersfromthenotationin[CW]where|λ|denotes(|λ•|,|λ◦|).
4 JONATHANCOMES
(ii) µ ∈ Rem◦(λ) (equivalently λ ∈ Add◦(µ)) if and only if the labels on x (δ)
λ
and x (δ) are all the same except an adjacent pair of vertices which are labelled as
µ
in one of the following four cases:
x (δ) (cid:13)× (cid:13)∨ ∧× ∧∨
µ
x (δ) ∨∧ ∨(cid:13) ×∧ ×(cid:13)
λ
UsingProposition2.2.1wecanprovethefollowing statementwhichallowsusto
recover δ from any given weight diagram whenever δ is an integer.
Corollary 2.2.2. For any integer δ and bipartition λ, δ is equal to the number of
×’s minus the number of (cid:13)’s in x (δ).
λ
Proof. Every bipartition can be constricted by adding a finite number of boxes to
(∅,∅),andbythepreviouspropositiontheprocessofaddingboxestoabipartition
does not change the number of ×’s minus the number of (cid:13)’s in the corresponding
weight diagram. The result now follows from (1). (cid:3)
2.3. Cap diagrams and the numbers D (δ). Given δ ∈ K and a bipartition
λ,µ
λ, the cap diagram c (δ) is constructed as follows:
λ
Step 0: Start with x (δ).
λ
Step n: Draw a cap connecting vertices i and j on the number line whenever (i)
i < j; (ii) i is labelled by ∨ and j is labelled by ∧ in x (δ); and (iii) each
λ
integer between i and j in x (δ) is either labelled by (cid:13), labelled by ×, or
λ
already part of a cap from an earlier step.
This process will terminate in a finite number of steps, leaving us with the cap
diagram c (δ). For an example see [CW, Example 6.3.2].
λ
Wesaythattwoverticesinx (δ)areconnected ifthereisacapconnectingthem
λ
in c (δ). Next, given two bipartitions λ and µ, we say x (δ) is linked to x (δ) if
λ µ λ
x (δ) if obtained from x (δ) by interchanging the labels on finitely many pairs of
µ λ
connected vertices in x (δ). Finally, set
λ
(cid:26)
1, if x (δ) is linked to x (δ);
D (δ)= µ λ
λ,µ 0, otherwise.
Remark 2.3.1. Ifδ (cid:54)∈Zandλisanybipartition,thennoneoftheverticesinx (δ)
λ
(cid:26)
1, λ=µ;
are labelled ∨, hence there are no caps in c (δ), hence D (δ)=
λ λ,µ 0, λ(cid:54)=µ.
The following propositions concerning the numbers D (δ) will be useful later.
λ,µ
Proposition 2.3.2. D (δ)=1, and for µ(cid:54)=λ we have D (δ)=0 unless µ⊂λ
λ,λ λ,µ
and |µ|≤|λ|−2.
Proof. Assume µ (cid:54)= λ is such that D (δ) = 1. Let k denote the number of pairs
λ,µ
of connected vertices in x (δ) whose labels differ from those in x (δ). Then I (µ)
λ µ ∧
(resp. I (µ,δ)) is obtained from I (λ) (resp. I (λ,δ)) by replacing k numbers
∨ ∧ ∨
a ,...,a withnumbersb ,...,b suchthatb <a (resp. b >a )forall1≤i≤k.
1 k 1 k i i i i
In particular, this implies µ• (cid:36)λ• (resp. µ◦ (cid:36)λ◦). The result follows. (cid:3)
Proposition 2.3.3. Fix δ ∈K and a bipartition λ.
(i) If for all µ∈Rem•(λ) there exists ν ∈Add◦(λ) with D (δ)=1, then x (δ)
ν,µ λ
has no adjacent labels of the form ∨∧, (cid:13)∧, or ∨×.
(ii)Ifforallµ∈Rem◦(λ)thereexistsν ∈Add•(λ)withD (δ)=1, thenx (δ)
ν,µ λ
has no adjacent labels of the form ∨∧, ∨(cid:13), or ×∧.
IDEALS IN DELIGNE’S TENSOR CATEGORY Rep(GLδ) 5
Proof. (i) Suppose x (δ) has adjacent labels of the form (cid:13)∧, ∨×, or ∨∧. Then
λ
by Proposition 2.2.1(i) there exists µ ∈ Rem•(λ) such that x (δ) has either a (cid:13)
µ
at a vertex where x (δ) has a ∧, or a × where x (δ) has a ∨. Now suppose ν is
λ λ
a bipartition with D (δ)=1. Then x (δ) must also have either a (cid:13) at a vertex
ν,µ ν
where x (δ) has a ∧, or a × where x (δ) has a ∨. Thus, by Proposition 2.2.1(ii),
λ λ
ν (cid:54)∈Add◦(λ). Statement (i) follows. The proof of (ii) is similar. (cid:3)
2.4. Almost (m|n)-cross bipartitions. Fix integers m,n ≥ 0. Following [CW]
we call a bipartition λ (m|n)-cross2 if there exists k with 0 ≤ k ≤ m such that
λ• +λ◦ ≤n. We call λ almost (m|n)-cross if it is not (m|n)-cross, but any
k+1 m−k+1
bipartitionstrictlycontainedinλis(m|n)-cross. Itiseasytoshowthatλisalmost
(m|n)-cross if and only if λ• =λ◦ =0 and λ• +λ◦ =n+1 whenever
m+2 m+2 k+1 m−k+1
0≤k ≤m. Fromthisperspectiveweseethatλisalmost(m|n)-crossifandonlyif
theYoungdiagramsofλ• andλ◦ canbegluedtogethertoforman(m+1)×(n+1)
rectangle as in the following picture:
(2)
The following proposition lists a couple useful properties of almost (m|n)-cross
bipartitions.
Proposition 2.4.1. Suppose λ is an almost (m|n)-cross bipartition.
(i) |λ| = (m+1)(n+1). In particular, all almost (m|n)-cross bipartition have
the same size.
(ii) For any integers m(cid:48),n(cid:48) ≥ 0 with m(cid:48) ≤ m and n(cid:48) ≤ n there exists an almost
(m(cid:48)|n(cid:48))-cross bipartition µ such that µ⊂λ.
Proof. (i) is immediate from (2). For (ii), remove the top m−m(cid:48) rows and the
leftmost n−n(cid:48) columns of (2); the bipartition which corresponds to the remaining
(m(cid:48)+1)×(n(cid:48)+1) rectangle is one such µ.3 (cid:3)
Next, we determine the weight diagrams of almost (m|n)-cross bipartitions.
Proposition 2.4.2. A bipartition λ is almost (m|n)-cross if and only if its weight
diagram is of the form
Proof. The condition λ• +λ◦ =n+1 whenever 0≤k ≤m is equivalent to
k+1 m−k+1
i−(m−n)−λ◦ =λ• −(m−i+1)whenever1≤i≤m+1. Hence,λisalmost
i m−i+2
(m|n)-cross if and only if I (λ)={λ•,λ•−1,...,λ• −m,−m−1,−m−2,...}
∧ 1 2 m+1
2See[CW,§8.7]forapictorialdescriptionof(m|n)-crossbipartitionswhichjustifiesitsname.
3Infact,onecouldremoveanym−m(cid:48) rowsandn−n(cid:48) columns.
6 JONATHANCOMES
andI (λ,m−n)={λ• −m,...,λ•−1,λ•,n+2,n+3,...}, whichisequivalent
∨ m+1 2 1
to x (m−n) having the desired form. (cid:3)
λ
Corollary 2.4.3. Suppose λ and µ are bipartitions.
(i) If λ is almost (m|n)-cross, then D (m−n)=0 whenever µ(cid:54)=λ.
λ,µ
(ii)Ifµisalmost(m|n)-cross,thenD (m−n)=0unlessλ=µor|λ|>|µ|+2.
λ,µ
Proof. If λ is almost (m|n)-cross, then by Proposition 2.4.2 c (m − n) has no
λ
caps which implies (i). To prove (ii), assume that µ is almost (m|n)-cross and
D (m−n)=1. Since µ is linked to λ, the ×’s and (cid:13)’s in x (m−n) are in the
λ,µ λ
same positions as those in x (m−n). Hence, by Proposition 2.4.2 applied to µ,
µ
there are no adjacent connected vertices in x (m−n). It follows that |λ|>|µ|+2
λ
(see for instance [CW, Proposition 6.3.6]). (cid:3)
2.5. One-box-bumps. We close this section by describing a sequence of moves
which transforms any given almost (m|n)-cross bipartition into any other almost
(m|n)-cross bipartition. To start, fix an almost (m|n)-cross bipartition λ. Notice
that if λ• (resp. λ◦) has a removable box in its ith row, then λ◦ (resp. λ•) has an
addable box in the (m+2−i)th row (see (2)). The procedure of removing a box
from the ith row of λ• (resp. λ◦) and adding a box to the (m+2−i)th row of
λ◦ (resp. λ•) will be called a rightward (resp. leftward) one-box-bump. From (2)
it is apparent that performing a one-box-bump to an almost (m|n)-bipartition will
yieldanotheralmost(m|n)-crossbipartition. Moreover,itiseasytoseethatwecan
get from any given almost (m|n)-cross bipartition to any other almost (m|n)-cross
bipartition by performing a finite sequence of of one-box-bumps. For example, the
bipartitions λ = ((3,3,1,0,...),(2,0,...)) and µ = ((2,2,2,0,...),(1,1,1,0,...))
are both almost (2|2)-cross, and we can get from λ to µ using 2 rightward one-
box-bumpsfollowedbyanleftwardone-box-bumpasillustratedusingtherectangle
description of almost (m|n)-cross bipartitions below:
Finally,althoughitisunnecessaryfortherestofthispaper,wenotethataone-box-
bumponanalmost(m|n)-crossbipartitionmodifiesitsweightdiagrambyswapping
a × with an adjacent (cid:13).
3. Decomposing tensor products in Rep(GL )
δ
As in §1.1, given δ ∈ K we let Rep(GL ) = Rep(GL ;K) denote the K-linear
δ δ
symmetric tensor category with parameter δ defined in [Del1] and [Del2, §10]4 (see
also [CW, §3]). In this section we will prove some technical results on decom-
posing tensor product in Rep(GL ). As we will see, while studying Rep(GL ) is
δ δ
sometimes useful to work with a “generic” parameter. To do so we let t denote
an indeterminate and write K(t) for the field of rational functions in t. Then
Rep(GL ) = Rep(GL ;K(t)) is a K(t)-linear symmetric tensor category with pa-
t t
rameter t. We begin with a brief account of the classification of indecomposable
objects in Rep(GL ) found in [CW].
δ
4DeligneusesthenotationRep(GL(δ),K)forRep(GL ). Wewillfollowthenotationin[CW].
δ
IDEALS IN DELIGNE’S TENSOR CATEGORY Rep(GLδ) 7
3.1. Indecomposable objects. Given a bipartition λ=(λ•,λ◦), in [CW, §4] an
indecomposable object L(λ) = L(λ•,λ◦) in Rep(GL ) is defined (up to isomor-
δ
phism). For example, L(∅,∅) denotes the unit object 1, L((cid:50),∅) is the “natural”
object of dimension δ mentioned in §1.1, and L(∅,(cid:50)) is the dual of L((cid:50),∅). We
will not use an explicit definition of L(λ) in this paper, so we refer the interested
reader to loc. cit. for more details on L(λ). However, the L(λ)’s have many useful
properties which we will exploit. For instance, the assignment
(cid:26) (cid:27) (cid:26) (cid:27)
bipartitions of bij. nonzero indecomposable objects in
−→
(3) arbitrary size Rep(GL ), up to isomorphism
δ
λ (cid:55)→ L(λ)
isabijection[CW, Theorem4.6.2]whichwewilluseimplicitlythroughouttherest
of this paper. It is important to note that while the object L(λ) depends on the
parameterδ,thefactthat(3)isabijectiondoesnot. Henceindecomposableobjects
in Rep(GL ) are parametrized by bipartitions regardless of the parameter δ. Our
δ
notation L(λ) for the indecomposable object labelled by λ is perhaps poor since it
does not keep track of the parameter δ. In particular, we will write L(λ) to denote
bothanobjectinRep(GL )andoneinRep(GL ). Consequently,wemustindicate
δ t
which category we are working in whenever we write about L(λ).
3.2. Decomposition formulae. Thefollowingdecompositionformulaefortensor
products in Rep(GL ) are found in [CW, Corollary 7.1.2]:
t
(cid:77) (cid:77)
(4) L((cid:50),∅)⊗L(λ)= L(µ)⊕ L(ν),
µ∈Add•(λ) ν∈Rem◦(λ)
(cid:77) (cid:77)
(5) L(∅,(cid:50))⊗L(λ)= L(µ)⊕ L(ν).
µ∈Add◦(λ) ν∈Rem•(λ)
Itisimportanttonotethat(4)and(5)holdinRep(GL )ratherthaninRep(GL ).
t δ
Itturnsoutthatthedecompositionoftensorproductinthelattercategorydepends
on the numbers D (δ) [CW, §7.2]. To make this dependency precise, given a bi-
λ,µ
partition λ and δ ∈ K, write Mδ(λ) = (cid:76)µL(µ)⊕Dλ,µ(δ) in Rep(GLt). As an
immediate consequence of Proposition 2.3.2 we can write
(cid:77)
(6) Mδ(λ)=L(λ)⊕ L(µ)⊕Dλ,µ(δ).
µ(cid:36)λ,
|µ|≤|λ|−2
TheobjectsM (λ)allowustocomparetensorproductdecompositionsinRep(GL )
δ δ
with those in Rep(GL ) via the following theorem:
t
Theorem 3.2.1. L(λ)⊗L(µ) = L(ν(1))⊕···⊕L(ν(k)) in Rep(GL ) if and only
δ
if M (λ)⊗M (µ)=M (ν(1))⊕···⊕M (ν(k)) in Rep(GL ).
δ δ δ δ t
Proof. See [CW, Theorem 6.2.3(1) and Corollary 6.4.2]. (cid:3)
Theorem 3.2.1 along with formulae (4), (5), and (6) will serve as our main tools
for proving the remaining results in §3. Indeed, the proof of the following useful
proposition will make use of each of these tools.
Proposition 3.2.2. Suppose λ and µ are bipartitions and δ ∈ K. If µ ⊂ λ then
L(λ) is a summand of A⊗L(µ) for some object A in Rep(GL ).
δ
8 JONATHANCOMES
Proof. If µ ⊂ ν ⊂ λ, L(µ) is a summand of A⊗L(ν), and L(ν) is a summand of
B⊗L(λ);thenL(µ)isasummandofA⊗B⊗L(λ). Hence,itsufficestoconsiderthe
case λ ∈ Add•(µ)∪Add◦(µ). In this case, λ is a bipartition of maximal size such
that L(λ) is a summand of either L((cid:50),∅)⊗L(µ) or L(∅,(cid:50))⊗L(µ) in Rep(GL )
t
(formulae(4)and(5)). HenceλisamaximallysizedbipartitionsuchthatL(λ)isa
summandofeitherM ((cid:50),∅)⊗M (µ)orM (∅,(cid:50))⊗M (µ)inRep(GL )(formulae
δ δ δ δ t
(4), (5), and (6)). It follows that L(λ) is a summand of either L((cid:50),∅)⊗L(µ) or
L(∅,(cid:50))⊗L(µ) in Rep(GL ) (Theorem 3.2.1 and formula (6)). (cid:3)
δ
3.3. Tensorproductsandalmost(m|n)-crossbipartitions. Inthissubsection
we combine the combinatorial properties of almost (m|n)-cross bipartitions (§2.4-
2.5)withourdecompositionformulaefortensorproductsinRep(GL )(§3.2)to
m−n
derivesometechnicalresultsontensorproductsofindecomposableobjectslabelled
by almost (m|n)-cross bipartitions. These results will be used in §4 to prove our
classification of ideals.
Proposition 3.3.1. Fix a bipartion λ (cid:54)= (∅,∅). Given another bipartition µ,
let a (resp. b ) denote the number of times L(µ) occurs in a decomposition of
µ µ
L((cid:50),∅)⊗L(λ) (resp. L(∅,(cid:50))⊗L(λ)) into indecomposable summands in the cat-
egory Rep(GL ). If a =b =0 whenever µ(cid:36)λ, then λ is almost (m|n)-cross for
δ µ µ
some m,n∈Z with m−n=δ.
≥0
Proof. Write L((cid:50),∅)⊗L(λ) = L(ν(1))⊕···⊕L(ν(k)) in Rep(GL ) and assume
δ
ν(i) is not strictly contained in λ for all i = 1,...,k. Then, by Theorem 3.2.1,
M ((cid:50),∅)⊗M (λ) = M (ν(1))⊕···⊕M (ν(k)) in Rep(GL ). Hence, by formulae
δ δ δ δ t
(6) and (4),
(7) |ν(i)|≤|λ|+1
foreachi=1,...,k. Moreover,equalityin(7)occursifandonlyifν(i) ∈Add•(λ).
Now, given µ ∈ Rem◦(λ), L(µ) is a summand of L((cid:50),∅)⊗L(λ) in Rep(GL )
t
(formula (4)), and is thus a summand of M ((cid:50),∅)⊗M (λ) in Rep(GL ) (formula
δ δ t
(6)). Hence, D (δ) (cid:54)= 0 for some j = 1,...,k. However, a = 0 implies
ν(j),µ µ
µ(cid:54)=ν(j), which implies δ ∈Z (Remark 2.3.1). Moreover, |ν(j)|≥|µ|+2=|λ|+1
(Proposition 2.3.2), which implies ν(j) ∈ Add•(λ). In particular, we have shown
that for every µ ∈ Rem◦(λ) there exists ν ∈ Add•(λ) with D (δ) (cid:54)= 0. Similarly,
ν,µ
assuming b = 0 for all µ (cid:36) λ, one can show that for every µ ∈ Rem•(λ) there
µ
exists ν ∈ Add◦(λ) with D (δ) (cid:54)= 0. It follows from Proposition 2.3.3 that the
ν,µ
label to the left (resp. right) of any ∧ (resp. ∨) in x (δ) is also a ∧ (resp. ∨).
λ
Thus, x (δ) must be of the form
λ
Letm∈Zbesuchthatthenumberof×’sinx (δ)ism+1. Notethatm≥0since
λ
λ (cid:54)= (∅,∅). Moreover, there are n+1 (cid:13)’s in x (δ) where n := m−δ (Corollary
λ
2.2.2). It follows from Proposition 2.4.2 that λ is almost (m|n)-cross. (cid:3)
Corollary 3.3.2. Suppose λ is a bipartition and δ ∈K. Either
(i) 1 is a summand of A⊗L(λ) for some object A in Rep(GL ); or
δ
IDEALS IN DELIGNE’S TENSOR CATEGORY Rep(GLδ) 9
(ii) δ ∈ Z and there exists a nonnegative integer m and an almost (m|m−δ)-
cross bipartition µ ⊂ λ such that L(µ) is a summand of A⊗L(λ) for some object
A in Rep(GL ).
δ
Proof. We induct on |λ|. If λ=(∅,∅) or λ is almost (m|m−δ)-cross for some m,
then take A=1. Otherwise, by Proposition 3.3.1 there exists a bipartition ν with
ν (cid:36) λ such that L(ν) is a summand of either L((cid:50),∅)⊗L(λ) or L(∅,(cid:50))⊗L(λ)
in Rep(GL ). By induction, either (i) 1 is a summand of A(cid:48) ⊗L(ν) and hence a
δ
summandofeitherA(cid:48)⊗L((cid:50),∅)⊗L(λ)orA(cid:48)⊗L(∅,(cid:50))⊗L(λ)forsomeobjectA(cid:48);or
(ii) δ ∈Z and there exists a nonnegative integer m and an almost (m|m−δ)-cross
bipartition µ ⊂ λ such that L(µ) is a summand of A(cid:48) ⊗L(ν) and hence of either
A(cid:48)⊗L((cid:50),∅)⊗L(λ) or A(cid:48)⊗L(∅,(cid:50))⊗L(λ) for some object A(cid:48). (cid:3)
Lemma 3.3.3. Suppose λ and µ are almost (m|n)-cross bipartitions.
(i) L(µ) is a summand of L((cid:50),∅)⊗2⊗L(λ) in Rep(GL ) if and only if L(µ)
m−n
is a summand of L((cid:50),∅)⊗2⊗L(λ) in Rep(GL ).
t
(ii) L(µ) is a summand of L(∅,(cid:50))⊗2⊗L(λ) in Rep(GL ) if and only if L(µ)
m−n
is a summand of L(∅,(cid:50))⊗2⊗L(λ) in Rep(GL ).
t
Proof. (i) Set δ = m−n and write L((cid:50),∅)⊗2 ⊗L(λ) = L(µ(1))⊕···⊕L(µ(k))
in Rep(GL ). Since M ((cid:50),∅) = L((cid:50),∅) and M (λ) = L(λ) (formula (6) and
δ δ δ
Corollary 2.4.3(i)), it follows that
(8) L((cid:50),∅)⊗2⊗L(λ)=M (µ(1))⊕···⊕M (µ(k))
δ δ
in Rep(GL ) (Theorem 3.2.1). In particular, |µ(i)| ≤ |λ| + 2 = |µ| + 2 for all
t
i=1,...,k (formulae (4), (6), and Proposition 2.4.1(i)). Now, L(µ) is a summand
of(8)inRep(GL )ifandonlyifD (δ)(cid:54)=0forsomei, whichoccursifandonly
t µ(i),µ
if µ=µ(i) for some i (Corollary 2.4.3(ii)). The proof of (ii) is similar. (cid:3)
Proposition3.3.4. Ifλandµarebothalmost(m|n)-cross, thenthereexistalmost
(m|n)-cross bipartitions ν(0),...,ν(k) such that ν(0) =µ, ν(k) =λ, and L(ν(i−1)) is
a summand of either L((cid:50),∅)⊗2⊗L(ν(i)) or L(∅,(cid:50))⊗2⊗L(ν(i)) in Rep(GL )
m−n
for each i=1,...,k.
Proof. Recall from §2.5 that λ can be obtained from µ by a finite sequence of one-
box-bumps. Let k denote the number of one-box-bumps and let ν(i) denote the
almost (m|n)-cross bipartition arising after the ith one-box-bump for i=0,...,k.
ByLemma3.3.3wemustshowL(ν(i−1))isasummandofeitherL((cid:50),∅)⊗2⊗L(ν(i))
orL(∅,(cid:50))⊗2⊗L(ν(i))inRep(GL )foralli=1,...,k. Ifν(i)isobtainedfromν(i−1)
t
byarightwardone-box-bump, thenL(ν(i−1))isasummandofL((cid:50),∅)⊗2⊗L(ν(i))
by (4). Similarly, if ν(i) is obtained from ν(i−1) by a leftward one-box-bump, then
L(ν(i−1)) is a summand of L(∅,(cid:50))⊗2⊗L(ν(i)) by (5). (cid:3)
Corollary 3.3.5. If λ and µ are both almost (m|n)-cross, then L(µ) is a summand
of A⊗L(λ) for some object A in Rep(GL ).
m−n
Proof. Let ν(0),...,ν(k) be as in Proposition 3.3.4. By induction on k, we can
assumeL(µ)isasummandofA(cid:48)⊗L(ν(k−1))forsomeA(cid:48). IfL(ν(k−1))isasummand
of L((cid:50),∅)⊗2⊗L(λ), set A=A(cid:48)⊗L((cid:50),∅)⊗2. Otherwise, set A=A(cid:48)⊗L(∅,(cid:50))⊗2.
(cid:3)
10 JONATHANCOMES
4. Ideals in Rep(GL )
δ
InthissectionweclassifytheidealsinRep(GL ),whichisthemainresultofthis
δ
paper. To start, notice that Definition 1.1.1 implies that ideals are always closed
undertakingdirectsummands: A⊕B ∈I impliesA,B ∈I. However,theconverse
implication does not follow from Definition 1.1.1 in an arbitrary tensor category.
WecallanidealI additive ifitisclosedundertakingdirectsums: A,B ∈I implies
A⊕B ∈I. Our first result on ideals of Rep(GL ) is that additivity is guaranteed.
δ
Proposition 4.0.6. Ideals in Rep(GL ) are additive for all δ ∈K.
δ
Proof. LetI beanidealinRep(GL ). ItsufficestoshowL(λ)⊕L(λ(cid:48))∈I whenever
δ
L(λ),L(λ(cid:48))∈I. SinceRep(GL )isadditive,wemayassumeI isnontrivialsothat
δ
1(cid:54)∈I. Hence, λ and λ(cid:48) do not satisfy part (i) of Corollary 3.3.2. Let m (resp. m(cid:48))
be a nonnegative integer and µ ⊂ λ (resp. µ(cid:48) ⊂ λ(cid:48)) be an almost (m|m−δ)-cross
(resp. (m(cid:48)|m(cid:48)−δ)-cross)bipartitionprescribedbyCorollary3.3.2(ii). Withoutloss
of generality assume m ≤ m(cid:48). In this case λ(cid:48) is not (m|m−δ)-cross, hence there
exists an almost (m|m−δ)-cross bipartition ν ⊂λ(cid:48).
Now, by Corollary 3.3.2 there exists an object A(cid:48) such that L(µ) is a summand
of A ⊗ L(λ). By Corollary 3.3.5 there exists an object A(cid:48)(cid:48) such that L(ν) is a
summand of A(cid:48)(cid:48)⊗L(µ). By Proposition 3.2.2 there exists an object A(cid:48)(cid:48)(cid:48) such that
L(λ(cid:48)) is a summand of A(cid:48)(cid:48)(cid:48) ⊗L(ν). Hence, setting A = A(cid:48)(cid:48)(cid:48) ⊗A(cid:48)(cid:48) ⊗A(cid:48), it follows
that L(λ(cid:48)) is a summand of A⊗L(λ). Therefore L(λ)⊕L(λ(cid:48)) is a summand of
(1⊕A)⊗L(λ)∈I. (cid:3)
4.1. The ideals I(m|n). Fix m,n ∈ Z . As in introduction, let I(m|n) denote
≥0
the ideal of Rep(GL ) consisting of all objects sent to zero by F (see §1.1).
m−n m|n
The indecomposable objects in I(m|n) have been worked out in [CW]:
Theorem 4.1.1. [CW, Theorem 8.7.6] L(λ) ∈ I(m|n) if and only if λ is not
(m|n)-cross.
Thetheoremaboveallowsustoprovethefollowinglemma,whichwillbecrucial
in the proof of the classification of ideals in Rep(GL ).
δ
Lemma 4.1.2. Suppose I is an ideal in Rep(GL ) and λ is an almost (m|n)-
m−n
cross bipartition. If L(λ)∈I, then I(m|n)⊂I.
Proof. If L(λ) ∈ I, then by Corollary 3.3.5 L(µ) ∈ I for every almost (m|n)-cross
bipartition µ. Since every non-(m|n)-cross bipartition contains an almost (m|n)-
bipartition, by Proposition 3.2.2 L(µ)∈I for every non-(m|n)-cross bipartition µ.
Hence we are done by Theorem 4.1.1 and Proposition 4.0.6. (cid:3)
4.2. Classification of ideals. We can now prove the main result of this paper:
Theorem 4.2.1. The set {I(m|n) | m,n ∈ Z ,m−n = δ} is a complete set of
≥0
pairwise distinct nontrivial ideals in Rep(GL ).
δ
Proof. It follows from Theorem 4.1.1 that the ideals I(m|n) are pairwise distinct
and nontrivial. Suppose I is a nontrivial ideal in Rep(GL ) and let λ be a biparti-
δ
tion which is minimal with respect to inclusion of bipartitions such that L(λ)∈I.
Since I is nontrivial, λ (cid:54)= (∅,∅). Hence, by Proposition 3.3.1 and the minimality
of λ, there exist m,n ≥ 0 with m−n = δ such that λ is almost (m|n)-cross. It
follows from Lemma 4.1.2 that I(m|n)⊂I.