CELL MODULES FOR TYPE AWEBS STUART MARTIN AND ROBERT A. SPENCER ABSTRACT .We examine the cell modules for the category of type Anwebs and their

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CELL MODULES FOR TYPE AWEBS

STUART MARTIN AND ROBERT A. SPENCER

ABSTRACT.

We examine the cell modules for the category of type

webs and their

natural cellular forms. We modify the bases of these modules, as described by Elias, to

obtain an orthogonal basis of each cell module. Hence, we calculate the determinant of the

Gram matrix with respect to such bases.

These Gram determinants are given in terms of intersection forms, computed from

certain traces of clasps — higher order Jones-Wenzl morphisms. Additionally, the modiﬁed

basis is constructed using these clasps, and each clasp is constructed using traces of smaller

clasps.

In [Eli15], Elias conjectures a value for these intersection forms and veriﬁes it in types

A1,A2and A3. This paper concludes with a proof of the conjecture in type An.

1. INTRODUCTION

The category of (tensor products of) fundamental representations of

Uq(sln)

comes

equipped with many structural features. Indeed, it is monoidal, cellular and diagrammatic.

These three structures intersect and interact in a manner which permits us to use one to

deduce facts about the other.

The monoidal structure comes from the ability to take tensor product of representations

Uq(sln)

. This gives us the structure of a monoid on the representations. It is associative

and has an identity: the trivial representation

. However, for a general choice of

, it is

not symmetric: X⊗Yis not isomorphic to Y⊗X.

The category is cellular because it satisﬁes the conditions of being an OACC [EMTW20]

or, more simply, that each algebra

End(X)

is cellular in the sense of Graham and Lehrer

[GL96].

We say it is diagrammatic, due to the existence of a presentation by planar diagram

generators and relations [CKM14], known as webs or spiders.

Of the three, the cellular structure is perhaps the most interesting, as it contains the

most information. However, as we shall see, its cellular structure can be built up out of

monoidal generators for the objects and diagrammatic generators for the morphisms. This

is the interplay we shall exploit.

Cellular categories are equipped with cell modules. These can be thought of as the

cell modules of each of the cellular endomorphism algebras,

End(X)

. If the category is

semisimple, these cell modules are indecomposable. If it is not, they provide us with a

complete list of indecomposable objects as their heads.

To be exact, each module inherits a bilinear pairing from the homomorphism space

structure and the quotients of the cell modules by this radical (where the quotient is not

zero) are a complete set of indecomposable modules.

These categories also admit an integral lattice, and through this, their modular represen-

tation theory can be studied. For example, we may take the indecomposable cell modules

arXiv:2210.09639v2 [math.RT] 29 Jan 2023

2 STUART MARTIN AND R. A. SPENCER

over characteristic zero

and study their decompositions in positive characteristic. This

decomposition is controlled, to the ﬁrst order, by the bilinear form on the cell modules.

For example, it gives a Jantzen-like ﬁltration.

Since the integral lattice forms a basis of these modules, even under specialisation, we

can study certain properties of the bilinear form over semisimple characteristic in order to

deduce properties over positive characteristic. One such property is the determinant of

the form, which this paper investigates.

In a semisimple cellular category, each object with a weight has an associated idempotent

known as a clasp. Indeed, if

is an object of weight

, there is a morphism

JWX∈End X

which is idempotent and which is killed by left- or right-composition by any morphism

factoring through an object of weight less than λ.

In this paper we will frequently refer to “simple” examples as both inspiration and

touchstones. The simplest example will be that of the Temperley-Lieb category,

. In this

category (which manifests when studying

Uq(sl2)

) the clasps are known as Jones-Wenzl

elements. They all exist, which is to say the category is semisimple iff

is not a root of

unity.

The diagrammatic category equivalent to the category of fundamental representations of

Uq(sln)

is known as the category of webs or spiders. The concepts were introduced by

Kuperberg for the rank 2 Cartan types [Kup96]. Type Awebs were developed by Cautis,

Kamnitzer and Morrison using a variation on Howe duality [CKM14]. Thereafter Bodish

et. al. extended the construction to type Cn[BERT21].

These presentations are in terms of generators and relations — a quotient of a certain

category by a tensor ideal. This doesn’t describe the category completely though, and

the problem of determining a basis for the morphism spaces was started by Elias (in type

) [Eli15], and continued by Bodish for type

[Bod20] and Bodish and Wu in type

G2[BW21].

Due to a very similar construction, it turns out that calculating clasps in the web

categories can be done nearly concurrently to ﬁnding bases. Indeed, we now have

constructions for clasps in type An[Eli15], C2[Bod21] and G2[BW21].

The familiar construction of Jones-Wenzl idempotents forms an important archetype

for these higher order clasps. Indeed, the Jones-Wenzl idempotents satisfy

(1.1) JWn+1

.=JWnJWn

.−[n]

[n+1]JWn

Two features of this construction generalise. Firstly, clasps are inductively deﬁned in terms

of smaller clasps. This is sometimes known as the “triple-clasp formula” (each of the two

morphisms in the right hand side of eq. (1.1) is hiding a further

JWn−1

) and comes from

plethysm rules.

Secondly, the factor [n]/[n+1]comes from a tracing rule:

(1.2)

JWn=[n+1]

[n]

JWn−1.

1In fact, the characteristic should be a ”semisimple mixed characteristic”.

CELL MODULES FOR TYPE AWEBS 3

Indeed, it is the reciprocal of the coefﬁcient of the identity diagram. In the more general

case, this trace is called an intersection form and calculating their numerical values can be

difﬁculy. Though these have been calculated for types

for

n≤

and

, in general

we must revert to a conjecture due to Elias for these constants.

In this paper we elucidate the connection between these intersection forms and the Gram

determinants of cell modules in the category. In particular we give expressions for all the

cell Gram determinants in terms of these intersection forms. We then go on to show that

Elias’ conjecture holds in type

, by ﬁnding pre-images of two distinguished bases across

the quantum skew Howe duality

While this paper concerns type

webs, it should be read with a more general as-

pect. Some results, such as lemma 3.1 hold in more general “object adapted cellular

categories” [Spe22]. These include web categories for other Coxeter types, and diagram-

matic Soergel bimodule categories. If such a category has a monoidal structure which

suitably interfaces with the category structure, then further constructions (such as those of

the ladder basis and triple-clasps) should generalise. However, a discussion of monoidal,

cellular categories is beyond the scope of this paper.

The remainder of this paper is laid out as follows. In section 2 we review the deﬁnition the

web category and its intersection forms, and introduce Elias’ conjecture for these forms.

Then, in section 3 we examine the cell module of this category and recount the ladder

basis. In section 4, new bases are found which give recurrant solutions to the determinant

of the cell module. We also determine an orthogonal basis. Finally, section 5 shows that a

previously-known basis of

Uq(glm)

is a pre-image of this orthogonal basis. Using this we

are able to conﬁrm Elias’ conjectured formula for the intersection forms.

2. THE CATEGORY OF TYPE AWEBS

Let us recall the deﬁnition of type

webs. This work was initiated by Kuperberg [Kup96]

who derived the diagrammatic form for all rank two Lie type, and completed by Cautis,

Kamnitzer and Morrison who gave diagrammatics for general type An[CKM14].

The aim is to study the representation theory of

Uq(sln)

over

Q(q)

. To do this, it will

be sufﬁcient to study the category of (tensor products of)

fundamental representations

Uq(sln)

, denoted

Fundn

. This category encodes the full category of ﬁnite dimensional

representations through its Karoubian envelope. The reader is directed to [Eli15,

3] for a

full description of the appropriate idempotents, known as clasps.

Remark 2.1.The reader may wish to overlook the quantum deformation on a ﬁrst reading.

The appropriate mental notational substitutions are

Uq(sln)7→ sln,q7→ 1, [n]7→ n.

The results that hold for undeformed

sln

often also hold for

Uq(sln)

under the above

notational substitution (see for example the deﬁnition of a Jones-Wenzl idempotent using

quantum numbers). However, this is not always straightforward. For example, the

representation theory of

sln

occurs as a limit of the theory of

Uq(sln)

q→

1. In this

paper, we make a point of keeping our results valid for the general case.

Recall that

Uq(sln)

has

n−

1 fundamental representations, namely

{Vi:=ViCn}n−1

i=1

Here

is given the standard action of

Uq(sln)

. In this way we can also consider

V0∼

4 STUART MARTIN AND R. A. SPENCER

Vn∼

. If

i+j=k

then there are natural

Uq(sln)

-maps

Vi⊗Vj→Vk

and

Vk→Vi⊗Vj

Note that these maps are not inverse.

The objects in the category

Fundn

are of the form

Vi1⊗Vi2⊗ · · · ⊗ Vik

and the morphisms

are generated (through composition and tensor product) by the aforementioned maps

between

Vi⊗Vj

and

. If

i=i1i2· · · ik

is a sequence of elements of

{

. . .

n−

}

we will

write Vifor Vi1⊗Vi2⊗ · · · ⊗ Vik.

Remark 2.2.Strictly, we should deﬁne the objects to also permit tensor factors of the duals

of the

, but since

V∗

i∼

=Vn−i

what we have constructed is sufﬁcient. In [Eli15] our

category is denoted by Fund+.

The composition of the map from

Vi⊗Vj

and the map from

Vi⊗Vj

again is

a multiple of the identity map on

. Indeed, let

[n]=(qn−q(n−1)/(q−q−1)

be the

-th

quantum number. Then if

[n]!= [n][n−1]· · · [1]

and

r]= [n]!/([r]![n−r]!)

, the multiple

is [k

j]=[k

i].

Remark 2.3.Our analysis will actually hold over any pointed ring wherein

is not a root of

unity. In these characteristics, all the quantum numbers are invertable and so the necessary

objects will be well deﬁned. When

is specialised to a root of unity, certain quantum

numbers and quantum binomial coefﬁcients will vanish, and the required clasps will not

be well deﬁned. See [MS21] for the correct idempotents in the A1case.

To provide an easier-to-work with category, we introduce the category of

webs

spiders

. This category denoted

Spn

is diagrammatic and is equivalent as a category to

Fundn. Thus we can think of these interchangeably.

The objects are taken from the set of ﬁnite words in letters

{

. . .

n−

}

and the object

Spn

corresponds to the object

Fundn

. As such, tensor product on words is simply

concatenation.

Morphisms are linear combinations of

diagrams2

. A diagram is an oriented, decorated,

planar graph with edges labelled by the letters

{

. . .

n−

}

. These graphs are taken up

to isotopy. Each nontrivial vertex is trivalent and if, when read left-to-right, one edge

splits into two, the sum of the outgoing labels equal that of the incoming. A similar result

holds for the reverse.

For example, the two classes of generating morphisms are represented as

(2.1) k

We take the precaution of removing all edges that are labelled with

or 0. Indeed, since

tensoring with

is a null operation, we lose no information on the object level. On the

morphism level, vertices with an nor 0 are turned into bivalent vertices (simply edges).

Technically we are working with the

ladder

incarnation of diagrams from [CKM14], however no generality

is lost

CELL MODULES FOR TYPE AWEBS 5

Example 2.1.

When

2so we are examining the representation theory of

Uq(sl2)

, the

well known category to consider is the Temperley-Lieb category. Here, the object set is simply

{1}∗' {•}∗'N.

Since

2, there is only a single label for edges and the only allowed vertex is bivalent.

However, the statement that graphs are taken up to isotopy means that we recover the usual

deﬁnition of the Temperley-Lieb category as planar matchings of points.

Example 2.2.

When considering

Uq(sl3)

there are two objects which are dual to each other. We

will often consider the set

−}∗

instead of

{

1, 2

}∗

. Instead of labeling the edges with

{

1, 2

}

we will orient them with the understanding that bivalent vertices have one incident edge of each

orientation (and hence can be taken up to isotopy or ignored) and trivalent vertices are all sources

or sinks.

An example diagram from (+ + −−−+) to (+ + +) might be

−

There are relations on the morphisms in this category, as suggested by the comment

after remark 2.3 but their exact forms are not important for our study. They can be found

in [CKM14].

2.1.

Weights.

Each fundamental representation

has highest weight a fundamental

dominant weight

. Viewed as a partition,

vi= (

, 0

n−i)

. For any object

Fund

, let

wt(x)

∑ivxi

. If this is viewed as a

1-part partition

(λ1

· · ·

λn

, 0

)

∑ixi

then

contains

λi−λi+1

copies of

. The set of all weights, denoted

, is equipped with the

dominance partial order.

2.2.

Compatible Families.

Let

λ∈Λ

and pick

ϕλ={ϕx,y:x→y}

be a set of morphisms

between objects of weight

. Let

J<λ

be the ideal spanned by morphisms that factor

through objects of weight less than λ. We say that ϕλis a compatible family if

•for each pair of objects, xand y, of weight λ, there is a unique ϕx,y:x→yin ϕλ,

•ϕx,y◦ϕz,x=ϕz,ymodulo J<λfor any three composable elements of ϕλ, and

•ϕx,x=idxmodulo J<λ.

The two examples of compatible families we will meet are clasps and neutral ladders.

Lemma 2.3. There is a compatible family of morphisms such that ϕx,x=idxexactly.

These morphisms are known as (families of) neutral ladders.

Proof.

We present the construction here and signpost the reader to the proof that it sufﬁces.

Set

ϕx,x=idx

. Now, suppose

and

differ in only two adjacent values, which is to say

x=x(0)· · · x(i−1)x(i)x(i+1)x(i+2)· · · x(k)

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CELLMODULESFORTYPEAWEBSSTUARTMARTINANDROBERTA.SPENCERABSTRACT.WeexaminethecellmodulesforthecategoryoftypeAnwebsandtheirnaturalcellularforms.Wemodifythebasesofthesemodules,asdescribedbyElias,toobtainanorthogonalbasisofeachcellmodule.Hence,wecalculatethedeterminantoftheGrammatrixwithrespecttosuchbases...

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CELL MODULES FOR TYPE AWEBS STUART MARTIN AND ROBERT A. SPENCER ABSTRACT .We examine the cell modules for the category of type Anwebs and their

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