Graded sum formula for A1-Soergel calculus and the nil-blob algebra.

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Graded sum formula for ˜

A1-Soergel calculus and the nil-blob algebra.

Marcelo Hern´andez Caro∗and Steen Ryom-Hansen†

Abstract

We study the representation theory of the Soergel calculus algebra ˜

w:= EndD(W,S)(w) over Cin type ˜

A1. We

generalize the recent isomorphism between the nil-blob algebra NBnand a diagrammatically deﬁnied subalgebra

wof ˜

wto deal with the two-parameter blob algebra. Under this generalization, the two parameters correspond

to the two simple roots for ˜

A1. Using this, together with calculations involving the Jones-Wenzl idempotents for

the Temperley-Lieb subalgebra of NBn, we obtain a concrete diagonalization of the matrix of the bilinear form

on the cell module ∆w(v) for ˜

w. The entries of the diagonalized matrices turn out to be products of roots for

A1. We use this to study Jantzen type ﬁltrations of ∆w(v) for ˜

w. We show that, at an enriched Grothendieck

group level, the corresponding sum formula has terms ∆w(sαv)[l(sαv)−l(v)], where [·] denotes grading shift.

1 Introduction

Cellular algebras were introduced by Graham and Lehrer in 1994 in the paper [15] as a framework for studying the

non-semisimple representation theory of many ﬁnite dimensional algebras,

The motivating examples for cellular algebras were the Iwahori-Hecke algebras of type Anand Temperley-Lieb

algebras, but it has since been realized that many other ﬁnite dimensional algebras ﬁt into this framework. For a

cellular algebra one has a family of cell modules {∆(λ)}, endowed with bilinear forms ⟨·,·⟩λ, that together control

the representation theory of the algebra in question. Unfortunately, the concrete analysis of these bilinear forms is

in general diﬃcult, but in this paper we give a non-trivial cellular algebra over Cfor which the bilinear forms ⟨·,·⟩λ

can in fact be diagonalized over an integral form of the algebra, thus solving all relevant questions concerning them,

and therefore, by cellular algebra theory, concerning the representation theory of the algebra itself.

Our cellular algebra has two origins. Firstly it arises in the diagrammatic Soergel calculus of the Coxeter system

(W, S) of type ˜

A1as the endomorphism algebra ˜

w:= End(w) of w:= stst· · · of length n, where S={s,t}. An

approach to Soergel calculus of universal Coxeter groups, in particular of type ˜

A1, has been developed recently by

Elias and Libedinsky in [10], see also [8]. For type ˜

A1this approach involves the two-colour Temperley-Lieb algebra

but unfortunately the two-colour Temperley-Lieb algebra only captures the degree zero part of ˜

w, whereas our

interests lie in the full grading on ˜

The second origin of our cellular algebra is as a certain idempotent truncation NBn−1of Martin and Saleur’s

blob-algebra from [28]. In [26] the algebras ˜

wand NBn−1were studied extensively and in particular presentations in

terms of generators and relations were found for each of them. Using this it was shown that there is an isomorphism

w∼

=NBn−1where AC

wis a natural diagrammatically deﬁned subalgebra of ˜

w, whose dimension is half the

dimension of ˜

w. On the other hand, we show in this paper that the representation theory of ˜

wcan be completely

recovered from the representation theory of AC

Similarly to the original blob-algebra, the diagrammatics for NBn−1is given by blobbed (marked) Temperley-

Lieb diagrams, as in the following examples

,(1.1)

although the rule for multiplying diagrams is diﬀerent. Following [26], we call NBn−1the nil-blob algebra, but in

fact NBn−1has also appeared in the literature under the name the dotted Temperley-Lieb algebra, see [35]. In [35]

it was shown that the associated dotted Temperley-Lieb category is equivalent to the Bar-Natan category of the

thickened annulus and this equivalence was used in the recent work [18] to construct a Kirby color for Khovanov

homology.

∗Supported in part by beca ANID-PFCHA/Doctorado Nacional/2019-21190827

†Supported in part by FONDECYT grant 1221112

arXiv:2210.03847v3 [math.RT] 8 Nov 2023

An important feature of ˜

wand NBn−1, and in fact of all cellular algebras appearing in this paper, is the fact

that they are Z-graded algebras, with explicitly given degree functions deﬁned in terms of the diagrams. They are

Z-graded cellular in the sense of Hu and Mathas, see [20].

In this paper our ﬁrst new result is a construction of integral forms Awand Bx,y

n−1for AC

wand NBn−1over the

two-parameter polynomial algebra R:= C[x, y] and a lift of the isomorphism AC

w∼

=NBn−1to Aw∼

=Bx,y

n−1. The

integral form Awis in fact already implicit in the setup for Soergel calculus, using the dual geometric realization

of the Coxeter group Wof type ˜

A1. Under this realization, the parameters xand ycorrespond to the two simple

roots for W. The integral form for NBn−1is also a well-known object, since it is simply the two-parameter blob-

algebra Bx,y

n−1with blob-parameter xand marked loop parameter y. Thus the novelty of our result lies primarily

in the isomorphism between these integral forms, which on the other hand has the quite surprising consequence of

rendering a Coxeter group theoretical meaning to the two blob algebra parameters for Bx,y

n−1, since they become

nothing but the simple roots for W.

Our main interests lies in the representation theory of Awwhich via the above isomorphism is equivalent to the

representation theory of Bx,y

n−1. For several reasons the representation theory of Bx,y

n−1is more convenient to handle.

Both algebras are cellular algebras with diagrammatically deﬁned cellular bases, but the straightening rules for

expanding the product of two cellular basis elements in terms of the cellular basis are easier in the Bx,y

n−1setting.

Secondly, there is a natural Temperley-Lieb subalgebra TLn−1of Bx,y

n−1whose associated restriction functor Res is

very useful for our purposes. We show in section 4 of our paper that Res maps a cell module ∆B

n−1(λ) for Bx,y

n−1to

a module with a cell module ﬁltration for TLn−1and that the sections of this ﬁltration induce a diagonalization of

the bilinear form ⟨·,·⟩B

n−1,λ on ∆B

n−1(λ).

This leaves us with the task of calculating the values of the bilinear form on these sections. For λ≥0 and

n−1 = 2k+λwe show that this task is equivalent to calculating the coeﬃcient of the identity

12λ

in the

expansion of

1k1

JWn−1

2 2

(1.2)

where JWn−1is the Jones-Wenzl idempotent for TLn−1. Similarly, for λ < 0 and n−1 = 2k+|λ|it is equivalent

to calculating the coeﬃcient of

12|λ|

in the expansion of

1k1

JWn−1

2 2 |λ|

(1.3)

The determination of these coeﬃcients constitutes the main calculatory ingredient of our paper and is done in

section 5. The result is given in Theorem 5.1 for λ≥0 and in Theorem 5.2 for λ < 0. Although one may possibly

not have expected Coxeter theory to appear in this calculation, the result turns out to be a nice product of positive

roots for W.

This work is partially motivated by the paper [36] by the second author in which a diagonalization of the

bilinear form for the cell module for ˜

Awis obtained, in fact the results of [36] are valid for a general Coxeter system.

Unfortunately, as was already mentioned in [36], the diagonalization process in that paper does not work over R

itself, but only over the fraction ﬁeld Qof R, since it relies on certain Jucys-Murphy elements for Awthat are of

degree 2, and not 0. As a consequence the Z-graded structure on the cell module for ˜

Awbreaks down under the

diagonalization process in [36]. The diagonalization process of the present paper, however, which is based on the

Jones-Wenzl idempotents that are of degree 0, resolves this problem at least for type ˜

A1.

In the ﬁnal section 7 of the paper we use the results of the previous sections to set up a version ∆gr,C

w(v)⊇

∆gr,1,C

w(v)⊇∆gr,2,C

w(v)⊇. . . of the Jantzen ﬁltration formalism for the graded cell module ∆gr,C

w(v), using the

bilinear form ⟨·,·⟩v

won ∆gr,C

w(v). Since ⟨·,·⟩v

wis a graded bilinear form, this ﬁltration consists of graded submodules

of ∆gr,C

w(v), and in our ﬁnal Corollary 7.12 we show that the following identity holds at enriched Grothendieck group

level X

k>0

⟨∆gr,k,C

W0(v)⟩q=X

α>0

v<sαv≤w

⟨∆gr,C

W0(sαv)[l(sαv)−l(v)]⟩q(1.4)

where α > 0 refers to the positive roots for Wand [·] to grading shift. This is our graded Jantzen sum formula.

Analogues of (ungraded) Jantzen ﬁltrations with associated sum formulas exist in many module categories of Lie

type and give information on the irreducible modules for the category in question, see for example [1], [2], [30]. But

although graded representation theories in Lie theory have been known since the beginning of the nineties, see for

example [3], [4], [12], [13], [19], [27] and [38], to our knowledge graded sum formulas in the sense of (1.4) have so

far not been available, even though they would be very useful for calculating decomposition numbers. We believe

(1.4) gives an interesting indication of the possible form of graded sum formulas in representation theory.

The layout of the paper is as follows. In the next section we introduce the notation that shall be used throughout

the paper and recall the various algebras that play a role throughout: the Temperley-Lieb algebra TLn, the blob

algebra Bx,y

n, the nil-blob algebra NBnand the Soergel algebra ˜

Aw. We also recall how each of them ﬁts into the

cellular algebra language. In section 3 we introduce the subalgebra Awof ˜

Awand show the isomorphism Bx,y

n−1∼

=Aw

that was mentioned above. We also show how the cellular algebra structure on ˜

Awinduces a cellular structure on

Awand that there is an isomorphism ∆B

n−1(λ)∼

=∆w(v) between the respective cell modules for Bx,y

n−1and Aw.

In section 4 we consider a natural ﬁltration of Res∆B

n(λ) where Res is the restriction functor from Bx,y

n−1-modules

to TLn−1-modules. We show that the Jones-Wenzl idempotents JWkfor TLkwhere k≤n−1 can be used to

construct sections for this ﬁltration and to diagonalize the bilinear form ⟨·,·⟩B

n−1,λ on ∆B

n−1(λ). In section 5 we prove

the key Theorems 5.1 and 5.2, that were already mentioned above. They allow us to give concrete expressions for

the diagonal elements of the matrix for ⟨·,·⟩B

n−1,λ, which, as already mentioned, turn out to be products of positive

roots αfor W. In section 6 we give a description of the reﬂections sαin Wthat correspond to the positive roots of

section 5. Finally, in section 7 we use the results of the previous sections to give the graded Jantzen ﬁltration with

corresponding graded sum formula.

The authors wish to express their gratitude to P. Wedrich for useful conversations and for pointing out that the

nil-blob algebra and the dotted Temperley-Lieb algebra are the same. They also wish to thank the two anonymous

referees for detailed reports that greatly helped improving the presentation and accuracy of the paper.

2 Blob algebras and Soergel calculus for ˜

Throughout we use as ground ﬁeld the complex numbers C, although several of our results hold in greater generality.

We set

R:= C[x, y].(2.1)

We consider Rto be a (non-negatively) Z-graded C-algebra via

deg(x) = deg(y)=2.(2.2)

In this paper we shall consider several diagram algebras. Possibly the oldest and most studied diagram algebra

is the Temperley-Lieb algebra. It arose in statistical mechanics in the seventies. In the present paper we shall use

the following variation of it.

Deﬁnition 2.1. The Temperley-Lieb algebra TLnwith loop-parameter −2is the R-algebra on the generators

U1,...,Un−1subject to the relations

i=−2Uiif 1≤i<n (2.3)

UiUjUi=Uiif |i−j|= 1 (2.4)

UiUj=UjUiif |i−j|>1.(2.5)

The blob algebra was introduced by Martin and Saleur in [28], as a a way of considering boundary conditions in

the statistical mechanical model of the Temperley-Lieb algebra. Since its introduction, the blob algebra has been

the subject of much research activity in mathematics as well as physics, see for example [14], [27], [29], [31], [32],

[33], [34]. In this paper, we shall use the following variation of it.

Deﬁnition 2.2. The two-parameter blob algebra Bx,y

n, or more precisely the blob algebra with loop-parameter −2,

marked loop parameter yand blob-parameter x, is the R-algebra on the generators U0,U1,...,Un−1subject to the

relations

i=−2Uiif 1≤i<n (2.6)

UiUjUi=Uiif |i−j|= 1 and i, j > 0 (2.7)

UiUj=UjUiif |i−j|>1 (2.8)

U1U0U1=yU1(2.9)

0=xU0.(2.10)

The nil-blob algebra NBn, that was introduced and studied extensively in [26], may be recovered from Bx,y

nvia

specialization, that is

NBn∼

=Bx,y

n⊗RC(2.11)

where Cis made into an R-algebra via x7→ 0 and y7→ 0. In other words, Bx,y

nmay be considered a deformation of

NBn, and in fact this shall be the point of view of the present paper.

Another interesting specialization of Bx,y

nis e

Tndeﬁned as e

Tn:= Bx,y

n⊗RCwhere Cis made into an R-algebra

via x7→ 1 and y7→ −2. Let I:= ⟨U0−1⟩be the two-sided ideal in e

Tngenerated by U0−1. Then

Tn:= e

Tn/I(2.12)

is the Temperley-Lieb algebra from Deﬁnition 2.1, but deﬁned over C.

Just as is the case for NBn, one easily checks that Bx,y

nis a Z-graded algebra.

Lemma 2.3. The rules deg(Ui)=0for i > 0and deg(U0) = 2 deﬁne a (non-negative) Z-grading on Bx,y

Proof. The relations are easily seen to be homogeneous with respect to deg.

As already indicated, TLnand Bx,y

nare diagram algebras. This fact plays an important role in our paper, and

let us brieﬂy explain it. The diagram basis for TLnconsists of Temperley-Lieb diagrams on npoints, which are

planar pairings between nnorthern points and nsouthern points of a rectangle. The diagram basis for Bx,y

nconsists

of blobbed (marked) Temperley-Lieb diagrams on npoints, or blob diagrams on npoints, which are marked planar

pairings between nnorthern points and nsouthern points of a rectangle, where only pairings exposed to the left

side of the rectangle may be marked, and at most once. There is thus a natural embedding of Temperley-Lieb

diagrams into blob diagrams. The multiplication D1D2of two diagrams D1and D2is given by concatenation of

them, with D2on top of D1. This concatenation process may give rise to internal marked or unmarked loops, as

well as diagrams with more than one mark. Internal unmarked loops are removed from a diagram by multiplying

it by −2, whereas internal marked loops are removed from a diagram by multiplying it by y. Finally, any diagram

with r > 1 marks on a diagram is set equal to the same diagram multiplied by xr−1, but with the (r−1) extra

marks removed. For example, for

D1=, D2= (2.13)

we have that

D1D2= = x2y

(2.14)

Later on, we shall give many more examples.

For the proof of the isomorphisms between Bx,y

nand its diagrammatic version, one may consult the appendix

of [9] or else adapt the more self-contained proof given in [26], and similarly for TLn. Under the isomorphism we

have that

17→ (2.15)

and that

U07→ ,Ui7→

(2.16)

The number of Temperley-Lieb diagrams and blob diagrams on npoints is the Catalan number 1

n+1 2n

nand 2n

n.

In particular TLnand Bx,y

nare free over Rof rank

rk TLn=1

n+ 12n

nand rk Bx,y

n=2n

n.(2.17)

All algebras considered in the present paper ﬁt into the general language of cellular algebras, introduced by

Graham and Lehrer.

Deﬁnition 2.4. Let Abe a ﬁnite dimensional algebra over a commutative ring kwith unity. Then a cellular basis

structure for Aconsists of a triple (Λ,Tab, C)such that Λis a poset, Tab is a function on Λwith values in ﬁnite

sets and C:`λ∈ΛTab(λ)×Tab(λ)→ A is an injection such that

{Cλ

st |s, t ∈Tab(λ), λ ∈Λ}

is a k-basis for A: the cellular basis for A. The rule (Cλ

st)∗:= Cλ

ts deﬁnes a k-linear antihomomorphism of Aand

the structure constants for Awith respect to {Cλ

st}satisfy the following condition with respect to the partial order:

for all a∈ A we have

aCλ

st =X

u∈Tab(λ)

rusaCλ

ut +lower terms

where lower terms means a linear combination of Cµ

ab where µ < λ and where rusa ∈k.

To make TLnﬁt into this language we choose k=R, Λ = Λn:= {n, n −2,...,1}(or Λ := Λn={n, n −

2,...,2,0}) if nis odd (or even), with poset structure inherited from Z. For λ∈Λnwe choose Tab(λ) to be

Temperley-Lieb half-diagrams with λpropagating lines, that is Temperley-Lieb diagrams on λnorthern and n

southern points in which each northern point is paired with a southern point. For s, t ∈Tab(λ) we deﬁne Cλ

st to be

the diagram obtained from gluing sand the horizontal reﬂection of t, with son the bottom. Here is an example of

this gluing process with n= 8 and s, t ∈Tab(2).

(s, t) =

!7→

(2.18)

Theorem 2.5. The above triple (Λn,Tab, C)makes TLninto a cellular algebra.

Proof: This follows directly from the deﬁnitions. □

To make Bx,y

nﬁt into the cellular algebra language we choose k=R, Λ = {Λ±n:= {±n, ±(n−2),...,±1}(or

Λ := Λ±n={±n, ±(n−2),...,±2,0}) if nis odd (or even), with poset structure given by λ<µif |λ|<|µ|or if

|λ|=|µ|and λ<µ. For example, for n= 6 we have

Λ±6:= {±6,±4,±2,0},0<−2<2<−4<4<−6<6 (2.19)

For λ∈Λ±nwe choose Tab(λ) to be blob half-diagrams with |λ|propagating lines, that is marked Temperley-Lieb

diagrams on |λ|northern and nsouthern points in which each northern point is paired with a southern point and

in which only non-propagating pairings exposed to the left side of the rectangle may be marked. For s, t ∈Tab(λ)

we deﬁne Cλ

st to be the diagram obtained from gluing sand the horizontal reﬂection of t, with son the bottom,

and marking the leftmost propagating line if λ < 0. Here is an example of this gluing process with n= 8 and

s, t ∈Tab(−2).

(s, t) =

!7→

(2.20)

With this notation we have the following Theorem.

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Gradedsumformulafor˜A1-Soergelcalculusandthenil-blobalgebra.MarceloHern´andezCaro∗andSteenRyom-Hansen†AbstractWestudytherepresentationtheoryoftheSoergelcalculusalgebra˜ACw:=EndD(W,S)(w)overCintype˜A1.Wegeneralizetherecentisomorphismbetweenthenil-blobalgebraNBnandadiagrammaticallydefiniedsubalgebraAC...

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Graded sum formula for A1-Soergel calculus and the nil-blob algebra.

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