Table Of ContentGentle introduction to Soergel bimodules I:
Thebasics
NICOLASLIBEDINSKY
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2 Abstract This paper is the first of a series of introductory papers on the
fascinatingworldofSoergelbimodules. Itiscombinatorialinnatureand
n
a shouldbeaccessibletoabroadaudience. Theobjectiveofthispaperisto
J help the reader feel comfortable calculating with Soergel bimodules and
1 toexplainsomeoftheimportantopenproblemsinthefield. Themotiva-
3 tions,historyandrelationstootherfieldswillbedevelopedinsubsequent
papersofthisseries.
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1 Introduction
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1.1 Declarationofintent
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9 As said in the abstract, this paper is an introduction to Soergel bimodules.
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We give many examples and show explicit calculations with the Hecke alge-
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0 braandtheHeckecategory(oneofitsincarnationsbeingSoergelbimodules).
0 Most of the other Hecke categories (categorifications of the Hecke algebra)
.
2 suchascategory O,Elias-Williamsondiagrammaticcategory,Sheavesonmo-
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ment graphs, 2-braid groups and Parity Sheaves over Schubert varieties, are
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1 leftforfollow-upsofthispaper. Theapplicationsofthistheoryarealsoleftfor
: thefollow-ups.
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SoergelbimoduleswereintroducedbyWolfgangSoergel[So2]intheyear92’,
r although many of the ideas were already present in his 90’ paper [So1]. In
a
thosepapersheexplaineditsrelationstorepresentationsofLiegroups. Inthe
year00’heproved[So4]alinkbetweenthem(atthattimehecalledthem“Spe-
cialbimodules”,andalthoughtheyarequitespecial,apparentlytheyaremore
Soergelthanspecial)andrepresentationsofalgebraicgroupsinpositivechar-
acteristicthatprovedtobeextremelydeep. Inhis07’paper[So5]hesimplified
manyargumentsandprovedsomenewthings. Afterthispaper...
WARNING: Thefollowingsectionisjustintendedtoimpressthereader.
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1.2 ...Bum! Explosionofthefield
Soergelbimoduleswere(andare)intheheartofanexplosionofnewdiscover-
ies in representation theory, algebraic combinatorics, algebraic geometry and
knot theory. We give a list of some results obtained using Soergel bimodules
inthelastfiveyears.
(1) Algebraicgroups: AdisproofofLusztig’sconjecturepredictingthesim-
ple characters of reductive algebraic groups (1980). This was probably
themostimportantopenconjectureinrepresentationtheoryofLie-type
objects.
(2) Lie algebras: An algebraic proof of Kazhdan-Lusztig conjectures pre-
dicting the multiplicities of simple modules in Verma modules (1979)
for complex semi-simple Lie algebras. A geometric proof was given in
theearly 80’s, butwe hadto wait 35years tohavean algebraicproofof
analgebraicproblem.
(3) Symmetricgroups: AdisproofofJamesconjecturepredictingthechar-
acters of irreducible modular representations for the symmetric group
(1990).
(4) Combinatorics: Aproofoftheconjectureaboutthepositivityofthecoef-
ficientsofKazhdan-LusztigpolynomialsforanyCoxetersystem(1979).
Thiswasamajoropencombinatorialproblem.
(5) Algebraicgeometry: AdisproofoftheBorho-BrylinskiandJosephchar-
acteristiccyclesconjecture(1984).
(6) Combinatorics: A proof of the positivity of parabolic Kazhdan-Lusztig
polynomialsforanyCoxetersystemandanyparabolicgroup.
(7) Knot theory: A categorification of Jones polynomials and HOMFLYPT
polynomials.
(8) Higherrepresentationtheory: AdisproofoftheanalogueforKLRalge-
brasofJamesconjecture,byKleschevandRam(2011).
(9) Liealgebras: AnalgebraicproofofJantzen’sconjectureabouttheJantzen
filtrationinLiealgebras(1979).
(10) CombinatoricsAproofoftheMonontonicityconjecture(1985aprox.)
(11) Combinatorics: A proof of the Unimodality of structure constants in
Kazhdan-Lusztigtheory.
Wewillstartthisstorybythefirstofthreelevels,theclassicalone.
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1.3 Acknowledgements
ThisworkissupportedbytheFondecytproject1160152andtheAnilloproject
ACT1415PIAConicyt. TheauthorwouldliketothankverywarmlyMacarena
Reyes for her hard work in doing most of the pictures. I would also like to
thank David Plaza, Paolo Sentinelli, Sebastia´n Cea and Antonio Behn for de-
tailedcommentsandsuggestions.
2 Classical level: Coxeter systems
2.1 Somedefinitions
ACoxetermatrixisasymmetricmatrixwithentriesin {1,2,...}∪{∞},diago-
nalentries 1 andoff-diagonalentriesatleast 2.
Definition 1 A pair (W,S), where W is a group and S is a finite subset of
W, is called a Coxeter system if W admits a presentation by generators and
relationsgivenby
(cid:104)s ∈ S |(sr)msr = e if s,r ∈ S andm is finite(cid:105),
sr
where (m ) isaCoxetermatrixand e istheidentityelement.
sr s,r∈S
We then say that W is a Coxeter group. One can prove that in the Coxeter
system defined above, the order of the element sr is m (it is obvious that it
sr
divides m ). The rank of the Coxeter system is the cardinality of S. If s (cid:54)= r,
sr
therelation (sr)msr = e isequivalentto
srs··· = rsr···
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
msr msr
This is called a braid relation. On the other hand, as m = 1, we have that
ss
s2 = e. This is called a quadratic relation. An expression of an element x ∈ W,
is a tuple x = (s,r,...,t) with s,r,...,t ∈ S such that x = sr···t. The
expressionisreducedifthelengthofthetupleisminimal. Wedenote l(x) this
length.
2.2 Babyexamples
We start with two baby examples of Coxeter systems. In these two cases (as
well as in examples (C) and (D)) we will calculate explicitly the two key ob-
jectsinthetheory,namelytheKazhdan-Lusztigbasisandtheindecomposable
Soergelbimodules.
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(A) The group Symm(∆) of symmetries of an equilateral triangle is isomor-
phictothegroupwith 6 elements
(cid:104)s,r |s2 = r2 = e, srs = rsr(cid:105).
Onepossibleisomorphismisgivenbythemap
(B) Thegroup Symm((cid:3)) ofsymmetriesofasquareisisomorphictothegroup
with 8 elements
(cid:104)s,r |s2 = r2 = e, srsr = rsrs(cid:105).
Oneisomorphismbetweenthesegroupsisgivenby
2.3 Generalizingthebabyexamples: (infinite)regularpolygons
Onenaturalwaytogeneralizethebabyexamplesistoconsiderthesymmetries
ofaregular n-sidedpolygon. ThisisalsoafiniteCoxetergroupdenoted I (n)
2
(the subindex 2 in this notation denotes the rank of the Coxeter system, as
definedinSection2.1). Apresentationofthisgroupisgivenby
(C) I (n) = (cid:104)s,r |s2 = r2 = e, (sr)n = e(cid:105),
2
4
so Symm(∆) ∼= I (3) and Symm((cid:3)) ∼= I (4). Again, one isomorphism here is
2 2
given by sending s to some reflection and r to any of the “closest reflections
toit”
Figure1: Examplewith n = 8
What is the Infinite regular polygon, the “limit” in n of the groups I (n)? A
2
reasonable way to search for a geometric limit of the sequence of n−sided
regular polygons, is to picture this sequence having “the same size” as in the
figure
Figure2: Theincorrectmentalimage
Ifwedoso,the(pointwise)limitisacircle. Ontheotherhand,thelimitwhen
n goestoinfinityof I (n) isalgebraicallyclearifweconsiderthepresentation
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giveninExample(C).Itistheinfinitegroup
U = (cid:104)s,r |s2 = r2 = e(cid:105).
2
But the geometric and algebraic descriptions given here do not coincide! The
group Symm((cid:13)) of symmetries of the circle (usually called the orthogonal
group O(2)) is not even finitely generated. We have passed to a continuous
group! Thebestwecandoistosee U asadensesubgroupof Symm((cid:13)) (just
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consider s and r tobetwo”random”reflectionsofthecircle).
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There is a beautiful way to solve this problem: there is a discrete geometric
limit of the sequence of n−sided regular polygons. Let us picture this se-
quencewithonefixedside(inthefigure,thedarkerone). Letussupposethat
thefixedsidehasverticesinthepoints (0,0) and (0,1) oftheplane.
Figure3: Anewsequence
Then, if we make a “close up” around the darker side, and we put all the
polygonstogether,wesee
Figure4: Thefigureopenslikeflower
R Z
The(pointwise)limitis withtheverticesconvergingtotheset . Thesym-
metriesofthisgeometricobject(thatwecall (R,Z))isisomorphicto U ! One
2
isomorphismisgivenbysending s tothereflectionthrough 0 and r tothere-
flectionthrough 1/2 (again“theclosestreflection”). Nowourgeometriclimit
andthealgebraiclimitcoincideandwecanregainourlostcalm.
Forourfourthexample,wejustjustraisetherankof U .
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(D) TheUniversalCoxetersystemofrank n isthegroup
U = (cid:104)s ,s ,...,s |s2 = s2 = ··· = s2 = e(cid:105)
n 1 2 n 1 2 n
This is the most complicated family of groups in which one can still compute
alloftheKazhdan-Lusztigtheoryexplicitly.
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2.4 Generalizingthebabyexamples: tesselations
We will denote by [n,p,q] the Coxeter system with simple reflections S =
{s,r,t} andwithCoxetermatrixgivenby m = n,m = p and m = q.
sr st rt
2.4.1 TesselationsoftheEuclideanplane
Anaturalwaytogeneralizetheequilateraltriangleistoconsiderthefollowing
tesselationoftheEuclideanplanebyequilateraltriangles
Figure5: [6,3,2]
This tesselation can be generalized by coloring this tiling as in Figure 6 or by
tilingtheplanewithother(maybecolored)regularconvexpolygonsasitisthe
checkboardofFigure7orthehoneycombofFigure8.
Figure6: [3,3,3] Figure7: [4,4,2] Figure8: [6,3,2]
Notethe(curious?) equalitiesobtainedbyaddingthecorrespondinginverses
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of n,p and q inthelastfigures:
1 1 1 1 1 1 1 1 1
+ + = + + = + + = 1
3 3 3 4 4 2 6 3 2
2.4.2 Tesselationsofthehyperbolicplane
It is quite fantastic the amount of tesselations by triangles of the hyperbolic
plane. HerewegivesomeexamplesusingthePoincare´ diskmodel.
Figure9: [7,3,2] Figure10: [6,6,6] Figure11: [∞,∞,∞,]
Figure10iscalledbysome(atleastbyme)“Devil’stesselation”.
Ingeneral,thereisatesselationbytrianglesofthehyperbolicplanewithgroup
ofsymmetries [n,p,q] ifandonlyif
1 1 1
+ + < 1. (1)
n p q
In particular, if n > 3 and p,q ≥ 3 then this inequality is satisfied. So most
rank three Coxeter groups are the group of symmetries of some hyperbolic
tiling. ThoseCoxetergroupswhicharenot,areeitherthegroupofsymmetries
of a tesselation by triangles of the Euclidean plane1 where inequality (1) is
changedbyanequalityortheyarethegroupofsymmetriesofatesselationby
trianglesofthespherelikethefollowing
1Essentially those that can be constructed from Figures 6, 7 and 8. The groups
ofsymmetriesappearinginthisfashionarejustthethreegroups [3,3,3],[4,4,2] and
[6,3,2].
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Figure12: [2,3,3] Figure13: [2,3,4] Figure14: [2,3,5]
Inthesphericalcasethereversedinequalityissatisfied
1 1 1
+ + > 1. (2)
n p q
sothetriples [n,p,q] thatappearherearethe“littlenumbers”(wejustneedto
addtoFigures12,13and14,thegroups [2,2,n] for n ≥ 2).
2.5 Generalizingthebabyexamples: moredimensions
If we raise the rank, we can generalize our baby examples in a different way.
Wegivethe n-analogueofthebabyexamples.
(1) TheWeylgroupoftype A . Itcanbedefinedasthesymmetriesofan n-
n−1
simplex,orequivalently,as S ,thesymmetricgroupin n elements. The
n
isomorphism between these two groups is obvious. It admits a Coxeter
presentationgivenbygenerators s ,1 ≤ i < n andrelations
i
• s2 = e forall i.
i
• s s = s s if |i−j| ≥ 2
i j j i
• s s s = s s s if |i−j| = 1 (i.e. i = j ±1)
i j i j i j
The isomorphism from this group to the symmetric group is given by
sending s tothetransposition (i,i+1) ∈ S .
i n
(2) The Weyl group of type BC . It is the group of symmetries of an n-
n−1
hypercube. Ithasorder 2nn!
(3) Moregenerally,allsymmetrygroupsofregularpolytopesarefiniteCox-
etergroups. Dualpolytopeshavethesamesymmetrygroup.
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2.6 Moreexamples
• Type A and type B groups are examples of Weyl groups. These groups
appearinthetheoryofLiealgebrasasthegroupsofsymmetriesofroot
systems associated to semisimple Lie algebras over the complex num-
bers (so they are examples of finite reflection groups). There are three
infinitefamiliesofWeylgroups,types A ,BC and D (n ∈ N)andthe
n n n
exeptional groups of type E , E , E , F , G . They are also symmetry
6 7 8 4 2
groups of regular or semiregular polytopes. For example, this figure is
aprojectionintheplaneofan 8-dimensionalsemiregularpolytopewith
symmetrygroup E
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• The complete list of finite Coxeter groups is also known. Apart from
the Weyl groups there are the groups H (symmetries of pentagon), H
2 3
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