RSK bases and Kazhdan-Lusztig cells

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by  K. N. Raghavan

RSK BASES AND KAZHDAN-LUSZTIG CELLS
K.N.RAGHAVAN,PREENASAMUEL,ANDK.V.SUBRAHMANYAM
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1
0 Abstract. From the combinatorial characterizations of the right, left, and
2 two-sidedKazhdan-Lusztigcellsofthesymmetricgroup,‘RSKbases’arecon-
structed for certain quotients by two-sided ideals of the group ring and the
l
u Heckealgebra. Applicationstoinvarianttheory,overvariousbaserings,ofthe
J generallineargroupandrepresentationtheory,bothordinaryandmodular,of
thesymmetricgrouparediscussed.
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2

]
T
1. Introduction: summary and organization of results
R
.
h The starting point of the work described in this paper is a question in classical
t invariant theory (§1.1). It leads naturally to questions about representations of
a
m the symmetric group over the complex numbers (§1.2, §1.3) and over algebraically
closedfields of positive characteristic(§1.5), and in turn to the computationof the
[
determinantofacertainmatrixencodingthemultiplicationofKazhdan-Lusztigba-

2 sis elements of the Hecke algebra (§1.6), using which one can recovera well-known
v
criterionfor the irreducibility of Specht modules over fields of positive characteris-
2
tic (§1.7).
4
8 For the sake of readability, we have tried, to the extent possible, to keep the
2 proofsofourresultsindependentofeachother. Sosections4–6canbereadwithout
.
2 reference to one another. This has of course come at the price of some repetition.
0
9 1.1. Motivation from invariant theory. We beginby recallinga basictheorem
0
of classical invariant theory. Let k be a commutative ring with identity and V a
:
v free k-module of finite rank d. Let GL(V) denote the group of k-automorphisms
i of V, and consider the diagonal action of GL(V) on V⊗n. Let S denote the
X n
symmetric group of bijections of the set {1,...,n} and kS the group ring of S
r n n
a with coefficients in k. There is a natural action of Sn on V⊗n by permuting the
factors: more precisely, (v ⊗···⊗v )·σ :=v ⊗···⊗v (all actions are on the
1 n 1σ nσ
right by convention). This action commutes with the action of GL(V), and so the
k-algebra map Θ : kS → End V⊗n defining the action of S has image in the
n n k n
space End V⊗n of GL(V)-endomorphisms of V⊗n.
GL(V)
We have the following result (see [5, Theorems 4.1, 4.2]):
Assumethefollowing: iff(X)isanelementofdegreenofthepoly-
nomial ring k[X] in one variable over k that vanishes as a function
on k, then f(X) is identically zero. (This holds for example when
2000 Mathematics Subject Classification. Primary: 05E10,05E15,20C08,20C30.
Key words and phrases. symmetric group, Hecke algebra, Kazhdan-Lusztig basis, RSK cor-
respondence, RSK-shape, Kazhdan-Lusztig cells, multilinear invariants, picture invariants, cell
module,Specht module,Gramdeterminant,Carterconjecture.
1
2 K.N.RAGHAVAN,PREENASAMUEL,ANDK.V.SUBRAHMANYAM
k is an infinite field, no matter what n is.) Then the k-algebra
homomorphism Θ maps onto End V⊗n and its kernel is the
n GL(V)
two-sided ideal J(n,d) defined as follows:
• J(n,d):=0 if d≥n;
• ifd<n,thenitisthetwo-sidedidealgeneratedbytheelement
y := (sgnτ)τ, where S is the subgroup of S
d τ∈Sd+1 d+1 n
consistPingofthepermutationsthatfixpoint-wisetheelements
d+2, ..., n, and sgnτ denotes the sign of τ.1
ThuskS /J(n,d)getsidentifiedwiththealgebraofGL(V)-endomorphismsofV⊗n
n
(under the mild assumptionon k mentioned above),andit is of invarianttheoretic
interest to ask:
Is there a natural choice of a k-basis for kS /J(n,d)?
n
Our answer:
Theorem 1. Let k be any commutative ring with identity. Those permutations σ
of S such that the sequence 1σ, ..., nσ has no decreasing sub-sequence of length
n
more than d form a basis for kS /J(n,d).
n
The proof of the theorem will be given in §4. It involves the Hecke algebra of the
symmetric group and its Kazhdan-Lusztig basis. Some further comments on the
proof can be found in §1.4.
The theorem enables us to:
• obtain a k-basis, closed under multiplication, for the subring of GL(V)-
invariants of the tensor algebra of V (§4.2).
• when k is a field of characteristic 0, to limit the permutations in the well-
known description ([24], [26]) of a spanning set for polynomial GL(V)-
invariants of several matrices (§4.2); or, more generally, to limit the per-
mutations in the description in [4] of a spanning set by means of ‘picture
invariants’ for polynomial GL(V)-invariants of severaltensors (§4.3).
1.2. A question about tabloid representations. Let us take the base ring k
in§1.1tobethefieldCofcomplexnumbers. ThentheidealJ(n,d)hasarepresen-
tation theoretic realization as we now briefly recall (see §5.2 for the justification).
Letλ(n,d)be the unique partitionofn withatmostdpartsthatis smallestinthe
dominanceorder(§2.2.1). Considerthe linearrepresentationofS onthe freevec-
n
torspaceCT generatedbytabloidsofshapeλ(n,d)(§2.5). TheidealJ(n,d)is
λ(n,d)
thekerneloftheC-algebramapCSn →EndCCTλ(n,d) definingthisrepresentation.
Replacing the special partition λ(n,d) above by an arbitrary one λ of n (§2.1)
and considering the C-algebra map ρλ : CSn → EndCCTλ defining the linear
representationof S on the space CT generated by tabloids of shape λ, we ask:
n λ
Is there a natural set of permutations that form a C-basis for the
group ring CS modulo the kernel of the map ρ ? Equivalently,
n λ
one could demand that the images of the permutations under ρ
λ
form a basis for the image.
1ThesubgroupSd+1 couldbetaken tobethat consistingofthepermutations thatfixpoint-
wiseanyarbitrarilyfixedsetofn−d−1elements.
RSK BASES AND KAZHDAN-LUSZTIG CELLS 3
Our answer:
Theorem 2. Permutations of RSK-shape µ, as µ varies over partitions that dom-
inate λ, form a C-basis of CS modulo the kernel of ρ :CS →EndCT .
n λ n λ
Thedominanceorderonpartitionsistheusualone(§2.2). TheRSK-shapeofaper-
mutation is defined in terms of the RSK-correspondence (§2.4). As follows readily
from the definitions, the shape of a permutation σ dominates the partition λ(n,d)
precisely when 1σ, ..., nσ has no decreasing sub-sequence of length exceeding d.
Thus, in the case when the base ring is the complex field, Theorem 1 follows from
Theorem 2.
The proof of the theorem will be given in §5. Like that of Theorem 1, it too
involves the Hecke algebra of the symmetric group and its Kazhdan-Lusztig basis.
Some further comments onthe proof canbe found in §1.4. The theoremholds also
over the integers and over fields of characteristic 0—as can be deduced easily from
the complex case (see §5.3)—but it is not true in general over a field of positive
characteristic: see Example 11.
1.3. A question regarding the irreducible representations of the symmet-
ric group. The question raised just above (in §1.2) can be modified to get one of
more intrinsic appeal. Given a partition λ of n, consider, instead of the action
ofSn ontabloids ofshape λ, the rightcellmodule R(λ)C inthe sense ofKazhdan-
Lusztig (§3.5), or, equivalently (see §8.3), the Specht module Sλ (§2.6). The right
C
cell modules are irreducible and every irreducible CS -module is isomorphic to
n
R(µ)C for some µ⊢n (§6.2).
TheirreducibilityofR(λ)C implies,byawell-knownresultofBurnside(see,e.g.,
[3, Chapter 8, §4, No. 3, Corollaire 1]), that the defining C-algebra map CS →
n
EndCR(λ)C is surjective. The dimension of R(λ)C (equivalently of SCλ) equals the
number d(λ) ofstandardtableaux of shape λ (§2.3.1, §6.2). Thus there exist d(λ)2
elements of CSn, even of Sn itself, whose images in EndCR(λ)C form a basis (for
EndCR(λ)C). We ask:
Is there is a natural choice of such elements of CS , even of S ?
n n
Our answer:
Theorem 3. Consider the Kazhdan-Lusztig basis elements of the group ring CS
n
indexedbypermutationsofRSK-shapeλ. TheirimagesunderthedefiningC-algebra
map CSn →EndCR(λ)C form a basis for EndCR(λ)C.
BytheKazhdan-LusztigbasiselementsofthegroupringCS ,wemeantheimages
n
inCS oftheKazhdan-LusztigbasiselementsoftheHeckealgebraofS underthe
n n
naturalmapsettingtheparametervalueto1(§3). TheRSK-shapeofapermutation
is defined using the RSK-correspondence (§2.4).
The theorem is proved in §6. Some comments on the proof of the theorem can
be found in §1.4.
We do not know a natural choice of elements of the group S itself whose
n
images in EndCR(λ)C are a basis. Permutationsof RSK-shape λ of course suggest
themselves, but they do not in general have the desired property (Example 16).
4 K.N.RAGHAVAN,PREENASAMUEL,ANDK.V.SUBRAHMANYAM
1.4. Comments on the proofs of Theorems 1–3. Properties of the Kazhdan-
LusztigbasisoftheHeckealgebraassociatedtothesymmetricgrouparethekeyto
the proofs, although the statements of Theorems 1 and 2 do not involve the Hecke
algebra at all. The relevant properties are recalled in two instalments: the first,
in §3, is the more substantial; the second, in §4.1, consists of further facts needed
more specifically for the proof of Theorem 1.
Theorem3followsbycombiningtheWedderburnstructuretheoryofsemisimple
algebras, as recalled in §6.1, with the following observation implicit in [14] and
explicitly formulated in §3.6:
AKazhdan-LusztigC-basiselementC killstherightcell(orequiv-
w
alentlySpecht)modulecorrespondingtoashapeλunlessλisdom-
inated by the RSK-shape of the indexing permutation w.
The observation in turn follows easily from the combinatorial characterizations of
the left, right, and two-sided Kazhdan-Lusztig cells in terms of the RSK-corres-
pondence and the dominance order on partitions (§3.4.1, §3.4.2). Our primary
source for these characterizations,which are crucial to our purpose, is [14].
Theorem 2 follows by combining the above observation with two well known
facts: the isomorphism of the right cell module with the Specht module and the
well-knowndecompositionintoirreduciblesofCT . AHeckeanalogueofTheorem2
λ
also holds: see Theorem 7 in the earlier version [25] of the present paper. A proof
of it parallel to the proof of Theorem 2 as in here can be given using results of [8].
The proof in [25] is different and more in keeping with the ideas developed here.
The main technical point in the proof of Theorem 1 is isolated as Lemma 7,
which is a two sided analogue of [21, Lemma 2.11]recalledbelow as Proposition6.
1.5. Analogue of Theorem 3 over fields of positive characteristic. The
HeckealgebraanditsKazhdan-Lusztigbasismakesenseoveranarbitrarybase(§3).
The cell modules and Specht modules are also defined and isomorphic over any
base (§3.5, §2.6, §8.3). Thus we can ask for the analogue of Theorem 3 over a field
ofarbitrarycharacteristic,keepinginmindofcoursethatthecellmodulesmaynot
be irreducible any longer. We prove:
Theorem 4. Let k be a field of positive characteristic p. Let λ be a partition of a
positive integer n no part of which is repeated p or more times. Suppose that the
right cell moduleR(λ) is irreducible. Consider theKazhdan-Lusztig basis elements
k
ofthegroupringkS indexedbypermutationsofRSK-shapeλ. Theirimagesunder
n
the defining k-algebra map kS →End R(λ) form a basis for End R(λ) .
n k k k k
The theorem is a special case of Theorem 23 proved in §11. Like the proof of
Theorem 3, that of Theorem 23 too uses the observation formulated in §3.6, but,
thegroupringkS beingnotnecessarilysemisimple,wecannotrelyonWedderburn
n
structure theory any more. Instead we take a more head-on approach:
Choosing a convenient basis of End R(λ) , we express as linear
k k
combinations of these basis elements the images in End R(λ) of
k k
the appropriate Kazhdan-Lusztig basis elements of kS . Denot-
n
ing by G(λ) the resulting square matrix of coefficients, we give an
explicit formula for its determinant detG(λ) .
k
RSK BASES AND KAZHDAN-LUSZTIG CELLS 5
In fact, we obtain a formula for detG(λ), where G(λ) is the analogous matrix of
coefficientsoveranarbitrarybaseandovertheHeckealgebra(ratherthanthegroup
ring): see §7 for details. We then need only specialize to get detG(λ) . Given the
k
formula,it is a relativelyeasy matterto geta criterionfor detG(λ) not to vanish,
k
thereby proving Theorem 23.
1.6. A hook length formula for the determinant of G(λ). To obtain the
formula for detG(λ), we first show that G(λ) has a nice form which enables us
to reduce the computation to that of the determinant of a matrix G(λ) of much
smaller size. We discuss how this is done.
The basis of EndR(λ) with respect to which the matrix G(λ) is computed sug-
gestsitself: R(λ) hasa basisconsistingofclassesofKazhdan-Lusztigelements C ,
w
where w belongs to a right cell of shape λ of S (§3.5); considering the endomor-
n
phisms which map one of these basis elements to another (possibly the same) and
kill the rest, we get the appropriate basis for EndR(λ). This means that the ma-
trix G(λ) encodes the multiplication table for Kazhdan-Lusztig basis elements C
w
indexed by permutations of RSK-shape λ, modulo those indexed by permutations
of lesser shape in the dominance order.
The special properties of the Kazhdan-Lusztig elements now imply that the
matrix G(λ), which is of size d(λ)2×d(λ)2 (where d(λ) is the number of standard
tableaux of shape λ), is a ‘block scalar’ matrix, i.e., when brokenup into blocks of
size d(λ)×d(λ), only the diagonalblocks are non-zero,and all the diagonalblocks
areequal. Denoting by G(λ) the diagonalblock,we arethus reducedto computing
the determinant of G(λ). The details of this reduction are worked out in §7.
The formula for the determinant of G(λ) is given in Theorem 18, the main
ingredients in the proof of which are formulas from [7] and [16]. The relevance of
thoseformulasto the presentcontextis notclearatfirstsight. They areaboutthe
determinant, denoted det(λ), of the matrix of a certain bilinear form, the Dipper-
James form, on the Specht module Sλ, computed with respect to the ‘standard
basis’ of Sλ; while G(λ) has to do with multiplication of Kazhdan-Lusztig basis
elements. The connection between det(λ) and detG(λ) is established in §9 (see
Equation (9.3.1)) using results of [21].
1.7. On the irreducibility of Specht modules. Finally,wediscussanotherap-
plication of the formula for the determinant of the matrix G(λ) introduced in §1.6.
Supposethatthe determinantdidnotvanishwhenthe Heckealgebrais specialized
to group ring and the scalars extended to a field k. Then, evidently, the images
in End R(λ) of the Kazhdan-Lusztig basis elements C , as w varies over permu-
k k w
tations of RSK-shape λ, form a basis for End R(λ) , which means in particular
k k
thatthedefiningmapkS →End R(λ) issurjective,andsoR(λ) isirreducible.
n k k k
In other words, the non-vanishing of detG(λ) in k gives a criterion for the ir-
reducibility of R(λ) (equivalently, of Sλ). The criterion thus obtained matches
k k
precisely the one conjectured by Carter and proved in [17, 16]. We thus obtain an
independent proof of the Carter criterion. The details are worked out in §11.
1.8. Acknowledgments. Thanks to the GAP program, computations performed
on which were useful in formulating the results; to the Abdus Salam International
Centre for Theoretical Physics, during a visit to which of one of the authors much
6 K.N.RAGHAVAN,PREENASAMUEL,ANDK.V.SUBRAHMANYAM
of this work was brought to completion; to the Skype program, using which the
authors were able to stay in touch during that visit.
2. Recall of some basic notions
We recall in this section the basic combinatorial and representation theoretic no-
tions that we need. We draw the reader’s attention to our definition of the RSK-
correspondence, which differs from the standard (as e.g. in [12, Chapter 4]) by a
flip. Throughout n denotes a positive integer.
2.1. Partitions and shapes. By a partition λ of n, written λ ⊢ n, is meant a
sequence λ ≥ ... ≥ λ of positive integers such that λ + ... +λ = n. The
1 r 1 r
integer r is the number of parts in λ. We often write λ = (λ ,...,λ ); sometimes
1 r
evenλ=(λ ,λ ,...). Whenthe latter notationis used, itis tobe understoodthat
1 2
λ =0 for t>r.
t
Partitionsofnareinbijectionwithshapes ofYoungdiagrams (orsimplyshapes)
with n boxes: the partition λ ≥... ≥λ corresponds to the shape with λ boxes
1 r 1
inthefirstrow,λ inthesecondrow,andsoon,theboxesbeingarrangedleft- and
2
top-justified. Here for example is the shape corresponding to the partition (3,3,2)
of 8:
Partitions are thus identified with shapes and the two terms are used interchange-
ably.
2.2. Dominance order on partitions. Given partitions µ = (µ ,µ ,...) and
1 2
λ=(λ ,λ ,...) of n, we say µ dominates λ, and write µDλ, if
1 2
µ ≥λ , µ +µ ≥λ +λ , µ +µ +µ ≥λ +λ +λ , ....
1 1 1 2 1 2 1 2 3 1 2 3
We write µ⊲λ if µDλ and µ6=λ. The partialorder D onthe set of partitions (or
shapes) of n will be referred to as the dominance order.
2.2.1. The partition λ(n,d). Given integers n and d, there exists a unique parti-
tion λ(n,d) ⊢ n that has at most d parts and is smallest in the dominance order
among those with at most d parts. For example, λ(8,2)=(3,3,2).
2.3. Tableaux and standard tableaux. A Young tableau, or just tableau, of
shape λ ⊢ n is an arrangement of the numbers 1, ..., n in the boxes of shape λ.
Thereare,evidently,n!tableauxofshapeλ. Atableauisrowstandard (respectively,
column standard) if in every row (respectively, column) the entries are increasing
left to right (respectively, top to bottom). A tableau is standard if it is both row
standardandcolumnstandard. Anexampleofastandardtableauofshape(3,3,2):
1 3 5
2 6 8
4 7
RSK BASES AND KAZHDAN-LUSZTIG CELLS 7
2.3.1. Thenumberofstandardtableaux. Thenumberofstandardtableauxofagiven
shapeλ⊢nisdenotedd(λ). Thereisawell-known‘hooklengthformula’forit[11]:
d(λ)=n!/ h , where β runs overall boxesof shape λ andh is the hook length
β β β
of the box βQwhich is defined as one more than the sum of the number of boxes to
the right of β and the number of boxes below β.
The hook lengths for the shape (3,3,2) are shown below:
5 4 2
4 3 1
2 1
Thus d(3,3,2)=8!/(5.4.2.4.3.1.2.1)=42.
2.4. The RSK-correspondenceand the RSK-shapeofa permutation. The
Robinson-Schensted-Knuth correspondence (RSK correspondence for short) is a
well-known procedure that sets up a bijection between the symmetric group S
n
andorderedpairsofstandardtableauxofthe same shapewithn boxes. We donot
recallhere the procedure, referringthe readerinstead to [12, Chapter 4]. It will be
convenient for our purposes to modify slightly the procedure described in [12].
Denoting by (A(w),B(w)) ↔ w the bijection of [12], what we mean by RSK
correspondence is the bijection (B(w),A(w)) ↔ w; since A(w) = B(w−1) and
A(w−1) = B(w) (see [12, Corollary on page 41]), we could equally well define our
RSK correspondenceas (A(w),B(w)) ↔w−1. The RSK-shape ofa permutation w
is defined to be the shape of either of A(w), B(w).
2.4.1. An example. The permutation (1542)(36) (written as a product of disjoint
cycles)hasRSK-shape(3,2,1). IndeeditismappedundertheRSKcorrespondence
in our sense to the ordered pair (A,B) of standard tableaux, where:
1 3 5 1 2 3
A= 2 4 B = 4 6
6 5
2.4.2. Remark. We have modified the standard definition of RSK-correspondence,
for it was the simplest way we could think up of reconciling the notationalconflict
amongthetwosetsofpapersuponwhichwerely:[6,21]and[14]. Permutationsact
ontherightintheformer—aconventionwhichwetoofollow—butontheleftinthe
latter. The direction in which they act makes a difference to statements involving
the RSK correspondence: most importantly for us, to the characterization of one-
sidedcells(see§3.4.1below). Thiscreatesaproblem: wecannotbequotingliterally
from both sets of sources without changing something. Altering the definition of
RSKcorrespondenceasaboveisthepathofleastresistance,andallowsustoquote
more or less verbatim from both sets.
2.5. Tabloids and tabloid representations. Letλ=(λ ,λ ,...)⊢n. Atabloid
1 2
of shape λ is a partition of the set [n] := {1,...,n} into an ordered r-tuple of
subsets, the first consisting of λ elements, the second of λ elements, and so on.
1 2
8 K.N.RAGHAVAN,PREENASAMUEL,ANDK.V.SUBRAHMANYAM
Depicted below are two tabloids of shape (3,3,2):
1 3 5 3 5 8
7 8 9 1 6
4 6 2 7
The members of the first subset are arranged in increasing order in the first row,
those of the second subset in the second row, and so on.
Given a tableau T of shape λ, it determines, in the obvious way, a tabloid of
shape λ denoted {T}: the first subset consists of the elements in the first row, the
second of those in the second row, and so on.
The defining action of S on [n] induces, in the obvious way, an action on the
n
setT oftabloidsofshapeλ. The free Z-module ZT withT as aZ-basisprovides
λ λ λ
therefore a linear representation of S over Z. By base change we get such a
n
representation over any commutative ring with unity k: kTλ :=ZTλ⊗Zk. We call
it the tabloid representation corresponding to the shape λ.
2.6. Specht modules. The Specht module corresponding to a partition λ ⊢ n
is a certain S -submodule of the tabloid representation ZT just defined. For a
n λ
tableau T of shape λ, define e in ZT by
T λ
e := sgn(σ){Tσ}
T
X
where the sum is taken over permutations σ of S in the column stabiliser of T,
n
sgn(σ) denotes the sign of σ, and {Tσ} denotes the tabloid corresponding to the
tableau Tσ in the obvious way (see §2.5). The Specht module Sλ is the linear span
ofthee asT runsoveralltableauxofshapeλ. ItisanS -submoduleofZT with
T n λ
Z-basise ,asT variesoverstandardtableaux(see,forexample,[12,§7.2]). Bybase
T
change we get the Specht module Sλ over any commutative ring with identity k:
k
Skλ :=Sλ⊗Zk. Evidently,Skλisafreek-moduleofrankthenumberd(λ)ofstandard
tableaux of shape λ (§2.3.1).
3. Set up: Hecke algebra and Kazhdan-Lusztig cells
Letn denotea fixedpositiveintegerandS the symmetricgrouponn letters. Let
n
S be the subset consisting of the simple transpositions (1,2), (2,3), ..., (n−1,n)
of the symmetric group S . Then (S ,S) is a Coxeter system in the sense of [3,
n n
Chapter 4]. Let A:=Z[v,v−1], the Laurent polynomial ring in the variable v over
the integers.
3.1. The Hecke algebra and its T-basis. Let H be the Iwahori-Hecke algebra
corresponding to (S ,S), with notation as in [14]. Recall that H is an A-algebra:
n
it is a free A-module with basis T , w ∈S , the multiplication being defined by
w n
T if ℓ(sw)=ℓ(w)+1
T T = sw
s w (cid:26) (v−v−1)Tw+Tsw if ℓ(sw)=ℓ(w)−1
for s∈S and w ∈S and ℓ is the length function. We put
n
ǫ(w):=(−1)ℓ(w) and v :=vℓ(w) for w ∈S .
w n
RSK BASES AND KAZHDAN-LUSZTIG CELLS 9
An induction on length gives (in any case, see [6, Lemma 2.1 (iii)] for (3.1.2)):
(3.1.1) TwTw′ =Tww′ if ℓ(w)+ℓ(w′)=ℓ(ww′).
(3.1.2) TuTu′ =Tuu′ + awTw for u, u′ in Sn
uuX′<w
Here,aselsewhere,<denotesthe Bruhat-ChevalleypartialorderonS . Inpartic-
n
ular, the coefficient of T1 in TuTu′ is non-zero if and only if u′ =u−1 and equals 1
in that case.
3.1.1. Therelationbetweenvandq. Wefollowtheconventionsof[14]. Inparticular,
to pass from our notation to that of [18], [6], or [21], we need to replace v by q1/2
and T by q−ℓ(w)/2T .
w w
3.1.2. Specializations of the Hecke algebra. Let k be a commutative ring with unity
and a an invertible element in k. There is a unique ring homomorphism A → k
definedbyv 7→a. WedenotebyH thek-algebraH⊗ kobtainedbyextendingthe
k A
scalars to k via this homomorphism. We have a natural A-algebra homomorphism
H → H given by h 7→ h⊗1. By abuse of notation, we continue to use the same
k
symbols for the images in H of elements of H as for those elements themselves.
k
If M is a (right) H-module, M ⊗ k is naturally a (right) H -module.
A k
An important special case is when we take a to be the unit element 1 of k. We
then have a natural identification of H with the groupring kS , under which T
k n w
maps to the permutation w in kS .
n
Regarding the semisimplicity of H , we have this result [7, Theorem 4.3]:
k
Assuming k to be a field, H is semi-simple except precisely when
k
• either a2 =1 and the characteristic of k is ≤n
• or a2 6=1 is a primitive rth root of unity for some 2≤r ≤n.
3.1.3. Two ring involutions and an A-antiautomorphism. We use the following two
involutions on H both of which extend the ring involution a 7→ a of A defined by
v 7→v :=v−1:
a T := a T−1 j( a T ):= ǫ a T (a ∈A)
w w w w−1 w w w w w w
X X X X
These commute with each other and so their composition, denoted h 7→ h†, is
an A-algebra involution of H. The A-algebra anti-automorphism ( a T )∗ =
w w
awTw−1 allows passing back and forth between statements aboutPleft cells and
oPrders and those about right ones (§3.4.1).
3.2. Kazhdan-Lusztig C′- and C-basis. Two types of A-bases for H are in-
troduced in [18], denoted {C′ |w ∈ S } and {C |w ∈ S }. They are uniquely
w n w n
determined by the respective conditions [20, Theorem 5.2]:
C′ =C′ and C′ ≡T mod H
w w w w <0
(3.2.1)
C =C and C ≡T mod H
w w w w >0
where
H := A T , A :=v−1Z[v−1] H := A T , A :=vZ[v]
<0 <0 w <0 >0 >0 w >0
wX∈Sn wX∈Sn
10 K.N.RAGHAVAN,PREENASAMUEL,ANDK.V.SUBRAHMANYAM
The anti-automorphismh7→h∗ and the ring involutionh7→h commute with each
other, so that, by the characterization(3.2.1):
(3.2.2) (Cx)∗ =Cx−1 (Cx′)∗ =Cx′−1
We have by [18, Theorem 1.1]:
(3.2.3) C′ =T + p T C =T + ǫ ǫ p T
w w y,w y w w y w y,w y
y∈SXn,y<w y∈SXn,y<w
where < denotes the Bruhat-Chevalleyorder onS , and p ∈A for all y <w,
n y,w <0
from which it is clear that
(3.2.4) C =ǫ j(C′ )
w w w
Combining (3.2.1) with (3.2.4), we obtain
(3.2.5) C =ǫ (C′ )†
w w w
3.2.1. Notation. For a subset S of S , denote by hC |y ∈ Si the A-span in H
n y A
of {C |y ∈ S}. For an A-algebra k, denote by hC |y ∈ Si the k-span in H of
y y k k
{C |y ∈S}. Similar meanings are attached to hT |y ∈Si and hT |y ∈Si .
y y A y k
3.2.2. A simple observation. From (3.2.3), we get T ≡ C modhT |x < wi .
w w x A
Fromthisinturnweget,byinductionontheBruhat-Chevalleyorder,thefollowing:
forasubsetS ofS , the (imagesof) elementsT , w ∈S \S, forma basisforthe
n w n
A-module H/hC |x ∈ Si . The same thing holds also in specializations H of H
x A k
(§3.1.2): the (images of) elements T , w ∈ S \S, form a basis for the k-module
w n
H /hC |x∈Si .
k x k
3.3. Kazhdan-Lusztig orders and cells. Let y and w in S . Write y← w if,
n L
forsome elements in S,the coefficientofC is non-zeroin the expressionofC C
y s w
as a A-linear combination of the basis elements C . Replacing all occurrences of
x
‘C’ by ‘C′’ in this definition would make no difference. The Kazhdan-Lusztig left
pre-order is defined by: y≤ w if there exists a chain y = y ← ···← y = w;
L 0 L L k
the left equivalence relation by: y∼ w if y≤ w and w≤ y. Left equivalence
L L L
classesarecalledleft cells. Notethat AC isaleftidealcontainingtheleft
x≤Lw x
ideal HCw. P
Right pre-order, equivalence, and cells are defined similarly. The two sided pre-
order is defined by: y≤ w if there exists a chain y = y , ..., y = w such that,
LR 0 k
for 0 ≤ j < k, either y ≤ y or y ≤ y . Two sided equivalence classes are
j L j+1 j R j+1
called two sided cells.
3.4. Cells and RSK Correspondence. We now recall the combinatorial char-
acterizations of one and two sided cells in terms of the RSK correspondence (§2.4)
andthedominanceorderonpartitions(§2.2). Thesestatementsarethefoundation
on which this paper rests. The ones in §3.4.2, 3.4.2 are used repeatedly, but the
more subtle one in §3.4.3 is used only once, namely in the proof of Theorem 1: it
is used in the proof of Lemma 7 which is the main ingredient in the proof of that
theorem.
Write (P(w),Q(w)) for the ordered pair of standard Young tableaux associated
to a permutation w by the RSK correspondence (in our sense—see §2.4). Call
P(w) the P-symbol andQ(w) the Q-symbol ofw. It will be convenientto use such