Table Of ContentRSK BASES IN INVARIANT THEORY AND
REPRESENTATION THEORY
By
Preena Samuel
The Institute of Mathemati
al S
ien
es, Chennai.
A thesis submitted to the
Board of Studies in Mathemati
al S
ien
es
In partial ful(cid:28)llment of the requirements
For the Degree of
DOCTOR OF PHILOSOPHY
of
HOMI BHABHA NATIONAL INSTITUTE
August 2010
Homi Bhabha National Institute
Re
ommendations of the Viva Vo
e Board
As members of the Viva Vo
e Board, we re
ommend that the dissertation prepared by
Preena Samuel entitled (cid:16)RSK Bases in invariant theory and representation theory(cid:17) may
be a
epted as ful(cid:28)lling the dissertation requirement for the Degree of Do
tor of Philoso-
phy.
Date :
Chairman : V. S. Sunder
Date :
Convener : K. N. Raghavan
Date :
Member : Parameswaran Sankaran
Date :
Member : Amritanshu Prasad
Date :
External Examiner : Dipendra Prasad
Final approval and a
eptan
e of this dissertation is
ontingent upon the
andidate's
submission of the (cid:28)nal
opies of the dissertation to HBNI.
I hereby
ertify that I have read this dissertation prepared under my dire
tion and
re
ommend that it may be a
epted as ful(cid:28)lling the dissertation requirement.
Date :
Guide : K. N. Raghavan
DECLARATION
I, hereby de
lare that the investigation presented in the thesis has
been
arried out by me. The work is original and the work has not
been submitted earlier as a whole or in part for a degree/diploma
at this or any other Institution or University.
Preena Samuel
ACKNOWLEDGEMENTS
A taskofthisnature
annot attain a
omplishment without thee(cid:27)orts ofmany people.
Yet, the mention of a few here does not atta
h (cid:16)order of preferen
e(cid:17).
My supervisor, K. N. Raghavan, brilliantly led me through the (cid:28)eld of Representation
Theory and the aspe
ts of resear
h in general. He has shown remarkable patien
e with me
all through these years and has always been a sour
e of en
ouragement for me. Inspite
of his other pressing obligations he
arefully went through various versions of this thesis
and in the pro
ess he has also instilled in me the importan
e of e(cid:27)e
tive writing. Grateful
a
knowledgement is made to his unparalleled guidan
e.
Myinstru
torsatIMS
(cid:21)J.Iyer,S.Kesavan,K.Maddaly, D.S.Nagaraj, K.Paranjape,
P. Sankaran and K. Srinivas (cid:21) who introdu
ed me to the unfamiliar fa
ets of mathemati
s
in my initial years of resear
h. I feel privileged to a
knowledge that the light they beamed
into my path has been immeasurable. I thank Prof. D. S. Nagaraj for the many hours he
most-willingly spent in an extremely prolonged reading
ourse that I took with him.
I would like to spe
ially mention my thanks to Dr. K. V. Subrahmanyam (CMI), who
I had the good fortune of working with. His a(cid:30)nity to mathemati
s has inspired me. I
also thank him for introdu
ing me to the GAP program whi
h led us to many interesting
results presented in this thesis.
I would like to thank the Dire
tor (IMS
.), Prof. R. Balasubramanian, for having en-
abled my stay at this institute. I also thank IMS
. for providing a
ondu
ive ambien
e for
the pursuit of resear
h. Mats
ien
e has also been instrumental in providing generous sup-
port for my various a
ademi
visits abroad. Thanks also to the library and administrative
sta(cid:27) of the institute, whose e(cid:27)orts have made my stay here so mu
h more
omfortable.
Ire
ordmygratitudetomyfriendsand
olleaguesatIMS
,whohavebeenmosthelpful
and patient. The
ompany of Pratyusha, Sunil, Gaurav, Anupam, Muthukumar, et al.
provided interesting interludes. Anusha, my wonderful friend, was partially instrumental
in my pursuing Mathemati
s.
I amindebted tomy familyin ways far morethan I
an express. Without theirsupport
and prayers, this thesis would not be.
Before
on
luding I respe
tfully remember Late Prof. C. Musili, my tea
her, who initi-
ated me topursue resear
h. He not only gave methe best
lasses Itook during my masters
programme but also the best memories I have of my days at the University of Hyderabad.
During the
ourse of my pilgrimage through the not-so polished terrains of mathe-
mati
s, I sin
erely a
knowledge what I have re
eived from whoever I
ame a
ross as well
as whoever was dire
tly in my res
ue, is mu
h more than I
an ever re
ompense. These
words of thankful a
knowledgement, I am
onvin
ed will not liberate me from my indebt-
edness to them.
Abstra
t
From the
ombinatorial
hara
terizations of theright, left, and two-sided Kazhdan-Lusztig
ells of the symmetri
group, `RSK bases' are
onstru
ted for
ertain quotients by two-
sided ideals of the group ring and theHe
ke algebra. Appli
ations to invariant theory, over
various base rings, of the general linear group and representation theory, both ordinary
and modular, of the symmetri
group are dis
ussed.
Contents
List of Publi
ations iii
1 Introdu
tion 1
2 Kazhdan-Lusztig
ells and their
ombinatori
s 8
W S
2.1 Coxeter System ( , ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
W S
2.2 He
ke Algebra
orresponding to ( , ) . . . . . . . . . . . . . . . . . . . . 9
H
2.2.1 Kazhdan - Lusztig bases of . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Kazhdan-Lusztig orders and
ells . . . . . . . . . . . . . . . . . . . . 14
S
n
2.3 Combinatori
s of
ells in . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Basi
notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 RSK-
orresponden
e . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.3 Cells and RSK-
orresponden
e . . . . . . . . . . . . . . . . . . . . . 19
2.3.4 Some notes and notations . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Spe
ht modules, Cell modules: An introdu
tion 26
3.1 A short re
ap of the stru
ture theory of semisimple algebras . . . . . . . . . 26
S
n
3.2 Some -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
H
3.3 Some -modules: De(cid:28)nitions and preliminaries . . . . . . . . . . . . . . . . 29
Mλ
3.3.1 Permutation modules . . . . . . . . . . . . . . . . . . . . . . . . 30
Sλ
3.3.2 Spe
ht modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Mλ ZT
λ
3.3.3 Relating with . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.4 Monomial module . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.5 Cell modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 M
Donough - Pallikaros Isomorphism . . . . . . . . . . . . . . . . . . . . . . 38
Sλ Sλ′
3.5 Interplay between and . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 RSK bases for
ertain quotients of the group ring 42
4.1 Key observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
C T
4.1.1 Moving from -basis to -basis of
ertain quotients . . . . . . . . . 43
C R(λ)
4.1.2 Images of the -basis elements in End . . . . . . . . . . . . . . 43
S
n
4.2 Tabloid representations of . . . . . . . . . . . . . . . . . . . . . . . . . . 43
i
C
4.2.1 Results over . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Z
4.2.2 Results over . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.3 Failure over (cid:28)elds of positive
hara
teristi
. . . . . . . . . . . . . . . 45
4.3 Remarks on the He
ke Analogue of Ÿ4.2.2 . . . . . . . . . . . . . . . . . . . 45
4.4 Certain rings of invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4.1 Multilinear invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4.2 A monomial basis for the tensor algebra . . . . . . . . . . . . . . . . 50
4.4.3 Rings of polynomial invariants . . . . . . . . . . . . . . . . . . . . . 51
5 Cell modules: RSK basis, irredu
ibility 56
R(λ)
k
5.1 RSK Bases for End . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.1 An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . 57
H
k
5.1.2 The
ase of semisimple . . . . . . . . . . . . . . . . . . . . . . . . 57
H
k
5.1.3 The
ase of arbitrary . . . . . . . . . . . . . . . . . . . . . . . . . 58
G(λ)
5.2 The matrix and its determinant . . . . . . . . . . . . . . . . . . . . . . 59
G(λ)
5.2.1 De(cid:28)nition of the matrix . . . . . . . . . . . . . . . . . . . . . . 59
G(λ) R(λ)
5.2.2 Relating the matrix to the a
tion on . . . . . . . . . . . . 60
R(λ)
5.2.3 The Dipper-James bilinear form on . . . . . . . . . . . . . . . 61
R(λ)
k
5.3 A
riterion for irredu
ibility of . . . . . . . . . . . . . . . . . . . . . . 62
5.4 Proofs of Theorems 5.1.3, 5.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . 62
e
5.4.1 Condition of -regularity in Theorem 5.1.3 . . . . . . . . . . . . . . . 62
G(λ)
6 A formula for det and an appli
ation 64
G(λ)
6.1 Relating det and the Gram determinant . . . . . . . . . . . . . . . . . 64
G(λ)
6.2 Hook Formula for the determinant of . . . . . . . . . . . . . . . . . . . 65
6.3 A new proof for Carter's
onje
ture . . . . . . . . . . . . . . . . . . . . . . . 66
6.4 Proof of the hook formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
(λ)
6.4.2 Computing the Gram determinant det . . . . . . . . . . . . . . . 69
6.4.3 Proof of Theorem 6.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 72
ii
List of Publi
ation(s)
•
K. N. Raghavan, P. Samuel, K. V. Subrahmanyam, RSK bases and Kazhdan-Lusztig
ells, preprint, 2010, URL http://arxiv.org/pdf/0902.2842
iii
Chapter 1
Introdu
tion
The starting point of the work
arried out in this thesis is a question in
lassi
al invariant
theory. It leads naturally to questions about representations of the symmetri
group over
the
omplex numbers and over (cid:28)elds of positive
hara
teristi
, and inturn to the
omputa-
tionofthedeterminant ofa
ertainmatrix en
oding themultipli
ation ofKazhdan-Lusztig
basis elements of the He
ke algebra, using whi
h one
an re
over a well-known
riterion
for the irredu
ibility of Spe
ht modules over (cid:28)elds of positive
hara
teristi
.
n S n k
n
Let denote a (cid:28)xed integer and be the symmetri
group on letters. Let denote
V d k
a
ommutative ring with unity and a free module of (cid:28)nite rank over .
k V GL(V)
Consider the group of -linear automorphisms of , denoted , a
ting diagonally
V⊗n n S V⊗n
n
on the spa
e of -tensors. The symmetri
group also a
ts on by permuting
σ ∈ S
n
the fa
tors: the a
tion of on pure tensors is given by
(v ⊗···⊗v )σ := v ⊗···⊗v ,
1 n 1σ nσ
iσ i σ
where denotes the image of under the a
tion of (whi
h we assume to a
t from
GL(V) φ : kS →
n n
the right). This a
tion
ommutes with the -a
tion and so, the map
V⊗n S V⊗n V⊗n
k n GL(V)
End de(cid:28)ning the a
tion of on has image in End (cid:21) the spa
e of
GL(V) V⊗n
-invariantendomorphismsof . A
lassi
alresultininvarianttheory(see[dCP76,
V⊗n
GL(V)
Theorems 4.1, 4.2℄) states that this map is a surje
tion onto End . The result
k k
further states that, under a mild
ondition on (whi
h holds for example when is an
J(n,d)
in(cid:28)nite (cid:28)eld), thekernel isequal to (cid:22)the two-sided ideal generated by the element
y := (τ)τ S S
d τ∈Sd+1sign where d+1 is the subgroup of n
onsisting of the permutations
d + 2 n n ≤ d J(n,d) 0
P
that (cid:28)x point-wise the elements , ..., ; when , is de(cid:28)ned to be .
kS /J(n,d) GL(V)
n
Thus, the quotient gets identi(cid:28)ed with the algebra of -endomorphisms
V⊗n
of .
kS /J(n,d)
n
Itis,therefore,ofinvariant-theoreti
interesttoobtainabasisforthequotient .
Indeed our (cid:28)rst main result provides su
h a basis (see Theorem 4.4.1):
k
Theorem 1.0.1 Let be an arbitrary
ommutative ring with unity. With notations as
1
σ S 1σ nσ
n
above, the permutations of su
h that the sequen
e , ..., has no de
reasing
d kS /J(n,d) 2
n
subsequen
e of length more than , form a basis for .
The proof of the theorem involves the He
ke algebra of the symmetri
group and its
C S A
n
Kazhdan-Lusztig -basis(seeŸ2.2 for de(cid:28)nitions). The He
ke algebra of over , where
A Z[v,v−1] S
n
is the Laurent polynomial ring , is a (cid:16)deformation(cid:17) of the group ring of over
A H k a k
. Wedenoteitas . Fora
ommutativering withunityand aninvertibleelementin ,
H k H⊗ k A→ k
k A
we denote by the -algebra de(cid:28)ned by the unique ring homomorphism
v 7→ a S
n
givenby . TheHe
kealgebra isadeformation ofthegroupalgebraof inthesense
k H v 7→ 1 kS
k n
that the -algebra , de(cid:28)ned by the map , is isomorphi
to the group algebra .
A
The main te
hni
al part of the proof of Theorem 1.0.1 is in des
ribing an -basis for
H v 7→ 1 J(n,d)
an appropriate two-sided ideal in that spe
ializes, under the map , to (see
Lemma 4.4.2). The key ingredient, in turn, in proving this is the
ombinatori
s asso
iated
to the Kazhdan-Lusztig
ells (Ÿ2.2.2).
k C
Ifwe take the base ring in the above dis
ussion to be the (cid:28)eld of
omplex numbers.
J(n,d)
Thentheideal asde(cid:28)nedabovehasarepresentationtheoreti
realizationnamely,let
λ(n,d) n d
be the unique partition of with at most parts that is smallest in the dominan
e
S CT
n λ(n,d)
order (Ÿ2.3.1);
onsider the linear representation of on the free ve
tor spa
e
λ(n,d) J(n,d) C
generated bytabloidsofshape (Ÿ3.2), theideal isthekernelofthe -algebra
CSn → CCTλ(n,d)
map End de(cid:28)ning this representation.
λ(n,d) n λ
Repla
ing the spe
ial partition above by an arbitrary partition of , say , and
C ρλ : CSn → CCTλ
onsidering the -algebra map End de(cid:28)ning the linear representation
S CT λ
n λ
of on the spa
e generated by tabloids of shape , we ask:
C
Is there a natural set of permutations that form a -basis for the group ring
CS ρ
n λ
modulo the kernel of the map ?
This question is addressed by the following theorem (see Theorem 4.2.1):
µ µ
Theorem 1.0.2 Permutations of RSK-shape , as varies over partitions that domi-
λ C CS ρ : CS → CT
n λ n λ
nate , form a -basis of modulo the kernel of End .
The dominan
e order on partitions is the usual one (Ÿ2.3.1). The RSK-shape of a permu-
tation is de(cid:28)ned in terms of the RSK-
orresponden
e (Ÿ2.3.2). As follows readily from the
σ λ(n,d)
de(cid:28)nitions, the shape of a permutation dominates the partition pre
isely when
1σ nσ d
, ..., has no de
reasing sub-sequen
e of length ex
eeding . Thus, in the
ase when
the base ring is the
omplex (cid:28)eld, Theorem 1.0.1 follows from Theorem 1.0.2.
The observation mentioned in Ÿ4.1.2 plays a key role in proving the above theorem.
The above result holds, as we observe in Ÿ4.2.2, even after extending s
alars to any (cid:28)eld
0 Z
of
hara
teristi
essentially owing to the fa
t that su
h a (cid:28)eld is (cid:29)at over . Further, by
an example (Ÿ4.2.3) we illustrate that these results are not true in general over (cid:28)elds of
positive
hara
teristi
.
2
