Rigged configurations and the $*$-involution

0.26 MB·

by  Ben Salisbury

RIGGED CONFIGURATIONS AND THE ∗-INVOLUTION
6 BENSALISBURYANDTRAVISSCRIMSHAW
1
0
2 Abstract. We give an explicit description of the ∗-involution on the rigged
configuration modelforB(∞).
r
a
M

2
1. Introduction

] The ∗-involution, sometimes referred to as Kashiwara’s involution, is an involu-
O
tiononthecrystalB(∞)thatisinducedfromasubtleinvolutiveantiautomorphism
C
of U (g). The importance of ∗ in the theory of crystal bases and their applications
q
.
h cannot be understated. Here are just a few of its applications.
t
a (1) Saito [35] used the involution during the proof that Lusztig’s PBW basis
m has a crystal structure isomorphic to B(∞), provided that g is a finite-
[ dimensional semisimple Lie algebra.
(2) Kamnitzer andTingley [12]generalizethe definition ofthe crystalcommu-
3
tor ofHenriques andKamnitzer[4]interms ofthe ∗-involution. This leads
v
to a proof by Savage [40] that the category of crystals forms a coboundary
7
3 category over any symmetrizable Kac-Moody algebra.
1 (3) In affine type A, Jacon and Lecouvey [8] prove that ∗ coincides with the
6 Zelevinsky involution [24, 48] on the set of simple modules for the affine
0
Hecke algebra.
.
1
Several combinatorial realizations of the ∗-involution are known in the literature.
0
Forexample,Lusztig[23]gaveadescriptionofthe behaviorof∗onLusztig’sPBW
6
1 basis in the finite types, Kamnitzer [10] showed that ∗ acts on an MV polytope
: by negation, Kashiwara and Saito [16] gave a description of ∗ in terms of quiver
v
varieties [16], and Jacon and Lecouvey [8] give a description of the involution in
i
X terms of the multisegment model. Such model-specific calculations of the ∗-crystal
r operators are important as, a priori, the algorithm for computing the action of
a
these operators is not efficient [14, Thm. 2.2.1] (see also [15, Prop. 8.1]).
Inthispaper,theauthorscontinuetheirdevelopmentoftheriggedconfiguration
model of B(∞) [37, 39]. Rigged configurations are sequences of partitions, one for
every node of the underlying Dynkin diagram, where each part is paired with an
integer, satisfying certain conditions. These objects arose as an important tool in
mathematicalphysicsfromthe studies ofthe Bethe Ansatz byKerov,Kirillov,and
Reshetikhin [17, 18], and they have been shown to correspond to the action and
angle variables of box-ball systems [19]. Additionally, rigged configurations have
been used extensively in the theory of Kirillov-Reshetikhin crystals [2, 3, 27, 29,
2010 Mathematics Subject Classification. 05E10,17B37.
Keywords and phrases. crystal,riggedconfiguration, ∗-involution.
B.S.waspartiallysupportedbyCMUEarlyCareergrant#C62847.
T.S.waspartiallysupportedbyRTGgrantNSF/DMS-1148634.
1
2 BENSALISBURYANDTRAVISSCRIMSHAW
30, 36, 41, 42, 43, 45, 46]. During the course of this study, a crystal structure was
given to rigged configurations [42, 43].
Our description of ∗ is as nice as one could hope: in contrast to the definition
of e and f on rigged configurations, one interchanges “label” and “colabel” to
a a
obtain a definition of e∗ and f∗ (see Definition 4.1). In turn, applying ∗ to a
a a
rigged configuration replaces all labels with its corresponding colabels and leaves
the partitions fixed (see Corollary 4.14).
The method of proof applied here is to use a classification theorem of B(∞) as-
sertedby TingleyandWebster[47]bytranslatingthe∗-involutiondirectlyintothe
classification theorem of Kashiwara and Saito [16] without the use of Kashiwara’s
embedding. This classification theorem requires several assertions to be satisfied,
and proving these assertions hold in RC(∞) with our new ∗-crystaloperators con-
sumes most of Section 4.
The (conjectural) bijection Φ between U′(g)-rigged configurations and tensor
q
products of Kirillov-Reshetikhin crystals [27, 28, 29, 30, 31, 41, 43, 44, 46] is given
roughly as follows. It removes the largest row with a colabel of 0, which is the
minimal colabel, for each e from b to the highest weight element in B(Λ ), where
a 1
b is the leftmost factor in the tensor product. Let θ be the involution on U′(g)-
q
riggedconfigurations which interchanges labels with colabels on classically highest
weight U′(g)-rigged configurations and let ∗L denote the involution which is the
q
compositionofLusztig’sinvolutionandthemapsendingtheresulttotheclassically
highest weight element [41, 44] (where it isealso denoted by ∗). It is known that
Φ◦θ =∗L◦Φ on classically highest weight elements. In particular,the latter map
reversesthe order of the tensor product. Thus, given the description of the crystal
commuteor,ourworksuggeststhereisastronglinkbetweenthe∗-involutionandthe
bijection Φ. We hope this could lead to a more direct description of the bijection
Φ, its related properties, and a (combinatorial) proof of the X = M conjecture
of [2, 3].
Another model for B(∞) uses marginally large tableaux, as developed by Hong
and Lee [6, 7]. It is known that the bijection Φ mentioned above can be extended
to a U (g)-crystalisomorphismbetween riggedconfigurationsandmarginallylarge
q
tableaux[38]whengisoffiniteclassicaltypeortypeG . Anambitioushopeofthis
2
paper is that it may lead to a description of the ∗-crystal structure on marginally
large tableaux. (In finite type A, this result is in [1].) However, this appears to be
ahardproblemasthe bijection Φ is highly recursiveanddepends onconditions on
colabels, many of which can change under applying the ∗-crystaloperators.
There is also a model for B(∞) using Littelmann paths constructed by Li and
Zhang [21]. From [32], natural virtualization maps arise to the embeddings on the
underlyinggeometricinformation. Thevirtualizationmaponriggedconfigurations
isalsoquite natural,givingevidencethatriggedconfigurationsencode moregeom-
etrythantheircombinatorialoriginsanddescriptionsuggests. Thisisalsoevidence
that there exists a straightforwardand natural explicit combinatorialbijection be-
tween rigged configurations and the Littelmann path model. Thus this work could
potentially lead to a description of the ∗-crystalon the Littelmann path model.
In a similar vein, the virtualization map is known to act naturally on MV poly-
topes [9, 25], also reflecting the geometric information of the root systems via the
Weyl group. This is evidence that there should be a natural explicit combinato-
rialbijectionbetweenMVpolytopesandriggedconfigurations(andtheLittelmann
RIGGED CONFIGURATIONS AND THE ∗-INVOLUTION 3
pathmodel). Moreover,consideringthe∗-involution,whichactsbynegationonMV
polytopes [10,11], this workgivesfurther evidence that suchabijection shouldex-
ist. Furthermore,thisbijectionwouldsuggestanaturalgeneralizationbeyondfinite
type, which the authors expect to recover the KLR polytopes of [47].
Thispaperisorganizedasfollows. InSection2,wegivethenecessarybackground
on crystals and the ∗-involution. In Section 3, we give background information on
the rigged configuration model for B(∞). In Section 4, we give the proof of our
maintheoremandsomeconsequences. InSection5,wegiveadescriptionofhighest
weight crystals using the ∗-crystal structure and describe the natural projection
from B(∞) in terms of rigged configurations.
2. Crystals and the ∗-involution
Let g be a symmetrizable Kac-Moodyalgebrawith quantizeduniversalenvelop-
ingalgebraU (g)overQ(q), indexsetI,generalizedCartanmatrixA=(A ) ,
q ij i,j∈I
weight lattice P, root lattice Q, fundamental weights {Λ : i ∈ I}, simple roots
i
{α : i ∈ I}, and simple coroots {h : i ∈ I}. There is a canonical pairing
i i
h , i: P∨ × P −→ Z defined by hh ,α i = A , where P∨ is the dual weight
i j ij
lattice.
An abstract U (g)-crystal is a set B together with maps
q
e ,f : B −→B⊔{0}, ε ,ϕ : B −→Z⊔{−∞}, wt: B −→P
i i i i
satisfying certain conditions (see [5, 15]). Any U (g)-crystal basis, defined in the
q
classical sense (see [13]), is an abstract U (g)-crystal. In particular, the negative
q
half U−(g) of the quantized universal enveloping algebra of g has a crystal basis
q
which is an abstract U (g)-crystal. We denote this crystal by B(∞) (rather than
q
the using the entire tuple (B(∞),e ,f ,ε ,ϕ ,wt)), and denote its highest weight
i i i i
element by u . As a set, one has
∞
B(∞)={f ···f f u :i ,...,i ∈I, d≥0}.
id i2 i1 ∞ 1 d
The remaining crystal structure on B(∞) is
wt(f ···f f u )=−α −α −···−α ,
id i2 i1 ∞ i1 i2 id
ε (b)=max{k ∈Z:e b6=0},
i i
ϕ (b)=ε (b)+hh ,wt(b)i.
i i i
We say that b∈B(∞) has depth d if b=f ···f f u for some i ,...,i ∈I.
id i2 i1 ∞ 1 d
There is a Q(q)-antiautomorphism ∗: U (g)−→U (g) defined by
q q
E 7→E , F 7→F , q 7→q, qh 7→q−h.
i i i i
ThisisaninvolutionwhichleavesU−(g)stable. Thus,themap∗inducesamapon
q
B(∞), which we also denote by ∗, and is called the ∗-involution or star involution
(and is sometimes known as Kashiwara’sinvolution). Denote by B(∞)∗ the image
of B(∞) under ∗.
Theorem 2.1 ([14, 22]). We have B(∞)∗ =B(∞).
This induces a new crystal structure on B(∞) with Kashiwara operators
e∗ =∗◦e ◦∗, f∗ =∗◦f ◦∗,
i i i i
and the remaining crystal structure is given by
ε∗ =ε ◦∗, ϕ∗ =ϕ ◦∗,
i i i i
4 BENSALISBURYANDTRAVISSCRIMSHAW
and weight function wt, the usual weight function on B(∞). Additionally, for
b∈B(∞) and i∈I, define
κ (b):=ε (b)+ε∗(b)+hh ,wt(b)i. (2.1)
i i i i
This was called the i-jump in [20].
We will appeal to the following statement from [1], which was proven in a
dual form in [47] based on Kashiwara and Saito’s classification theorem for B(∞)
from [16]. First, a bicrystal is a set B with two abstract U (g)-crystal structures
q
(B,e ,f ,ε ,ϕ ,wt) and (B,e⋆,f⋆,ε⋆,ϕ⋆,wt) with the same weight function. In
i i i i i i i i
such a bicrystal B, we say b∈B is a highest weight element if e b=e⋆b=0 for all
i i
i∈I.
Proposition 2.2. Fix a bicrystal B with highest weight b such the crystal data is
0
determined by setting wt(b ) = 0. Assume further that, for all i 6= j in I and all
0
b∈B,
(1) f b, f⋆b6=0;
i i
(2) f⋆f b=f f⋆b;
i j j i
(3) κ (b)≥0;
i
(4) κ (b)=0 implies f b=f⋆b;
i i i
(5) κ (b)≥1 implies ε⋆(f b)=ε⋆(b) and ε (f⋆b)=ε (b);
i i i i i i i
(6) κ (b)≥2 implies f f⋆b=f⋆f b.
i i i i i
Then
(B,ei,fi,εi,ϕi,wt)∼=(B,e⋆i,fi⋆,ε⋆i,ϕ⋆i,wt)∼=B(∞),
with e⋆ =e∗ and f⋆ =f∗.
i i i i
However, we will need to slightly weaken the assumptions of Proposition 2.2.
Proposition 2.3. Let (B,e ,f ,ε ,ϕ ,wt) and (B⋆,e⋆,f⋆,ε⋆,ϕ⋆,wt) be highest
i i i i i i i i
weight abstract U (g)-crystals with the same highest weight vector b ∈ B ∩B⋆,
q 0
where the remaining crystal data is determined by setting wt(b )=0. Suppose also
0
that (1)–(6) are satisfied. Then
(B,ei,fi,εi,ϕi,wt)∼=(B⋆,e⋆i,fi⋆,ε⋆i,ϕ⋆i,wt)∼=B(∞),
with e⋆ =e∗ and f⋆ =f∗.
i i i i
Proof. We prove that B∩B⋆ is closed under f and f⋆ by using induction on the
i i
depthandmakingrepeateduseofconditions(1)–(6)above. Thebasecaseisdepth
0,where we just haveb . Suppose allelements of depth atmostd in B and B⋆ are
0
in B∩B⋆. Next, fix some b ∈B∩B⋆ at depth d. If κ (b)=0, then f b =f⋆b for
i i i
all i∈I. If i6=j, then f⋆f b′ =f f⋆b′. Hence f∗b∈B∩B⋆, and f b∈B∩B⋆ by
i j j i i j
our induction assumption and that B (resp., B⋆) is closed under f (resp., f⋆). A
j i
similar argument shows that f b ∈ B∩B⋆ if κ (b)≥ 1 since κ (b′) ≥ 2. Therefore
i i i
B = B ∩B⋆ = B⋆ since B ∩B⋆ is closed under f and f⋆ and generated by b
i j 0
(along with B and B⋆). Thus the claim follows by Proposition 2.2.
3. Rigged configurations
Let H =I ×Z . A rigged configuration is a sequence of partitions ν =(ν(a) :
>0
a ∈ I) such that each row ν(a) has an integer called a rigging, and we let J =
i
J(a) :(a,i)∈H , where J(a) is the multiset of riggingsof rows of length i in ν(a).
i i
We consider there to be an infinite number of rows of length 0 with rigging 0; i.e.,
(cid:0) (cid:1)
RIGGED CONFIGURATIONS AND THE ∗-INVOLUTION 5
J(a) = {0,0,...} for all a ∈ I. The term rigging will be interchanged freely with
0
the term label. We identify two rigged configurations (ν,J) and (ν,J) if
J(a) =J(a)
i i e e
for any fixed (a,i)∈H. Let (ν,J)(a) denote the rigged partition (ν(a),J(a)).
e
Define the vacancy numbers of ν to be
p(a)(ν)=p(a) =− A min(i,j)m(b), (3.1)
i i ab j
(b,Xj)∈H
where m(a) is the number of parts of length i in ν(a). The corigging, or colabel, of
i
arowin(ν,J)(a) with riggingxis p(a)−x. Inaddition, wecanextendthe vacancy
i
numbers to
p(a) = lim p(a) =− A |ν(b)|
∞ i ab
i→∞
b∈I
X
since ∞ min(i,j)m(b) = |ν(b)| for i ≫ 1. Note this is consistent with letting
j=1 j
i=∞ in Equation (3.1).
P
Let RC(∞) denote the set of rigged configurations generated by (ν ,J ), where
∅ ∅
ν(a) =0 for all a∈I, and closed under the crystal operators as follows.
∅
Definition 3.1. Fix some a∈I, and let x be the smallest rigging in (ν,J)(a).
e : If x = 0, then e (ν,J) = 0. Otherwise, let r be a row in (ν,J)(a) of
a a
minimal length ℓ with rigging x. Then e (ν,J) is the rigged configuration
a
which removesa box from rowr, sets the new rigging of r to be x+1, and
changes all other riggings such that the coriggings remain fixed.
f : Letrbearowin(ν,J)(a) ofmaximallengthℓwithriggingx. Thenf (ν,J)
a a
is the riggedconfigurationwhich adds a box to row r, sets the new rigging
of r to be x−1, and changes all other riggings such that the coriggings
remain fixed.
We define the remainder of the crystal structure on RC(∞) by
ε (ν,J)=max{k∈Z:ek(ν,J)6=0}, ϕ (ν,J)=hh ,wt(ν,J)i+ε (ν,J),
a a a a a
wt(ν,J)=− |ν(a)|α .
a
a∈I
X
From this structure, we have p(a) =hh ,wt(ν,J)i for all a∈I.
∞ a
Theorem 3.2 ([37, 39]). Let g be of symmetrizable type. Then RC(∞) ∼= B(∞)
as U (g)-crystals.
q
Proposition 3.3 ([37, 42]). Let (ν,J)∈RC(∞) and fix some a∈I. Let x denote
the smallest label in (ν,J)(a). Then we have
ε (ν,J)=−min(0,x) ϕ (ν,J)=p(a)−min(0,x).
a a ∞
It is a straightforwardcomputation from the vacancy numbers to show that
hh ,λi− A m(b) =−p(a) +2p(a)−p(a). (3.2)
a ab i i−1 i i+1
b∈I
X
From this, we obtain the well-knownconvexity properties of the vacancy numbers.
6 BENSALISBURYANDTRAVISSCRIMSHAW
Lemma 3.4 (Convexity). If m(a) =0, then we have
i
2p(a) ≥p(a) +p(a).
i i−1 i+1
Moreover, p(a) ≥p(a) ≤p(a) if and only if p(a) =p(a) =p(a).
i−1 i i+1 i−1 i i+1
Inthesequel,wewillrefertothislemmasimplyasconvexityaswewillfrequently
use it.
4. Star-crystal structure
Definition 4.1. Fix some a∈I, and let x be the smallest corigging in (ν,J)(a).
e∗: Ifx=0,then e (ν,J)=0. Otherwiseletr be arowin(ν,J)(a) ofminimal
a a
length ℓ with corigging x. Then e (ν,J) is the rigged configuration which
a
removes a box from row r and sets the new corigging of r to be x+1.
f∗: Let r be a row in (ν,J)(a) of maximal length ℓ with corigging x. Then
a
f (ν,J) is the riggedconfiguration which adds a box to row r and sets the
a
new colabel of r to be x−1.
Ife∗ removesaboxfromarowoflengthℓin(ν,J),thenthethevacancynumbers
a
change by the formula
p(b) if i≤ℓ,
p(b) = i (4.1)
i (pi(b)+Aab if i>ℓ.
Ontheotherhand,iffa∗ adedsaboxtoarowoflengthℓ,thenthevacancynumbers
change by
p(b) if i<ℓ,
p(b) = i (4.2)
i (pi(b)−Aab if i≥ℓ.
Similar equations hold foreea and fa respectively. So the riggings of unchanged
rowsare changedaccordingto Equation(4.1) and Equation(4.2) under e and f ,
a a
respectively.
Remark 4.2. By Equation (4.1) and Equation(4.2), the crystaloperatorse and
a
f preserve all colabels of (ν,J) other than the row changed in (ν,J)(a).
a
Example 4.3. Consider type D with Dynkin diagram
4
3
1 2
4.
Let (ν,J) be the rigged configuration
(ν,J)=f∗f∗f∗f∗f∗f∗f∗f∗f∗(ν ,J )
2 3 1 2 2 4 3 1 2 ∅ ∅
= −1 0 −3 −2 −1 0 0 0 .
−1 −1
Then
f∗(ν,J)= −1 0 −5 −3 −1 0 0 0 .
2
−1 −1
RIGGED CONFIGURATIONS AND THE ∗-INVOLUTION 7
Let RC(∞)∗ denote the closure of (ν ,J ) under f∗ and e∗. We define the
∅ ∅ a a
remaining crystal structure by
ε∗(ν,J)=max{k∈Z:(e∗)k(ν,J)6=0}, ϕ∗(ν,J)=hh ,wt(ν,J)i+ε∗(ν,J),
a a a a a
wt(ν,J)=− |ν(a)|α .
a
a∈I
X
Remark 4.4. We willsayanargumentholds bydualitywhenwe caninterchange:
• “label” and “colabel”;
• e and e∗;
a a
• f and f∗.
a a
For an example, compare the proof of Proposition 4.6 with [36, Thm. 3.8].
Lemma 4.5. The tuple (RC(∞)∗,e∗,f∗,ε∗,ϕ∗,wt) is an abstract U (g)-crystal.
a a a a q
Proof. The proof that (RC(∞)∗,e∗,f∗,ε∗,ϕ∗,wt) is an abstract U (g)-crystal is
a a a a q
dualtothatRC(∞)isanabstractU (g)-crystalundere andf in[37,Lemma3.3].
q a a
Proposition 4.6. Let (ν,J) ∈ RC(∞) and fix some a ∈ I. Let x denote the
smallest colabel in (ν,J)(a). Then we have
ε∗(ν,J)=−min(0,x), ϕ∗(ν,J)=p(a)−min(0,x).
a a ∞
Proof. The following argument for ε∗ is essentially the dual to that given in [36,
a
Thm. 3.8]. We include it here as an example of Remark 4.4.
It is sufficient to prove ε∗(ν,J) = −max(0,x) since p(a) = hh ,wt(ν,J)i. If
a ∞ a
x ≥ 0 = ε∗(ν,J), then e∗(ν,J) = 0 by definition. Thus we proceed by induction
a a
on ε∗(ν,J) and assume x < 0. Let (ν′,J′) = e∗(ν,J) and y′ denote the resulting
a a
colabel from a colabel y. In particular, we have x′ = x+1 and all other colabels
followEquation(4.1). Next,lety denotethecolabelofarowoflengthj. Forj <ℓ,
wehavey >x(equivalentlyy ≥x−1)becausewechoseℓaslargeaspossible. Thus
y′ =y, and hence y′ =y ≥x+1=x′. For j ≥ℓ, we have y ≥x by the minimality
ofx andy′ =y+2. Hence, y′ =y+2≥x+1=x′, andso ε∗(ν′,J′)=ε∗(ν,J)−1
a a
as desired.
TherestofthissectionwillamounttoshowingthatConditions(1)–(6)ofPropo-
sition2.3hold. Note that usingProposition4.6and[37, Prop. 4.2],wecanrewrite
Equation (2.1) as
κ (ν,J)=−min(0,x )−min(0,x )+hh ,wt(ν,J)i,
a ℓ c a
(4.3)
=−min(0,x )−min(0,x )+p(a),
ℓ c ∞
where x and x are the smallest label and colabel, respectively, in (ν,J)(a).
ℓ c
Lemma 4.7. Fix (ν,J) ∈ RC(∞) and a ∈ I. Assume κ (ν,J) = 0. Then
a
f (ν,J)=f∗(ν,J).
a a
Proof. Suppose that f adds a box to a row of length i with rigging x. Recall that
a
x= −ε (ν,J). Suppose the longest row of ν(a) has length ℓ > i and let x denote
a ℓ
any rigging of the longest row. Therefore, we have x > x by the definition of f ,
ℓ a
and we have p(a) ≤p(a) by convexity. Thus from the definition of ε∗, we have
ℓ ∞ a
ε∗(ν,J)≥x −p(a) >x−p(a) =−ε (ν,J)−p(a). (4.4)
a ℓ ℓ ∞ a ∞
8 BENSALISBURYANDTRAVISSCRIMSHAW
This implies ε∗(ν,J)+ε (ν,J)+p(a) > 0, which is a contradiction. Therefore f
a a ∞ a
must add a box to one of the longest rows of ν(a). Moreover, if p(a) < p(a), then
ℓ ∞
Equation (4.4) would still hold and result in a contradiction. Similar statements
holds for f∗ by duality.
a
Therefore f and f∗ act on the longest row of ν(a) and p(a) = p(a) = p(a). Let
a a i i+1 ∞
x and x∗ denote the label of the row on which f and f∗ act, respectively. Both of
a a
these labels decreaseby 1 after applying f andf∗, respectively,by Equation(4.6)
a a
and Equation (4.10), respectively. So it is sufficient to show x = x∗. Note that
x≤x∗ as the smallest colabel is the one with the largestrigging. Suppose x<x∗,
then we have
ε∗(ν,J)≥x∗−p(a) >x−p(a) ≥−ε (ν,J)−p(a),
a i ∞ a ∞
which is a contradiction. Therefore we have f =f∗.
a a
Example 4.8. Let(ν,J)betheriggedconfigurationoftypeD fromExample4.3.
4
Then κ (ν,J)=0, and and
2
f2(ν,J)= −1 0 −5 −3 −1 0 0 0 .
−1 −1
One can check that this agrees with f∗(ν,J) from Example 4.3.
2
Lemma 4.9. Fix (ν,J)∈RC(∞) and a∈I. Assume κ (ν,J)≥1. Then
a
ε∗ f (ν,J) =ε∗(ν,J), ε (f∗(ν,J))=ε (ν,J).
a a a a a a
Proof. Let xc d(cid:0)enote th(cid:1)e smallest colabel of (ν,J)(a). Let x and i denote the
rigging and length of the row on which f acts. By the minimality of xc, we have
a
x := p(a) −x ≥ xc. Note that the colabel of the row after the application of f
c i a
becomes
x :=p(a) −2−(x−1)=p(a) −x−1, (4.5)
c i+1 i+1
which implies that
e x =x +p(a) −p(a)−1. (4.6)
c c i+1 i
The remainder of the proof will be split into two cases: xc = p(a) − x and
i
xc <p(a)−x. e
i
xc =p(a)−x:
i
First, consider the case p(a) ≤ p(a). We also assume there exists a row of
i+1 i
length ℓ > i of ν(a), and let x denote the rigging of that row. Thus x < x and
ℓ ℓ
xc ≤p(a)−x by the definition of f and the minimality of xc. Hence
ℓ ℓ a
p(a)−x=xc ≤p(a)−x <p(a)−x,
i ℓ ℓ ℓ
which is equivalent to p(a) < p(a). It must be the case, then, that p(a) > p(a) by
i ℓ i+1 i
convexity, which is impossible. Thus the longest row of ν(a) must be of length i.
Convexity implies p(a) =p(a) =p(a), which results in
i i+1 ∞
κ (ν,J)=p(a)−min(x,0)−min(xc,0)
a ∞
=p(a)−min(x,0)−min p(a)−x,0 .
i i
(cid:0) (cid:1)
RIGGED CONFIGURATIONS AND THE ∗-INVOLUTION 9
Sincexwastheriggingchosenbyf ,wemusthavex≤0. Additionally,ifp(a)−x=
a i
xc ≤0, we have
1≤κ (ν,J)=p(a)−x−(p(a)−x)=0,
a i i
which is a contradiction. Thus xc ≥ 1, which implies ε∗(ν,J) = 0 = ε∗ f (ν,J))
a a a
since x ≥0.
c
Next if p(a) = p(a) +1, then ε∗ f (ν,J) = ε∗(ν,J) since x = xc,(cid:0)all other
i+1 i a a a c
coriggeings are fixed, and xc =xc by Equation (4.6) .
(cid:0) (cid:1)
So we now assume p(i+a)1 ≥ p(ia) +2, which implies xc > xc. Ief there is another
row with a corigging of xec or xc ≥ 0, then ε∗ f (ν,J) = ε∗(ν,J). So assume f
a a a a
acts on the only row with a corigging of xc < 0. Noteethat p(a)−x = xc < 0 and
(cid:0) (cid:1) i
x≤0 implies p(a) <0.
i
We have m(a) = 1 as, otherwise, we would either have a second corigging of xc
i
or a smaller corigging from the minimality of x. Thus, by Equation (3.2),
−2− A m(b) =−p(a) +2p(a)−p(a)
ab i i−1 i i+1
b6=a
X
≤−p(a) +2p(a)−p(a)−2
i−1 i i
=−p(a) +p(a)−2.
i−1 i
Since m(b) ≥0 and −A ≥0 for all a6=b, we then have
i ab
0≤−p(a) +p(a),
i−1 i
or,equivalently, p(a) ≤p(a). If i is the length of the smallest row,then 0≤p(a) ≤
i−1 i i−1
p(a) < 0 by convexity, which is a contradiction. Thus let x denote rigging of the
i ℓ
longest row in ν(a) such that ℓ < i, and by convexity, we have p(a) ≤ p(a). By the
ℓ i
definitionoff ,wehavex ≥x. Thus,wehavep(a)−x ≤p(a)−x. However,bythe
a ℓ ℓ ℓ i
unique minimality ofxc, we havep(a)−x >xc =p(a)−x. This is a contradiction.
ℓ ℓ i
Therefore ε∗ f (ν,J) =ε∗(ν,J).
a a a
xc <p(a)−x(cid:0): (cid:1)
i
Assume ε∗ f (ν,J) 6=ε∗(ν,J). Then
a a a
(cid:0) (cid:1)p(a) −p(a)−1<xc+x−p(a) <0, (4.7)
i+1 i i
as, otherwise, the new corigging is not smaller than the minimal corigging (i.e.,
x < xc), which occurs on a different row and does not change under f . We
c a
rewrite Equation (4.7) as
e p(a) −1<xc+x<p(a). (4.8)
i+1 i
Suppose there exists a row of length ℓ>i in ν(a). Then x >x and xc ≤p(a)−x .
ℓ ℓ ℓ
Therefore,
p(a) −ℓ<p(a)−x +x<p(a), (4.9)
i+1 ℓ ℓ ℓ
which implies p(a) ≤ p(a) for all ℓ ≥ j > i by convexity. Note that Equation (4.9)
i+1 j
implies that ℓ > i+1 since, otherwise, we would have p(a) < p(a). We also have
i+1 i+1
p(a) ≤ p(a) for all j > i if there does not exist a row of length ℓ > i. Since there
i+1 j
10 BENSALISBURYANDTRAVISSCRIMSHAW
does not exist a rowof length i+1, we must have p(a) ≤p(a) ≤p(a) by convexity.
i i+1 i+2
Yet, Equation (4.8) implies
p(a) <p(a) ≤p(a),
i+1 i i+1
but this is a contradiction. Therefore ε∗ f (ν,J) =ε∗(ν,J).
a a a
Example 4.10. Again, let (ν,J) be t(cid:0)he rigged(cid:1) configuration of type D from
4
Example 4.3. Then ε (ν,J)=0, ε∗(ν,J)=1, and κ (ν,J)=1. We have
3 3 3
f3(ν,J)= −1 0 −1 −1 −2 −1 0 0
−1 −1
f∗(ν,J)= −1 0 −2 −2 −2 0 0 0 .
3
−1 −1
Then ε∗ f (ν,J) =1 and ε f∗(ν,J) =0.
3 3 3 3
Lemma(cid:0)4.11. F(cid:1)ix (ν,J)∈R(cid:0)C(∞) an(cid:1)d a∈I. Assume κ (ν,J)≥2. Then
a
f f∗(ν,J)=f∗f (ν,J).
a a a a
Proof. Suppose f (resp., f∗) acts on row r of length i (resp., row r∗ of length
a a
i∗) with rigging x (resp., x∗). Without loss of generality, let r (resp., r∗) be the
northernmostsuchrowinthediagramofν(a). Letx∗ =p(a)−x∗ andx =p(a)−x.
c i∗ c i
Notethatx≤x∗ andx∗ ≤x . Applyingf∗,thenewrigging(andtheonlychanged
c c a
rigging) is
x∗ =x∗+p(a) −p(a)−1. (4.10)
i∗+1 i∗
Recall that Equation (4.6) gives the new corigging (and only changed corigging)
e
x =x +p(a) −p(a)−1
c c i+1 i
after applying f . We split the proof into three cases: the first two are cases in
a
which r 6=r∗ and the last is wehen r =r∗.
f acts on row r 6=r∗ in f∗(ν,J):
a a
Suppose f f∗(ν,J)6=f∗f (ν,J). This is equivalent to f∗ acting on row r′ 6=r∗
a a a a a
in f (ν,J). Note that we must have r′ = r, since f preserves all other colabels.
a a
From Equation (4.6), we must have p(a) < p(a) +2, as otherwise x > x ≥ x∗,
i+1 i c c c
which would imply r = r′ = r∗ and be a contradiction. Next, consider when
p(i+a)1 = p(ia) +1. Thus we have xc = xc. Since r′ 6= r∗, we must haeve i = i∗ and
x∗ =x =x . However,this contradicts the assumption r6=r∗ as we have x=x∗.
c c c
Hence p(a) ≤ p(a). Supposeei is the length of the longest row of ν(a). Then
i+1 i
p(a) =e p(a) = p(a) by convexity. Moreover, we have x = x − 1 = x∗ since
i i+1 ∞ c c c
r,r′ 6=r∗. Note that since f (resp., f∗) acts on r (resp., r∗), we must have x≤0
a a
(resp., x∗c ≤0). Therefore, we have e
2≤κ (ν,J)=p(a)−min(x,0)−min(x∗,0)
a i c
=p(a)−x−x∗
i c
=p(a)−x−(p(a)−x−1)=1,
i i
which is a contradiction.