Table Of ContentA MULTIPLICATIVE PROPERTY OF QUANTUM FLAG
MINORS II
3
0
0
PHILIPPE CALDEROANDROBERTMARSH
2
n
a Abstract. LetU+ bethepluspartofthequantized envelopingalgebraofa
J simpleLiealgebraandletB∗ bethedual canonical basisofU+. Let b,b′ be
inB∗ andsuppose that one ofthe twoelements isa q-commutingproduct of
4
quantum flag minors. We show that band b′ aremultiplicativeifand onlyif
2
theyq-commute.
]
T
R
.
h Contents
t
a
1. Introduction 1
m
2. Backgroundand notation 2
[
3. PBW-strings 5
1 4. Properties of PBW i-linearity domains 8
v 5. q-commuting products of quantum flag minors 10
0
References 12
8
2
1
0
3
1. Introduction
0
h/ Let B∗ denote the dual of the canonical basis [8], [14, §14.4.6] of a quantized
t enveloping algebra of a simple Lie algebra g. Two elements of B∗ are said to be
a
multiplicative if their product also lies in B∗ up to a power of q. They are said
m
to q-commute if they commute up to a power of q. The Berenstein-Zelevinsky
:
v conjecture states that two elements of B∗ are multiplicative if and only if they q-
i commute. WhileReineke[17,4.5]hasshownthatiftwoelementsaremultiplicative
X
then they q-commute, the conjecture is now known to be false: a counter-example
r
a was provided by Leclerc [10], who showed that for all but finitely many types of
simple Lie algebra, there exist elements b ∈ B∗ whose square does not lie in B∗
even up to a power of q (such elements obviously q-commute with themselves).
Such elements are called imaginary; elements of B∗ whose square does lie in B∗ up
to a power of q are known as real. We consider the following question:
Question 1.1. Let b, b′ be in B∗ and suppose that b is real. Is it the case that b
and b′ are multiplicative if and only if they q-commute?
Date:24thJanuary2003.
The authors would like to thank EC grant of the TMR network “Algebraic Lie Representa-
tions”,contractno. ERBFMTX-CT97-0100,forsupportofthevisitofbothauthorstoWuppertal
in October 2001 and for support of the first author’s visit to Leicester in August 2002, and the
UniversityofLeicester forstudyleaveforthesecondauthorinAutumn2002.
1
2 PHILIPPECALDEROANDROBERTMARSH
Leclerchasmadeaconjecture[10,Conjecture1]concerningtheexpansionofthe
productoftwodualcanonicalbasiselements,andhasremarkedthatiftrue,itwould
imply that the answer to this question is yes in general. In this paper, we prove
that the answer to this question is yes if one of the two elements involved is a q-
commutingproductofquantumflagminors(intypeA). Wenotethatsuchproducts
are known to be real [11]. A key part of our proof involves results concerning
the piecewise-linear reparametrizationfunction Ri′ associated by Lusztig to a pair
i
i, i′ of reduced decompositions for the longest word w in the Weyl group of g.
0
LusztighasdefinedthecanonicalbasisviathebasesofPoincar´e-Birkhoff-Witttype
associated to such reduced decompositions, and such reparametrization functions
arise from taking two of the Lusztig parametrizations of the canonical basis. We
showthatthesefunctionssharesomeofthepropertiesknowntobepossessedbythe
reparametrization functions arising from string parametrizations for the canonical
basis[1]. Inparticularweshowthatiftwodualcanonicalbasiselementsq-commute,
then their PBW parametrizations (with respect to a fixed reduced decomposition
i)lie in a single PBWi-linearitydomain;in other words,if their parametersarem
and m′, then Ri′(m+m′) = Ri′(m)+Ri′(m′) for all reduced decompositions i′
i i i
for w . The corresponding result for string parametrizations is already known [1,
0
2.9]. Such behaviour hints at an explanation for the compatibility of examples in
which the canonical basis has been computed explicitly [6], [15], [21] with respect
to linearity domains. As a consequence, we obtain that the set of PBW i-linearity
domains forms a fan in RN, where N is the length of w .
0
This enables us to provethatthe answerto Question1.1is yes intype A when
n
b is a q-commuting product of quantum flag minors. This generalizes a theorem
of [5].
In the paper, the Lie algebra g is supposed to be of type A . Note that results
n
in section 1-4 can be easily generalized to all simply-laced types. Note also that
Lemma 5.3 is only true in type A . Hence, Theorem 5.5 needs this assumption.
n
2. Background and notation
We use the set-upof[4]. Letg=sl (C)denote the simple Lie algebraoftype
n+1
A . LethbeaCartansubalgebraandletg=n−⊕h⊕n+beacompatibletriangular
n
decomposition. Let α ,α ,...,α be the corresponding simple roots, and ∆+ the
1 2 n
correspondingsetofpositiveroots. LetW betheWeylgroupassociatedtotheroot
system,P betheweightlatticegeneratedbythefundamentalweights̟ ,1≤i≤n,
i
and h, i be the W-invariant form on P. Let U = U (g) be the simply connected
q
Drinfel’d-Jimbo quantized enveloping algebra over Q(q) associated to g as defined
in[7]. LetU−,U0andU+bethesubalgebrasassociatedtothesub-Liealgebrasn−,
h and n+ respectively; we have the triangular decomposition U ∼=U−⊗U0⊗U+.
The subalgebra U+ is generated over Q(q) by canonical generators E ,E ,...E ,
1 2 n
subject to the quantized Serre relations; the subalgebra U− is isomorphic to U+,
withcorrespondinggeneratorsF ,F ,...F ,andthesubalgebraU0isisomorphicto
1 2 n
Q(q)[P],theelementcorrespondingtoλ∈P beingdenotedbyK . Letb+ =h⊕n+
λ
and b− =h⊕n− and we define U(b+)=U0U+ and U(b−)=U−U0. The algebra
U isa Hopfalgebrawithcomultiplication∆,antipodeS andaugmentationεgiven
by:
∆(E )=E ⊗1+K ⊗E , ∆(F )=F ⊗K +1⊗F , ∆(K )=K ⊗K ,
i i αi i i i −αi i λ λ λ
A MULTIPLICATIVE PROPERTY OF QUANTUM FLAG MINORS II 3
S(E )=−K E , S(F )=−F K , S(K )=K ,
i −2αi i i i 2αi λ −λ
ε(E )=ε(F )=0, ε(K )=1.
i i λ
The root lattice Q is defined to be Q = Zα , with Q+ = Z α . Recall
Pi i Pi ≥0 i
that if α = m α ∈ Q+, then an element in the subspace of U generated by
Pi i i
{En1En2···Enk : n α +n α +···+n α =α}(respectively,{Fn1Fn2···Fnk :
i1 i2 ik 1 i1 2 i2 k ik i1 i2 ik
n α +n α +···+n α =α}), is saidto have weightα (respectively,−α). For
1 i1 2 i2 k ik
all α ∈ Q+, let U+ (respectively, U− ) denote the subspace of U (respectively,
α −α +
U−) consisting of elements of weight α (respectively, −α). We write the weight of
an element X as wt(X) (if it exists), and tr(X) (or tr(wt(X))) for the sum m
Pi i
if wt(X) = m α . For u ∈ U, set ∆(u) = u ⊗u ∈ U ⊗U. There exists a
Pi i i (1) (2)
unique bilinear form (, ) [18], [20] on U(b+)×U(b−) satisfying
(E ,F )=δ (1−q2)−1,
i i ij
(u+,u−u−)=(∆(u+),u−⊗u−), u+ ∈U(b+),u−,u− ∈U(b−),
1 2 1 2 1 2
(u+u+,u−)=(u+⊗u+,∆(u−)), u− ∈U(b−),u+,u+ ∈U(b+),
1 2 2 1 1 2
(K ,K )=q−(λ,µ), (K ,F )=0, (E ,K )=0, λ,µ∈P.
λ µ λ i i λ
The form (, ) is nondegenerate on U+ ⊗ U− for all α ∈ Q . Since U+ and
α −α +
U− are isomorphic algebras (with isomorphism preserving their weight spaces),
we can naturally identify U− with U+, so for each element u ∈ U+ there is a
−α α α
corresponding element u∗ ∈ U+ (corresponding to u using the form). Let w be
α 0
the longest element of W; denote by R(w ) the set of all reduced decompositions
0
for w . Fix a reduced decomposition i = (i ,i ,...,i ) of w , and for 1 ≤ t ≤ N,
0 1 2 N 0
let β = s s ···s (α ); we get an ordering β < β < ··· < β of ∆+. For
t i1 i2 it−1 it 1 2 N
1 ≤ i ≤ n, let T denote the Lusztig braid automorphism of U+ [14, 37.1.3], [19]
i
associated to i. For 1 ≤ t ≤ N let E = T T ···T (E ). The Poincar´e-
βt i1 i2 it−1 it
Birkhoff-Witt basis of U+ is the basis
B ={E(m) : m∈ZN },
i ≥0
where
N
1
E(m)=Ei(m)=Y[m ]!Eβmtt,
t
t=1
with the product taken in the ordering given above. Here [m]!=[m][m−1]···[1],
where [m]= qm−q−m. By [12] we have that
q−q−1
N
E(m)∗ =Yψmt(q2)E(m),
t=1
whereψ (z)= m (1−zk). LetLbethesub-Z[q]-latticeofU+ generatedbyB,
m Qk=1 i
and let L∗ be the sub-Z[q]-lattice of U+ generated by B∗ ={E(m)∗ : m∈ZN }.
i i ≥0
Lusztig has shown that both L and L∗ are independent of the choice of reduced
decomposition i. Let η be the Q-algebra automorphism of U fixing the generators
E and F , with η(K )=K and η(q)=q−1. Lusztig [13] also shows:
i i λ −λ
Theorem 2.1. (Lusztig) Fix a reduced decomposition i of w . Then, for each m∈
0
ZN , there is a unique element B(m) = B (m) ∈ U+ such that η(B(m)) = B(m)
≥0 i
and B(m) ∈ E(m)+qL. The set B = {B(m) : m ∈ Z } is a basis (called the
≥0
canonical basis) of U+ which does not depend on i.
4 PHILIPPECALDEROANDROBERTMARSH
We’ll call the parametrizationZN →B, m7→B(m), Lusztig’s parametrization
≥0
of B (arising from i). If b ∈ B, we denote by L (b) its Lusztig parameter (so L
i i
is the inverse of B ). For any i,i′ ∈ R(w ), we also have Lusztig’s piecewise-linear
i 0
reparametrizationfunctionRii′ =Li′L−i 1 (whichcanberegardedasafunctionfrom
RN →RN via the same formula in terms of coordinates). The canonical basis was
discovered independently by Kashiwara [8], who called it the global crystal basis.
Let B∗ ={b∗ : b∈B} denote the dual canonical basis of U+. For b∈B we define
L(b∗) to be L(b). Let σ be the antihomomorphism of U (as Q(q)-algebra) taking
i i
E to E , F to F and K to K . As in [11, Proposition 16] we have:
i i i i λ −λ
Proposition 2.2. Fix a reduced decomposition i of w . Then, for each m∈ZN ,
0 ≥0
the element B(m)∗ is the unique element X of U+ with weight m α such that
Pi i i
(2.1) η(X)=(−1)tr(X)q−hwt(X),wt(X)i/2q−1σ(X), X ∈E(m)∗+qL∗,
X
where qX =Qiqmi, wt(X)=Pimiαi.
Fix a reduced decomposition i in R(w ). Let e , 1 ≤ k ≤ N, be the canonical
0 k
generatorsofZN . We candefine anorderingonthe semigroupZN associatedtoi
≥0 ≥0
in the following way. The ordering ≺i is generated by m≤ek+ek′, 1≤k <k′ ≤
N ⇔ Ei(m) is a term of the PBW decomposition of EβkEβk′ −q(βk′,βk)Eβk′Eβk.
It will be denoted by ≺ if no confusion occurs.
Proposition 2.3. Fix i in R(w ). Then, for all m, n in ZN :
0 ≥0
(i) if m≺ n then m is lower than n for the lexicographical ordering,
i
(ii) B(m)=E (m)+ dn E(n), dn ∈qZ[q],
i i Pm≺n m i m
(iii) B(m)∗ =E(m)∗+q cnE (n)∗, cn ∈Z[q].
i i Pn≺m m i m
Proof: See [5, 2.1]. 2
Let qZB∗ denote the set {qnb∗ : b∈B,n∈Z}. We therefore have:
Corollary 2.4. Let α∈Q+, and suppose u∈U+. Fix i∈R(w ). Then u∈qZB∗
α 0
if and only if
(i) There is f ∈Z[q,q−1] such that ση(u)=fu, and
(ii) u = qk(E(m)∗ +q cn E(n)∗), where k ∈ Z, cn ∈ Z[q] and < denotes
Pn<m m i m
the lexicographic ordering.
Proof: It is immediate from Proposition 2.2 that if u ∈ qZB∗ then it satisfies (i)
and (ii). If k = 0 in part (ii), this ensures that, in the expansion of u in terms
of the dual canonical basis, B(m)∗ occurs with coefficient 1 (see Proposition 2.3).
i
It follows that the eigenvalue f in part (i) must be the same as that appearing
in (2.1), and it follows from Proposition 2.2 that u ∈ B∗. The result for arbitrary
k follows. 2
Remark 2.5. We note that, from the proof, we can see that u∈B∗ if and only if
u satisfies (i) and (ii) with k =0.
Twoelementsofthe dualcanonicalbasisB∗ aresaidto bemultiplicative iftheir
product also lies in the dual canonical basis up to a power of q. They are said
to q-commute if they commute up to a power of q. The following Corollary of
Proposition 2.2 is due to Reineke [17, 4.5].
Corollary 2.6. If two elements of the dual canonical basis are multiplicative then
they q-commute.
A MULTIPLICATIVE PROPERTY OF QUANTUM FLAG MINORS II 5
Definition 2.7. There is a natural action of U+ on itself [1], in which each gener-
ator E of U+ acts on U+ as a q-differential operator δ . This is the unique action
i i
of U+ on itself satisfying the following properties:
(a) (Homogeneity) If E ∈U+, x∈U+, then E(x)∈U+ .
α γ γ−α
(b) (Leibnitz formula)
δ (xy)=δ (x)y+q−(γ,αi)xδ (y), for x∈U+, y ∈U+.
i i i γ
(c) (Normalization) δ (E )=(1−q2)−1δ for i,j =1,2,...,n.
i j ij
We remark that formula (b) implies that for all i and for r ∈ Z , we have
≥0
δ (E(r))=q−r+1(1−q2)−1E(r−1), which is easily checked by induction on r (E(0)
i i i i
is interpreted as 1 and E(−1) as zero). For u ∈ U+, set ϕ (u) = max{r,δr(u) 6=
i i i
0}, and for r ∈ Z , let δ(r) denote the divided power δir . Define δ(max)(u) :=
≥0 i [r]! i
δ(ϕi(b∗))(u). It is known that for b∈B, δ(max)(b∗)∈B∗ (see [1, §1]). If i∈R(w ),
i i 0
set a = ϕ (b∗), a = ϕ (δ(a1)(b∗)),...,a = ϕ (δ(aN−1)···δ(a1)(b∗)); then
1 i1 2 i2 i1 N iN iN−1 i1
(a ,a ,...,a ) is known as the string of b (or b∗) in direction i [1]; this coincides
1 2 N
with the string of b arising from Kashiwara’sapproachto B (see [9] and the end of
Section 2 in [16]).
3. PBW-strings
Inthis sectionwewillshowhowthe Lusztigparametrizationofadualcanonical
basis element can also be regarded as a string (in a similar sense to the above).
We define new operators ∆ , i = 1,2,...,n, depending on a choice of reduced
i
decomposition for w , which play the role of the operators δ(max) for the PBW
0 i
parametrization. We firstofallnote that δ(max) is awell-definedoperatoronallof
i
U+.
Definition 3.1. Let w = s s ···s be a reduced decomposition for w ∈ W. Let
i1 i2 it
∆ = δ(max), ∆ = Te δ(max)T−1,...,∆ = (T ···T )δ(max)(T ···T )−1,
1 i1 2 i1 i2 i1 t i1 it−1 it i1 it−1
operators on U+ (with codomain U).
We will use the following:
Lemma 3.2. (Saito) Let i∈{1,2,...,n}. Then T (U+)∩U+ =kerδ .
i i
Proof: See [19]. 2
Lemma 3.3. Let i∈R(w ), and m=(m ,m ,...,m )∈ZN . Then
0 1 2 N ≥0
δ(max)(E (m)∗)=E (0,m ,m ,...,m )∗.
i1 i i 2 3 N
Proof: Let γ = N ψ (q2), so that E(m)∗ = γ E (m). Then Em =
m Qt=1 mt i m i i
E(m1)T (y), where y = E(m2)T (E(m3))···T T ···T (E(mN)) ∈ U+. Us-
i1 i1 i2 i2 i3 i2 i3 iN−1 iN
ing Lemma 3.2, the formula for γ and the fact that δ (E(m)) = q−m+1(1 −
m i1 i1
q2)−1E(m−1) the result follows. 2
i1
We can now prove a Lemma giving basic properties of the Lusztig parametriza-
tion with respect to the operators ∆ .
i
6 PHILIPPECALDEROANDROBERTMARSH
Lemma 3.4. Let b∈B, and suppose i∈R(w ) and ∆ ,∆ ,...,∆ are as above.
0 1 2 N
Let L (b)=(m ,m ,...,m )∈ZN . Then, for k =1,2,...,N, we have:
i 1 2 N ≥0
(i) ∆ ∆ ···∆ (b∗)∈B∗,
k k−1 1
(ii) ϕ ((T ···T )−1∆ ···∆ (b∗))=m ,
ik i1 ik−1 k−1 1 k
(iii) L(∆ ∆ ···∆ (b∗))=(0,0,...,0,m ,m ,...,m ).
i k k−1 1 k+1 k+2 N
Proof: Since δ(max) preserves B∗, we see that ∆ (b∗) lies in B∗, so (i) holds for
i1 1
k = 1, but we need more precise information. Since b∗ ∈ B∗, we know by [5] (see
also Proposition 2.3) that
b∗ =E (m)∗+qx ,
i 0
where x is a Z[q]-linear combination of dual PBW-basis elements E (n)∗ with
0 i
n < m (and in particular, satisfying n ≤ m ). Here < denotes the lexicographic
1 1
ordering. By Lemma 3.3, we see that
δ(max)(b∗)=E(0,m ,m ,...,m )∗+qx ,
i1 i 2 3 N 1
where x is a Z[q]-linear combination of elements of the form E(n)∗ with n =
1 i
(0,n ,n ,...,n ) < m for all n occurring in the sum. By Corollary 2.4, we have
2 3 N
that L(∆ (b∗))=(0,m ,m ,...,m ), so (ii) holds for k =1. It is also clear that
i 1 2 3 N
ϕ (b∗)=m , so(iii) holdsfor k =1. We thus see that(i), (ii) and(iii) allholdfor
i1 1
k =1.
¿From the above form for ∆ (b∗), we have that
1
Ti−11(∆1(b∗))=Ei′(m2,m3,...,mN,0)∗+qy1,
wherei′ =si2si3···siNsi∗1,i∗1istheindexofthesimpleroot−w(αi1)(theChevalley
automorphismappliedtoi ),andy isaZ[q]-linearcombinationofdualPBW-basis
1 1
elements Ei′(n)∗ with n= (n2,n3,...,nN,0)<(m2,m3,...,mN,0) and therefore
satisfies Corollary 2.4(ii) (with k =0). Since for all i, T−1ση and σηT−1 differ on
i i
weight spaces by plus or minus a power of q, it follows that T−1(∆ (b∗)) satisfies
i1 1
Corollary 2.4(i), and therefore (see the remark after Corollary 2.4(i)) lies in B∗.
Since δ(max) preserves B∗, we thus obtain that δ(max)T−1∆ (b∗)∈B∗. Arguing as
i2 i2 i1 1
above, with i playing the role of i , we obtain that
2 1
δi(2max)Ti−11∆1(b∗)=Ei′(0,m3,m4,...,mN,0)∗+qy2,
where y is a Z[q]-linear combination of dual PBW-basis elements of the form
2
Ei′(n)∗ with n = (0,n3,n4,...,nN,0) < (0,m3,m4,...,mN,0). We also obtain
that ϕ (T−1∆ (b∗))=m (part (ii) for k =2). Applying T we obtain that
i2 i1 1 2 i1
∆ ∆ (b∗)=T δ(max)T−1∆ (b∗)=E (0,0,m ,m ,...,m )∗+qx ,
2 1 i1 i2 i1 1 i 3 4 N 2
where x is a Z[q]-linear combination of dual PBW-basis elements of the form
2
E(n)∗ with n = (0,0,n ,n ,...,n ) < (0,0,m ,m ,...,m ). Arguing as above
i 3 4 N 3 4 N
for T−1 and applying Corollary 2.4, we obtain that ∆ ∆ (b∗) ∈ B∗ and that
i1 2 1
L(∆ ∆ (b∗)) = (0,0,m ,m ,...,m ). We thus see that (i) and (iii) hold for
i 2 1 3 4 N
k = 2. It is now clear that an inductive argument gives (i),(ii) and (iii) for
k =1,2,...,N. 2
Givenareduceddecompositionw =s s ···s foranelementw ∈W,wedefine
i1 i2 it
thePBW-stringL (b∗)ofadualcaenonicalbasiselementb∗indirectionwasfollows.
w
Leti=s s ···ses ···s be anycompletionofw toa reduceddeceomposition
i1 i2 it it+1 iN
for w . Then let L (b∗) = (m ,m ,...,m ) where Le(b∗) = (m ,m ,...,m ). It
0 w 1 2 t i 1 2 N
is clear from Lemmea 3.4 that this is well-defined.
A MULTIPLICATIVE PROPERTY OF QUANTUM FLAG MINORS II 7
WemakedefinitionsforPBW-stringsinthesamewayasBerensteinandZelevin-
sky [1] for usual strings.
Definition 3.5. Let i∈R(w ). A PBW i-wall is defined to be a hyperplane in RN
0
given by the equation a = a for some index k such that i = i = i ±1.
k k+2 k k+2 k+1
Let m ∈ RN . We say that m is PBW i-regular if for every i′ ∈ R(w ) the point
>0 0
Ri′(m) does not lie on any PBW i′-wall. We define the PBW i-linearity domains
i
to be the closures of the connected components of the set of PBW i-regular points.
As in the string case [1, 2.8], we have the following justification for the ter-
minology; the proof is basically the same, as Ri′ is very similar to the string
i
reparametrizationfunction.
Proposition 3.6. Every PBW i-linearity domain is a polyhedral convex cone in
RN . Two points m,m′ in RN lie in a single PBW i-linearity domain if and only
≥0 ≥0
if
Ri′(m+m′)=Ri′(m)+Ri′(m′)
i i i
for every i′ ∈R(w ).
0
We can now prove the analogue of [1, 2.9]:
Theorem 3.7. Let b∗,b′∗ be elements of the dual canonical basis that q-commute.
Then, for every i ∈ R(w ), the PBW strings m = L (b∗) and m′ = L(b′∗) belong
0 i i
to a single i-linearity domain.
Proof: We follow the proof of Berenstein and Zelevinsky [1]; however there will
be some differences, so we include the details. We know by Proposition 3.6 that
it is enough to show that, for every i′ ∈R(w ), Ri′(m+m′)= Ri′(m)+Ri′(m′).
0 i i i
Thus, it is enough to show that m and m′ are not separated by any PBW i-wall.
Suppose that a i-wall corresponds to the move
i=s ···s s s s s ···s 7→s ···s s s s s ···s =i′.
i1 it i j i it+4 iN i1 it j i j it+4 iN
By [1, 3.6] and the fact that the T are algebra automorphisms, the elements
i
(T T ···T )−1 ∆ ∆ ···∆ (b∗) and (T T ···T )−1 ∆ ∆ ···∆ (b′∗)
ik ik−1 i1 t t−1 1 ik ik−1 i1 t t−1 1
q-commute. Replacing b∗ and b′∗ by (T T ···T )−1 ∆ ∆ ···∆ (b∗) and
ik ik−1 i1 t t−1 1
(T T ···T )−1 ∆ ∆ ···∆ (b′∗), we can assume that t = 0 (using Lemma
ik ik−1 i1 t t−1 1
3.4). Let L = L (b∗)= (m ,m ,m ) and L′ =L (b′∗)= (m′,m′,m′)
iji sisjsi 1 2 3 iji sisjsi 1 2 3
be the PBW-strings of b∗ and b′∗ in direction s s s . It is sufficient to show that
i j i
theintegersm −m andm′ −m′ areofthe samesign. Nowlet∆ ,∆ ,∆ be the
1 3 1 3 1 2 3
operatorsonB∗ associatedtothereduceddecompositions s s asinDefinition3.1.
i j i
Let b∗ = ∆ ∆ ∆ (b∗), and let b′∗ = ∆ ∆ ∆ (b′∗). Suppose that b′∗b∗ = qnb∗b′∗
0 3 2 1 0 3 2 1
and b′0∗b∗0 =qn0b∗0b′0∗ for integers n,n0. Then we have
∆ ∆ ∆ (b∗b′∗) = T T δ(max)T−1T−1δ(max)T−1δ(max)(bb′)
3 2 1 i j i j i j i i
= qr1∆ ∆ ∆ (b∗)∆ ∆ ∆ (b′∗)
3 2 1 3 2 1
= qr1b∗b′∗,
0 0
where
r =Φ (m ,m′)+Φ (m ,m′)+Φ ,
1 i,γ 1 1 j,si(γ−m1αi) 2 2 i,sjsi(γ−m1αi)−m2αj
and γ is the degree of b∗. Here Φ (n,m) = nm−(µ,mα ) = m(n−(µ,α ))
k,µ k k
is defined as in [1, Proposition 3.1] and we are using [1, (3.6)]. We also use the
8 PHILIPPECALDEROANDROBERTMARSH
fact that for all α ∈Q+, T±1(U+)∩U+ ⊂U+ . Expanding and simplifying, we
i α si(α)
obtain
r = m m′ +m m′ +m m′ +m m′ +m m′ −m m′ −
1 1 1 2 2 3 3 1 2 2 3 1 3
m′(γ,α )−m′(γ,α +α )−m (γ,α ).
1 i 2 i j 3 j
Similarly, we obtain that ∆3∆2∆1(b∗b′∗)=qr2∆3∆2∆1(b∗)∆3∆2∆1(b′∗), where
r = m′m +m′m +m′m +m′m +m′m −m′m −
2 1 1 2 2 3 3 1 2 2 3 1 3
m (γ′,α )−m (γ′,α +α )−m (γ′,α ),
1 i 2 i j 3 j
and γ′ is the weight of b′∗. We thus have:
(3.1) n −n=r −r .
0 1 2
Let L = L (b∗) = (n ,n ,n ) and L′ = L (b′∗) = (n′,n′,n′) be the
jij sjsisj 1 2 3 jij sjsisj 1 2 3
PBW-stringsofb∗ andb′∗ indirections s s . ThenweknowthatR(m ,m ,m )=
j i j 1 2 3
(n1,n2,n3) and that R(m′1,m′2,m′3) = (n′1,n′2,n′3), where R = Rss12ss21ss12 is Lusztig’s
piecewise reparametrization function associated to the canonical basis in type A
2
(see [13]). Let ∆′,∆′,∆′ be the operators on B∗ associated to the reduced de-
1 2 3
composition s s s as in Definition 3.1. It follows from Lemma 3.4 that b∗ =
j i j 0
∆′∆′∆′(b∗), and that b′∗ = ∆′∆′∆′(b′∗). Hence, a similar argument to that
3 2 1 0 3 2 1
given above shows that
(3.2) n −n=r′ −r′,
0 1 2
where
r′ = n n′ +n n′ +n n′ +n n′ +n n′ −n n′ −
1 1 1 2 2 3 3 1 2 2 3 1 3
n′(γ,α )−n′(γ,α +α )−n (γ,α )
1 i 2 i j 3 j
and
r′ = n′n +n′n +n′n +n′n +n′n −n′n −
2 1 1 2 2 3 3 1 2 2 3 1 3
n (γ′,α )−n (γ′,α +α )−n (γ′,α ).
1 i 2 i j 3 j
Suppose now that m > m and m′ < m′. Then (n ,n ,n ) = R(m ,m ,m ) =
1 3 1 3 1 2 3 1 2 3
(m ,m ,m +m −m )and(n′,n′,n′)=R(m′,m′,m′)=(m′+m′−m′,m′,m′).
2 3 1 2 3 1 2 3 1 2 3 2 3 1 1 2
Equating3.1and3.2weobtain: (m −m )(m′ −m′)=0,whichisacontradiction,
1 3 3 1
as each of m −m and m′ − m′ has been assumed to be positive. A similar
1 3 3 1
argument shows that we cannot have m < m and m′ > m′. It follows that m
1 3 1 3
and m′ cannot be separated by a PBW i-wall, and the Theorem is proved. 2
4. Properties of PBW i-linearity domains
In this section we show that the set of i-linearity domains forms a fan. We
startwith a key proposition(which weshallalso needin section5when we discuss
q-commutingpropertiesofthedualcanonicalbasis). Wecalltheconnectedcompo-
nentsofthesetofPBWi-regularpointsPBWi-chambers,sothatPBWi-linearity
domains are the closures of PBW i-chambers (see [1, §8]).
Lemma 4.1. Let m,m′ ∈ RN . Then m and m′ lie in a single PBW i-linearity
≥0
domain if and only if for all i′ ∈ R(w ), Ri′(m) and Ri′(m′) are weakly on the
0 i i
same side of all PBW i′-walls.
A MULTIPLICATIVE PROPERTY OF QUANTUM FLAG MINORS II 9
Proof: Bydefinition,mandm′ lieinasinglePBWi-chamberifandonlyifforall
i′ ∈ R(w ), Ri′(m) and Ri′(m′) are strictly on the same side of all PBW i′-walls.
0 i i
SincePBWi-linearitydomainsaretheclosuresofPBWi-chambers,itfollowsfrom
the continuity of the functions Ri′ that if m andm′ lie in a single PBWi-linearity
i
domain,thenforalli′ ∈R(w ),Ri′(m)andRi′(m′)areweaklyonthe samesideof
0 i i
all PBW i′-walls. Conversely, if for all i′ ∈R(w ), Ri′(m) and Ri′(m′) are weakly
0 i i
on the same side of all PBW i′-walls, then for all i′ ∈R(w ),
0
Ri′(m+m′)=Ri′(m)+Ri′(m′).
i i i
This can be proved by induction on the number of braid relations needed to take i
to i′ (see the proof of [1, 8.1]). By Proposition 3.6, this implies that m and m′ lie
in a single PBW i-linearity domain. 2
Proposition 4.2. Let i ∈ R(w ). Suppose that m , 1 ≤ i ≤ k and k m all
0 i Pi=1 i
ie in a single PBW i-linearity domain X. Suppose also that k m and q both
Pi=1 i
lie in a single PBW i-linearity domain (not necessarily the same as X). Then m ,
i
i=1,2,...,k, k m and q all lie in a single PBW i-linearity domain.
Pi=1 i
Proof: We use Lemma 4.1 throughout. By the assumptions in the Proposition,
we have:
(i) Ri′(m ), 1 ≤ i ≤ k and Ri′( k m ) are weakly on the same side of all PBW
i i i Pi=1 i
i′-walls, and:
(ii) Ri′(q) and Ri′( k m ) are weakly on the same side of all PBW i′-walls. Let
i i Pi=1 i
H be a PBW i′-wall. Firstly, let i′ ∈ R(w ) be such that Ri′( k m ) does not
0 i Pi=1 i
lie on H. Then Ri′(m ), i = 1,2,...,k, Ri′( k m ) and Ri′(q) are all weakly
i i i Pi=1 i i
on the same side of H. Next, suppose that i′ ∈ R(w ) is such that Ri′( k m )
0 i Pi=1 i
does lie on H. By (i) and Proposition 3.6, Ri′(m ) lies on H for 1≤i≤k. Hence
i i
Ri′(m ), 1≤i≤k, Ri′( k m ) andRi′(q) are allweaklyonthe same side of H.
i i i Pi=1 i i
It follows (using Lemma 4.1 again) that the m , i=1,2,...,k, k m and q all
i Pi=1 i
lie in a single PBW i-linearity domain. 2
Remark4.3. If,intheassumptionsandconclusionoftheproposition,theproperty
”lie in a single PBW i-linearity domain” is replaced by the property ”q-commute”
(and the tuples involved are assumed to lie in ZN ), the result is not clear.
≥0
Recall that a strongly convexpolyhedralcone is a convex polyhedralcone C for
which v ∈ C implies −v 6∈ C. A set of strongly convex polyhedral cones is said to
form a fan if the face of every cone in the set lies in the set and if the intersection
of any two cones in the set lies again in the set.
Corollary 4.4. Let i∈R(w ). Then the set of PBW i-linearity domains (together
0
with all their faces) forms a fan in RN.
Proof: WefirstofallnotethateachPBWi-linearitydomainisaconvexpolyhedral
cone (by Proposition 3.6) and thus is in fact a strongly convex polyhedral cone as
it contains no point with negative coordinates. The same is therefore true for all
of its faces. We now show that if C and C′ are distinct PBW i-linearity domains,
then C ∩C′ is a face of C and of C′. Suppose that C ∩C′ is not a face of C.
Then C ∩C′ ( F, where F is a face of C (since C and C′ are convex polyhedral
cones). Let (C′)◦ denote the interior of C′; we will use similar notation for the
10 PHILIPPECALDEROANDROBERTMARSH
interiorsofother cones. Fix p∈F\C. Then, we canchoosem∈C∩C′ suchthat
m+p∈C∩C′, since C∩C′ andF are convexpolyhedralcones. Now, let q be in
(C′)◦. We havethat qandm+p lie ina singlePBWi-linearitydomain,andm,p
and m+p lie in a single PBW i-linearity domain. So by Proposition 4.2, q and
p both lie in a single PBW i-linearity domain. But, as q ∈ (C′)◦, C′ is the only
PBW i-linearity domain containing q. Since p6∈C′, we have a contradiction, and
we see that C∩C′ is a face of C (and similarly, we can see that it is a face of C′).
We next consider the case where C,C′ are faces of distinct PBW-linearity do-
mains, P and P′. We show that C ∩C′ is a face of C and of C′. Suppose that
C ∩C′ is not a face of C. Then, as above, C ∩C′ ( F, where F is a face of C.
As above, we can then choose q ∈ (P′)◦, p ∈ F \C′ and m ∈ C ∩C′ such that
m+p ∈ C ∩C′, since C ∩C′ and F are convex polyhedral cones. Note that the
first two properties imply that p 6∈ P′. This is because p lies in the linear span
of C′ ∩C, so p lies in the linear span of C′. Since P′ is convex, if p was in P′
and in the linear span of its face C′, it would have to lie in C′, a contradiction to
the assumption that p ∈ F \C′. So q and m+p lie in a single PBW i-linearity
domain P′, while m,p and m+p all lie in a single PBW i-linearity domain P.
Proposition 4.2 asserts that q and p both lie in a single PBW i-linearity domain,
butasq∈(P′)◦,P′ istheonlyPBWi-linearitydomaincontainingq,whilep6∈P′,
so we have a contradiction. Hence, C∩C′ is a face of C, and similarly, we can see
thatitisafaceofC′. Finally,supposethatC,C′ aredistinctfacesofasinglePBW
i-linearitydomainP. ThenC∩C′ isaface ofC, andofC′,since P isapolyhedral
cone. We have therefore shown that the set of PBW i-linearity domains, together
with all their faces, forms a fan in RN as required. 2
5. q-commuting products of quantum flag minors
For any reduced decomposition w˜ = s ...s of an element w in W, we define
i1 ik
as in [5], see also [2], the quantum flag minors ∆∗. Roughly speaking, ∆∗ is the
w˜ w˜
element of the dual canonical basis corresponding to the extremal vector of weight
w̟ in the Weyl module with highest weight ̟ . By [4], we have:
ik ik
Proposition 5.1. Fix a reduced decomposition i = (i ,i ,...,i ) in R(w ). Set
1 2 N 0
∆∗ = ∆∗ , where w˜ = s ...s , 1 ≤ k ≤ N. Then, the algebra A generated
k w˜k k i1 ik i
by the ∆∗ is a q-polynomial algebra, spanned (as a space) by a part of the dual
k
canonical basis.
Let d be the form on ZN defined by
i
di(m,n)=X(βi,βj)minj +Xmini,
j<i i
where the β ’s are the roots associated to i. We denote it by d if no confusion
k
occurs. The form d encodes the q-commutations in the graded algebra Gr (U+)
i i
(see [4]) associated to i. To be more precise, set
nk = X et.
t,t≤k,it=ik
Then:
Proposition 5.2. Fix i in R(w ). For k, 1 ≤ k ≤ N, we have ∆∗ = B(n )∗.
0 k i k
Moreover, for all m, n in ZN :
≥0
