QUIVER VARIETIES AND THE B(∞) CRYSTAL FOR
SYMMETRIZABLE KAC-MOODY ALGEBRAS
VINOTH NANDAKUMAR, PETER TINGLEY
Abstract. Kashiwara and Saito give a geometric construction of the B(∞) crystal
for a simply-laced Kac-Moody algebra by using irreducible components of Lusztig’s
quiver varieties (which are representation varieties of the corresponding pre-projective
algebra). We generalize their construction to symmetrizable Kac-Moody algebras.
The main idea of our work is to replace that preprojective algebra with a more general
one, first studied by Dlab and Ringel. The varieties that we use are defined over fields
which are not algebraically closed.
Contents
1. Introduction
2. Recollections
2.1. Symmetrizable Kac-Moody algebras
2.2. Crystals
3. Preprojective Algebras
4. Quiver varieties
4.1. The crystal operators
4.2. Proof of the Main Theorem
References
1
4
4
4
5
6
7
12
16
1. Introduction
Fix a symmetrizable Kac-Moody algebra g. Kashiwara’s crystal B(−∞) is a combinatorial object (a discrete set along with some partial-permutations f˜i ) that encodes
a lot of information about g and its integrable representations. This crystal is usually
defined using the corresponding quantized universal enveloping algebra, but it can also
be realized by other methods. In [2], Kashiwara and Saito developed a very useful
geometric realization, where the underlying set consists of the collection of irreducible
components of some algebraic varieties, and the partial permutations f˜i are geometric
operations.
The varieties used in Kashiwara and Saito’s construction are those developed by
Lusztig in [3, §12], and often called Lustzig’s nilpotent varieties. They are the varieties
of nilpotent representations of an algebra Λ acting on a fixed (graded) vector space.
1
2
VINOTH NANDAKUMAR, PETER TINGLEY
The algebra Λ, usually called the preprojective algebra, is associated to an acyclic
oriented graph. By choosing an appropriate orientation of the Dynkin diagram, one
can associate a preprojective algebra to any symmetric Cartan datum, but not, with
Lusztig’s definition, to a merely symmetrizable cartan datum. This is why Kashiwara
and Saito’s construction is restricted to the symmetric case.
However, even before Lusztig’s work, Dlab and Ringel ([1]) define the preprojective
algebra of a “modulated graph”. There is a natural way to associate a symmetrizable
(but not necessarily symmetric) Cartan datum to any modulated graph, and all symmetriable Cartan matrices arise this way. If that Cartan datum happens to be symmetric
then, by making the appropriate choices in Dlab and Ringel’s construction, one recovers
Lusztig’s preprojective algebra (see [4]).
Our main result is to generalize Kashiwara and Saito’s realization of B(−∞) by replacing Lusztig’s preprojective algebra with certain cases of Ringel and Dlab’s preprojective algebras. This gives a realization of B(−∞) for any symmetrizable Kac-Moody
algebra. Before stating our result more precisely, let us expain the special case of Dlab
and Ringle preprojectve algebras that we use in some more detail.
Begin with an underlying undirected graph Γ, and denote the set of vertices by I and
the set of edges by E. Let A be the set of directed edges, which we will call arrows;
so there are two arrows in A for each edge in E. We do not allow edges connecting a
vertex to itself, or multiple edges. Denote the tail of an arrow a by t(a) and the head
by h(a). Fix a field F with Q ⊂ F ⊂ C. We need the additional data of:
• A choice of Fi for each vertex i of the graph, such that Fi is a finite extension
of F.
• For each arrow a, an (Fi , Fj ) bimodule i Mj (which is a left Fi module and a right
Fj module), where i = h(a), j = t(a).
• For each arrow a, a non-degenerate Fi -bilinear form ji : i Mj ⊗Fj j Mi → Fi , where
i = h(a), j = t(a).
We will refer to all of this data as a modulated graph, and will often denote it by M .
The analogue of the path algebra for a modulated graph M is the tensor algebra TM
generated by all the Ma . That is,
TM =
M
k Mk−1
⊗Fk−1 · · · ⊗Fi3 i2 Mi2 ⊗Fi2 i2 Mi1
i1 i2 ···ik a path in Γ
as a vector space over C, and multiplication is give by tensor product, provided the end
of one path agrees with the beginning of the next, and is 0 otherwise.
For each i, j connected by an arrow a, the bilinear
form ji defines a canonical element
P
rji in j Mi ⊗Fi i Mj , which can be taken to be k vk ⊗ v k for any pair of dual Fi bases
{vk } ⊂ j Mi , {v k } ⊂ i Mj with respect to ji . It is well known that this does not depend
on the choice of dual bases.
QUIVER VARIETIES AND PREPROJECTIVE ALGEBRAS
3
For each i ∈ I, define
ri :=
X
h(a)
ri
t(a)=i
Definition 1.1. The preprojective algebra ΛM is the quotient of the tensor algebra
T (M ) by the ideal generated by {ri }i∈I .
One can associate a symmetrizable (but not necessarily symmetric) Cartan matrix
C = (cij ) to such a modulated graph as follows:


2 if i = j
ci,j = − dimFi i Mj if there is an arrow from i to j

0
otherwise.
As in [1], ΛM is finite dimensional over F if and only if C is a finite type Cartan matrix.
If C is symmetric then Fi = C for all i, for the obvious choice of bimodules and bilinear
forms, one recovers the preprojective algebra as it appears in Lusztig’s work [3] (see
[4]). Note however that, even in this case, different choices of bilinear form do give
non-isomorphic algebras (see [4]). To generalize Kashiwara and Saito’s construction we
use the representation varieties of ΛM .
P
There is a natural partition of the identity e ∈ ΛM as e = i ei , where ei is the
lazy L
path at node i. Given a representation of ΛM on a vector space V , notice that
V = i ei V , and in fact each ei V is naturally a left Fi module. Given a dimension vector
v = (vi )i∈I , fix a vi dimensional left Fi module Vi for each i. Define the representation
variety Λ(ν) to be the variety of representations of ΛM on V = ⊕i Vi such that ei V = Vi ,
and the induced left Fi module structure on Vi agrees with the original Fi module
structure. Note that, up to isomorphism, Λ(ν) does not depend on any choice.
The representation variety Λ(ν) is contained in
M
HomFj (j Mi ⊗Fi Vi , Vj ),
(i,j)∈A
and is cut out by the polynomial equations stating that each ri acts as 0. Although Λ is
only an algebra over F, our construction ensures that Λ(ν) is an algebraic variety over
F.
We will show that the union over all ν of the set of irreducible components of Λ(ν)
realizes B(−∞), where the crystal operators are natural analogues of those used by
Kashiwara and Saito [2]. There are a few technicalities to deal with, but, to a large
extent, Kashiwara and Saito’s original proof goes through.
It is well-known how to realize the infinity crystal for a symmetrizable Kac-Moody
algebra by a “folding” procedure on B(−∞) for a corresponding symmetric Kac-Moody
algebra. For this reason, Kashiwara and Saito’s work can already be used to study the
non-symmetric cases. However, in non-symmetric types, that construction does not
realize the crystal as the irreducible components of a representation variety, as we do
here.
4
VINOTH NANDAKUMAR, PETER TINGLEY
2. Recollections
2.1. Symmetrizable Kac-Moody algebras. Let g be a symmetrizable Kac-Moody
algebra with Cartan matrix C = (cij )i∈I , and let D = diag{di }i∈I be such that DC
is symmetric, with the di are relatively prime positive integers. Let P be the weight
lattice of g and Q the root lattice, and denote by {αi } be the simple roots. Let Q+ =
spanZ≥0 {αi }i∈I . Recall the usual bilinear form on Q defined by (αi , αj ) = di cij .
2.2. Crystals. Here we summarize some of the exposition in Section 3 of [2] regarding
crystals.
Definition 2.1. Let C be a set, with maps wt : C → P ; for each i ∈ I, maps
i : C → Z ∪ −∞, ẽi , f˜i : C → C ∪ {0}. Let φi (b) = i (b) + hα̌i , wt(b)i. C is a crystal if
these maps satisfying the following properties:
(1) If b ∈ B, ẽi b ∈ B, wt(ẽi b) = wt(b) + αi , i (ẽi b) = i (b) − 1.
(2) If b ∈ B, f˜i b ∈ B, wt(f˜i b) = wt(b) − αi , i (f˜i b) = i (b) + 1.
(3) Given b, b0 ∈ B, i ∈ I, then b0 = ẽi b ⇐⇒ b = f˜i b0 .
(4) If φi (b) = −∞, ẽi b = f˜i b = 0.
Definition 2.2. Given two crystals C1 and C2 , a map ψ : C1 → C2 is a strict embedding
if it satisfies the following properties:
(1) ψ is an injective map satisfying ψ(ẽi c1 ) = ẽi ψ(c1 ) and ψ(f˜i c1 ) = f˜i ψ(c1 ) for
c1 ∈ C1 (where let ψ(0) := 0).
(2) We have wt(ψ(c1 )) = wt(c1 ) and i (ψ(c1 )) = i (c1 ).
Definition 2.3. Given two crystals C1 and C2 , define a crystal structure on the tensor
product C1 ⊗ C2 = {c1 ⊗ c2 |c1 ∈ C1 , c2 ∈ C2 } as follows:
(1) Let i (c1 ⊗ c2 ) = max(
i (c1 ), i (c2 ) − hα̌i , wt(c1 )i), wt(c1 ⊗ c2 ) = wt(c1 ) + wt(c2 ).
(
ẽi b1 ⊗ b2 , if φi (b1 ) ≥ i (b2 )
(2) Let ẽi (b1 ⊗ b2 ) =
b ⊗ ẽi b2 , if φi (b1 ) i (b2 )
(1
f˜i b1 ⊗ b2 , if φi (b1 ) i (b2 )
(3) Let f˜i (b1 ⊗ b2 ) =
b1 ⊗ f˜i b2 , if φi (b1 ) ≤ i (b2 )
Theorem 2.4. Let B be a crystal. If the following conditions are satisfied, we have
B ' B(∞).
(1) For all b ∈ B, i ∈ I we have −wt(b) ∈ P + , i (b) ∈ Z.
(2) There exists a unique element b0 ∈ B with wt(b) = 0. It also satisfies i (b0 ) = 0
for all i ∈ I.
(3) For each i, we have a strict embedding of crystals Ψi : B → B ⊗ Bi , such that for
each b ∈ B, Ψi (b) = b0 ⊗ f˜in bi for some b0 ∈ B, n ≥ 0. Further for each b 6= b0 ,
there exists i with Ψi (b) = b0 ⊗ f˜in bi for some b0 ∈ B, n 0.
Proof. See Proposition 3.2.3 in [2].
QUIVER VARIETIES AND PREPROJECTIVE ALGEBRAS
5
3. Preprojective Algebras
Definition 3.1 ([1]). Let M be a graph with vertices indexed by I, and the set of
oriented edges given by H ⊂ I × I. Suppose that an edge τ ∈ H goes from the vertex
out(τ ) to in(τ ); let τ denote the edge from the vertex in(τ ) to out(τ ) (assume that
τ ∈ H ⇒ τ ∈ H). We allow multiple edges between two vertices. Say that the graph
M is “modulated” if we are given the data of a field Fi for each i ∈ I, a (Fout(τ ) ,Fin(τ ) )
bi-module Mτ with a non-degenerate bilinear pairing τ : Mτ ⊗Fin(τ ) Mτ → Fout(τ ) for
each τ ∈ H.
Consider the following modulated graph M , with di,j edges from node i to node j;
let τ be one such edge, with out(τ ) = i, in(τ ) = j. Here let F be any characteristic 0
field, with Q ⊆ F ⊆ C.
(1) Let the vertices of M be indexed by I. Given i, j ∈ I, join the corresponding
vertices by aij edges.
(2) Assign Fi to be a degree ki field extension of F; let Fi,j be the smallest subfield
of F containing Fi and Fj . Let Mτ = Fi,j considered as an (Fi , Fj ) bi-module
and let k i,j = dimF (Fi,j ).
(3) Let Fi,j = Fi ∩ Fj . Let πi,j : Fi,j → Fi be a non-zero map, and define τ :
Fi,j ⊗Fj Fi,j → Fi by τ (a, b) = πi,j (ab).
Let g be a symmetrizable Kac-Moody algebra defined by the following datum. Let
the simple roots be indexed by I, for each simple root αi with i ∈ I, suppose it has
d ki,j
length ki ; and let hα̌i , αj i = i,jki .
Q
Let M0 = i∈I Fi , M1 = ⊕τ ∈H Mτ and Mi+1 = M1 ⊗M0 Mi (here note Mτ ⊗M0 Mτ 0 = 0
unless in(τ ) = out(τ 0 )). Define the path algebra of the Dynkin diagram D to be FM =
⊕i≥0 Mi , with product being given by tensor product of bi-modules.
Assuming that dimFout(τ ) (Mτ ) < ∞, dimFin(τ ) (Mτ ) < ∞ , given τ : Mτ ⊗Fin(τ ) Mτ →
Fout(τ ) we will construct an element ∗τ ∈ Mτ ⊗Fout(τ ) Mτ as follows. Pick a basis
(a1 , · · · , ad ) of Mτ and aPbasis (b1 , · · · , bd ) of Mτ , as Fout(τ ) -vector spaces, such that
τ (ai , bj ) = δij . Let ∗τ = 1≤i≤d bi ⊗ ai . Then we have:
Lemma 3.2. The element ∗τ ∈ Mτ ⊗Fout(τ ) Mτ defined above is independent of the choice
of bases (a1 , · · · , ad ) and (b1 , · · · , bd ).
Proof. See Lemma 1.1 of [1].
Example 3.3. For each τ , let cτ ∈ Mτ = Fi,j denote the unit. Given τ with out(τ ) =
i, in(τ ) = j, suppose [Fi,j : Fi ] = k and that Fi,j = Fi [α] for some α with αk ∈ Fi . Then
we can describe ∗τ as follows. The Fi -basis {cτ , αji cτ , · · · , (αji )
i,j
1− kk
j
ki,j
kj
−1
cτ } of Mτ = Fi,j , is
dual to the Fi -basis {cτ , (αji )−1 cτ , · · · , (αji )
cτ } of Mτ = Fi,j , with respect to the
X
i t
∗
form τ . Thus we have: τ =
(αj ) cτ ⊗ (αji )−t cτ .
0≤t≤
ki,j
kj
−1
6
VINOTH NANDAKUMAR, PETER TINGLEY
P
Definition 3.4. Let ∗i ∈ M2 be defined by ∗i = τ,in(τ )=i ∗τ . Let I be the ideal in FM
generated by ∗i for i ∈ I. Define Λg = FM/I to be the preprojective algebra associated
to g.
Example 3.5. Let us give a description of the pre-projective algebra in type C2 . Suppose v1 is the vertex of length 1, and v2 the vertex of length 2; to the first vertex we
attach the field R, and to the second vertex we attach the field C (more generally, we
1
1
can use the fields F and F[p 2 ] where p ∈ F, p 2 ∈
/ F). Let τ (resp. τ ) be the edge
connecting v1 to v2 (resp. v2 to v1 ); let Mτ (resp. Mτ ) be C considered as a (R, C)
(resp. (C, R)) bi-module. We have idempotents e1 and e2 at these two vertices, and
paths cτ , icτ ∈ Mτ , cτ , cτ i ∈ Mτ . Then
Λ = Re1 + Ce2 + Rcτ + Ricτ + Rcτ + Rcτ i + Rcτ icτ + Ricτ cτ + Rcτ cτ
with the relations cτ cτ = 0 and cτ cτ + imτ mτ i = 0.
4. Quiver varieties
Let P = ⊕i∈I Zαi denote the
P root lattice of g, where αi denote the simple roots. Let
P = ⊕i∈I Z≥0L
αi , and ν = i∈I νi αi . Let V (ν)i be a νi -dimensional Fi -vector space;
define V (ν) = i∈I V (ν)i , considered as a F vector space.
M
X(ν) =
HomFin(τ ) ∩Fout(τ ) (Vout(τ ) , Vin(τ ) )
+
τ ∈H
Note that any B ∈ X(ν) gives a representation of the path algebra FM . Let Λ(ν)
denote the variety of all B ∈ X(ν), such that B gives a nilpotent representation of the
pre-projective algebra Λg. Given B ∈ Λ(ν), denote the corresponding Λg module by
MB . Let IrrΛ(ν) denote the irreducible components of Λ(ν), considered as an algebraic
variety over F.
Example 4.1. We will continue the example of type C2 . Let us compute the variety
Λ(ν) when ν = (1 − 2 ) + n(22 ) (where the two simple roots are 1 − 2 and 22 ).
Accordingly, we have V , a R-vector space of dimension 1 attached to the vertex v1 , and
W , a C-vector space of dimension n attached to the vertex v2 . Λ(ν) consists of R-linear
maps α : V → W , β : W → V which satisfy the pre-projective relations.
Fix isomorphisms V ' R, W ' Cn , and suppose α and β are determined as follows:
α(1) = [x1 + x2 i : x3 + x4 i : · · · : x2n−1 + x2n i]
β([t1 + t2 i : t3 + t4 i : · · · : t2n−1 + t2n i]) =
2n
X
ti yi
i=1
The first pre-projective relation tells us that βα = 0, which means that
2n
X
i=1
xi y i = 0
QUIVER VARIETIES AND PREPROJECTIVE ALGEBRAS
7
The second pre-projective relation tells us that αβ + iαβi = 0, i.e. that




y1
−y2
 y2 




 y1 
 · · ·  x1 x2 · · · x2n−1 x2n +  · · ·  x2 −x1 · · · x2n −x2n−1 = 0




 y2n−1 
 −y2n 
y2n−1
y2n
Looking at the block corresponding to {y2i−1 , y2i }, {x2j−1 , x2j }, we obtain that
y2i−1 x2j−1 = y2i x2j and y2i−1 x2j = −y2i x2j−1 . This implies that
2
2
2
2
y2i−1 (x22j−1 + x22j ) = y2i (x22j−1 + x22j ) = 0 = x2j−1 (y2i−1
+ y2i
) = x2j (y2i−1
+ y2i
)
Since this holds for each i, j, if x22i−1 +x22i 6= 0 for some i, then y2j−1 = y2j = 0 for each j.
Since x2i−1 , x2i ∈ R, x22i−1 +x22i = 0 implies x2i−1 = x2i = 0. So either x2i−1 = x2i = 0 for
each i, or y2j−1 = y2j = 0 for each j. Thus as a variety Λ(ν) consists of 2 n-dimensional
vector spaces inside a 2n-dimensional vector space, that intersect at 0; in particular,
Λ(ν) has 2 irreducible components.
4.1. The crystal operators. `
We will show that, as was shown in the symmetric case
in [2, theorem 5.3.2], the union ν IrrΛ(ν) along with some natural geometric operators
realized the infinity crystal B(∞). Before stating this precisely we must first define the
operators.
Definition 4.2. Given ν, ν 0 ∈ P , let ν = ν + ν 0 define Λ(ν, ν 0 ) to be the set of triples
φ
φ0
i
i
→
V (ν)i −
→
(B, φ, φ0 ) with φ = (φi )i∈I , φ0 = (φ0i )i∈I , with exact sequences 0 → V (ν)i −
V (ν 0 )i → 0 for each i ∈ I, such that Im(φ) is a B-stable subspace of V (ν).
Given (B, φ, φ0 ) ∈ Λ0 (ν, ν 0 ), we obtain points B ∈ Λ(ν), B 0 ∈ Λ(ν 0 ). Accordingly define ω1 : Λ(ν, ν 0 ) → Λ(ν), ω2 : Λ(ν, ν 0 ) → Λ(ν)×Λ(ν 0 ) by ω1 (B, φ, φ0 ) = B, ω2 (B, φ, φ0 ) =
(B, B 0 ).
Definition 4.3. Let Si denote the unique simple Λg module, with dimension vector αi .
Define i (B) = dimFi Hom(MB , Si ).
Let ν 0 = cαi ; since Λ(cαi ) is a point, we have ω2 : Λ0 (ν, cαi ) → Λ(ν).
Definition 4.4. For ν ∈ P, i ∈ I, k ∈ Z≥0 , let Λ(ν)i,k := {B ∈ Λ(ν) | i (B) =
k}. Define Λ0 (ν, cαi )0 = ω1−1 (Λ(ν)i,c ) = ω2−1 (Λ(ν)i,0 ). Let B(∞, ν)i,c = {Λ ∈
B(∞, ν) | Λ(ν)i,k ∩ Λ is open in Λ}.
It will be convenient for us to define auxiliary vector spaces as follows. Given τ with
in(τ ) = i, out(τ ) = j, denote V (ν)τi = V (ν)i ⊗Fi Fi,j .
Given the map Bτ : V (ν)i → V (ν)j , we can construct a map Bτ : V (ν)τi → V (ν)j , by
defining Bτ (αv) = αBτ (v) for v ∈ V (ν)i , α ∈ Fj ; it is clear that Bτ is well-defined, and
Fj -linear.
8
VINOTH NANDAKUMAR, PETER TINGLEY
fτ : V (ν)j → V (ν)τ as
Given the map Bτ : V (ν)j → V (ν)i , we will construct maps B
i
follows. Pick a set of bases {β1 , · · · , βk } and {α1 , · · · , αk } of Fi,j over Fi , with αi ∈ Fj ,
which are dual with respect to the form τ (see the beginning of Section 3) and let:
fτ (v) = β1 Bτ (α1 v) + · · · + βk Bτ (αk v)
B
The following lemma is a simple computation:
fτ is a map of Fj -vector spaces, and does not depend on the choice of
Lemma 4.5. B
X
dual bases. The pre-projective relation
∗i = 0 is satisfied iff the composite map
τ,in(τ )=i
fτ )
(B
V (ν)i −−→
L
(Bτ )
out(τ )=i
V (ν)τin(τ ) −−→ V (ν)i is 0.
fτ does not depend on the choice of dual bases, note that
Proof. To show that B
if we change the first element of the basis (i.e. replacing {α1 , · · · , αi , · · · } by
{α10 , · · · , αi , · · · }), then it is easy to check that the corresponding dual basis changes
from {β1 , · · · , βi , · · · }, to { βc11 , · · · , βi − βc11 ci , · · · } provided that α10 = c1 α1 + · · · + ck αk .
fτ (v) also doesn’t change:
Under this transformation, the value B
fτ (v) = β1 B(α1 v) + · · · + βk B(αk v)
B
α 0 − c2 α 2 − · · · − ck α k
v) + · · · + βk B(αk v)
= β1 B( 1
c1
β1
ci
= B(α10 v) + · · · + (βi − β1 )B(αi v) + · · ·
c1
c1
fτ
Since any two dual bases can be linked by a sequence of such mutations, the map B
does not depend on the choice of dual bases. Now given c ∈ Fj ,
fτ (cv) = β1 B(α1 cv) + · · · + βk B(αk cv)
B
fτ (v) = cβ1 B(α1 v) + · · · + cβk B(αk v)
cB
0
0
Note that under the form given by πi,j
: Fi,j → Fi , πi,j
(a) = πi,j (c−1 a) (see the beginning
of the Section 3), {β1 , · · · , βk } is dual to {α1 c, · · · , αk c}, and {cβ1 , · · · , cβk } is dual to
fτ (cv) = cB
fτ (v) since the map does not depend on the
{α1 , · · · , αk }. It follows that B
choice of dual bases.
X
It follows from the definitions that pre-projective relation
∗i = 0 is equivalent
τ,in(τ )=i
eτ = 0 (see the example following Lemma 3.2).
to B τ ◦ B
P
For all i, we have i (B) = dimFi V (ν)i / in(τ )=i V (ν)τ , where V (ν)τ denotes the
smallest Fin(τ ) -vector space in V (ν)in(τ ) containing Im(Bτ ). Alternatively, i (B) =
(Bτ )
dimFi Coker(⊕out(τ )=i V (ν)τin(τ ) −−→ V (ν)i ).
QUIVER VARIETIES AND PREPROJECTIVE ALGEBRAS
9
Remark 4.6. Below we use the fact that if π : X → Y is a (locally trivial) fiber bundle
with irreducible fiber F , then there is a bijection between the irreducible components
of X and Y ; note that this statement is true even when X and Y are defined over fields
which are not algebraically closed.
First assume that Y is irreducible; we will show that X is irreducible. Suppose that
X = X1 ∪ X2 , where X1 and X2 are closed in X. For each y ∈ Y , π −1 (y) = (π −1 (y) ∩
X1 ) ∪ (π −1 (y) ∩ X2 ); since π −1 (y) ' F is irreducible, π −1 (y) ⊆ X1 or π −1 (y) ⊆ X2 . Let
Y1 = {y|π −1 (y) ⊆ X1 }, Y2 = {y|π −1 (y) ⊆ X2 }. We will show that Y1 and Y2 are closed
sets; then since Y = Y1 ∪ Y2 is irreducible it will follow that Y = Y1 or Y = Y2 ; and
hence X = X1 or X = X2 as required.
Trivialize π over an open cover {Y i } of Y . We will prove that Y1 ∩ Y i is closed in Y i
for each i; this implies that Y1 is closed (since then Y˜1 = ∪(Y˜1 ∩ Y i ) is open, where Y˜1
is the complement of Y1 ). Now identifying π −1 (Y i ) ' Y i × F , we have
Y1 ∩ Y i = {y ∈ Y i |y × F ⊆ X1 } = ∩f ∈F {y ∈ Y i |y × f ∈ X1 }
This is closed, since {y ∈ Y i |y × f ∈ X1 } is closed (it is the pre-image of X under the
map Y i → Y i × F, y → (y, f )). Similarly Y2 is closed in Y .
Now if Y is not irreducible, suppose Y = ∪i Yi is it’s decomposition into irreducible
components; then using the above argument, π −1 (Yi ) is irreducible, so X = ∪i π −1 (Yi )
is its decomposition into irreducible components.
PropositionQ
4.7.
(1) The map ω1 : Λ0 (ν, cαi )0 → Λ(ν)i,c is a fibre bundle, with
fibres j∈I GLFj (νj ) × GLFi (c).
(2) The map ω2 : Λ0 (ν, cαi )0 → Λ(ν)i,0 is a fibre bundle with irreducible fibres.
(3) We have a bijection ẽmax
: B(∞, ν)i,c ' B(∞, ν)i,0 .
i
Proof. First note that the (3) follows from (1) and (2): B(∞, ν)i,c is in bijection with
the irreducible components of Λ(ν)i,c , and B(∞, ν)i,0 is in bijection with the irreducible
components of Λ(ν)i,0 . The irreducible components of Λ(ν)i,c are in bijection with the
irreducible components of Λ0 (ν, cαi )0 by (1), and the irreducible components of Λ(ν)i,0
are also in bijection with the irreducible components of Λ0 (ν, cαi )0 by (2); so the result
follows.
To show (1), consider the fibre of Λ0 (ν, cαi ) above a point B ∈ Λ(ν)i,c . For each j 6= i,
we must choose an isomorphism V (ν)j ' V (ν)j ; which gives us a fibre of GLνj (Fj ). We
φ
φ0
i
i
also need to choose P
an exact sequence 0 → V (ν)i −
→
V (ν)i −
→
V (cαi )i → 0. Since
i (B) = 0, Im(φi ) = in(τ )=i V (ν)τ ; which gives us a fibre of GLν i (Fi ) for choosing φi .
P
Choosing φ0i is equivalent to choosing an isomorphism V (ν)i / in(τ )=i V (ν)τ ' V (cαi )i ,
which gives us a fibre of GLFi (c). For each τ , the map Bτ : V (ν)out(τ ) → V (ν)in(τ )
determines a map B τ : V (ν)out(τ ) → V (ν)in(τ ) . The result now follows.
To show that ω1 is locally trivial, consider the following open cover of the base Λ(ν)i,c .
Pick a basis {e1 , · · · , eνi } of V (ν)i , and for each S ⊆ {1, · · · , νi } with |S| = c, let
10
VINOTH NANDAKUMAR, PETER TINGLEY
Λ(ν)Si,c ⊆ Λ(ν)i,c be the open subset defined by the condition that
X
i
(
V (ν)τ ) ⊕ Fi {ej }j∈S = V (ν)i
in(τ )=i
It suffices to show that ω1 |Λ(ν)Si,c is the trivial bundle. To see this, for each j 6= i,
there is aPcanonical choice of isomorphism V (ν)i ' V (ν). We have a natural map
V (ν)i ' in(τ )=i V (ν)τ : identify V (ν)i with Fi {ej }j ∈S
/ , look at it’s image under i and
P
project to in(τ )=i V (ν)τ . Thus we have a canonical choice of exact sequence
φ0
φ
i
i
0 → V (ν)i −
→
V (ν)i −
→
V (cαi )i → 0
It follows that we can trivialize the bundle ω1 |Λ(ν)Si,c .
To show (2), consider the fibre of Λ0 (ν, cαi ) above a point B 0 ∈ Λ(ν)i,0 . For each
j 6= i, we must choose an isomorphism V (ν)j ' V (ν)j ; this gives us a fibre of GLνj (Fj ).
φ0
φ
i
i
Next we must choose an exact sequence 0 → V (ν)i −
→
V (ν)i −
→
V (cαi )i → 0. Choice of
the map φi gives us a fibre of GLFi (ν i ) × Gr(ν i , V (ν)i ). If out(τ ) 6= i, Bτ is determined
fτ )
(B
by B τ ; it remains to choose Bτ when out(τ ) = i (or equivalently, the map V (ν)i −−→
L
τ
f
τ,out(τ )=i V (ν)in(τ ) ). Choose a splitting V (ν)i = Im(φi ) ⊕ Wi . The map (Bτ )|Im(φi ) is
fτ )|W subject to the pre-projective
determined; we have freedom in choosing the map (B
i
fτ ) L
(Bτ )
(B
τ
relation that the composition Wi −−→ τ,out(τ )=i V (ν)in(τ ) −−→ V (ν)i is zero. Thus we
L
(Bτ )
get a fiber of Hom(Wi , Ker( τ,out(τ )=i V (ν)τin(τ ) −−→ V (ν)i )). This is an affine space of
Q
fixed dimension, as we compute below; thus the fibre of ω2 is a product of j6=i GLνj (Fj ),
a Grassmanian, and an affine space; and hence irreducible.
dim Wi = c
dim Ker (
M
(Bτ )
V (ν)τin(τ ) −−→ V (ν)i ) = dim (
τ,out(τ )=i
M
V (ν)τin(τ ) ) − dim Im(Bτ )
τ,out(τ )=i
=
X
νin(τ )
out(τ )=i
kin(τ ),i
− νi + c
ki
To show that ω2 is locally trivial, first let P → Λ(ν)i,0 be the fiber bundle determined
by the data V (ν)j ' V (ν)j for each j 6= i, and an exact sequence
φ
φ0
i
i
0 → V (ν)i −
→
V (ν)i −
→
V (cαi )i → 0
By the above arguments, this bundle is locally trivial.
QUIVER VARIETIES AND PREPROJECTIVE ALGEBRAS
11
Now we have a map θ : Λ0 (ν, cαi ) → P , and ω2 is the composition Λ0 (ν, cαi ) → P →
Λ(ν)i,0 ; so it is sufficient to show that θ is locally trivial. Recall we have chosen a basis
{e1 , · · · , eνi } of V (ν)i ; for each S ⊆ {1, · · · , νi } with |S| = c, let PS ⊆ P be the open
set where
Im(φi ) ⊕ Fi {ej }j∈S = V (ν)i
L
τ
Also choose a basis of
τ,out(τ )=i V (ν)in(τ ) : {f1 , · · · , fN }; for each T ⊆ {1, · · · , N }
with |T | = ν i , let PT ⊆ P be the open subset where
M
M
(Bτ )
j
Ker(
V (ν)τin(τ ) −−→ V (ν)i ) ⊕ Fi {ej }j∈T =
V (ν)τin(τ )
τ,out(τ )=i
τ,out(τ )=i
L
(Bτ )
Denote W1 = Fi {ej }j∈S , and W2 = Ker( τ,out(τ )=i V (ν)τin(τ ) −−→ V (ν)i ). Now it
suffices to show that θ|PS ∩PT is the trivial bundle, for each S and T . Note that W2 has
a distinguished basis (look at the images of {ek }k∈T
/ under j). Now the fiber of θ|PS ∩PT
consists of an element of Hom(W1 , W2 ); since these two spaces have distinguished bases,
θ|PS ∩PT is the trivial fiber bundle.
` Using the above proposition, we now ˜define the structure of a crystal on
ν∈P + B(∞, ν) by defining the operators ẽi , fi , i , wt and φi .
Definition 4.8. Given Λ ∈ B(∞, ν)i,c :
• Let ẽi (Λ) ∈ B(∞, ν − αi )i,c−1 be the image of Λ under the composite bijection
B(∞, ν)i,c ' B(∞, ν)i,0 ' B(∞, ν − αi )i,c−1 if c ≥ 1; if c = 0, let ẽi (Λ) = 0.
• Let f˜i (Λ) ∈ B(∞, ν + αi )i,c+1 be the image of Λ under the composite bijection
B(∞, ν)i,c ' B(∞, ν)i,0 ' B(∞, ν + αi )i,c+1 .
• Let i (Λ) = i if Λ ∈ B(∞, ν)i,c .
• Let wt(Λ) = −ν if Λ ∈ B(∞, ν).
• Let φi (Λ) = i (Λ) + hα̌i , wt(Λ)i.
We can now state the main result of this section:
`
Theorem 4.9. ν∈P + B(∞, ν) along with the operators ẽi , f˜i from Definition 4.8 is a
realization of B(∞).
As a first step towards proving Theorem 4.9, note that:
`
Lemma 4.10. The above maps endow ν∈P + B(∞, ν) with the structure of a combinatorial crystal.
Proof. The conditions in Definition 2.1 are straightforward to check:
• If Λ ∈ B(∞, ν)i,c , then wt(Λ) = −ν, wt(ẽi Λ) = −ν + αi , i (Λ) = c, i (ẽi Λ) = c − 1;
so the first condition is satisfied.
• If Λ ∈ B(∞, ν)i,c , then wt(Λ) = −ν, wt(f˜i Λ) = −ν − αi , i (Λ) = c, i (f˜i Λ) = c + 1;
so the second condition is satisfied.
12
VINOTH NANDAKUMAR, PETER TINGLEY
• To see that f˜i ẽi b = b, the operation f˜i ẽi corresponds to the composite bijection
B(∞, ν)i,c ' B(∞, ν)i,0 ' B(∞, ν − αi )i,c−1 ' B(∞, ν)i,0 ' B(∞, ν)i,c ; similarly
ẽi f˜i b = b.
• The fourth condition is vacuously true, since φi (b) = −∞ is not possible.
Example 4.11. While it is difficult to check Theorem 4.9 in type C2 by explicit computation, we can at least check that it is consistent with some of the data that we have.
Denoting the Kostant partition function by kpf, it is well-known that
|b ∈ B(∞) : wt(b) = ν| = kpf(ν)
Thus we expect that |B(∞, ν)| = kpf(ν). When ν = (1 − 2 ) + n(22 ), this statement
follows from the calculation in Example 4.1, since there we show that B(∞, ν) has 2
irreducible components. On the other hand,
ν = (1 − 2 ) + n(22 ) = (1 + 2 ) + (n − 1)(22 )
so kpf(ν) = 2. This example also illustrates the need to work over varieties that are
not algebraically closed. If we based-changed to C, then the number of irreducible
components would increase, and the result would no longer be true. For instance if
n = 1, then we have the subvariety of A4 cut out by the two equations y1 x1 = y2 x2 and
y1 x2 = −y2 x1 . This has 4 irreducible components: the two components that we have
seen before ({x1 = x2 = 0} and {y1 = y2 = 0}) as well as {x1 = x2 i, y1 = y2 i} and
{x1 = −x2 i, y1 = −y2 i}.
ω
ω
2
1
4.2. Proof of the Main Theorem. Recall the diagram Λ(ν) × Λ(ν 0 ) ←−
Λ0 (ν, ν 0 ) −→
Λ(ν), where ν = ν 0 +ν. If we set ν = cαi (as opposed to ν 0 = cαi ); then since Λ(cαi ) = 0
ω2
ω1
we have maps Λ(ν 0 ) ←−
Λ0 (cαi , ν 0 ) −→
Λ(ν).
Definition 4.12. Given B ∈ Λ(ν), define ∗i (B) = dimFi Hom(Si , MB ).
Note that given fields F1 ⊂ F2 and a V a vector space over F2 , HomF1 (V, F1 ) is an F2 vector space (where given a ∈ F2 , φ : V → F1 , let (aφ)(v) = φ(av)). For each i ∈ I, fix
an isomorphism
HomF (V (ν)i , F) ' L
V (ν)i ; this givesLan isomorphism
L of Fi -vector spaces
L
θ : HomF ( i∈I V (ν)i , F) '
i∈I V (ν)i . Given
i∈I V (ν)i →
i∈I V (ν)i , B ∈
L B :
L
Λ(ν), we obtain a natural map B t : HomF ( i∈I V (ν)i , F) → HomF ( i∈I V (ν)i , F).
Using the identification θ, we may consider B t ∈ Λ(ν). This map extends to give a
bijection t : B(∞, ν) → B(∞, ν) (i.e. given Λ ∈ B(∞, ν), there exists Λt ∈ B(∞, ν)
such that B ∈ Λ ⇐⇒ B t ∈ Λt ).
(B̃τ )
Note that for all i, we have ∗i (B) = dimFi Ker(V (ν)i −−→ ⊕out(τ )=i V (ν)τin(τ ) ).
Definition 4.13. Let ẽ∗max
(Λ)
i
t t
∗
t
˜
(fi (Λ )) , φi (Λ) = φi (Λ ).
=
(ẽmax
(Λt ))t , ẽ∗i (Λ)
i
=
(ẽi (Λt ))t , f˜i∗ (Λ)
=
Lemma 4.14. Suppose that for all i ∈ I, ∗i (Λ) = 0. Then Λ is the zero component
(i.e. wt(Λ) = 0).
QUIVER VARIETIES AND PREPROJECTIVE ALGEBRAS
13
Proof. Pick a generic B ∈ Λ. By definition, it follows that for all i, Hom(Si , MB ) = 0.
This will imply that MB is the trivial module for Λg (and hence wt(Λ) = 0) provided we
know that {Si |i ∈ I} is a complete set of simple modules for the pre-projective algebra
Λg.
Lemma 4.15. We have i (Λ) = max(i (ẽ∗max
(Λ)), ∗i (Λ) + hα̌i , ν ∗ i).
i
∗
∗
Proof. Let Λ = ẽ∗,max
(Λ), c = ∗i (Λ). Given a generic point B ∈ Λ , we will construct
i
a generic point B ∈ Λ below. Pick a surjective Fi -linear map φ0i : V (ν)i → V (ν ∗ )i , and
define:
M
fτ )
φ0i
(B
V (ν)τin(τ ) )
Ni = Coker(V (ν)i −
→ V (ν ∗ )i −−→
out(τ )=i
Note that if i 6= j, and there are di,j edges connecting nodes i and j, then hα̌i , αj i =
di,j ki,j
, so:
ki
X di,j k i,j
hα̌i , ν ∗ i = 2(νi − c) −
νj
ki
j6=i
X
= 2 dim V (ν ∗ )i −
dim V (ν)τin(τ )
out(τ )=i
dim Ni = dim(
=
M
V (ν)τin(τ ) ) − dimV (ν ∗ )i
out(τ )=i
dimV (ν ∗ )i −
hα̌i , ν ∗ i
The maps (Bτ ) gives us a map ψi : Ni → V (ν ∗ )i . Choose a generic map ψi : Ni → V (ν)i
such that ψi ◦ φ0i = ψ i . Since ψi was chosen generically,
dim Ker ψi = max (dim Ker ψi − c, 0)
Given an edge τ , define Bτ : V (ν)out(τ ) → V (ν)in(τ ) as follows. If in(τ ), out(τ ) 6= i, let
φ0
B
τ
i
→
V (ν ∗ )i −→
Bτ = Bτ . If out(τ ) = i, let Bτ be the composition V (ν)i −
V (ν)in(τ ) . If
L
τ0
in(τ ) = i, let Bτ be the composition V (ν)out(τ ) ,→ out(τ 0 )=i V (ν)in(τ 0 ) → Ni → V (ν)i .
L
(Bτ )
ψi
Noting that Im( out(τ )=i V (ν)τin(τ ) −−→ V (ν)i ) = Im(N −→ V (ν)i ), compute (where
the dimension is over Fi ):
i (B) = dimV (ν)i − dim Im(ψ i ) = dimV (ν)i − dimNi + dim Ker(ψ i )
= dimV (ν)i − dimNi + max (dim Ker ψi − c, 0)
= max(dimV (ν)i − dimNi + dim Ker ψi − c, dimV (ν)i − dimNi )
= max(dimV (ν ∗ )i − dim Im ψi , dimV (ν)i − dimV (ν ∗ )i + hα̌i , ν ∗ i)
∗
= max(Coker(ψ i ), c + hα̌i , ν ∗ i) = max(i (Λ ), c + hα̌i , ν ∗ i)
14
VINOTH NANDAKUMAR, PETER TINGLEY
Lemma 4.16. We have:
(
ẽi (ẽ∗,max
Λ), if i (ẽ∗,max
Λ) ≥ hα̌i , ν ∗ i + ∗i (Λ)
i
i
ẽ∗,max
(ẽ
Λ)
=
i
i
ẽ∗,max
Λ, if i (ẽ∗,max
Λ) < hα̌i , ν ∗ i + ∗i (Λ)
i
i
(
Λ) ≥ hα̌i , ν ∗ i + ∗i (Λ)
∗ (Λ), if i (ẽ∗,max
i
∗i (ẽi Λ) = i∗
i (Λ) − 1, if i (ẽ∗,max
Λ) < hα̌i , ν ∗ i + ∗i (Λ)
i
Proof. First we show that i (ẽ∗,max
Λ) ≥ hα̌i , ν ∗ i + ∗i (Λ) ⇔ dim Ker ψi ≥ c.
i
∗
ψi
i (Λ ) = dim Coker(Ni −→ V (ν ∗ )i )
= dim V (ν ∗ )i − dim Im ψi
= dim V (ν ∗ )i − dim Ni + dim Ker ψi
= hα̌i , ν ∗ i + dim Ker ψi
∗
dim Ker ψi = i (Λ ) − hα̌i , ν ∗ i
If dim Ker ψi < c, then Im ψi 6⊇ Ker φ0i ; while if dim Ker ψi ≥ c, due to our generic
choice of the map ψi , Im ψi ⊇ Ker φ0i . Now consider a generic hyperplane H in V (ν) such
that H ⊇ Im ψi ; taking V (ν−αi ) = H we obtain a point B 0 in the irreducible component
∗
∗
ẽi (Λ). Similarly, considering a generic hyperplane H in V (ν ∗ ) such that H ⊇ Im ψi
∗
0
∗
and taking V (ν ∗ − αi ) = H we obtain a point B in the irreducible component ẽi (Λ ).
∗
∗
Recall that given a point B in a component Λ of Λ(ν), a point B in Λ = ẽ∗max
Λ can
i
f
L
(Bτ )
be constructed by replacing V (ν)i by Im(V (ν)i −−→ out(τ )=i V (ν)τin(τ ) ).
(ẽi Λ) corresponds to replacing H by
If dim Ker ψi < c, Im ψi 6⊇ Ker φ0i , and ẽ∗max
i
fτ ) L
fτ ) L
(B
(B
the vector space Im(H −−→ out(τ )=i V (ν)τin(τ ) ) = Im(V (ν)i −−→ out(τ )=i V (ν)τin(τ ) ).
∗
Thus ẽ∗max
(ẽi Λ) = Λ . Also
i
M
M
fτ )
fτ )
(B
(B
dim Ker(H −−→
V (ν)τin(τ ) ) = dim Ker(V (ν)i −−→
V (ν)τin(τ ) )) − 1
out(τ )=i
out(τ )=i
∗i (ẽi Λ)
= c − 1.
If dim Ker ψi ≥ c, Im ψi ⊇ Ker φ0i , and ẽ∗max
(ẽi Λ) corresponds to replacing H by the
i
fτ )
fτ ) L
(B
(B
vector space Im(H −−→ out(τ )=i V (ν)τin(τ ) ), which is a hyperplane in Im(V (ν)i −−→
L
∗
∗max
τ
(ẽi Λ) = ẽi Λ . Also,
out(τ )=i V (ν)in(τ ) ). Thus ẽi
M
M
fτ )
fτ )
(B
(B
dim Ker(H −−→
V (ν)τin(τ ) ) = dim Ker(V (ν)i −−→
V (ν)τin(τ ) )
so
out(τ )=i
so ∗i (ẽi Λ) = c.
out(τ )=i
QUIVER VARIETIES AND PREPROJECTIVE ALGEBRAS
15
Lemma 4.17. For i 6= j, ∗i (ẽj (Λ)) = ∗i (Λ) and ẽ∗,max
(ẽj Λ) = ẽj (ẽ∗,max
(Λ)).
i
i
Proof. Given a point B ∈ Λ, we may a point B 0 ∈ ẽi (Λ) if we replace V (ν)j by a generic hyperplane H =: V (ν − αj )j containing the vecfτ )
L
(B
tor space Im( out(τ )=i V (ν)τin(τ ) → V (ν)j ). Since we have dim Ker(V (ν)i −−→
fτ ) L
L
(B
τ
τ
out(τ )=i V (ν)in(τ ) ) due to the generic
out(τ )=i V (ν − αj )in(τ ) ) = dim Ker(V (ν)i −−→
It is also clear that ẽ∗,max
(ẽj Λ) =
choice of H, it follows ∗i (ẽj Λ) = ∗i Λ.
i
∗,max
ẽj (ẽi
(Λ)).
`
Proof of theorem 4.9. We will use Theorem 2.4 to show that ν B(ν) ' B(∞). Define
`
`
∗ (Λ)
(Λ)⊗ f˜i i bi . Note that the first two
Ψi : ( ν B(ν)) → ( ν B(ν))⊗Bi by Ψi (Λ) = ẽ∗max
i
conditions in Theorem 2.4 are vacuously satisfied: if Λ ∈ B(∞, ν), −wt(Λ) = ν, i (Λ) ∈
Z≥0 , and if wt(Λ) = ν, Λ is the zero component and i (Λ) = 0 for all i. By definition we
have Ψi (Λ) = Λ0 ⊗ f˜in bi for n ≥ 0; by Lemma 4.14 for each non-zero component Λ there
exists i such that ∗i (Λ) 6= 0, so that Ψi (Λ) = Λ0 ⊗ f˜in bi for n 0. To see that the third
condition in Theorem 2.4 is satisfied, it suffices to check that Ψi is a strict embedding
(Λt )
of crystals. The injectivity of the map Ψi is clear, since knowing ẽ∗max
(Λ) = ẽmax
i
i
∗
t
t
and i (Λ) = i (Λ ) uniquely determines Λ , and hence Λ. Now we will check that Ψi
commutes with j , wt, ẽj and f˜j . For convenience, define ν ∗ = ν − ∗i (Λ)αi .
∗
(Λ)
wt(Ψi (Λ)) = wt(ẽ∗max
(Λ) ⊗ f˜i i bi )
i
∗
(Λ)
= wt(ẽ∗max
(Λ)) + wt(f˜i i bi ) = (−ν + ∗i (Λ)αi ) − ∗i (Λ)αi
i
= −ν = wt(Λ)
∗
(Λ)
i (Ψi (Λ)) = i (ẽ∗max
(Λ) ⊗ f˜i i bi )
i
(Λ)), ∗i (Λ) + hα̌i , ν − ∗i (Λ)αi i)
= max(i (ẽ∗max
i
= i (Λ)
(from Lemma 4.15)
16
VINOTH NANDAKUMAR, PETER TINGLEY
∗
(Λ)
(Λ)) = −ν ∗ )
φi (ẽ∗,max
(Λ)) − i (f˜i i ) = i (ẽ∗,max
Λ) − hα̌i , ν ∗ i − ∗i (Λ) (wt(ẽ∗,max
i
i
i
∗
(Λ)
(Λ) ⊗ f˜i i bi )
ẽi (Ψi (Λ)) = ẽi (ẽ∗max
i
(
∗ (Λ)
ẽi ẽi∗max (Λ) ⊗ f˜i i bi ,
=
∗ (Λ)
(Λ) ⊗ ẽi f˜i i bi ,
ẽ∗max
i
(
∗ (Λ)
ẽi ẽ∗max
(Λ) ⊗ f˜i i bi ,
i
=
∗ (Λ)
ẽ∗max
(Λ) ⊗ ẽi f˜i i bi ,
i
∗
(Λ)
if φi (ẽ∗max
(Λ)) ≥ i (f˜i i bi )
i
∗ (Λ)
(Λ)) i (f˜i i bi )
if φi (ẽ∗max
i
if i (ẽ∗,max
Λ) ≥ hα̌i , ν ∗ i + ∗i (Λ)
i
if i (ẽ∗,max
Λ) < hα̌i , ν ∗ i + ∗i (Λ)
i
∗
(ẽ Λ)
= ẽ∗,max
(ẽi Λ) ⊗ f˜i i i bi = Ψi (ẽi (Λ)) (see Lemma 4.16)
i
∗
(Λ)
For i 6= j , j (Ψi (Λ)) = j (ẽ∗max
(Λ) ⊗ f˜i i bi )
i
= j (ẽ∗max
(Λ)) (since j (f˜ik bi ) = −∞ ∀k)
i
= j (Λ) (using Lemma 4.15)
∗
(Λ)
ẽj (Ψi (Λ)) = ẽj (ẽ∗max
(Λ) ⊗ f˜i i bi )
i
∗ (Λ)
= ẽj ẽ∗max
(Λ) ⊗ f˜i i
i
∗i (ẽj (Λ))
= ẽ∗max
(ẽj (Λ)) ⊗ f˜i
i
∗ (Λ)
bi (since j (f˜i i
bi ) = −∞)
= Ψi (ẽj Λ) (using Lemma 4.17)
Note that f˜j (Ψi (Λ)) = Ψi (f˜j (Λ)) follows from ẽj (Ψi (Λ)) = Ψi (ẽj (Λ)).
References
[1] V. Dlab, C. M. Ringel, The preprojective algebra of a modulated graph, Representation theory
II, Lecture Notes in Math 832, Springer, Berlin 1980, 216-231
[2] M. Kashiwara, Y. Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), 9-36.
[3] G. Lusztig. Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math. Soc.,
4(2):365–421, 1991.
[4] Ringel, Claus Michael. The preprojective algebra of a quiver. Algebras and modules, II (Geiranger,
1996), 467480, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998.
[5] P. Baumann and J. Kamnitzer, Preprojective algebras and MV polytopes, Represent. Theory 16
(2012), 152-188; math.RA/1009.2469
[6] Pierre Baumann, Joel Kamnitzer, and Peter Tingley. Ane Mirkovic-Vilonen polytopes. Preprint.
arXiv:1110.3661.
[7] P. Tingley and B. Webster, A characterization of Mirkovic-Vilonen polytopes, preprint