BALANCED PARABOLIC QUOTIENTS AND BRANCHING RULES

BALANCED PARABOLIC QUOTIENTS AND BRANCHING RULES FOR
DEMAZURE CRYSTALS
JOHN M. DUSEL
Abstract. We study a subset of a parabolic quotient in a simply-laced Weyl group W —stable under
an automorphism σ—which we call the balanced parabolic quotient. This subset describes the interaction
between the branching rule for a Levi subalgebra, Demazure modules, and σ-invariant weight spaces in
σ-stable simple modules for the corresponding Lie algebra. The balanced quotients in types A and D
are enumerated by forest digraphs, whose components possess a remarkable self-similarity property. We
characterize an element of a balanced quotient on the level of the root system of W , and find that the
subalgebras of the Borel associated with these elements decompose into the direct sum of two subalgebras:
one contained in the Borel for a Levi subalgebra, and another consisting of σ-invariants.
Introduction
It is well-known that many aspects of the representation and structural theory of a semisimple Lie algebra g
and its quantum analogue are closely related to and connected by the combinatorial properties of the Weyl
group W of g (see [4, 5, 7, 12, 14] to name but a few). In this paper we establish a relationship between certain
properties of a simple module of a semisimple Lie algebra g of simply-laced type and a specific subset of the
Weyl group W of the corresponding Cartan datum, comprising elements of a parabolic quotient J W ⊂ W
whose reduced decompositions are balanced with respect to a diagram automorphism σ. We call this set
the balanced quotient and denote it by J Wσ . In the context of a simple σ-stable g-module, a balanced
quotient describes the interaction between the branching rule for a Levi subalgebra, Demazure modules, and
σ-invariant weight spaces. Our results are expressed in terms of the Kashiwara-Littelmann crystals B(∞)
and B(λ), for dominant integral weights λ. Although our results are applicable to representation theory,
our methods use only the combinatorial properties of the Kashiwara-Littelmann crystals along with the
Kashiwara ? operation.
We give a canonical reduced expression for each element of J Wσ , and give a presentation of these expressions using a forest digraph. (This is different than the normal forest of [1, 3.4] in that we are considering a
subset of W , and the edges are labeled by elements of the type A parabolic subgroup.) By design a balanced
quotient is a proper subset of set of fully commutative elements in a Coxeter group. As such, our normal
forms can be viewed as refinements of results of Stembridge [18].
Our first main result is a characterization of an element of J Wσ by a property of its inversion set. In
[2] it is shown—in the context of a simply-laced Coxeter group—that w is a fully commutative element if
and only if the inverse of w yields an abelian subalgebra of the Borel. Combining our result on inversion
sets with this property, we find that an element of the balanced quotient gives an abelian subalgebra of the
Borel that decomposes into the direct sum of two subalgebras: one contained in the Borel for the Jth Levi
subalgebra, and one consisting of σ-invariants. Formulae for the dimensions of these subalgebras are given,
based on the structure of a reduced decomposition of w.
Turning to the crystal theory, recall for all w ∈ W the Kashiwara-Littelmann crystal B(λ) contains a
unique subset Bw (λ), stable under all e˜i , called a Demazure crystal. In particular, when W is finite the subset
Bw◦ (λ) corresponding to the longest element coincides with B(λ). A Demazure crystal is a combinatorial
analogue of a Demazure module for an extremal vector in a simple finite-dimensional g-module of highestweight λ. On the other hand, a branching rule is a combinatorial analogue of the decomposition of a simple
finite-dimensional g-module into a direct sum of simple modules for a Levi subalgebra. The crystal B(λ)
is graded by the weight lattice, on which σ acts, and elements whose degree is preserved by σ are called
Date: revision 9, last modified May 20, 2015.
This research was partially supported by the J.W. and Ida M. Jameson foundation.
1
weight-invariant. Our second main result is that a Demazure crystal contains a weight-invariant element
that is highest-weight for the Jth branching rule if and only if w ∈ J Wσ . We prove this using a combination
of the Kashiwara embedding and ? involution [7], Kashiwara-Nakashima-Zelevinsky polyhedral realization
[7, 14], Littelmann path model [10], and our normal forms for J Wσ .
This paper is organized as follows. Section 1 contains the statements of our main results along with the
necessary technical definitions. In Section 2 we present the normal forms for the elements of a balanced
quotient, along with the associated forest digraphs, and give cardinality formulae in types A, D. Section 3
is devoted to the proof of Theorem 1.3. Finally, in Section 4 we give a proof of Theorem 1.5.
Acknowledgement. The author thanks J. Greenstein for his mentorship throughout the preparation of
this paper.
1. Main results
1.1. Preliminaries. We denote the set of nonnegative integers by N. Let I be a finite set, and A = (aij )i,j∈I
be an integral matrix such that aii = 2 for all i, aij ∈ {−1, 0}, and aij = 0 if and only if aji = 0. Remark
that these conditions imply that A is symmetric. Let Λ be the free abelian group on {$i | i ∈ I}, and let
Λ∨ := HomZ (Λ, Z). Let h·, ·i : Λ∨ ⊗Z Λ → Z denote the evaluation pairing, and denote by Π∨ = {αi∨ : i ∈ I}
the dual basis of Λ∨ . Let Λ+ := {λ ∈ Λ | hαi∨ , λi ≥ 0 for all i} denote the set of dominant weights. We choose
a linearly independent set Π := {αi | i ∈ I} ⊂ Λ, called the set of simple roots, such that hαi∨ , αj i = aij for
all i, j. The quintuple C = (A, Λ, Λ∨ , Π, Π∨ ) is called a simply-laced Cartan datum.
We call a bijection σ : I → I a diagram automorphism of C if σ(A) = A. Following Lusztig [12, 12.1.1]
we refer to σ as admissible if aij = 0 whenever j belongs to the σ-orbit of i. An admissible diagram
automorphism permutes the sets {$i | i ∈ I} and Π∨ , and these permutations induce automorphisms of the
∨
lattices Λ and Λ∨ such that hαi∨ , λi = hασ(i)
, σ(λ)i. Furthermore, σ permutes the set of simple roots.
The Weyl group W of C is generated by the set {si | i ∈ I} of simple reflections modulo the relations
s2i = 1 for all i and
(1.1)
si sj = sj si
(1.2)
si sj si = sj si sj
if aij = 0
if aij = −1
We refer to relations (1.1) and (1.2) as commutation and braid relations respectively. (This terminology is
justified by the absence of any other types of relations in the simply-laced case.) The Weyl group acts on Λ
via
(1.3)
si λ := λ − hαi∨ , λiαi .
On the other hand, an admissible diagram automorphism acts on the Weyl group by extending si 7→ sσ(i) .
Let W σ denote the set of all w ∈ W such that σ(w) = w.
An expression w = si1 · · · sit is called reduced when t takes its minimal value amongst all possible expressions for w as products of simple reflections, and the tuple (i1 , . . . , it ) ∈ I t is called a reduced word for w.
The length of w is `(w) := t, and the set of all reduced words for w is denoted R(w).
According to our choice of Π, we define the root system Φ ⊂ Λ of C
Π ⊂ Φ, and given α, β ∈ Φ
∨as follows:
P
P
∨
we have α + β ∈ Φ if and only if hα , βi = −1, where
= i∈I ni αi∨ . It is well-known (see
i∈I ni αi
+
[3, Section 10], for example) that Π induces a decomposition Φ = Φ t Φ− of the root system into subsets of
positive and negative roots, and Φ± = Φ ∩ ±NΠ. The root poset of C is the set of positive roots Φ+ under
the partial order given by β ≺ γ if and only if γ − β ∈ NΠ. Note that ≺ also gives a partial ordering of Λ.
The Weyl group permutes Φ ([3, Section 9.1]), as does σ (A.2), and we denote
Φσ := {β ∈ Φ | σ(β) = β}.
A subset
P J ( I determines a subdatum CJ ⊂ C with simple roots ΠJ := {αj | j ∈ J}, weight lattice
ΛJ := j∈J Z$j , root system ΦJ = Φ ∩ ZΠJ and Weyl group WJ := hsj | j ∈ Ji. The latter is called the
−
±
±
Jth parabolic subgroup of W . As before, we have a decomposition ΦJ = Φ+
J t ΦJ , where ΦJ := Φ ∩ ΦJ . It
is well-known (see, e.g., [4, A.1.19]) that
(1.4)
W J := {w ∈ W | wΠJ ⊂ Φ+ }
2
coincides with the set of minimal-length left coset representatives for W/WJ . The image J W of W J under
the antiautomorphism of W defined by w 7→ w−1 is the set of minimal-length right coset representatives for
WJ \W . The sets J W, W J are known as parabolic quotients of W modulo J.
1.2. The balanced quotient. A well-known theorem of Tits [19, Theor`eme 3] states that two reduced
expressions for w ∈ W can be obtained from one another by applying a sequence of commutation/braid
relations. We refer to w as fully commutative if any two reduced decompositions are obtained from one
another using only commutation relations. A subset of W is called fully commutative when each of its
elements is fully commutative.
In this paper we focus our attention on those Cartan data that correspond to the simple finite-dimensional
Lie algebras of simply-laced type. Thus, the matrix A is assumed to be positive definite and cannot be written
in a block-diagonal form, and W is finite. The subdata we use are related to the action of an admissible
diagram automorphism; they are maximal proper subsets J ( I that meet each σ-orbit in I nontrivially.
•
•
•
•
Type
Type
Type
Type
A2r+1 with |J| = |I| − 1.
Dr with CJ ∼
= Ar−1 .
E6 with CJ ∼
= D5 .
D4 under the triality with CJ ∼
= A3 .
More details on these data, including indexing schemes and the corresponding automorphisms are shown in
(2.1), (2.2), (2.3), and (2.4).
Theorem 1.1 ([17, Theorem 6.1]). In the cases listed above the parabolic quotient J W is fully commutative.
Given i = (i1 , . . . , it ) ∈ I t , let `i (i) := |{1 ≤ r ≤ t | ir = i}|. A reduced decomposition of a fully
commutative w admits no braid relation. Accordingly, `i (i) is independent of i ∈ R(w) and will be denoted
by `i (w). Following Theorem 1.1 we make the following definition.
Definition 1.2. The σ-balanced quotient
i ∈ I.
J
Wσ is the set of all w ∈ J W such that `i (w) = `σ(i) (w) for all
The reference to σ is omitted when the context is clear.
1.3. Inversion sets and the balanced quotient. Given w ∈ W , Φ+ (w) := Φ+ ∩ w−1 Φ− is called the
inversion set of w. It is well-known (cf. [3, Section 10]) that `(w) = |Φ+ (w)| and W contains a unique
element w◦ , called the longest element, with `(w◦ ) = |Φ+ |. The map w 7→ Φ+ (w) is injective ([4, A.1.1])
and provides a way to recognize w on the level of the root system (that is, without appealing to a reduced
decomposition). Our first main result is the following, which we prove in section 3.
Theorem 1.3. In types Dn (2.1), E6 (2.3) and D4 under the triality (2.4), for all w 6=
w ∈ J Wσ if and only if
(1.5)
J
w◦ we have
σ
−1
Φ+ (w) r Φ+
Π J ⊂ Φ+ .
J ⊂ Φ and w
In type A2r+1 (2.2) we have w ∈ J Wσ if and only if
(1.6)
Φ+ (w) = {β ∈ Φ+ (Ir ) | β is supported on ασ(i) }
or (1.5) holds.
Associated to C is the simple complex Lie algebra g(C ), (cf. [3, Section 18]), which has a presentation
dependingL
only on the matrix A. The choice of L
Cartan subalgebra h := CΠ∨ provides a decomposition
g = h ⊕ β∈Φ gβ with Borel subalgebra b :=
β∈Φ+ gβ ; recall that dimC gβ = 1 for all β ∈ Φ. A
result of Fan [2, Proposition 7] L
implies that the inversion set Φ+ (w) of a fully commutative element gives a
commutative subalgebra aw := β∈Φ+ (w) gβ of the Borel; these are known (cf. [9]) to be related to abelian
ideals in the Borel subalgebra.
Let gJ be the Levi subalgebra of g associated with J; thus, gJ is generated as a Lie algebra by gα ,
±α ∈ ΠJ . Let bJ := b ∩ gJ be the Borel subalgebra of gJ .
3
Corollary 1.4. Assume C is of type Dr+2 (2.1), A2r+1 (2.2), E6 (2.3) and D4 under the triality (2.4), and
let w 6= J w◦ . Then the abelian subalgebra aw ⊂ b decomposes as
aw = aw,J ⊕ aσw
where aw,J ⊂ bJ and aσw consists of σ-invariants. The dimensions of these algebras are as follows, where m
and k are uniquely determined by w per Propositions 2.3 and 2.4
(
m+1
C = Dr+2
dimC aw,J =
m(m + 1) − k(k − 1) + 1 C = A2r+1
and dimC aσw = `(w) − dimC aw,J .
Full details are given in Section 3.
The exceptional cases. The exceptional cases E6 (2.3) and D4 under the triality (2.4) are treated by hand
in in Appendix 5, where we exhibit the normal forms, directly check that Theorem 1.3 holds, and calculate
the dimensions of aw,J and aσw for each w ∈ J Wσ .
1.4. Demazure crystals, branching rules, and σ-invariant weight spaces. We use the terminology
and notation of [4, Chapter 5] for crystals, except we index the set of simple roots by I. Let {B(λ) | λ ∈ Λ+ }
denote the Kashiwara-Littelmann family of highest-weight normal crystals [6, 10] with the limit crystal
B(∞). (A purely combinatorial construction of B(∞) is given in [4, 6.4.21].) More information on crystals
is included in Appendix A.3.
Given a subset J ⊂ I we can regard the C -crystal B(λ) as a CJ -crystal; put
HWJ B(λ) := {b ∈ B(λ) | e˜j b = 0 for all j ∈ J}.
That is, HWJ B(λ) comprises the elements of B(λ) that are highest-weight with respect to its CJ -crystal
structure. Let πJ : Λ ΛJ be the canonical projection homomorphism, whereby $i 7→ $i if i ∈ J and
$i 7→ 0 otherwise. The Jth branching rule (cf. [8, 4.6]) states that B(λ) is isomorphic, as a CJ -crystal, to
a direct sum of the CJ -crystals BJ (µ), for µ ∈ Λ+
J ∪ {∞}, as follows:
M
L
B(λ) ∼
BJ (πJ (wt b)) for λ ∈ Λ+ ,
B(∞) ∼
=
= BJ (∞) HWJ B(∞) .
b∈HWJ B(λ)
+
Note that for λ ∈ Λ , this is a combinatorial analogue of the decomposition of a simple finite-dimensional
g(C )-module V (λ) of highest weight λ into a direct sum of simple finite dimensional g(CJ )-modules.
A crystal is graded by the weight lattice Λ. Now, σ acts on Λ and we can ask whether the weight wt(b) ∈ Λ
of an element b ∈ HWJ B(∞) is σ-invariant.
P
Let Λ++ ⊂ Λ+ be the semigroup of regular weights, that is Λ++ =
i∈I Z>0 $i , and suppose λ ∈
Λ++ ∪ {∞}. For each w ∈ W , the Kashiwara-Littelmann crystal B(λ) contains a unique subset Bw (λ),
stable under all e˜i , called a Demazure crystal [7]. Recall that the Bruhat order is the transitive closure of the
relation u < usi , where `(usi ) = `(u) + 1. If w ≤ v in the Bruhat order, then Bw (λ) ⊂ Bv (λ). It turns out
that the balanced quotients characterize the Demazure crystals in which weight-invariant J-highest-weight
elements appear.
Theorem 1.5. Let λ ∈ Λ++ ∪ {∞}. For (C , J) of type Dn (2.1), A2r+1 (2.2), E6 (2.3),
and D4 under the
¯w (λ) := Bw (λ) r S
triality (2.4) an element w ∈ J W is σ-balanced if and only if B
v≤w Bv (λ) contains a
weight-invariant element of HWJ B(λ).
Further details and a proof are given in section 4.
2. Normal forms and the enumeration of J Wσ in types A and D
Recall the weak right ordering ≤R on W , defined by u ≤R uv if and only if `(uv) = `(u) + `(v). This is a
partial-order relation on W , which restricts to a partial ordering on the balanced quotient J Wσ . Recall that
a subset O of a poset (X, ≤) is an order ideal if {x ∈ O | x ≤ a} ⊂ O for all a ∈ O.
Lemma 2.1 ([17, Proposition 2.5]). For any J ⊂ I the set J W is an order ideal of W under the right weak
ordering. In particular, we can regard J W = [1, J w◦ ]R .
4
In § 2.1 we give a set of normal forms for the elements of J Wσ , and in § 2.3 we present the corresponding
self-similar forests, which in turn yield an enumeration of J Wσ .
2.1. Normal forms for J Wσ . The key ingredient in our proofs of Theorems 1.3 and 1.5 is a set of normal
forms for the balanced quotient. This set can be viewed as a refinement of the one given in [18] for the fully
commutative elements of W .
The parabolic subdata used in this paper are related to the action of an admissible diagram automorphism.
Thus we take C , I, σ, and J as follows, with I r J = {•}. The D-series
C = Dr+2
(2.1)
Ir := {−1, 0, 1, . . . , r}
Jr := Ir r {−1}
σ := (−1, 0)
0
◦
1
◦
···
r−1
r
◦
◦
•
−1
The odd-ranked A series A2r+1 , with some 1 ≤ k ≤ r
C = A2r+1
(2.2)
σ=
r
Y
Ir := {1, . . . , 2r + 1}
Jr,k := Ir r {2k + 1}
(2i, 2i + 1)
i=1
2
◦
···
◦
···
2k
◦
···
•
···
2r
◦
◦
1
3
2k+1
◦
2r+1
The exceptional type
C = E6
(2.3)
I = {1, 2, 3, 4, 5, 6}
J = {1, 2, 3, 4, 5}
σ := (1, 6)(2, 5)
4
◦
◦
◦
1
2
◦
◦
3
•
5
6
The triality of D4
C = D4
(2.4)
I = {1, 2, 3, 4}
J = {1, 2, 3}
3
◦
σ := (1, 3, 4)
◦
1
◦
2
•
4
Remark 2.2. The cases covered in this paper correspond to the cases in [17, Theorem 6.1] such that W J is
minuscule, C admits an admissible diagram automorphism, and J is adapted to the σ-action as described
above. Up to an application of σ, (2.1)–(2.4) exhaust these cases.
In types D and A it is most convenient to state our normal forms in the injective limits W (D∞ ) and
W (A∞ ), which are also partially ordered by ≤R . That is, we specify a set of normal forms for J∞ W (D∞ )σ
and J∞,k W (A∞ )σ (k ≥ 1) from which the normal forms for Jr W (Dr )σ and Jr,k W (A2r+1 )σ are obtained by
an appropriate restriction of parameters.
In type D (2.1), given 1 ≤ j ≤ k let
(j)
sk :=
j−1
Y
sk−t ∈ W σ ,
t=0
Then our set of normal forms is as follows.
5
(j)
`(sk ) = j.
Proposition 2.3. In the injective limit J∞ W (D∞ ) of type D as in (2.1) we have
is σ-balanced if and only if m is odd, in which case
Jm
(2.5)
J1
w◦ = s−1 s1 s0 and
Jm
w◦
w◦ = Jm−2 w◦ sm−1 sm · · · s2 s3 s1 s2 J1 w◦
Every w ∈ J∞ W (D∞ )σ not of the form
Jm
w◦ with odd m has a reduced expression
(j )
(j )
1
n
· · · sm+n
w = Jm w◦ sm+1
(2.6)
with 2`−1 (w) − 1 = m ≥ j1 ≥ · · · ≥ jn ≥ 1. Taking all elements of the form (2.6) such that m + n ≤ r along
with Jm w◦ for odd m ≤ r gives a set of normal forms for Jr W (Dr )σ .
In type A (2.2), given 0 ≤ j < k let
(j)
sˆk :=
j
Y
s2(k−j) s2(k−j)+1 ,
(j)
`(ˆ
sk ) = 2j,
i=0
(0)
and declare sˆk := sk . Then our set of normal forms is as follows.
Proposition 2.4. In the injective limit J∞,k W (A∞ ) of type A as in (2.2), the elements
σ-balanced and have reduced expressions
(2.7)
Jk,k
(2.8)
Jm+1,k
Jr,k
w◦ , r ≥ k are all
w◦ = s2k+1 · · · s3 s1 s2 · · · s2k
w◦ = Jm,k w◦ sˆm+1 · · · sˆk+1 Jk,k w◦ , m ≥ k.
Every w ∈ J∞,k W (A∞ )σ not of the form
Jm,k
(2.9)
1
n
w = Jm,k w◦ sˆm+1
· · · sˆm+n
w◦ with m ≥ k has a reduced decomposition
(j )
(j )
with m − k ≥ j1 ≥ · · · ≥ jn ≥ 1. Taking all elements of the form (2.9) such that r ≥ m + n along with
Jm,k
w◦ for k ≤ m ≤ r gives a set of normal forms for Jr,k W( A2r+1 ).
2.2. Proofs of Propositions 2.3 and 2.4. The longest element w◦ ∈ W factors uniquely as w◦ = w◦ (J) ·
J
w◦ , with w◦ (J) being the longest element of WJ . Accordingly
`(J w◦ ) = |Φ+ | − |Φ+
J |.
(2.10)
As J w◦ is the unique element of J W with this length, J w◦ = w◦J and (1.4) gives
Φ+ (J w◦ ) = Φ+ r Φ+
J.
(2.11)
We begin with type D, and use the notation (2.1). Equation (2.10) indicates
(r + 1)(r + 2)
2
Jr
Jr−1
hence `( w◦ ) − `(
w◦ ) = r + 1. That is to say, for each r ≥ 0 there is a reduced expression
`(Jr w◦ ) =
Jr
w◦ = Jr−1 w◦ τr ,
`(τr ) = r + 1.
More precisely:
Lemma 2.5. We have τ0 = s−1 and, for all r ≥ 1,
(
sr · · · s1 s0
(2.12)
τr =
sr · · · s1 s−1
In particular the longest element of
expression
(2.13)
Jr
Jr
r odd
r even.
Wσ is σ-balanced if and only if r is odd, in which case it has a reduced
w◦ = Jr−2 w◦ sr−1 sr · · · s2 s3 s1 s2 J1 w◦ , odd r
and if r is even then
(2.14)
w = Jr−1 w◦ sr · · · s−1 , even r.
6
Proof. We argue by induction on r. Since J1 w◦ = s−1 s1 s0 and J2 w◦ = s−1 s1 s0 s2 s1 s−1 , induction begins.
Take an even n ≥ 1. By the induction hypothesis there is a reduced expression
Jn−1
w◦ sn · · · s1 s−1 = Jn−2 w◦ sn−1 · · · s1 s0 sr · · · s1 s−1 .
Then
1
n(n + 1) + (n + 1) = `(Jn w◦ ).
2
are fully commutative, hence
`(Jn−1 w◦ sn · · · s1 s−1 ) =
Both
Jn−2
w◦ and sn−1 · · · s1 s0 sr · · · s1 s−1
Jn−1
Therefore
Jn−1
w◦ sn · · · s1 s−1 coincides with
w◦ sn · · · s1 s−1 ∈ Jn−1 W.
Jn
w◦ . The case for odd n is proved similarly.
Proof of Proposition 2.3. We argue by induction on r. The induction begins because J2 Wσ = {J1 w◦ s2 , J1 w◦ s2 s1 }.
For 1 < m ≤ r let
Jm W
:= Jm W r Jm−1 W
F
and J1 W := J1 W so that Jr W = 1≤m≤r Jm W . A reduced expression for w ∈ Jm W contains sr and does
not contain any sp with p > m. Accordingly w = vu with v ∈ Jm−1 W of maximal length, `(w) = `(v) + `(u),
and u ≤R τr by Lemma 2.5. That is, u ≤R sr · · · s1 si for i ∈ {0, −1} depending on the parity of r.
If `(us−1 ) < `(u) then w is balanced if and only if v is balanced. In this case, using the induction
hypothesis we obtain
(j )
(jn ) (j)
w = J2t−1 w◦ s2t1 · · · s2t+n
sm
where m − 1 = 2t + n. If j > jn then a reduced expression for w contains sm−jn −1 sm−jn sm−jn −1 as a
subword, contradicting full commutativity. Therefore in this case w has the desired form.
If u = τr and v <R Jr−1 w◦ then `(vsj ) < `(v) for some 1 ≤ v ≤ r − 1. But then a reduced expression for
w contains the expression sj sj+1 sj , contradicting full commutativity. And so if u = τr then w = Jr w◦ . Remark 2.6. The alternating occurrences of −1, 0 in a reduced expression of w ∈ Jr W is one of several
equivalent conditions for an element to be Ar -stable in the sense of [18]. In fact Jr W is a proper subset of
the set of Ar -stable elements of W . Besides the central role played by σ, our perspective differs in that we
focus on the subdatum Ar+1 ⊂ Dr+2 , while Dr+1 ⊂ Dr+2 is studied in [18].
00
0
00
Next we treat type A, using the notation (2.2). Put Jr,k
:= {2i + 1 | k < i ≤ r} and Jr,k
:= Jr,k r Jr,k
so
that
∼ Ar+k × Ar−k .
(2.15)
CJ = CJ 0 × CJ 00 =
r,k
r,k
r,k
Now w◦ (Jr,k ) ∼
= w◦ (Ar+k ) · w◦ (Ar−k ) and equation (2.10) indicates
`(Jr,k w◦ ) = (r + 1)2 − k 2
and also that `(Jr+1,k w◦ ) − `(Jr,k w◦ ) = 2r + 3.
Lemma 2.7. For all 1 ≤ k ≤ r
(2.16)
Jk,k
(2.17)
Jr+1,k
w◦ = s2k+1 · · · s3 s1 s2 · · · s2k
w◦ = Jr,k w◦ sˆr+1 · · · sˆk+1 Jk,k w◦ .
Proof. This proof is similar to the proof of Lemma 2.5, and is omitted.
Proof of Proposition 2.4. Each
in this proof. Note that
Jm,k
w◦ with k ≤ m ≤ r is balanced, so we need not consider these elements
Jk,k
(2.18)
Wσ = {Jk,k w◦ }
by (2.16). For k < m ≤ r let
:= Jm,k W r Jm−1,k W
and
=
W so that W = k≤m≤r Jm,k W . A reduced expression for w ∈ Jm,k W contains s2m
or s2m+1 and contains no sp with p > 2m + 1. Accordingly w = vs2m · · · s2(m−j) s2m+1 · · · s2(m−j 0 )+1 with
v ∈ Jm−1,k W of maximal length.
Jm,k W
Jk,k W
Jk,k
r,k
F
7
(j )
(j ) (j)
1
n
Now w is balanced if and only if j = j 0 and v is balanced. By induction v = Jt,k w◦ sˆt+1
· · · sˆt+n
sˆm (where
m = t + n + 1). If j > jn then a reduced expression for w contains the expression st+n−jn st+n−jn +1 st+n−jn ,
contradicting full commutativity.
2.3. Enumeration of (J Wσ , ≤R ). For each positive integer k define an infinite digraph Tk := (Vk , Ek ) as
follows
Vk := {(k; ∅)} ∪ {(k; j1 , . . . , jn ) | k ≥ j1 ≥ · · · ≥ jn }
0
(k; j1 , . . . , jn ) → (k; j10 , . . . , jm
) ⇐⇒ m = n + 1 and jt0 = jt ∀ t = 1, . . . , n.
It is easy to see Tk is an infinite tree. Indeed, first note that there is a unique path between any vertex
(k; j1 , . . . , jn ) and (k; ∅). Now suppose (k; j1 , . . . , jn ) and (k; l1 , . . . , lm ) are arbitrary. Then the unique path
between these two vertices is given as follows: Let i := max{p ≥ 1 | jt = lt for all 1 ≤ t ≤ p}. If no such i
exists, then take the unique path from (k; j1 , . . . , jn ) to (k; ∅) followed by the unique path from (k; ∅) to
(k; l1 , . . . , lm ). Otherwise, we have
(k; j1 , . . . , jn ) ← · · · ← (k; j1 , . . . , jp ) = (k; l1 , . . . , lp ) → · · · → (k; l1 , . . . , lm )
uniquely.
Observe that Tk is isomorphic to a subgraph of each Tr , r > k via the natural inclusions of vertices
and edges. Furthermore, Tk is isomorphic to infinitely many subgraphs of itself via (k; j1 , . . . , jn ) 7−→
(k; k, . . . , k , j1 , . . . , jn ), m ≥ 0.
| {z }
m
(0)
(N )
(N )
(N ) Declare Tk := {(k; ∅)}, and for a fixed positive integer N define the N th truncation Tk := Vk , Ek
by
(N )
(N )
and let Tk
Vk
:= {(k; ∅), (k; j1 , . . . , jn ) | k ≥ j1 ≥ · · · ≥ jn and n ≤ N }
(N )
Ek
:= {x → y | x, y ∈ Vk
(N )
},
(3)
(3)
be the empty graph when N < 0. Figures 2.1, 2.2 show respectively T2 , T3 .
•
•
•
•
(3)
Figure 2.1. T2
•
•
◦
•
•
•
with vertex (2; 2, 2, 1) indicated by ◦.
(r)
Lemma 2.8. For all r ≥ 1, k ≥ 0 we have |Tk | =
r+k
k
.
Proof. We prove that
(2.19)
(r)
|Vk | =
r+k
k
for all r ≥ 0
(r)
by induction on k. When k = 1, (2.19) is true by the definition of V1 , thus induction begins. Now
L let k > 1
and assume for induction that (2.19) holds for k − 1. Observe that there exists a bijection Vk ∼
= n∈N Vk−1
whereby
Vk−1 3 (k − 1; j1 , . . . , jm ) 7−→ (k; k, . . . , k , k − 1, j1 , . . . , jm ) ∈ Vk , n ≥ 0
| {z }
n
8
•
•
•
•
y
•

•
•
{
•
•
•
•
•
◦
$
•
#
•
&
•
•
•
&
•
&
•
(3)
Figure 2.2. T3 with vertex (3; 3, 2, 1) indicated by ◦. The subgraph obtained by deleting
(3)
(2)
(1)
(0)
the dashed arrows is isomorphic to T2 ⊕ T2 ⊕ T2 ⊕ T2 , as indicated in the proof of
Lemma 2.8
Accordingly, for all r ≥ 0 there exists a bijection
(r)
Vk
∼
=
r
M
(n)
Vk−1 .
n=0
(r)
|Vk |
Pr
n+k−1
n=0
k−1
Now, by the induction hypothesis
=
.
Pa
To complete the inductive step it remains to prove that for all a ≥ 0, b ≥ 1 we have n=0 n+b−1
= a+b
b−1
b
Ps
This claim is clear for a = 0 and for all b ≥ 1. Assuming n=0 n+b−1
= s+b
b−1
b , for all s ≤ a and b ≥ 1, we
have
a+1
X n + b − 1 a + b a + b a + b + 1
=
+
=
.
b−1
b
b−1
b
n=0
Thus,
r X
n+k−1
(r)
|Vk | =
k−1
n=0
=
r+k
,
k
which completes the inductive step.
Corollary 2.9. The L
balanced parabolic quotient J∞ W (D∞ ) is in one-to-one correspondence with the vertices
of the directed forest k>0 T2k , and thus
M (r−2k+1)
H(Jr W (Dr+2 ), ≤R ) ∼
T2k
.
=
k≥1
r
Jr
Therefore, in type D we have | W (Dr+2 )σ | = 2 − 1.
Proof. By Proposition 2.3, the assignments
Jm
(j1 )
w◦ sm+1
Jm
w◦ 7→ (m + 1; ∅) and
(j )
n
· · · sm+n
7→ (m + 1; j1 , . . . , jn )
L
place the elements of J∞ W (D∞ ) in one-to-one correspondence with the vertices of k>0 T2k .
L
P
(r−2k+1)
Using the convention that ab = 0 if b > a, we have | k≥1 T2k
| = k≥1 r+1
by Lemma 2.8. It
2k
P
P
r+1
r+1
j r+1
is well-known that j≥0 j = 2 , and also that j≥0 (−1) j = 0. Adding these identities gives
P
P
r+1
r+1
r
, whence k≥1 r+1
k≥0 2 2k = 2
2k = 2 − 1.
L
There is a map J∞,k W (A∞ )σ → k>0 Tk given by Jm,k w◦ 7−→ (m − k; ∅) and w 7−→ (m − k; j1 , . . . , jn )
for w of the form (2.9).LHowever, owing to the value m − k dominating the sequence j1 ≥ · · · ≥ jn , the
preimage of a vertex of k>0 Tk is infinite. Indeed, for a fixed (t; j1 , . . . , jn ) we have
(j )
(j )
1
n
{w ∈ J∞,k W (A∞ )σ | w 7→ (t; j1 , . . . , jn )} = {Jm,k w◦ sˆm+1
· · · sˆm+n
| m − k = t}
However, for fixed (r, k) the Hasse diagram of (Jr,k W (A2r+1 )σ , ≤R ) is as follows.
9
Corollary 2.10. In type A, suppose r ≥ k ≥ 1. Then we have a bijection
M
Jr,k
(r−k+1−m)
W (A2r+1 )σ ∼
Tm
=
m≥1
Hence
J
r,k W (A2r+1 )σ = 2r−k+1 − 1.
(2.20)
(r)
Remark 2.11. The graphs Tk
natorial properties. Let
may be combined in different ways to make forests with interesting combi-
(n)
Fl
:=
c
b n+l−1
l
M
(n+l−1−lk)
Tk
.
k=1
(n)
Then, for example, |F2 | = fn − 1, where fn are the Fibonacci numbers.
Lb n+l−1
c (n+l−1−lk)
(n)
l
Tk
; that is, G (n) equals F2 with the 1-branching
On the other hand, let G (n) :=
k=2
(n)
∞
component deleted. Then |G | = fn+5 − n − 4. Now, define sequences (dt )t=0 and (bt )∞
t=0 by
dt := |Jt W (Dt+2 )σ |
and bt := the tth nonnegative integer having no consecutive zeros or no consecutive ones in its binary
representation. For example, bt = t for 0 ≤ t ≤ 11, but b12 = 13 because (12)2 = 1100. Then one can show
(see, e.g., [15, A107909]) that
b|G (t−1) | = dt .
Taking the direct sum of Tk with k ≥ 3 odd, and a generational lag of 2 gives a sequence of forests with
orders equal to the Eulerian numbers [15, A000295].
3. Inversion sets and the balanced quotient
In [2], the following characterization of fully commutative elements in the simply-laced case was given.
Theorem 3.1 ([2, Proposition 7]). Suppose W is a simply-laced Weyl group. Then an element w ∈ W is
fully commutative if and only if there do not exist three roots α, β, α + β ∈ Φ+ (w).
Recall the following standard facts. See [4, A.1.1], for example, for proofs.
Lemma 3.2. Take a reduced expression w = si1 · · · si` and define βj := si` · · · sij+1 αij for 1 ≤ j ≤ `. Then
Φ+ (w) = {βj | 1 ≤ j ≤ `}.
Lemma 3.3. For all w ∈ W and i ∈ I we have
(3.1)
Φ+ (si w) = Φ+ (w) t {w−1 αi } in case `(si w) > `(w)
(3.2)
Φ+ (w−1 ) = −wΦ+ (w).
These yield a right-insertion analogue that we failed to find in the literature, a proof is given in §A.1.
Lemma 3.4. For all v, w ∈ W such that `(vw) = `(v) + `(w), we have Φ+ (vw) = w−1 Φ+ (v) t Φ+ (w).
3.1. Inversion set of a long element’s coset. Suppose C is of type Dr+2 (2.1) or A2r+1 (2.2). A given
0
w ∈ J Wσ has the form w = J w◦ sˆ, with sˆ ∈ W σ and
(
Jm
w◦ , odd 1 ≤ m ≤ r if C = Dr+2
J0
w◦ = J
m,k
w◦ , k ≤ m ≤ r
if C = A2r+1
by Propositions 2.3 and 2.4. The inclusion of Cartan data ι : Dm+2 ,→ Dr+2 (respectively ι : A2m+1 ,→
A2r+1 ) induces a monomorphism ι : W (Dm+2 ) ,→ W (Dr+2 ) and an inclusion ι : Φ+ (Dm+2 ) ,→ Φ+ (Dr+2 )
(respectively ι : W (A2m+1 ) ,→ W (A2r+1 ) and ι : Φ+ (A2m+1 ) ,→ Φ+ (A2r+1 )) that commute with σ and the
map w 7−→ Φ+ (w). Denote C 0 := Dm+2 , A2m+1 when C = Dr+2 , A2r+1 respectively.
10
Now, by Lemma 3.4 we have
0
Φ+ (w) = sˆ−1 ι[Φ+ (J w◦ )] t Φ+ (ˆ
s)
0
0
= (ˆ
s−1 ι[Φ+ (J w◦ )] r Φ(C )σ ) t (ˆ
s−1 ι[Φ+ (J w◦ )] ∩ Φ(C )σ ) t Φ+ (ˆ
s),
hence
0
0
|Φ+ (w)| = |ˆ
s−1 ι[Φ+ (J w◦ )] r Φ(C )σ | + |ˆ
s−1 ι[Φ+ (J w◦ )] ∩ Φ(C )σ | + `(ˆ
s).
Because sˆ ∈ W σ we have sˆ−1 β ∈ Φσ if and only if β ∈ Φσ ; hence
0
0
0
0
|ˆ
s−1 ι[Φ+ (J w◦ )] r Φ(C )σ | = |ι[Φ+ (J w◦ )] r Φ(C )σ |
|ˆ
s−1 ι[Φ+ (J w◦ )] ∩ Φ(C )σ | = |ι[Φ+ (J w◦ )] ∩ Φ(C )σ | + `(ˆ
s).
0
0
In W (C 0 ) we have σ(J w◦ ) = w◦J , thus by (2.11) we have
(3.3)
0
0
0
ι[Φ+ (J w◦ )] = ι[Φ+ (σ(w◦J ))] = σ ◦ ι[Φ+ (w◦J )] = σ ◦ ι[Φ+ (C 0 ) r Φ+
J 0 ].
This relationship will be used to handle the inversion sets of the elements
and 3.4 below.
Jm
w◦ and
Jm,k
w◦ in §§ 3.2, 3.3,
3.2. Type D.
Proof of Theorem 1.3. We use induction on r. Lemma 3.2 gives Φ+ (J1 w◦ ) = {α0 , α0 + α1 , α−1 + α0 + α1 },
and so induction begins.
(j1 )
(j )
Suppose w ∈ Jr W r {Jm w◦ | odd m ≤ r}. Then w is balanced if and only if w = Jm w◦ sm+1
· · · sr n for
some odd m < r by Lemma 2.3. Lemma 3.4 gives
−1 + Jm
(jn−1 )
(j1 )
n)
n)
Φ+ (w) = s(j
Φ ( w◦ sm+1
· · · sr−1
) t Φ+ (s(j
)
r
r
(jn−1 )
(j1 )
(jn ) −1
(j )
(j )
· · · sr−1
γ for some γ ∈ Φ+ (Jm w◦ st+1
).
but Φ+ (sr n ) ⊂ Φσ since sr n ∈ W σ . If β ∈ Φ+ (w)rΦ+
J then β = sr
(jn )
+
σ
σ
σ
/ ΦJ as well. By induction γ ∈ Φ and it follows that β ∈ Φ .
As sr ∈ W , it must be that γ ∈
Suppose instead that w = Jm w◦ for odd 1 ≤ m < r. Recalling (3.3), we have
Φ+ (w) = σ ◦ ι[Φ+ (Dm+2 ) r Φ+
Jm ]
= σ[{β ∈ ι[Φ+ (Dm+2 )] | β is supported on α−1 }]
= {β ∈ ι[Φ+ (Dm+2 )] | β is supported on α0 }.
Now, if γ ∈ Φ+ (w) is supported on α−1 then it must be that γ ∈ Φσ .
Example 3.5. In type D4 we have I2 = {−1, 0, 1, 2}, J2 = {0, 1, 2}, J1 = {0, 1} and σ = (−1, 0). To
illustrate the case w = Jt w◦ (odd t) of the above proof, let us examine the inversion set of the coset of the
long element J1 w◦ from type D3 in the context of Φ+ (D4 ).
Figure 3.1 shows the Hasse diagram of the root poset Φ+ (D4 ), decorated as follows:
∼ +
• To visually separate the parabolic subsystem Φ+
J2 = Φ (A3 ) from the set of roots supported on α−1 ,
+
/ .
edges corresponding to adding α−1 to β ∈ ΦJ2 are indicated by
• Elements of Φσ (those supported on α1 + α0 + α−1 ) are indicated by #.
• Elements of Φ+ (J1 w◦ ) are indicated by ∗.
• Elements of Φ+ (J1 w◦ ) ∩ Φσ are indicated by ~.
• All other elements of Φ+ (w) are indicated by •.
In accordance with Theorem 1.3, each element of Φ+ (J1 w◦ ) that is supported on α−1 is symmetric.
Regarding the conclusion of Corollary 1.4, this calculation indicates that aJ1 w◦ ,J2 has dimension 2 and aσJ1 w◦
has dimension 1.
Example 3.6. In type D6 we have I4 = {−1, 0, 1, 2, 3, 4}, J4 = {0, 1, 2, 3, 4}, J3 = {0, 1, 2, 3}, and σ =
(−1, 0). Let us now examine the inversion set of the coset of the long element J3 w◦ from type D5 in the
context of Φ+ (D6 ).
Figure 3.2 shows the Hasse diagram of the root poset Φ+ (D6 ) , decorated in the same manner as Example
3.5.
11
#O
8 #O g
•O g
7•g
7 ~O
@•^
@∗^
5•`
•α2
•α1
∗α0
•α−1
Figure 3.1. Root poset of type D4 , decorated as described in Example 3.5.
#O
@#^
•α4
#O g
~O
8 #O g
~O f
•O f
8•g
7 ~O f
~O
@•^
@∗^
5•^
5@ ~ ^
•O f
•O g
∗O g
7•g
7 ~O
@•^
@•^
@•^
@∗^
5•`
•α3
•α2
•α1
∗α0
•α−1
Figure 3.2. Root poset of type D6 , decorated as described in Example 3.5.
In accordance with Theorem 1.3, each element of Φ+ (J3 w◦ ) that is supported on α−1 is symmetric.
Regarding the conclusion of Corollary 1.4, this calculation indicates that aJ3 w◦ ,J4 has dimension 3 and aσJ3 w◦
has dimension 7.
3.3. Type A.
Proof of Theorem 1.3. First we verify (1.6) holds if and only if w =
immediate consequence of (3.3), indeed, in this case
Jm,k
w◦ for k ≤ m ≤ r. But this is an
Φ+ (w) = σ ◦ ι[Φ+ (A2m+1 ) r Φ+
Jm,k ]
= σ[{β ∈ ι[Φ+ (A2m+1 )] | β is supported on α2k+1 }]
= {β ∈ ι[Φ+ (A2m+1 )] | β is supported on α2k+2 }.
12
In case r = k condition (1.5) holds (vacuously) by (2.18). Assume that condition (1.5) holds for some pair
(r, k) with r ≥ k. We only need to check the elements of Jr+1,k W excluding Jr+1,k w◦ . Now w ∈ Jr+1,k W σ if
and only if it has the form provided by Theorem 2.4 with m + n = r + 1.
Lemma 3.2 gives
(jn ) −1 +
(jn ) −1
Φ+ (w) = (ˆ
sr+1
) Φ (w0 ) t Φ+ ((ˆ
sr+1
) )
(j )
(j )
n
n
for w0 = w(ˆ
sr+1
)−1 ∈ Jr,k W . Because m − k ≥ jn it follows that Φ+ ((ˆ
sr+1
)−1 ) ⊂ Φ+
J . Accordingly if
(j
)
(j
)
+
n
n
+
−1
+
0
−1
β ∈ Φ (w) r ΦJ then β = (ˆ
sr+1 ) γ for some γ ∈ Φ (w ). Since `2k+1 ((ˆ
sr+1 ) ) = 0 it follows that β ∈
/ Φ+
J
+
σ
σ σ
σ
if and only if γ ∈
/ ΦJ . By induction, this happens if and only if γ ∈ Φ . Therefore β ∈ W Φ ⊂ Φ .
3.4. Proof of Corollary 1.4. Retain the notation of §3.1. Recalling (3.3), since σ, ι are injective it follows
that
0 σ
dimC aw,J = |(Φ+ (C 0 ) r Φ+
J 0 ) r Φ(C ) |
0 σ
dimC aσw = |(Φ+ (C 0 ) r Φ+
s).
J 0 ) ∩ Φ(C ) | + `(ˆ
The cardinalities of these sets are readily determined. Because dimC aw = `(w), which is known by virtue of
Propositions 2.3 and 2.4, it suffices to determine one of them.
(j )
(j )
1
n
; then
3.4.1. Type D. Suppose w = Jm w◦ sm+1
· · · sm+n
σ
(Φ+ (Dm+2 ) r Φ+
Jm ) r Φ(Dm+2 ) = {α−1 , α−1 + α1 , α−1 + α1 + · · · + αm },
whence
Jm
Now, since `(
w◦ ) =
1
2 (m
dimC aw,Jr = m + 1.
+ 1)(m + 1) it follows that
dimC aσw =
(j )
n
X
1
m(m + 1) +
ji .
2
i=1
(j )
n
; then
3.4.2. Type A. Suppose w = Jm,k w◦ sˆm11 · · · sˆm+n
t
n
o
X
(Φ+ (A2m+1 ) r Φ+ (Jm,k )) ∩ Φ(A2m+1 )σ = α1 +
(α2i + α2i+1 ) k ≤ t ≤ m ,
i=1
whence
dimC aσw = k − m + 1 + 2
n
X
ji .
i=1
Now, since `(Jm,k w◦ ) = (m + 1)2 − k 2 , it follows that
dimC aw,Jr,k = m(m + 1) − k(k − 1) + 1.
4. Demazure crystals, branching rules, and σ-invariant weight-spaces
We assume familiarity with the crystal theory, including the Littelmann path crystals Pλ ∼
= B(λ) [10]
and the Kashiwara-Nakashima-Zelevinsky polyhedral realization Σι ∼
= B(∞) [7, 13, 14]. A review of the
essentials of these topics, along with all the necessary references, is included in Appendix A.3. We use the
terminology and notation of [4, Chapter 5] for crystals in general, except we index the set of simple roots
by I. The notation we use for the Kashiwara-Nakashima-Zelevinsky polyhedral realization is defined in
Appendix A.3.4.
When working with the Littelmann path crystals Pλ ∼
= B(λ), λ ∈ Λ+ we use the notation of [10] and
assume σ(λ) = λ. Recall that π ∈ Pλ has the form π = (τ , a) where τ = τ1 > · · · > τr is a sequence of
linearly (Bruhat) ordered elements of W and a = a0 := 0 < a1 · · · ar := 1 is a sequence of rational numbers
satisfying certain conditions [10, Sections 2.1, 2.2]. We regard π as the concatenation of the straight line
paths πk (t) := τk λt living in Λ ⊗Z R, modulo reparameterization.
Recall the surjection φ : Pλ → W/StabW λ [10, 5.2] whereby a path π = (τ , a) is mapped to its “first
direction” τ1 ; we may assume im φ ⊂ W by taking λ ∈ Λ++ . In the same paper path analogues Pλ,w :=
{π ∈ Pλ | φ(π) ≤ w} of the Demazure crystal are defined. For each w ∈ W there is a bijection Pλ,w ∼
= Bw (λ)
13
commuting with the e˜i -action, where Pλ,w := {π ∈ Pλ | φ(π) ≤ w}. Under this bijection {π ∈ Pλ | φ(π) =
¯w (λ).
w} ∼
=B
Elements of HWJ B(λ)—that is, the elements of B(λ) that are highest-weight for the Jth branching
rule—are readily seen to be related to J W by the following, which we failed to find in the literature.
S
Lemma 4.1. For any J ⊂ I, we have HWJ B(λ) ⊂ w∈J W Bw (λ), λ ∈ Λ++ ∪ {∞}
Proof. It suffices to prove the claim using the path crystal. Immediately from [10, Proposition 1.5], we have
that
(4.1)
e˜j π = 0 ⇐⇒ hαj∨ , im πi ⊂ [0, ∞).
We have π ∈ HWJ Pλ if and only if (4.1) holds for all j ∈ J. In particular hαj∨ , τ1 λi > 0 for all j ∈ J. Now
hτ1−1 αj∨ , λi > 0 for all j ∈ JLimplies that τ1−1 ∈ W J by (1.4).
P
Observe that, Bw (λ) = ν∈NΠ Bw (λ)λ−ν , where Bw (λ)λ−ν := {fin`` · · · fin11 bλ | 1≤k≤` nk αik = ν}. Now
|Bw (λ)λ−ν | are increasing inPλ for the ordering ≺ of Section 1.1 and uniformly bounded by the number of
monomials fin`` · · · fin11 with 1≤k≤` nk αik = ν. Thus there exists λ ∈ Λ++ such that |Bw (λ)λ−ν |. The set
Bw (∞) is defined to be the disjoint union of these Bw (λ)λ−ν , hence the results of Bw (λ) imply the result on
Bw (∞).
Several important properties of the Demazure crystals are given in [7, Propositions 3.2.4, 3.2.5]. The
relevant ones for our purposes are the following.
(1) If v ≤ w in the Bruhat order, then Bv (λ) ⊂ Bw (λ).
S
(2) Bw (∞) = n∈N` f˜in`` · · · f˜in11 b∞ , where (i` , . . . , i1 ) indexes a reduced expression for w. Observe that,
¯w (∞) if and only if b = f˜n` · · · f˜n1 b∞ and all ni > 0.
in particular, b ∈ B
i`
i1
The proof of Theorem 1.5 is technical and uses a combination of the polyhedral realization, the combinatorics
of the reduced expressions of w, and Kashiwara’s ? involution. We begin by outlining the steps.
Proof outline of Theorem 1.5. We must show that
¯w (∞) ∩ HWJ B(∞) contains a weight-invariant element ⇐⇒ w ∈ J Wσ
(4.2)
B
We restrict our attention to type D as in (2.1), type A is proved similarly and the exceptional cases are
checked by hand.
¯w (∞) contains a weightFirst, we shall prove that if a representative w ∈ J W is σ-balanced, then B
invariant element of HWJ B(∞), by constructing such an element. Given a reduced word (i` , . . . , i1 ) ∈ R(w)
P`
let b := f˜i` · · · f˜i1 b∞ . It follows immediately that b is weight-invariant, since wt(b) = − k=1 αik and w is
¯w (∞), by (2) above.
σ-balanced. Note also that b ∈ B
Proposition 4.2. b ∈ HWJ B(∞)
Proposition 4.2 is proved using the polyhedral realization Ψι : Bw (∞) ∼
= Σι,w [13]. Using a relationship
between the operators f˜i , f˜j when aij = −1 (Lemma 4.5 below), we compute Ψι (b). Next, we apply a result of
Littelmann (Lemma 4.6 below), which describes the structure of a reduced expression of a fully commutative
¯w (∞) ∩ HWJ B(∞)
element, to conclude that e˜j b = 0 for all j ∈ J. This shows that b is an element of B
having σ-invariant weight, which proves the “only if” direction of (4.2).
Remark 4.3. It does not follow immediately from the definitions of (B(∞), e˜i , f˜i ) alone that b is J-highestweight.
The following Lemma completes the proof of (4.2).
¯w (∞) does not contain a weight-invariant element of
Lemma 4.4. If w ∈ J W is not σ-balanced, then B
HWJ B(∞).
To prove Lemma 4.4 we first observe that by Lemma 2.5, if w ∈ J W is not σ-balanced then it has a
unique reduced factorization w = us−1 s1 s2 · · · sk with u ∈ J Wσ of maximal length and `(w) = `(u) + k + 1.
If w = us−1 we check that Lemma 4.4 holds by a direct calculation in Σι,w . Otherwise, if w = us−1 s1 · · · si
with i ≥ 1 we can use Kashiwara’s ?-operation along with the combinatorial properties of Demazure crystals
from Appendix A.3.3.
14
Proof of Proposition 4.2. First, we need a lemma about the interaction of the operators f˜i , f˜j when aij = −1.
The notations appearing below are defined in Appendix A.3.4
ι
Lemma 4.5. Let w ∈ J Wσ and x ∈ Σw
and i, j be such that aij = −1. If f˜j acts on x at position k, then
0
˜
˜
fj acts on fi x at position k ≥ k.
Proof. We show that min M (j) (f˜i x) ≥ min M (j) (x). Let ` = `(w). By definition (see Appendix A.3.4),
f˜i x = x + emin M (i) (x) . Take k ≥ 1 such that ιk = j; then
γk (f˜i x) = γk (x) + γk (δk,min M (i) (x) )
(
γk (x),
k > min M (i) (x)
=
γk (x) − 1, k < min M (i) (x)
by [13, 2.8]. There are three cases to consider:
(1) If M (j) (x) ⊂ (min M (i) (x), `] then εj (f˜i x) = εj (x) and min M (j) (f˜i x) = min M (j) (x) by the above
display.
(2) If M (j) (x) ⊂ [1, min M (i) (x)) then, similarly as before, εj (f˜i x) = εj (x) − 1 and min M (j) (f˜i x) =
min M (j) (x).
(3) If M (j) (x) ∩ [1, M (i) (x)) and M (j) (x) ∩ (M (i) (x), `] are nonempty then there exists k ∈ M (j) (x) such
that k > min M (i) (x) and γk (x) = εj (x). Thus εj (f˜i x) = εj (x) and min M (j) (f˜i x) > min M (j) (x).
These calculations complete the proof.
It follows from this Lemma and Theorems 2.3, 2.4, and the definition of f˜i , f˜j (see Appendix A.3.4) that
Ψι (b) is the sequence (· · · , 0, 1, . . . , 1).
| {z }
`(w)
To prove that Ψι (b) ∈ HWJ Σι we use combinatorial information about the parameterizing sequence ι,
which is based on (i` , . . . , i1 ) ∈ R(w), via the following.
Lemma 4.6 ([11, Lemma 3.4]). Let (ir , . . . , i1 ) index a reduced expression of a fully commutative w ∈ W .
Then
(1) If j ≥ 2 is such that ik 6= ij for all k < j then there exists exactly one l ∈ {1, . . . , j − 1} such that
ajl 6= 0.
(2) Suppose i = ip = iq but ij 6= i for all p < j < q. Then there exist two (not necessarily different)
ij , ik with p < j < k < q such that ai,ij , ai,ik = −1.
We are now in a position to show that e˜j b = 0 for all j ∈ J. It suffices to show εj (b) = 0 for all j ∈ J, by
to the upper-normality of B(∞). Let x := Ψι (b). By [13, (2.13)] it suffices to show γl (x) ≤ 0 for all l such
that ιl = j, where j ∈ J is arbitrary. Recall
X
γl (x) = 1 +
aιt ,j
l<t≤`
and note that only those t with ιt ∈ {2, −1} contribute to the sum.
If there exists a unique l such that ιl = j then εj (x) = max{0, γl (x)}. By Lemma 4.6(1) there exists a
unique t > l such that aιt ,j = −1. Thus γl (x) = xl + (−1)xt = 0.
It there exist more than one l such that ιl = j, say there are n such indices ln > · · · > l1 . Then Lemma
4.6(2) provides two indices lk+1 > ir (k) > is (k) > lk such that aιr ,j = aιs ,j = −1. By Lemma 4.6(1) there
exists a unique l > ln such that aιl ,j = −1. Thus
X
γlk (x) = xlk − xln +
2xlt − (xir (k) + xis (k))
k<t≤n
=1−1+
X
2 − (1 + 1) = 0.
t
Hence in either case γl (x) ≤ 0, which completes the proof of Proposition 4.2.
15
Proof of Lemma 4.4. Recall that we use the indexing scheme (2.1) To begin, observe that Lemma 2.5 implies
that w ∈ J W that is not σ-balanced has a unique reduced factorization w = us−1 s1 s2 · · · sk with u ∈ J Wσ
of maximal length.
¯w (∞) does not contain
In the case w = us−1 it follows from Lemma 2.5 that w = 2k w◦ . We check that B
a weight-invariant element of HWJ B(∞) by a direct calculation. Fix n = 2k > 0 and take the reduced
decomposition of w = Jn w◦ = τ0 · · · τn afforded by Lemma 2.5. Let (i` , · · · , i1 ) be the reduced word for
this decomposition, and let ι = (· · · , ιk , · · · , ι2 , ι1 ) be an infinite sequence from I such that ιk = ik for all
1 ≤ k ≤ `; combinatorial properties of ι will be used frequently and without reference in the sequel.
ι
ι
We claim that HWJ (Σw
)σ = ∅. Indeed, let x ∈ HWJ (ΣW
)σ . Given i ∈ I, set Ki := {k ≥ 1 | ιk = i},
ki := min Ki , Ki := max Ki . Then
[
x ∈ HWJ Σ ι ⇐⇒ γk (x) ≤ 0 for all k ∈
Kj .
j∈J
and
x ∈ Σσι ⇐⇒
(4.3)
X
xk =
k∈K0
X
xk
k∈K−1
ι
We are considering only those x lying in the interior of the cone Σw
, which means all xk > 0. We will show
that
X
X
(4.4)
xk ≤
xk ,
k∈K−1 r{k−1 }
k∈K0
which contradicts equation (4.3). It will be useful to use the following form of the definition of γk , which is
adapted to our simply-laced case
X
X
[
γk (x) = xk + 2
xm −
xm , K(i) :=
Kj
m∈Kιk :m>k
m∈K(ιk ) :m>k
j:aij <0
∗
Let K := {k ≥ 1 | k0 ≤ k ≤ Kn−1 }, this range of indices corresponds to τn−1 in our preferred reduced
decomposition of w. In the following we describe how to systematically reduce the set of inequalities {γk (x) ≤
0 | k ∈ K∗ } to the desired contradictory inequality. Consider two inequalities at a time, beginning with the
largest index: In case k = Kn−1 we have
xKn−1 ≤ xKn−2
(4.5)
whereas in case k = Kn−1 − 1, because ιKn−1 −1 = n − 2 and K(n−2) = Kn−1 ∪ Kn−3 we have
(4.6)
xmax(Kn−2 r{Kn−2 }) + 2xKn−2 ≤ xKn−1 + xmax(Kn−3 r{Kn−3 }) + xKn−3
Combining (4.5) and (4.6) gives
(4.7)
xmax(Kn−2 r{Kn−2 }) + xKn−2 ≤ xmax(Kn−3 r{Kn−3 }) + xKn−3
In case k = Kn−1 − 2, because ιk = n − 3 and K(n−3) = Kn−2 ∪ Kn−1 (go to the final steps if k ≤ 4) we have
(4.8)
xmax(Kn−3 r{Kn−3 ,max(Kn−3 r{Kn−3 })}) + 2xmax(Kn−3 r{Kn−3 }) + 2xKn−3
≤ xn1 + xn2 + xn3 + xn4 + xn5
where
n1 = max(Kn−2 r {Kn−2 })
n2 = max(Kn−4 r {Kn−4 , max(Kn−4 r {Kn−4 })})
n3 = Kn−2
n4 = max(Kn−4 r {Kn−4 })
n5 = Kn−4
Combining (4.7) and (4.8) gives
xmax(Kn−3 r{max Kn−3 ,max(Kn−3 r{max Kn−3 })}) + xmax(Kn−3 r{max Kn−3 }) + xmax Kn−3
≤ xk2 + xk4 + xk5
16
This process eventually terminates; the final step is the case k = min K0 in which we have
X
X
xk0 + 2
xm ≤
xn .
m∈K0 r{k0 }
n∈K1 r{k1 }
In case k + 1 we found that
X
xn ≤
n∈K1 r{k1 }
X
r∈K0 r{k0 }
X
xr +
xs .
s∈K−1 r{k−1 }
ι
Combining these inequalities yields (4.4). Thus, HWJ (Σw
)σ = ∅.
¯w (∞)σ =
In case w = us−1 s1 · · · si with i ≥ 1 we use Kashiwara’s ?-operation. We will show that b ∈ HWJ B
k
k
∅. Assume that b is an element of that set, using the notation of (A.3.3) we can express b = f˜i f˜i b∞ , with
k, min kj > 0, where i ∈ R(wsi ). Then Lemma A.1 provides b0 ∈ Bwsi (∞) such that b = f˜i?k b0 . Now
wt b = wt b0 − kαi and εj (b) = εi (b0 ) for all j 6= i by [7, Corollary 2.2.2].
In case σ(i) = i we have i ∈ J. Since w is not σ-balanced, we see that wsi is also not σ-balanced.
Furthermore, b0 ∈ Bwsi (∞)σ . By induction there exists j ∈ J such that εj (b0 ) > 0. If j 6= i then b ∈
/
HWJ Bw (∞), contradicting the choice of b. Therefore it must be that εj (b0 ) = 0 for all j ∈ J r {i}. Writing
Ψi (b0 ) = b0 ⊗ ci (−k), it follows from (A.1) that Ψi (b) = b0 ⊗ ci (−m − k). Now because Ψi is an isomorphism
the tensor product axioms give
0 < εi (b0 ) = max{εi (b0 ), εi (ci (−m)) − wti b0 }
= max{εi (b0 ), −(wti b0 + m)}
and
0 = εi (b) = max{εi (b0 ), εi (ci (−m − k)) − wti b0 }
= max{εi (b0 ), k − (wti b0 + m)}.
Accordingly εi (b0 ) = 0, forcing −(wti b0 + m) > 0. But now k − (wti b0 + m) ≤ 0, which is impossible.
5. Exceptional types
5.1. Type D4 with the triality. In this case J Wσ = {s4 s2 s3 s1 , s4 s2 s3 s1 s2 } and
Φ+ (s4 s2 s3 s1 ) r Φ+
J = {α1 + α2 + α3 + α4 }
Φ+ (s4 s2 s3 s1 s2 ) r Φ+
J = {α1 + 2α2 + α3 + α4 }
while J W r J Wσ = {s4 , s4 s2 , s4 s2 s1 , s4 s2 s3 , s4 s2 s3 s1 s2 s4 } and
Φ+ (s4 ) r Φ+
J = {α4 }
Φ+ (s4 s2 ) r Φ+
J = {α2 + α4 }
Φ+ (s4 s2 s1 ) r Φ+
J = {α1 + α2 + α4 }
Φ+ (s4 s2 s3 ) r Φ+
J = {α2 + α3 + α4 }
Φ+ (s4 s2 s3 s1 s2 ) r Φ+
J = {α2 + α3 + α4 }
thus condition (1.5) holds.
5.2. Type E6 . In this case J Wσ = {s6 s5 s4 s2 s3 s1 , s6 s5 s4 s2 s3 s1 s4 , s6 s5 s4 s2 s3 s1 s4 s3 , s6 s5 s4 s2 s3 s1 s4 s3 s5 s4 s2 }
and
Φ+ (s6 s5 s4 s2 s3 s1 ) r Φ+
J = {α1 + α2 + α3 + α4 + α5 + α6 }
Φ+ (s6 s5 s4 s2 s3 s1 s4 ) r Φ+
J = {α1 + α2 + α3 + 2α4 + α5 + α6 }
Φ+ (s6 s5 s4 s2 s3 s1 s4 s3 ) r Φ+
J = {α1 + α2 + 2α3 + 2α4 + α5 + α6 }
Φ+ (s6 s5 s4 s2 s3 s1 s4 s3 s5 s4 s2 ) r Φ+
J = {α1 + 2α2 + 2α3 + 3α4 + 2α5 + α6 }
while Φ+ (w) r Φ+
J for each of the 22 remaining w contains an asymmetric root supported on α6 . Thus
condition (1.5) holds.
17
Appendix A.
This Appendix contains necessary background material, along with some supplementary proofs of results
from the body of this paper.
A.1. Preliminaries.
Proof of Lemma 3.4. We use induction on the length of w. If `(v) = 1 then (3.1) and (3.2) give
Φ+ (vsi ) = −si v −1 Φ+ (si v −1 )
= −si v −1 [Φ+ (v −1 ) t {vαi }]
= si (−v −1 Φ+ (v −1 )) t {−si v −1 vαi }
= si Φ+ (v) t {αi }
hence the induction begins.
Assume Φ+ (vw) = w−1 Φ+ (v) t Φ+ (w) for all w of length `(w) ≤ n and take i such that `(wsi ) = `(w) + 1.
By the above, we have
Φ+ (vwsi ) = si Φ+ (vw) t {αi }.
Then, using the induction hypothesis and Lemma 3.2, we obtain
Φ+ (vwsi ) = si (w−1 Φ+ (v) t Φ+ (w)) t {αi }
= (wsi )−1 Φ+ (v) t (si Φ+ (w) t {αi })
= (wsi )−1 Φ+ (v) t Φ+ (wsi ).
A.2. Action of σ on Φ. The action of σ on the set of simple roots Π extends to an action on the root
system Φ as follows. Suppose β, σ(β) ∈ Φ and i ∈ I satisfies β + αi ∈ Φ; we must show that σ(β + αi ) ∈ Φ.
By definition σ(β + αi ) = σ(β) + ασ(i) . Now, we know that hβ, αi i = −1, and since hβ, αi i = hσ(β), ασ(i) i it
follows that σ(β) + ασ(i) ∈ Φ.
A.3. Demazure crystals, branching rules, and symmetric weight spaces.
A.3.1. Elementary crystals. For i ∈ I the elementary crystal Ci has underlying set Ci := {ci (n) | n ∈ Z} and
C -crystal structure given by wt(ci (n)) := nαi and
(
−n, j = i
εj (ci (n)) :=
−∞, j 6= i
(
ci (n + 1), j = i
e˜j ci (n) :=
0,
j 6= i
(
ci (n − 1), j = i
f˜j ci (n) :=
0,
j 6= i
A.3.2. Tensor product rule. It is shown in [7] that the tensor product B ⊗ B 0 of two C -crystals is also a
C -crystal. The underlying set of B ⊗ B 0 is the Cartesian product B × B 0 , and the operators e˜i , f˜i act
according to the rules
(
f˜i b ⊗ b0 if ϕi (b) > εi (b0 )
f˜i (b ⊗ b0 ) :=
b ⊗ f˜i b0 if ϕi (b) ≤ εi (b0 )
(
e˜i b ⊗ b0 if ϕi (b) ≥ εi (b0 )
e˜i (b ⊗ b0 ) :=
b ⊗ e˜i b0 if ϕi (b) < εi (b0 ).
We will also use
εi (b ⊗ b0 ) := max{εi (b), εi (b0 ) − hαi∨ , wt(b)i},
see [7, 1.13] for more details on the tensor product of crystals.
18
A.3.3. Kashiwara’s involution and Demazure crystals. Following [7], let ? : B(∞) → B(∞) denote the
Kashiwara involution of B(∞) and define operators x?i : B(∞) → B(∞) for x ∈ {e, f } by x?i := ?xi ?. For
each i ∈ I there exists a strict C -crystal embedding Ψi : B(∞) ,→ B(∞) ⊗ Ci carrying b∞ 7→ b∞ ⊗ ci (0),
also introduced in [7], with the property
(A.1)
Ψi (b) = b0 ⊗ ci (−m) =⇒ Ψi (fi? b) = b0 ⊗ ci (−m − 1).
Given i = (i` , . . . , i1 ) ∈ I n denote f˜i := f˜i` · · · f˜i1 ; for k = (k` , . . . , k1 ) ∈ N` denote f˜ik := f˜ik`` · · · f˜ik11 . It is
well-known that
[
f˜ik b∞
(A.2)
Bw (∞) =
k∈Nn
for a reduced expression si = si` · · · si1 of w, indeed this can be taken as the definition of Bw (∞). Let
i−1 := (i1 , . . . , i` ) and k−1 = (k1 , . . . , k` ) denote the reversals of i and k. It is shown in [16] that
[
−1
f˜i?k
b∞ .
(A.3)
Bw (∞) =
−1
k∈Nn
Now we have the following easy consequence of (A.2) and (A.3).
Lemma A.1. Take u ∈ W and i ∈ I such that usi < u. Then Bu (∞) =
S
k≥0
f˜i∗k Busi (∞)
A.3.4. Kashiwara-Nakashima-Zelevinsky polyhedral realization. The underlying set of the polyhedral realization is the abelian group Z∞ , comprising those infinite sequences x = (. . . , xk , . . . , x2 , x1 ) such that
xk ∈ Z, xk = 0 for k 0. Suppose that (i` , . . . , i1 ) indexes a reduced expression for w ∈ J W , and let
ι = (. . . , ιk , . . . , ι2 , ι1 ) be an infinite sequence such that ιk = ik for 1 ≤ k ≤ ` and |{k | ιk = i}| = ∞ for all
i ∈ I. We equip Z∞ with a C -crystal structure according to ι as follows:
f˜i x := x + emin M (i) (x)
(
x − emax M (i) (x) , |M (i) (x)| < ∞
e˜i x :=
0,
otherwise
εi (x) := max{γk (x) | k ≥ 1, ιk = i}
where
γk (x) := xk +
X
aιk ,ιl xl
l>k
M (i) (x) := {k ≥ 1 | ιk = i, γk (x) = εi (x)}
where ek , k ≥ 1 are the standard basis vectors of Z∞ . We say that f˜i , e˜i act on x at positions min M (i) (x),
max M (i) (x) respectively.
There is a strict embedding of C -crystals Ψι : B(∞) → Z∞
ι [7, §2.2], and a sequence x belongs to
Ψι [B(∞)] if and only if it satisfies a finite set of linear inequalities [14, Theorem 3.1]. Note that this result
depends on a certain technical assumption [14, (3.5)] that is verified to hold when C is of finite or affine
type. The highest-weight element b∞ ∈ B(∞) corresponds to the zero vector 0, not to be confused with
ι
the special ghost element 0. Given w ∈ W , the image Σw
:= Ψι [Bw (∞)] is described by the inequalities of
[14, Theorem 3.1] with the additional requirement that xk = 0 for k > ` [13, (2.21)].
References
[1] A. Bj¨
orner and F. Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York,
2005. MR2133266 (2006d:05001)
[2] C. K. Fan, A Hecke algebra quotient and properties of commutative elements of a Weyl group, ProQuest LLC, Ann Arbor,
MI, 1995. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR2716583
[3] J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, SpringerVerlag, New York-Berlin, 1972. MR499562 (81b:17007)
[4] A. Joseph, Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in
Mathematics and Related Areas (3)], vol. 29, Springer-Verlag, Berlin, 1995. MR1315966 (96d:17015)
[5] V. G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR1104219
(92k:17038)
19
[6] M. Kashiwara, On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2,
465–516, DOI 10.1215/S0012-7094-91-06321-0. MR1115118 (93b:17045)
[7]
, The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839–858,
DOI 10.1215/S0012-7094-93-07131-1. MR1240605 (95b:17019)
[8]
, On crystal bases, Representations of groups (Banff, AB, 1994), CMS Conf. Proc., vol. 16, Amer. Math. Soc.,
Providence, RI, 1995, pp. 155–197. MR1357199 (97a:17016)
[9] B. Kostant, The set of abelian ideals of a Borel subalgebra, Cartan decompositions, and discrete series representations,
Internat. Math. Res. Notices 5 (1998), 225–252, DOI 10.1155/S107379289800018X. MR1616913 (99c:17010)
[10] P. Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994), no. 1-3,
329–346, DOI 10.1007/BF01231564. MR1253196 (95f:17023)
[11]
, Cones, crystals, and patterns, Transform. Groups 3 (1998), no. 2, 145–179, DOI 10.1007/BF01236431. MR1628449
(99e:17009)
[12] G. Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkh¨
auser Boston, Inc., Boston, MA,
1993. MR1227098 (94m:17016)
[13] T. Nakashima, Polytopes for crystallized Demazure modules and extremal vectors, Comm. Algebra 30 (2002), no. 3, 1349–
1367, DOI 10.1081/AGB-120004876. MR1892604 (2002k:17048)
[14] T. Nakashima and A. Zelevinsky, Polyhedral realizations of crystal bases for quantized Kac-Moody algebras, Adv. Math.
131 (1997), no. 1, 253–278, DOI 10.1006/aima.1997.1670. MR1475048 (98m:17023)
[15] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences.
[16] S. Ryom-Hansen, Littelmann’s refined Demazure character formula revisited, S´
em. Lothar. Combin. 49 (2002/04), Art.
B49d, 10 pp. (electronic). MR2036354 (2004k:17014)
[17] J. R. Stembridge, On the fully commutative elements of Coxeter groups, J. Algebraic Combin. 5 (1996), no. 4, 353–385,
DOI 10.1023/A:1022452717148. MR1406459 (97g:20046)
[18]
, Some combinatorial aspects of reduced words in finite Coxeter groups, Trans. Amer. Math. Soc. 349 (1997), no. 4,
1285–1332, DOI 10.1090/S0002-9947-97-01805-9. MR1389789 (97h:05181)
[19] J. Tits, Le probl`
eme des mots dans les groupes de Coxeter, Symposia Mathematica (INDAM, Rome, 1967/68), Academic
Press, London, 1969, pp. 175–185 (French). MR0254129 (40 #7339)
Department of Mathematics, University of California, Riverside, California 92521
E-mail address: [email protected]
URL: http://math.ucr.edu/~jmd
20