1 Introduction
Let
$(W,S)$
be a Coxeter system with length function
$\ell :W\to \mathbb N$
. Let V be the
$ \mathbb {R}$
-vector space with basis the simple system
$\Delta =\{\alpha _s~|~s \in S\}$
. Let B be the symmetric bilinear form on V defined by
$$ \begin{align*} B(\alpha_s,\alpha_t)= \left\{ \begin{array}{ll} -\text{cos}(\frac{\pi}{m_{st}}), & \text{if}~~ m_{st} < \infty, \\ ~~~~-1, & \text{if}~~m_{st} = \infty. \end{array} \right. \end{align*} $$
Denote by
$O_B(V)$
the orthogonal group for the pair
$(V,B)$
. For each
$s \in S$
, we consider the reflection
$ \sigma _s : V \rightarrow V$
by
$\sigma _s(x) = x - 2B(\alpha _s,x)\alpha _s$
. The map
$\sigma :W \hookrightarrow O_B(V)$
defined by
$s \mapsto \sigma _s$
is a geometrical representation of
$(W,S)$
. The orbit
$\Phi =W(\Delta )$
is a root system of
$(W,S)$
, with positive root system
$\Phi ^+=\operatorname {cone}(\Delta )\cap \Phi $
, where
$\operatorname {cone}(X)$
is the set of nonnegative linear combinations of vectors in X. The inversion set of
$w\in W$
is the set
$$ \begin{align*}N(w)=\Phi^+\cap w(\Phi^-), \end{align*} $$
where
$\Phi ^-=-\Phi ^+$
. It is well known that
$\ell (w)=|N(w)|$
.
1.1 Tits cone and Coxeter arrangement
The contragredient representation
$\sigma ^*:W\to \text {GL}(V^*)$
of
$\sigma $
is defined as follows: for
$w \in W$
and
$f \in V^*$
, we have
$wf := \sigma ^*(w)(f) = f\circ \sigma (w^{-1}).$
For
$\alpha \in \Phi $
, we consider the following hyperplane
$H_\alpha $
and open half-space
$H_\alpha ^+$
:
$$ \begin{align*}H_{\alpha} := \{ f \in V^*~| ~f(\alpha) = 0 \}~\text{~and~}~H_\alpha^+:= \{ f \in V^*~| ~f(\alpha)> 0 \}. \end{align*} $$
The intersection C of all
$H_\alpha ^+$
for
$\alpha \in \Delta $
is called the fundamental chamber. Let
$D =\overline {C}$
be the topological closure of C, the Tits cone
$ \mathcal {U}(W)$
of W is the following cone in
$V^*$
:
$$ \begin{align*}~\mathcal{U}(W) := \bigcup\limits_{w \in W}wD. \end{align*} $$
If
$w \in W$
,
$wC$
is called a chamber of
$\mathcal {U}(W)$
. The action of W on
$\{wC, w \in W \}$
is simply transitive; the chambers of
$\mathcal {U}(W)$
are, therefore, in bijection with the elements of W. The fundamental chamber C corresponds to the identity element e of W. The hyperplane arrangement
$\mathcal A_{(W,S)}=\{H_{\alpha }~|~\alpha \in \Phi \}$
is called the Coxeter arrangement of
$(W,S)$
(see [Reference Humphreys19] for more details).
1.2 Small roots, Shi arrangement, and minimal elements
The Shi arrangement of an affine Weyl group
$(W,S)$
is an affine hyperplane arrangement introduced by Shi in [Reference Shi23, Chapter 7] to study Kazhdan–Lusztig cells in affine Weyl groups of type A and later extended to other affine types in [Reference Shi25] (see also Fishel’s survey [Reference Fishel15] for more information).
Shi arrangements are generalized to any Coxeter system as follows. Introduced by Brink and Howlett [Reference Brink and Howlett3] to prove the automaticity of Coxeter groups, the dominance order is the partial order
$\preceq _{\mathrm {dom}}$
on
$\Phi ^+$
defined by
$$ \begin{align*}\alpha \preceq_{\mathrm{dom}} \beta \Longleftrightarrow \forall w \in W, \beta \in N(w) \Longrightarrow \alpha \in N(w). \end{align*} $$
We say that
$\beta $
is a small root if
$\beta $
dominates no other root than itself. The set of small roots is denoted by
$\Sigma (W,S)$
. One of the remarkable results of Brink and Howlett in [Reference Brink and Howlett3] is that the set
$\Sigma (W,S)$
is finite (if S is finite); we refer the reader to the book [Reference Björner and Brenti2, Chapter 4] for more details.
The Shi arrangement of W is the subarrangement of the Coxeter arrangement
$\mathcal A_{(W,S)}$
defined by
$$ \begin{align*}{\operatorname{Shi}}(W,S) = \{ H_\alpha ~|~ \alpha\in \Sigma(W,S) \}. \end{align*} $$
The Shi regions of W are the connected components of
${\operatorname {Shi}}(W,S)$
in the Tits cone
$\mathcal U(W)$
, that is, the connected components of
$$ \begin{align*}\mathcal{U}(W) ~\backslash \bigcup_{\alpha \in \Sigma(W,S)} H_{\alpha}. \end{align*} $$
Each Shi region is uniquely determined by a union of chambers in the Tits cone, leading to an equivalence relation
$\sim _{\operatorname {Shi}}$
on W defined by:
$u \sim _{\operatorname {Shi}} w$
if and only if
$uC$
and
$wC$
are in the same Shi region. Let us define
$$ \begin{align*}L_{\operatorname{Shi}} (W,S) := \{w\in W\mid u\sim_{\operatorname{Shi}} w \implies \ell(w)\leq \ell(u) \}. \end{align*} $$
1.3 Low elements
A conjectural characterization of
$L_{\operatorname {Shi}}(W,S)$
is given in terms of low elements. Low elements were introduced by Dehornoy, Dyer, and the second author [Reference Dehornoy, Dyer and Hohlweg12] in relation to Garside theory [Reference Dehornoy11] in the case of Artin–Tits (braid) groups. Low elements are defined as follows:
$w\in W$
is a low element if
$$ \begin{align*}N(w)=\operatorname{cone}(\Sigma(W,S)\cap N(w))\cap \Phi. \end{align*} $$
Denote by
$L(W,S)$
the set of low elements of
$(W,S)$
. Dyer and the second author show in [Reference Dyer and Hohlweg13] that
$L(W,S)$
is a finite Garside shadow, i.e., a subset of W containing S and closed under taking suffixes and under taking joins in the right weak order.
The following conjecture is a reformulation of [Reference Dyer and Hohlweg13, Conjecture 2] in the case of low elements (see also [Reference Hohlweg, Nadeau and Williams18, Section 3.6]).
Conjecture 1 We have
$L_{\operatorname {Shi}}(W,S)=L(W,S)$
. Moreover, any Shi region
$\mathcal R$
contains a unique low element, which is the unique element of minimal length in
$\mathcal R$
.
Both statements in the above conjecture are equivalent (see Section 4.3). This conjecture implies also, if true, that the set
$L_{\operatorname {Shi}}(W,S)$
is a Garside shadow, since the set of low elements is a Garside shadow, as shown in [Reference Dyer and Hohlweg13]. In [Reference Hohlweg, Nadeau and Williams18], the authors show that any Garside shadow is the set of states of an automaton that recognizes the language of reduced words for
$(W,S)$
. They also show that the Brink–Howlett automaton, whose states are indexed by
$L_{\operatorname {Shi}}(W,S)$
, projects onto the Garside shadow automaton associated with
$L(W,S)$
.
Therefore, Conjecture 1 is known to be true in the cases where the Brink–Howlett automaton is minimal. For instance, this conjecture is true for Coxeter systems with complete Coxeter graphs and right-angled Coxeter groups (see [Reference Hohlweg, Nadeau and Williams18, Theorem 1.3]); all cases of a minimal Brink–Howlett automaton are characterized by Parkinson and Yau in [Reference Parkinson and Yau21]. However, the sole affine Weyl groups for which the Brink–Howlett automaton is minimal are of type A. Recently, Charles [Reference Charles9] proved Conjecture 1 for any rank 3 Coxeter system.
1.4 Main results
In this article, we prove Conjecture 1 for any affine Coxeter systems.
Theorem 1.1 If
$(W,S)$
is an affine Coxeter system, then
$L_{\operatorname {Shi}}(W,S)=L(W,S)$
.
Corollary 1.2 If
$(W,S)$
is an affine Coxeter system, then
$L_{\operatorname {Shi}}(W,S)$
is a Garside shadow for
$(W,S)$
.
The above corollary shows that the Brink–Howlett automaton is the Garside shadow automaton, in the sense of [Reference Hohlweg, Nadeau and Williams18], associated with
$L(W,S)$
.
In the case of an (irreducible) affine Coxeter system
$(W,S)$
, Shi shows in [Reference Shi25, Proposition 7.2] that each equivalence class under
$\sim _{\operatorname {Shi}}$
contains exactly one element of minimal length, which means, by Theorem 1.1, that
$|L(W,S)|=|L_{\operatorname {Shi}}(W,S)|$
counts the number of regions of
${\operatorname {Shi}}(W,S)$
. Writing
$(W_0,S_0)$
for the underlying finite Coxeter system of
$(W,S)$
, Shi provides the following remarkable formula in [Reference Shi25, Theorem 8.1]:
$$ \begin{align*}|L_{\operatorname{Shi}}(W,S)|= (h+1)^{n}, \end{align*} $$
where
$n=|S_0|$
is the rank of
$W_0$
and h is the Coxeter number of
$W_0$
. This formula also appears in relation to diagonal invariants in the work of Haiman [Reference Haiman17, Section 7]: Haiman conjectured that there exist a
$W_0$
-stable quotient ring of the covariant ring
$\mathbb C[V\oplus V^*]^{co\, W}$
of dimension
$(h+1)^{n}$
; this conjecture was confirmed by Gordon in [Reference Gordon16]. As a corollary of Theorem 1.1, we obtain the enumeration of the set of low elements in the case of affine Coxeter systems.
Corollary 1.3 If
$(W,S)$
is an affine Coxeter system with underlying finite Weyl group
$W_0$
of rank n and Coxeter number h, then the number of low elements in
$(W,S)$
is
$|L(W,S)| = (h+1)^{n}$
.
1.5 Outline of the article
The proof of Theorem 1.1 can be easily reduced to the case of irreducible affine Coxeter systems by standard technics (see, for instance, [Reference Dyer and Lehrer14]). Therefore, all Coxeter systems considered in this article are irreducible.
In order to prove Theorem 1.1, we first recollect in Section 2 well-known facts about affine Coxeter systems and affine Weyl groups. In Section 3, we survey the notions of low elements. In Section 4, we survey Shi arrangements and Shi’s admissible signs. In particular, in Section 4.2, we show that
$L_{\operatorname {Shi}}(W,S)$
is closed under taking suffixes (Proposition 4.5), and in Section 4.3, we reduce the proof of Theorem 1.1 to showing that
$L_{\operatorname {Shi}}(W,S)\subseteq L(W,S)$
.
Then we tackle the core of the proof of Theorem 1.1 in Sections 5–7. First, we characterize the so-called descent walls of a Shi region
$\mathcal R$
, that is, the walls that separate
$\mathcal R$
from the fundamental alcove (see Definition 5.1); this characterization is given in terms of Shi’s admissible sign types and Shi coefficients (Proposition 5.3). This leads to the first key result, which is Lemma 5.9, which describes some descent walls of a Shi region.
Second, we observe that Cellini and Papi’s set of minimal elements, which appears in their works [Reference Cellini and Papi4, Reference Cellini and Papi5] on
$ad$
-nilpotent ideals of a Borel subalgebra, is precisely the set
$L^0(W,S)$
of low elements in the dominant region of
$\mathcal A(W,S)$
, i.e., the fundamental region of the Coxeter arrangement of
$(W_0,S_0)$
(Corollary 6.4). This covers the initial case of our proof by induction. In order to conclude our inductive proof, we prove our second key result, the descent-wall theorem: the descent walls of
$w\in L_{\operatorname {Shi}}$
are precisely those of its corresponding Shi region (Theorem 7.1). We conclude the proof of Theorem 1.1 in Section 7.4.
The connection with Cellini and Papi’s results implies, in particular, the following corollary where the second equality is already known via the last term, understood [Reference Cellini and Papi5, Theorem 1] as the number of ad-nilpotent ideals of a Borel subalgebra
$\mathfrak {b}$
of a simple Lie algebra
$\mathfrak {g}$
.
Corollary 1.4 If
$(W,S)$
is an irreducible affine Coxeter system with underlying finite Weyl group
$W_0$
, then the number of low elements in the fundamental chamber of
$\mathcal A(W,S)$
is the
$W_0$
-Catalan number:
$$ \begin{align*}|L^0(W,S)|={\operatorname{Cat}}(W_0) = \frac{1}{|W_0|}\prod_{i=1}^n (h+e_i+1), \end{align*} $$
where
$e_1,\dots ,e_n$
are the exponents of
$W_0$
.
For a survey of the history of Coxeter–Catalan numbers, we refer the reader to [Reference Athanasiadis1, p. 181]. For more details on Coxeter numbers and exponents, see [Reference Humphreys19, Chapter 3].
2 Affine Coxeter systems
For a Euclidean vector space
$V_0$
with inner product
$\langle \cdot ,\cdot \rangle $
, we denote by
$O(V_0)$
the group of isometries on
$V_0$
and
${\operatorname {Aff}}(V_0)$
the group of affine isometries on
$V_0$
. The aim of this section is to survey some well-known facts on affine Coxeter systems along the lines of [Reference Dyer and Lehrer14, Section 3] (see also [Reference Kac20, Chapters 6 and 7]).
2.1 Weyl group
Let
$\Phi _0$
be an irreducible crystallographic root system in
$V_0$
with simple system
$\Delta _0$
; set
$n=|\Delta |$
. Let
$\Phi _0^+$
be the associated positive root system. The root poset on
$\Phi _0^+$
is the poset
$(\Phi _0^+,\preceq )$
defined by
$\alpha \preceq \beta $
if and only if
$\beta -\alpha \in \operatorname {cone}(\Delta _0)$
.
For any
$\alpha \in \Phi _0$
, let
$\alpha ^\vee =\frac {2}{\langle \alpha , \alpha \rangle }\alpha $
be the coroot of
$\alpha $
. The reflection
$s_\alpha $
associated with
$\alpha $
is defined by
$$ \begin{align*}\begin{array}{ccccc} s_{\alpha} & : & V_0 & \longrightarrow & V_0 \\ & & x& \longmapsto & x-\langle \alpha, x \rangle\alpha^\vee. \end{array} \end{align*} $$
Notice that, for all
$\alpha \in \Phi _0$
, we have
$$ \begin{align} \langle \alpha^{\vee}, \alpha \rangle = \langle \frac{2\alpha}{\langle\alpha, \alpha \rangle}, \alpha \rangle = 2. \end{align} $$
The set of fixed points of
$s_\alpha $
is the hyperplane
$H_\alpha :=\{x\in V_0 \mid \langle x,\alpha \rangle = 0\}$
. Let
$W_0$
be the Weyl group associated with
$\Phi _0$
. We identify the root lattice
$Q:= \mathbb {Z}\Phi _0$
and the coroot lattice
$Q^{\vee } := \mathbb {Z}\Phi _0^{\vee }$
with their groups of associated translations and we denote by
$\tau _x$
the translation corresponding to
$x \in Q^{\vee }$
. With
$S_0 = \{s_{\alpha }\mid \alpha \in \Delta _0\}$
, the pair
$(W_0,S_0)$
is a finite Coxeter system. The set of reflections
$T_0$
of
$W_0$
is
$$ \begin{align*}T_0 = \bigcup\limits_{w \in W_0}wS_0w^{-1}. \end{align*} $$
The fundamental chamber of
$W_0$
, also called the dominant region, is the set:
$$ \begin{align*}C_\circ := \{x \in V_0~|~\langle x,\alpha \rangle> 0, ~\forall \alpha \in \Delta_0\}.\end{align*} $$
2.2 Affine Weyl group and affine Coxeter arrangement
The affine reflection
$s_{\alpha ,k} \in \text {Aff}(V_0)$
is defined by
$$ \begin{align} s_{\alpha,k}(x) &=x-(\langle \alpha, x \rangle-k)\alpha^{\vee} = \tau_{k\alpha^{\vee}}s_{\alpha}. \end{align} $$
The affine Weyl group W associated with
$\Phi _0$
is the following discrete reflection group of affine isometries:
$$ \begin{align*}W = \langle s_{\alpha,k}~|~\alpha \in \Phi_0, ~k \in \mathbb{Z \rangle} = Q^{\vee} \rtimes W_0. \end{align*} $$
Let
$\alpha _0$
be the highest root of
$\Phi _0$
. The set
$ S := S_0 \cup \{s_{\alpha _0,1}\}$
is a set of simple reflections for W. The set of fixed points of the reflection
$s_{\alpha ,k}$
is the affine hyperplane
$$ \begin{align*} H_{\alpha,k} = \{ x \in V~|~ \langle x, \alpha \rangle = k\}. \end{align*} $$
The collection of hyperplanes
$H_{\alpha ,k}$
, for
$ \alpha \in \Phi _0$
and
$k \in \mathbb {Z}$
, is denoted by
$\mathcal {A}(W,S)$
and is called the affine Coxeter arrangement of
$(W,S)$
. An alcove of
$(W,S)$
in
$V_0$
is a connected component of
$$ \begin{align*}V_0 ~\backslash \bigcup\limits_{H \in \mathcal{A}(W,S)} H. \end{align*} $$
The group W acts transitively on the set of alcoves, i.e., the map
$w\mapsto w\cdot A_\circ $
is a bijection between W and the set of alcoves of
$(W,S)$
in
$V_0$
where
$A_\circ $
is the alcove associated with the identity element.
Remark 2.1 The affine Coxeter arrangement is, in fact, the intersection of the usual Coxeter arrangement with an affine hyperplane in the Tits cone. The fundamental alcove is the intersection of the fundamental chamber with this affine hyperplane. For more details, see [Reference Humphreys19, Section 6.5], [Reference Dyer and Lehrer14, Proof of Proposition 3], or [Reference Chapelier-Laget8, Section 1.3.5].
2.3 Shi coefficients and Shi parameterization
In [Reference Shi24, Proposition 5.1], Shi describes a correspondence
$ w \mapsto (k(w,\alpha ))_{\alpha \in \Phi _0^+} $
between W (or equivalently the set of alcoves of
$(W,S)$
) and some
$\Phi _0^+$
-tuples over
$\mathbb Z$
. We refer to some of Shi’s results from [Reference Shi24, Reference Shi25]. However, there are three differences in the conventions between Shi’s articles and our article:
-
(1) In Shi’s articles, W is denoted by
$W_a$
and the underlying Weyl group is denoted by W. -
(2) In Shi’s articles, the root system of the affine Coxeter system
$(W,S)$
is associated with
$\Phi _0^\vee $
instead of
$\Phi _0$
in this article. Since
$(\Phi _0^\vee )^\vee = \Phi _0$
, it is enough to apply the involution
$\alpha \mapsto \alpha ^\vee $
when applying Shi’s results. -
(3) The action of W on the root system is from the right in Shi’s articles, where we adopt the convention of acting by the left; for more details, see [Reference Chapelier-Laget7, Remark 2.1].
By convention, we have the following formula [Reference Shi24, Section 1] relating Shi coefficients on negative roots and Shi coefficients on positive roots:
$$ \begin{align} k(w,-\alpha) = -k(w,\alpha) ~\mathrm{~~for ~~any~} \alpha \in \Phi_0. \end{align} $$
Now the following statements give very important relations between Shi coefficients on positive roots.
Lemma 2.2 [Reference Shi24, Lemma 3.1]
Let w be an element of
$W_0$
and
$\alpha \in \Phi ^+$
. Then
$$ \begin{align*} k(w,\alpha) = \left\{ \begin{array}{rl} 0, & \text{if}~~ \alpha \notin N(w), \\ -1, & \text{if}~~\alpha \in N(w). \end{array} \right. \end{align*} $$
The following proposition is, in general, only stated for positive roots, but turns out to stand also for any root thanks to equation (3).
Proposition 2.3 Let
$w \in W$
,
$s \in S_0$
, and
$t \in T_0$
. For all
$\alpha \in \Phi _0$
, one has
-
(1)
$k(sw, \alpha ) = k(w, s(\alpha )) + k(s, \alpha )$
[Reference Shi24, Proposition 4.2]. -
(2)
$k(tw, \alpha ) = k(w, t(\alpha )) + k(t, \alpha )$
(reformulation of [Reference Chapelier-Laget7, Proposition 3.2]).
Lemma 2.4 [Reference Shi24, Proposition 4.3]
Let
$w \in W$
and
$\beta \in \Delta _0$
. If
$s_{\beta } \in D_L(w)$
, then
$k(w,\beta ) \leq -1$
.
Remark 2.5
-
(1) Shi does not give a proof of Lemma 2.4 in [Reference Shi24]; instead, he refers the reader to [Reference Shi23, Chapter 7]. We point out that a direct proof of this result can be obtained using an analog of [Reference Dyer and Hohlweg13, Proposition 2.16] for “p-inversions sets
$N^p(w)$
.” -
(2) It is very useful to notice that, for an arbitrary
$w\in W$
and
$\alpha \in \Phi ^+_0$
,
$|k(w,\alpha )|$
is given by the number of hyperplanes parallel to
$H_{\alpha }$
that separates
$w\cdot A_\circ $
from
$A_\circ $
; the sign is then negative if
$H_{\alpha }$
is one of these hyperplanes.
Example 2.6 (Type
$\tilde A_2$
)
Consider
$(W,S)$
of type
$\tilde A_2$
. Alcoves and their labeling are shown in Figure 1. Write
$S_0=\{s_1,s_2\}$
, so
$\Delta _0=\{\alpha _1,\alpha _2\}$
, and
$\Phi _0^+=\{\alpha _1,\alpha _2,\alpha _1+\alpha _2\}$
(with
$s_i=s_{\alpha _i}$
).
The Coxeter arrangement of type
$\tilde A_2$
. Each alcove is labeled with its Shi parameterization. The labels
$k(w,\alpha )$
for
$\alpha \in \Phi _0^+$
are indicated in each alcove with the parameterization given at the left-hand side of the figure. The shaded region is the dominant region of
$\mathcal A(W,S)$
, which is also the fundamental chamber for the finite Weyl group
$W_0$
. The fundamental alcove
$A_\circ $
, which corresponds to e, is parameterized by
$0$
’s.
Example 2.7 (Type
$\tilde B_2$
)
Consider
$(W,S)$
of type
$\tilde B_2$
, with underlying finite Weyl group of type
$B_2$
. We consider here
$V=\mathbb R_2$
with orthonormal basis
$\{e_1,e_2\}$
. Set
$\alpha _1=e_1$
and
$\alpha _2=e_2-e_1$
. Write
$S_0=\{s_1,s_2\}$
, so
$\Delta _0=\{\alpha _1,\alpha _2\}$
, and
$\Phi _0^+=\{\alpha _1,\alpha _2,\alpha _1+\alpha _2,2\alpha _1+\alpha _2\}$
(with
$s_i=s_{\alpha _i}$
). The long roots are
$\alpha _2$
and
$2\alpha _1+\alpha _2$
. Alcoves and their labeling are shown in Figure 2.
The Coxeter arrangement of type
$\tilde B_2$
. Each alcove is labeled with its Shi parameterization. The labels
$k(w,\alpha )$
for
$\alpha \in \Phi _0^+$
are indicated in each alcove with the parameterization given at the left-hand side of the figure. The shaded region is the dominant region of
$\mathcal A(W,S)$
, which is also the fundamental chamber for the finite Weyl group
$W_0$
. The fundamental alcove
$A_\circ $
, which corresponds to e, is parameterized by
$0$
’s.
The example go type
$\tilde G_2$
is given in Figure 7.
2.4 Affine crystallographic root systems
Set
$V=V_0\oplus \mathbb R \delta $
to be a real vector space with basis
$\Delta _0\sqcup \{\delta \}$
, where
$\delta $
is an indeterminate. We define a symmetric bilinear form
$B:V\times V\to \mathbb R$
by extending
$\langle \cdot ,\cdot \rangle $
to V as follows: for
$\alpha ,\beta \in \Delta _0$
, we set
$ B(\alpha ,\beta ) = \langle \alpha ,\beta \rangle ,\ B(\alpha ,\delta )=0 \textrm { and } B(\delta ,\delta )=0. $
The pair
$(V,B)$
is a quadratic space for which the isotropic cone is
$\mathbb R\delta = \{x\in V\mid B(x,x)=0\}$
.
We describe now a geometric representation of
$(W,S)$
on
$(V,B)$
. A simple system in
$(V,B)$
for
$(W,S)$
is
$ \Delta = \Delta _0\sqcup \{\delta - \alpha _0\}. $
The root system
$\Phi $
and the positive root system
$\Phi ^+$
for
$(W,S)$
in
$(V,B)$
are defined by
$\Phi =\Phi ^+\sqcup \Phi ^-$
, where
$\Phi ^-=-\Phi ^+$
and
$\Phi ^+ =( \Phi _0^+ + \mathbb N\delta ) \sqcup ( \Phi _0^-+ \mathbb N^*\delta )$
. The pair
$(\Phi ,\Delta )$
is called a (crystallographic) based root system of
$(W,S)$
in
$(V,B)$
. For
$\alpha +k\delta \in \Phi ^+$
, the reflection
$s_{\alpha +k\delta }:V\to V$
is defined as follows. Let
$x=x_0+a\delta \in V$
, then
$$ \begin{align} s_{\alpha+k\delta} (x):= x - 2\frac{B(\alpha+k\delta,x)}{B(\alpha+k\delta,\alpha+k\delta)}(\alpha+k\delta) = s_\alpha(x_0) + (a - k\langle \alpha^\vee,x_0 \rangle)\delta. \end{align} $$
Observe that
$s_{\alpha +0\delta } = s_\alpha $
. It follows that
$ S=S_0\sqcup \{s_{\delta -\alpha _0}\}. $
The correspondences between the positive root system
$\Phi ^+$
, the reflections on V, and those on
$V_0$
defined in Section 2.2 are as follows:
$$ \begin{align*}\begin{array}{ccccc} \Phi^+ & \longrightarrow & \{ \text{reflections on} ~V \} & \longrightarrow & \{ \text{reflections on} ~V_0 \} \\ \alpha+k\delta & \longmapsto & s_{\alpha +k\delta} & \longmapsto & s_{-\alpha,k}. \end{array} \end{align*} $$
Therefore, the hyperplane associated with
$\alpha +k\delta \in \Phi ^+$
is
$ H_{\alpha +k\delta }:= H_{-\alpha ,k}. $
3 Small roots and low elements in affine Weyl groups
Let
$(W,S)$
be an affine Coxeter system with underlying Weyl group
$W_0$
. We now survey the notions of small roots and low elements (see [Reference Dyer and Hohlweg13] for more details). We gave general definitions of small roots and low elements in the introduction. In the case of affine Weyl groups, these notions have easier interpretations, which we discuss below.
3.1 Small roots and low elements
A positive root
$\alpha +k\delta \in \Phi ^+$
(resp. a hyperplane
$H_{\alpha +k\delta }\in \mathcal A(W,S)$
) is small if there is no hyperplane parallel to
$H_{\alpha +k\delta }$
that separates
$H_{\alpha +k\delta }$
from
$A_\circ $
. Denote by
$\Sigma :=\Sigma (W,S)$
the set of small roots. It turns out that
$$ \begin{align*}\Sigma =\Phi_0^+\sqcup (\delta-\Phi_0^+) = \{\alpha,\delta-\alpha\mid \alpha\in\Phi_0^+\}, \end{align*} $$
which is of cardinality
$|\Phi _0|$
. It is well known (see, for instance, [Reference Humphreys19, Proposition 1.4]) that
$s\in S_0$
is a bijection on
$\Phi _0^+\setminus \{\alpha _s\}$
, where
$\alpha _s$
denotes the root in
$\Delta _0$
such that
$s= s_{\alpha _s}$
. The next proposition is an analog of this result for
$\Sigma $
.
Proposition 3.1 Let
$s\in S_0$
, then
$s(\Sigma \setminus \{\alpha _s,\delta -\alpha _s\})=\Sigma \setminus \{\alpha _s,\delta -\alpha _s\}$
.
Proof Let
$\alpha \in \Phi ^+_0\setminus \{\alpha _s\}$
, then
$s(\alpha )\in \Phi ^+_0\setminus \{\alpha _s\} \subseteq \Sigma \setminus \{\alpha _s,\delta -\alpha _s\}$
. We also have by equation (4)
$$ \begin{align*}s(\delta-\alpha)=\delta-s(\alpha) \in \delta - (\Phi^+_0\setminus\{\alpha_s\})\subseteq \Sigma\setminus\{\alpha_s,\delta-\alpha_s\}. \end{align*} $$
Therefore,
$s(\Sigma \setminus \{\alpha _s,\delta -\alpha _s\})\subseteq \Sigma \setminus \{\alpha _s,\delta -\alpha _s\}$
; the equality follows since s is injective.
An element
$w\in W$
is a low element of
$(W,S)$
if there is
$X\subseteq \Sigma $
such that
$$ \begin{align*}N(w)=\operatorname{cone}_\Phi(X):=\operatorname{cone}(X)\cap\Phi. \end{align*} $$
We denote by
$L(W,S)$
the set of low elements of
$(W,S)$
. If there is no possible confusion, we write
$L:=L(W,S)$
. Low elements were introduced in order to prove that there is a finite Garside shadow, i.e., a finite subset of W containing S, closed by taking suffixes and closed under the join operator of the right weak order. It is proven in [Reference Dyer and Hohlweg13, Theorem 1.1].
Example 3.2 We continue Example 2.6. The small roots are
$$ \begin{align*}\Sigma=\{\alpha_1,\alpha_2,\alpha_1+\alpha_2, \delta-\alpha_1,\delta-\alpha_2, \delta-(\alpha_1+\alpha_2)\}. \end{align*} $$
Obviously,
$k(e,\alpha _1)=k(e,\alpha _2)=k(e,\alpha _1+\alpha _2)=0$
. Now, for an arbitrary
$w\in W$
,
$|k(w,\alpha _1)|$
is given by the number of hyperplanes parallel to
$H_{\alpha _1}$
that separates
$w\cdot A_\circ $
from
$A_\circ $
; the sign is then negative if
$H_{\alpha _1}$
is one of these hyperplanes (see Remark 2.5). For instance, the w indicated in Figure 1 has
$k(w,\alpha _1)=1$
since only
$H_{\delta -\alpha _1}$
separates the alcove
$w\cdot A_\circ $
from
$A_\circ $
. We proceed similarly for
$\alpha _2$
and the highest root
$\alpha _1+\alpha _2$
. So, for the particular w indicated in Figure 1, we have:
$k(w,\alpha _2)=0$
, since no parallel to
$H_{\alpha _2}$
separates
$w\cdot A_\circ $
from
$A_\circ $
;
$k(w,\alpha _1+\alpha _2)=2$
, since
$H_{\delta -(\alpha _1+\alpha _2)}$
and
$H_{2\delta -(\alpha _1+\alpha _2)}$
separates
$w\cdot A_\circ $
from
$A_\circ $
.
Example 3.3 We continue Example 2.7. The small roots are
$$ \begin{align*}\Sigma=\{\alpha_1,\alpha_2,\alpha_1+\alpha_2, 2\alpha_1+\alpha_2,\delta-\alpha_1,\delta-\alpha_2, \delta-(\alpha_1+\alpha_2), \delta-(2\alpha_1+\alpha_2)\}. \end{align*} $$
Now, for an arbitrary
$w\in W$
,
$|k(w,\beta )|$
is, as in the preceding example, given by the number of hyperplanes parallel to
$H_{\beta }$
that separates
$w\cdot A_\circ $
from
$A_\circ $
; the sign is then negative if
$H_{\beta }$
is one of these hyperplanes. For instance, the w indicated in Figure 2 has
$k(w,\alpha _1)=1$
since only
$H_{\delta -\alpha _1}$
separates
$w\cdot A_\circ $
from
$A_\circ $
. We proceed similarly for
$\alpha _2$
,
$\alpha _1+\alpha _2$
and the highest root
$2\alpha _1+\alpha _2$
. So we obtain
$k(w,\alpha _2)=1$
, since only
$H_{\delta -\alpha _2}$
separates
$w\cdot A_\circ $
from
$A_\circ $
,
$k(w,\alpha _1+\alpha _2)=0$
, since no parallel to
$H_{\alpha _1+\alpha _2}$
separates the alcove
$w\cdot A_\circ $
from
$A_\circ $
, and finally
$k(w,2\alpha _1+\alpha _2)=-1$
, since only
$H_{2\alpha _1+\alpha _2}$
separates
$w\cdot A_\circ $
from
$A_\circ $
.
3.2 Small inversion sets
In order to check if an element
$w\in W$
is a low element, it is often convenient to consider all small roots in the inversion set
$N(w)$
.
The small inversion set of
$w\in W$
is
$\Sigma (w):=N(w)\cap \Sigma $
. Therefore, an element
$w\in W$
is a low element if and only if
$N(w)=\operatorname {cone}_\Phi (\Sigma (w))$
. The set of all small inversion sets is
$ \Lambda (W,S)=\{\Sigma (w)\mid w\in W\} \subseteq 2^\Sigma. $
We write
$\Lambda :=\Lambda (W,S)$
if there is no possible confusion. Notice that, since
$\Sigma $
is finite, the set
$2^\Sigma $
of subsets of
$\Sigma $
is finite, hence
$\Lambda $
is finite.
The following result, which defines the transitions in the so-called Brink–Howlett automaton, is [Reference Dyer and Hohlweg13, Lemma 3.21(1)]; we refer the reader to [Reference Hohlweg, Nadeau and Williams18, Reference Parkinson and Yau21, Reference Parkinson and Yau22] for more information on the Brink–Howlett automaton.
Lemma 3.4 Let
$w\in W$
and
$s\in S$
such that
$\ell (w)>\ell (sw)$
. Then
$ \Sigma (w)=\{\alpha _s\}\sqcup s(\Sigma (sw))\cap \Sigma. $
The set of low elements is finite since the map
$\sigma :L\to \Lambda $
, defined by
$w\mapsto \Sigma (w)$
, is injective [Reference Dyer and Hohlweg13, Proposition 3.26(ii)]. It is conjectured by Dyer and the second author that this map is surjective, which turns out to be equivalent to Conjecture 1 (see Section 4.3 for more details).
Conjecture 2 [Reference Dyer and Hohlweg13, Conjecture 2]
The map
$\sigma :L\to \Lambda $
is a bijection.
Remark 3.5 The map
$\sigma $
is denoted by
$\Sigma $
in [Reference Dyer and Hohlweg13], we change the notation to avoid confusion with the set of small roots
$\Sigma $
.
3.3 Basis of inversion sets
The most efficient way to check if an element
$w\in W$
is low is to consider a basis for the inversion set
$N(w)$
(see [Reference Dyer and Hohlweg13, Section 4.2] for details).
Let
$w\in W$
, and consider the set
$$ \begin{align*} N^1(w)&:=\{\alpha+k\delta\in \Phi^+\mid \mathbb R(\alpha+k\delta) \textrm{ is an extreme ray of }\operatorname{cone}(N(w))\}\\ &=\{\alpha+k\delta\in \Phi^+\mid \ell(s_{\alpha+k\delta}w)=\ell(w)-1\}. \end{align*} $$
The set
$N^1(w)$
is called the basis of the inversion set
$N(w)$
and is the inclusion-minimal subset of
$N(w)$
such that
$N(w)=\operatorname {cone}_\Phi (N^1(w)).$
Proposition 3.6 Let
$w\in W$
, then
$w\in L$
if and only if
$N^1(w)\subseteq \Sigma $
.
Proof If
$N^1(w)\subseteq \Sigma $
, then
$w\in L$
by definition of low elements. Assume now that
$w\in L$
, so
$ N(w)=\operatorname {cone}_\Phi (\Sigma (w))=\operatorname {cone}_\Phi (N^1(w)). $
Since
$N^1(w)$
is the inclusion-minimal subset with that property, we have
$ N^1(w)\subseteq \Sigma (w)\subseteq \Sigma. $
3.4 Left and right descent sets
The basis of inversion sets gives a useful interpretation of the right and left descent sets. The left descent set of
$w\in W$
is the set:
$$ \begin{align*}D_L(w)=\{s\in S\mid \ell(sw)=\ell(w)-1\} = \{s\in S\mid \alpha_s\in \Delta\cap N(w)\}, \end{align*} $$
where
$\alpha _s\in \Delta $
is the simple root such that
$s=s_{\alpha _s}$
in equation (4). The set of left descent roots of w is
$$ \begin{align*}ND_L(w)=\Delta\cap N(w)\subseteq N^1(w). \end{align*} $$
In other words,
$\alpha +k\delta \in ND_L(w)$
if and only if
$H_{\alpha +k\delta }$
is a wall of
$A_\circ $
that separates
$A_\circ $
from
$w\cdot A_\circ $
. Similarly, the right descent set of
$w\in W$
is the set:
$$ \begin{align*}D_R(w)=\{s\in S\mid \ell(ws)=\ell(w)-1\} = \{s\in S\mid w(\alpha_s)\in \Phi^-\}. \end{align*} $$
Let
$s\in D_R(w)$
, then there is a reduced word for w ending with s:
$w=us$
with
$u\in W$
and
$\ell (w)=\ell (u)+1$
. Therefore,
$$ \begin{align*}N(u)=N(w)\setminus\{u(\alpha_s)\}. \end{align*} $$
The set of right descent roots of w, which is a subset of
$N^1(w)$
, is
$$ \begin{align*} ND_R(w)&= -w(\{\alpha_s\mid s\in D_R(w)\}) \\ &= \{\alpha+k\delta\in N(w)\mid \exists u\in W,\ N(u)=N(w)\setminus \{\alpha+k\delta\}\}. \end{align*} $$
In other words,
$\alpha +k\delta \in ND_R(w)$
if and only if
$H_{\alpha +k\delta }$
is a wall of
$w\cdot A_\circ $
that separates
$A_\circ $
from
$w\cdot A_\circ $
.
The following proposition gives a useful relationship between the sets of right descent roots of an element
$w\in W$
and of one of its maximal suffixes.
Proposition 3.7 We have
$ND_R(sw)= s(ND_R(w)\setminus \{\alpha _s\})$
for all
$w\in W$
and
$s\in D_L(w)$
.
Proof Since
$s\in D_L(w)$
, we have
$w=su$
for some
$u\in W$
and
$\ell (su)=\ell (u)+1$
. We first show that
$D_R(u) = D_R(w)\setminus (\{w^{-1}sw\}\cap S)$
. Let
$r\in D_R(u)$
, then
$u=vr$
for some
$v\in W$
and
$\ell (u)=\ell (v)+1$
. So
$w=svr$
with
$\ell (w)=\ell (v)+2$
. Thus,
$r\in D_R(w)$
. If
$r=w^{-1}sw$
, then
$wr=sw=s(su)=u$
. Therefore,
$w=ur$
and
$r\notin D_R(u)$
, a contradiction. So
$D_R(u) \subseteq D_R(w)\setminus (\{w^{-1}sw\}\cap S)$
. Now let
$r\in D_R(w)\setminus (\{w^{-1}sw\}\cap S)$
. By the exchange condition, we either have
$wr=sur=u$
or
$r\in D_R(u)$
. The first case implies
$w^{-1}sw=(u^{-1}s)sw=u^{-1}w=r\in S$
, a contradiction. So
$r\in D_R(u)$
and we have the desired equality. Now, by definition of
$ND_R(\cdot )$
, we have
$$ \begin{align*} ND_R(sw)&= ND_R(u)\\ &= -u(\{\alpha_r\mid r\in D_R(u)\}) = -u(\{\alpha_r\mid r\in D_R(w)\}\setminus(\{w^{-1}(\alpha_s)\}\cap \Delta))\\ &= -sw(\{\alpha_r\mid r\in D_R(w)\}\setminus(\{w^{-1}(\alpha_s)\}\cap \Delta))\\ &= s(ND_R(w)\setminus\{\alpha_s\}).\\[-33pt] \end{align*} $$
4 Shi arrangements in affine Weyl groups
Let
$(W,S)$
be an affine Coxeter system with underlying Weyl group
$W_0$
. The Shi arrangement of
$(W,S)$
is the affine hyperplane arrangement constituted of the small hyperplanes of
$(W,S)$
:
$$ \begin{align*}{\operatorname{Shi}}(W,S) = \{H_\beta\mid \beta \in \Sigma\}. \end{align*} $$
A Shi region of
$(W,S)$
is a connected component of the complement of the Shi arrangement in
$V_0$
, i.e., a connected component of
$$ \begin{align*}V_0\setminus \bigcup_{H\in {\operatorname{Shi}}(W,S)} H. \end{align*} $$
4.1 Separation sets of Shi regions
Shi regions correspond to equivalence classes of the relation
$\sim _{\operatorname {Shi}}$
over W given by
$u\sim _{\operatorname {Shi}} v$
if and only if both
$u\cdot A_\circ $
and
$v\cdot A_\circ $
are contained in the same Shi region. These equivalence classes have a useful interpretation in terms of separation sets.
Definition 4.1 The separation set of a Shi region
$\mathcal R$
is the set of hyperplanes in
${\operatorname {Shi}}(W,S)$
that separate
$\mathcal R$
from
$A_\circ $
. We denote the inversion set of the Shi region
$\mathcal R$
by
$$ \begin{align*}\Sigma(\mathcal R) = \{\alpha \in \Sigma\mid H_\alpha \textrm{ separates } \mathcal R \textrm{ from } A_\circ\}. \end{align*} $$
Proposition 4.2 Let
$w \in W$
and
$s \in S.$
Let
$\mathcal R$
be a Shi region such that
$w\cdot A_\circ \subseteq \mathcal R$
. Then
-
(1)
$\Sigma (\mathcal R)=\Sigma (w)$
. In particular,
$u \sim _{\operatorname {Shi}} v$
if and only if
$\Sigma (u) = \Sigma (v)$
. -
(2) If
$\alpha _s\in \Sigma (\mathcal R)$
, then there exists a Shi region
$\mathcal R'$
such that
$s\cdot \mathcal R' \subseteq \mathcal R$
.
Proof
(1) Since
$w\cdot A_\circ \subseteq \mathcal R$
, any hyperplane that separates
$\mathcal R$
from
$A_\circ $
also separates
$w\cdot A_\circ $
from
$A_\circ $
. Therefore,
$\Sigma (\mathcal R)\subseteq N(w)\cap \Sigma =\Sigma (w)$
. Conversely, assume that
$H_\alpha $
,
$\alpha \in \Sigma $
, separates
$w\cdot A_\circ $
from
$A_\circ $
. Since
$w\cdot A_\circ \subseteq \mathcal R$
and
$\mathcal R$
is a connected component of
$V_0\setminus \bigcup _{H\in {\operatorname {Shi}}(W,S)} H$
,
$H_\alpha $
separates
$w\cdot A_\circ $
from
$\mathcal R$
.
(2) Let
$w\in W$
such that
$w\cdot A_\circ \subseteq \mathcal R$
. Then, by (1),
$\alpha _s\in \Sigma (\mathcal R)=\Sigma (w)\subseteq N(w)$
. Therefore,
$\ell (sw)<\ell (w)$
. Let
$\mathcal R'$
be the Shi region such that
$sw\cdot A_\circ \subseteq \mathcal R'$
. Let
$u\in W$
such that
$u\sim _{\operatorname {Shi}} sw$
, and we need to show that
$su\sim _{\operatorname {Shi}} w$
. Since
$\alpha _s\notin N(sw)$
, we have
$\alpha _s\notin \Sigma (sw)=\Sigma (u)$
. In other words,
$\ell (su)>\ell (u)$
. Then, by Lemma 3.4, we have
$\Sigma (w)=\{\alpha _s\}\sqcup s(\Sigma (sw))\cap \Sigma $
; moreover, since
$u\sim _{\operatorname {Shi}} sw$
, it follows that
$\{\alpha _s\}\sqcup s(\Sigma (sw))\cap \Sigma = \{\alpha _s\}\sqcup s(\Sigma (u))\cap \Sigma $
. Then we obtain
$ \{\alpha _s\}\sqcup s(\Sigma (u))\cap \Sigma = \{\alpha _s\}\sqcup s(\Sigma (s(su)))\cap \Sigma = \Sigma (su)$
since
$\ell (su)>\ell (u).$
Thus,
$su\sim _{\operatorname {Shi}} w$
.
4.2 Minimal elements in Shi regions
In [Reference Shi25, Proposition 7.1], Shi shows that any Shi region
$\mathcal R$
on an affine Weyl group W contains a unique minimal element, that is, there is a unique element
$w\in W$
with
$w \cdot A_\circ \subseteq \mathcal {R}$
and such that
$ \ell (w) \leq \ell (u)$
, for all
$u\sim _{\operatorname {Shi}} w .$
We denote by
$L_{\operatorname {Shi}}$
the set of minimal elements in Shi regions. Shi shows in [Reference Shi25, Proposition 7.2] that
$w\in L_{\operatorname {Shi}}$
if and only if for all
$\alpha \in \Phi _0^+$
we have
$$ \begin{align} |k(w,\alpha)|=\min\{|k(g,\alpha)|\mid g\sim_{\operatorname{Shi}} w\}. \end{align} $$
In [Reference Shi25, Proposition 7.3], Shi characterizes the minimal elements in Shi regions by
$$ \begin{align} w\in L_{\operatorname{Shi}} \iff w\nsim_{\operatorname{Shi}} ws, \ \forall s\in D_R(w), \end{align} $$
where
$u\nsim _{\operatorname {Shi}} v$
means that u and v are not in the same Shi region. This characterization implies that any element in a given Shi region is greater or equal in the right weak order than its corresponding minimal element, i.e., the minimal element in a Shi region is the prefix of any other element in that Shi region.
Proposition 4.3 Let
$w\in L_{\operatorname {Shi}}$
and
$g\in W$
such that
$g\sim _{\operatorname {Shi}} w$
, then
$N(w)\subseteq N(g)$
.
Proof We proceed by induction on
$m=\ell (g)-\ell (w)$
. Since
$w\in L_{\operatorname {Shi}}$
is of minimal length in its Shi region,
$m\in \mathbb N$
. If
$m=0$
, then
$g=w$
by unicity of
$w\in L_{\operatorname {Shi}}$
, so
$w=g$
and
$N(g)=N(w)$
. If
$m>0$
, then
$g\notin L_{\operatorname {Shi}}$
, so there is
$s\in D_R(g)$
such that
$gs\sim _{\operatorname {Shi}} w$
by equation (6). Since
$\ell (gs)-\ell (w)=\ell (g)-1-\ell (w)=m-1$
, we conclude by induction that
$N(w)\subseteq N(gs)$
. The result follows from the fact that
$N(gs) = N(g)\setminus \{-g(\alpha _s)\}$
(see Section 3.4).
Proposition 4.4 We have
$L\subseteq L_{\operatorname {Shi}}$
.
Proof Let
$w\in L$
, then
$ND_R(w)\subseteq N^1(w)\subseteq \Sigma $
, by Proposition 3.6. Let
$s\in D_R(w)$
, then
$-w(\alpha _s)\in ND_R(w)\subseteq \Sigma $
and
$N(ws)=N(w)\setminus \{-w(\alpha _s)\}$
. Hence,
$\Sigma (ws)=\Sigma (w)\setminus \{-w(\alpha _s)\}$
. Therefore, by Proposition 4.2,
$w\nsim _{\operatorname {Shi}} ws$
. We conclude that
$w\in L_{\operatorname {Shi}}$
by equation (6).
The next proposition shows that
$L_{\operatorname {Shi}}$
is closed under taking suffixes.
Proposition 4.5 Let
$w\in L_{\operatorname {Shi}}$
, then
$sw \in L_{\operatorname {Shi}}$
for all
$s\in D_L(w)$
. Moreover, if
$g\sim _{\operatorname {Shi}} sw$
, then
$sg\sim _{\operatorname {Shi}} w$
.
Proof Let
$s\in D_L(w)$
. Let
$w'\in L_{\operatorname {Shi}}$
be such that
$w'\sim _{\operatorname {Shi}} sw$
. Then
$\Sigma (w')=\Sigma (sw)$
. By Lemma 3.4, since
$\ell (w)>\ell (sw)$
, we have
$ \Sigma (w) = \{\alpha _s\}\cup (\Sigma \cap s(\Sigma (sw)) = \{\alpha _s\}\cup (\Sigma \cap s(\Sigma (w')) =\Sigma (sw'). $
So
$w\sim _{\operatorname {Shi}} sw'$
by Proposition 4.2. But
$\ell (sw')=\ell (w')+1\leq \ell (sw)+1= \ell (w)$
implying
$sw'=w$
by minimality of w. Hence,
$sw\in L_{\operatorname {Shi}}$
. Now let
$g\sim _{\operatorname {Shi}} sw$
, then
$\Sigma (g)=\Sigma (sw)$
. Proceeding as above, we obtain
$ \Sigma (w)= \{\alpha _s\}\cup (\Sigma \cap s(\Sigma (sw)) = \{\alpha _s\}\cup (\Sigma \cap s(\Sigma (g))=\Sigma (sg), $
which means that
$sg\sim _{\operatorname {Shi}} w$
.
4.3 Theorem 1.1 implies Conjectures 1 and 2
To prove Theorem 1.1, we only have to prove the converse of Proposition 4.4, that is, to prove that
$L_{\operatorname {Shi}} \subseteq L$
. Furthermore, this statement is also enough to show both conjectures.
First, as announced in the introduction, Theorem 1.1 implies Conjecture 1.
Proposition 4.6 Assume that
$L_{\operatorname {Shi}} \subseteq L$
, then
$L=L_{\operatorname {Shi}}$
and any region
$\mathcal R$
of
${\operatorname {Shi}}(W,S)$
contains a unique low element, which is the unique element of minimal length in
$\mathcal R$
.
Proof The first statement follows from Proposition 4.4. Let
$\mathcal R$
be a Shi region. Since
$L_{\operatorname {Shi}} \subseteq L$
,
$\mathcal R$
contains a low element w, which is of minimal length. The fact that w is the unique low element in
$\mathcal R$
follows by definition and Proposition 4.2. Assume that
$g\in L$
is another low element in
$\mathcal R$
, then
$g\sim _{\operatorname {Shi}} w$
. Therefore,
$\Sigma (w)=\Sigma (g)$
and by definition of low elements:
$ N(g)=\operatorname {cone}_\Phi (\Sigma (g))= \operatorname {cone}_\Phi (\Sigma (w))=N(w). $
Therefore,
$g=w$
, since any element of W is uniquely determined by its inversion set.
Second, to prove Conjecture 2, it is enough to show that any Shi region contains a low element, which is implied by the statement
$L_{\operatorname {Shi}} \subseteq L$
.
Proposition 4.7 Assume that any Shi region contains a low element, then the map
$\sigma : L \to \Lambda $
is a bijection.
Proof We know already that
$\sigma $
is injective by [Reference Dyer and Hohlweg13, Proposition 3.26(ii)]. Let
${\Sigma (w)\in \Lambda }$
with
$w\in W$
. Since any Shi region contains a low element, there is
$u\in L$
such that
$u\sim _{\operatorname {Shi}} w$
. Therefore, by Proposition 4.2,
$\Sigma (u)=\Sigma (w)$
. The map
$\sigma $
is, therefore, surjective.
4.4 Shi’s admissible sign types and Shi regions
In order to show that
$L_{\operatorname {Shi}}\subseteq L$
and then to prove Theorem 1.1 (as explained in Section 4.3), we need now to survey Shi’s admissible sign types.
In [Reference Shi25], Shi uses the parametrization of the alcoves of
$(W,S)$
in order to describe the Shi regions of
${\operatorname {Shi}}(W,S)$
. We follow here Shi’s notations from [Reference Shi25]. Let
$\overline {\mathscr S}$
be the set of
$\Phi _0^+$
-tuples over the set
$\{-,0,+\}$
; its elements are called sign types. For
$w\in W$
, the function
$\zeta : W\to \overline {\mathscr S}$
is defined as follows:
$\zeta (w)=(X(w,\alpha ))_{\alpha \in \Phi _0^+}$
where
$$ \begin{align} X(w,\alpha) = \left\{ \begin{array}{cc} +, &\textrm{if } k(w,\alpha)>0,\\ 0, &\textrm{if } k(w,\alpha)=0,\\ -, &\textrm{if } k(w,\alpha)<0. \end{array} \right. \end{align} $$
Definition 4.8 The sign type
$X(\mathcal R)=(X(\mathcal R,\alpha ))_{\alpha \in \Phi _0^+}$
of a Shi region
$\mathcal R$
is
$X(\mathcal R)=\zeta (w)$
for some
$w\in W$
such that
$w\cdot A_\circ \subseteq \mathcal R$
. A sign type
$X=(X_\alpha )_{\alpha \in \Phi _0^+}$
is said to be admissible if X is in the image of
$\zeta $
. We denote by
$\mathcal {S}_{\Phi }$
the set of admissible sign types of
$\Phi $
.
The following theorem sums up results from [Reference Shi25, Theorem 2.1 and Section 6].
Theorem 4.9 (Shi, 1987)
Let W be an irreducible affine Weyl group.
-
(1) The rank
$2$
admissible sign types are precisely those in Figures 3, 4, and 8.Figure 3:The Shi arrangement of type
$\tilde A_2$
. Each Shi region is labeled with its admissible sign type. The labels
$X(\mathcal R,\alpha )$
for a Shi region
$\mathcal R$
are indicated in each alcove with the parameterization given at the left-hand side of the figure. The signs colored in red indicate the descent roots (see Definition 5.1) of the corresponding Shi region.Figure 4:The Shi arrangement of type
$\tilde B_2$
. Each Shi region is labeled with its admissible sign type. The labels
$X(\mathcal R,\alpha )$
for a Shi region
$\mathcal R$
are indicated in each alcove with the parameterization given at the right-hand side of the figure. The signs colored in red indicate the descent roots (see Definition 5.1) of the corresponding Shi region. -
(2) A sign type
$X=(X_\alpha )_{\alpha \in \Phi _0^+}$
is admissible if and only if for any irreducible root subsystem
$\Psi $
of rank
$2$
in
$\Phi _0$
, the restriction
$(X_\alpha )_{\alpha \in \Psi ^+}$
of X to
$\Psi ^+$
is one of the rank
$2$
admissible sign types. -
(3) For
$u,v\in W$
, we have
$u\sim _{\operatorname {Shi}} v$
if and only if
$\zeta (u)=\zeta (v)$
.
Example 4.10 For
$(W,S)$
of type
$\tilde A$
, the irreducible root subsystems of rank
$2$
are easy to describe. Set
$V=\mathbb {R}^{n+1}$
with the usual orthonormal basis
$\{e_1,\dots , e_{n+1}\}$
. We abbreviate
$e_i - e_j$
by
$e_{ij}:=e_i - e_j$
. A way to describe the roots of
$A_n$
is by
$ \Phi =\{\pm (e_{ij}) ~| ~ 1\leq i < j \leq n+1 \} $
with simple system
$\Delta = \{ e_{i,i+1} ~| ~ 1\leq i < n +1\},$
and positive roots
$ \Phi ^+=\{e_{ij} ~| ~ 1\leq i < j \leq n+1 \}. $
Thus, the irreducible root subsystems of rank 2 of
$\Phi ^+$
are of the form
$ \{e_{ik}, e_{kj}, e_{ij}~|~1\leq i<k<j\leq n+1\}. $
A convenient way to write the Shi coordinates
$k(w,e_{ij})$
of an element
$w\in W$
(or the admissible signs
$X(\mathcal R,e_{ij})$
of a Shi region
$\mathcal R$
) is by placing them in a triangular shape as shown in Figure 5 for Type
$A_4$
.
Triangular presentation of the positive root system in Type
$A_4$
. The blue roots show a irreducible root subsystem of rank 2.
Then, using this presentation, it is easy to check if a sign is admissible using Theorem 4.9(1): we only have to look at all the subtriangles corresponding to irreducible root subsystems of rank
$2$
and check in Figure 3 if the sign type is admissible using Theorem 4.9. An example in rank
$5$
is given in Figure 6: on the left-hand side, the sign type is admissible, while on the right-hand side, the sign type is not admissible because the triplet in red does not belong to the rank
$2$
admissible sign types given in Figure 3.
On the left side, the sign type is admissible, while on the right side, it is not because the triplet in red does not belong to the rank
$2$
admissible sign types given in Figure 3.
Remark 4.11 The game of finding the admissible sign types in types
$B,C, D$
is more complicated because there is no triangular presentation of the positive roots. An alternative was recently provided by Charles [Reference Charles10].
Proposition 4.12 If
$\mathcal R$
is a Shi region, then the separation set of
$\mathcal R$
is
$$ \begin{align*}\Sigma(\mathcal R) = \{\alpha\in \Phi_0^+ \mid X(\mathcal R,\alpha)=-\}\sqcup\{\delta-\alpha\mid \alpha\in \Phi_0^+,\ X(\mathcal R,\alpha)=+\}. \end{align*} $$
Proof Let
$w\in W$
such that
$w\cdot A_\circ \subseteq \mathcal R$
. By Proposition 4.2, we have
$\Sigma (\mathcal R)=\Sigma (w)\subseteq N(w)$
. Let
$\alpha \in \Phi _0^+\subseteq \Sigma $
. By Lemma 2.2, we have
$\alpha \in N(w)\cap \Sigma =\Sigma (w)$
if and only if
$k(w,\alpha )<0$
. From equation (7), we get that
$\alpha \in \Sigma (w)$
if and only if
$X(\mathcal R,\alpha )=-$
. The same line of reasoning shows that
$\delta -\alpha \in \Sigma (w)$
if and only if
$X(\mathcal R,\alpha )=+$
, which concludes the proof.
Example 4.13 We continue Example 2.6. Consider the Shi region
$\mathcal R$
in Figure 3. This region is the union of all the alcoves in Figure 1 bounded by the small hyperplanes
$H_{\alpha _1}$
and
$H_{\delta -(\alpha _1+\alpha _2)}$
. This region contains, for instance, the alcove
$w\cdot A_\circ $
such that
$k(w,\alpha _1) = -1$
,
$k(w,\alpha _2)=2$
and
$k(w,\alpha _1+\alpha _2)=2$
. Therefore,
$X(\mathcal R,\alpha _1)$
is the sign of
$k(w,\alpha _1) = -1$
, that is,
$X(\mathcal R,\alpha _1)=-$
. Similarly, we obtain
$X(\mathcal R,\alpha _2)=+$
and
$X(\mathcal R,\alpha _1+\alpha _2)=+$
. Finally, we have
$ \Sigma (\mathcal R) = \{\alpha _1,\delta -\alpha _2, \delta -(\alpha _1+\alpha _2)\}. $
Example 4.14 (Type
$\tilde B_2$
)
We continue Example 2.7. Consider the Shi regions
$\mathcal R_1$
and
$\mathcal R_2$
in Figure 4. We have
$ \Sigma (\mathcal R_1) = \{\alpha _1,\delta -\alpha _2, \delta -(\alpha _1+\alpha _2), \delta -(2\alpha _1+\alpha _2)\} $
together with
$ \Sigma (\mathcal R_2) = \{\alpha _1,\delta -\alpha _2, \alpha _1+\alpha _2, \delta -(2\alpha _1+\alpha _2)\}. $
5 Descent roots and descent walls of Shi regions
We are now interested in descent wall of a Shi region, that is, those walls of a Shi region
$\mathcal R$
that separates
$\mathcal R$
from the Shi region
$A_\circ $
. We show at the end of this section the first of the two core results (Lemma 5.9) for proving
$L_{\operatorname {Shi}}\subseteq L$
and therefore Theorem 1.1.
5.1 Walls and descents of Shi regions
Following classical terminology, a wall of a Shi region
$\mathcal R$
is a hyperplane
$H\in {\operatorname {Shi}}(W,S)$
such that H supports a facet of
$\mathcal R$
.
Definition 5.1 (Descent wall and descent root)
Let
$\mathcal R$
be a Shi region. We say that a wall H of
$\mathcal R$
is a descent wall of
$\mathcal R$
if H separates
$\mathcal R$
from
$A_\circ $
. In other words, they represent the last hyperplane crossed in a gallery from
$A_\circ $
to
$\mathcal R$
. The set of descent roots of a Shi region
$\mathcal R$
is
$$ \begin{align*}\Sigma D(\mathcal R)=\{\alpha\in \Sigma(\mathcal R)\mid H_{\alpha} \textrm{ is a descent-wall of } \mathcal R\}. \end{align*} $$
Definition 5.2 Let
$\mathcal {R}$
be a Shi region, and let
$\alpha \in \Phi _0^+$
. We define
$$ \begin{align*}X_{\mathcal{R}}^\alpha :=(X^\alpha_\beta)_{\beta \in \Phi_0^+},\end{align*} $$
where
$X^\alpha _{\alpha }=0$
and
$X^\alpha _\beta =X(\mathcal R,\beta )$
for all
$\beta \in \Phi _0^+\setminus \{\alpha \}$
. If there is no possible confusion, we often omit the
$\mathcal {R}$
as subscript of
$X_{\mathcal {R}}^\alpha $
and we just write
$X^{\alpha }$
.
The following statement, which is an immediate consequence of the definitions, gives a characterization of the descent walls of a Shi region in terms of walls, sign types, and Shi coefficients: a wall
$H_{\theta }$
is a descent wall of a Shi region
$\mathcal {R}$
if and only if there is a Shi region
$\mathcal {R}'$
sharing a common wall with
$\mathcal {R}$
such that one sign of
$\mathcal R$
changes to a
$0$
in the sign type of
$\mathcal {R}'$
. We point out that the points (a) and (b) of Proposition 5.3 are of the utmost importance in the proof of out two key results: Lemma 5.9 and the descent-wall theorem in Section 7. Recall also that
$\mathcal {S}_{\Phi }$
is the set of Shi admissible sign types.
Proposition 5.3 Let
$\mathcal {R}$
be a Shi region with minimal element w. Let
$\alpha \in \Phi _0^+$
, and let
$\theta \in \{\alpha ,\delta -\alpha \}$
. The following statements are equivalent:
-
(1)
$H_\theta $
is a descent wall of
$\mathcal R$
. -
(2) There is a Shi region
$\mathcal R'$
such that
$\Sigma (\mathcal R')=\Sigma (\mathcal R)\setminus \{\theta \}$
. -
(3) The sign type
$X_{\mathcal {R}}^{\alpha }$
is admissible, i.e.,
$X_{\mathcal {R}}^{\alpha }\in \mathcal {S}_{\Phi }$
.
In particular, the set of descent roots of
$\mathcal R$
is
$$ \begin{align*} \Sigma D(\mathcal R)&=\{\alpha\in \Phi_0^+ \mid X(\mathcal R,\alpha)=-, \ X^{\alpha}_{\mathcal R} \in \mathcal{S}_{\Phi} \} \\ &\quad \sqcup\ \{\delta-\alpha\mid \alpha\in \Phi_0^+,\ X(\mathcal R,\alpha)=+, \ X^{\alpha}_{\mathcal R} \in \mathcal{S}_{\Phi}\}. \end{align*} $$
Remark 5.4 In Proposition 5.3, since
$X(\mathcal R)$
is admissible, it follows from Theorem 4.9 that the sign type
$X^\alpha _{\mathcal R}=(X^\alpha _\beta )_{\beta \in \Phi _0^+}$
is admissible if and only if
$(X^\alpha _\beta )_{\beta \in \Psi ^+}$
for any irreducible root subsystem
$\Psi \subseteq \Phi _0$
of rank
$2$
such that
$\alpha \in \Psi $
.
Example 5.5 (Type
$\tilde A_2$
– continuation of Example 4.4)
For each admissible sign type of a Shi region in Figure 3, we colored in red the signs that correspond to descent roots in
$\Sigma (\mathcal R)$
. For instance, for the Shi region
$\mathcal R$
in Figure 3, we have
$\Sigma D(\mathcal R) = \{\alpha _1, \delta -(\alpha _1+\alpha _2)\}$
, thanks to Proposition 5.3.
Example 5.6 (Type
$\tilde B_2$
– continuation of Example 4.14)
For each admissible sign type of a Shi region in Figure 4, we colored in red the signs that correspond to descent roots in
$\Sigma (\mathcal R)$
. For instance, for the Shi regions
$\mathcal R_1$
and
$\mathcal R_2$
in Figure 4, we have, thanks to Proposition 5.3,
$$ \begin{align*}\Sigma D(\mathcal R_1) = \{\alpha_2, \delta-(\alpha_1+\alpha_2)\}\text{~}\text{~and~}\text{~} \Sigma D(\mathcal R_2) = \{\alpha_1+\alpha_2, \delta-(2\alpha_1+\alpha_2)\}. \end{align*} $$
Proposition 5.7 Let
$w\in W$
, with Shi region
$\mathcal R$
, then
$w \in L_{\operatorname {Shi}}$
if and only if
$ND_R(w)\subseteq \Sigma D(\mathcal R)$
. In particular, the set of descent walls of a Shi region contains those of its minimal element.
Proof Assume first that
$w \in L_{\text {Shi}}$
. Let
$\alpha \in ND_R(w)$
. We show that
$\alpha \in \Sigma D(\mathcal {R})$
. Since
$\alpha \in ND_R(w)$
, there is by definition
$s \in D_R(w)$
such that
$\alpha = -w(\alpha _s)$
. Moreover, we have
$N(ws) = N(w) \setminus \{\alpha \}$
(see Section 3.4). Let
$\mathcal {R}'$
be the Shi region containing
$ws$
. By Proposition 4.2(2), we know that
$\Sigma (\mathcal {R}) = \Sigma (w)$
and
$\Sigma (\mathcal {R}') = \Sigma (ws)$
. Then it follows that
$$ \begin{align*} \Sigma(\mathcal{R})\setminus \{\alpha\} & = \Sigma(w) \setminus \{\alpha\} = (N(w) \cap \Sigma) \setminus \{\alpha\} = (N(w) \setminus \{\alpha\}) \cap \Sigma \\ &= N(ws) \cap \Sigma = \Sigma(ws) = \Sigma(\mathcal{R}'). \end{align*} $$
Since w is minimal in
$\mathcal {R}$
and since
$s \in D_R(w)$
, it follows by equation (6) that
$\mathcal {R'} \neq \mathcal {R}$
. So
$\alpha \in \Sigma $
and, therefore,
$\alpha $
is a descent root of
$\mathcal {R}$
by Proposition 5.3. Thus, we have
$ND_R(w) \subseteq \Sigma D(\mathcal {R})$
. The other direction of the statement follows immediately from equation (6) and the definitions of descent roots.
5.2 On the descent roots
$\delta -\alpha _s$
for
$s\in S_0$
We end this section with a result (Lemma 5.9), which we need in order to prove Theorem 1.1. We start by discussing the case of rank
$2$
affine Weyl groups.
Example 5.8 (On some descent roots in rank
$2$
)
Let
$\Phi _0$
be an irreducible finite crystallographic root system of rank
$2$
, that is,
$\Phi _0$
is of type
$A_2$
,
$B_2$
, or
$G_2$
. Set
$\Delta _0=\{\alpha _1,\alpha _2\}$
. We are particularly interested in the descent roots corresponding to
$\delta - \alpha _i$
, for
$i=1,2$
. Let
$\mathcal R$
be a Shi region of W. We know by Proposition 4.12 that
$\delta -\alpha _i \in \Sigma (\mathcal R)$
if and only if
$X(\mathcal R,\alpha _i)=+$
. By Proposition 5.3, we know that
$\delta -\alpha _i \in \Sigma D(\mathcal R)$
if and only if
$X(\mathcal R,\alpha _i)=+$
and the sign type
$X^{\alpha _i}$
, obtained from
$X(\mathcal R)$
by replacing
$X(\mathcal R,\alpha _i)=+$
by
$0$
, is admissible.
(Type
$A_2$
): for the Shi region
$\mathcal R$
in Figure 3,
$\delta -\alpha _2\notin \Sigma D(\mathcal R)$
since the sign type
$X^{\alpha _2}=(X^{\alpha _2}_{\alpha _1},X^{\alpha _2}_{\alpha _1+\alpha _2},X^{\alpha _2}_{\alpha _2})=(-,+,0)$
obtained from
$X(\mathcal R)$
by replacing
$X(\mathcal R,\alpha _2)=+$
by
$0$
, is not admissible; recall that all admissible sign types for type
$A_2$
are given in Figure 3.
More generally, let
$\mathcal R$
be any Shi region in Figure 3 with
$X(\mathcal R,\alpha _2)=+$
, i.e.,
$\delta -\alpha _2\in \Sigma (\mathcal R)$
by Proposition 4.12. It is easy to check – case by case – that
$$ \begin{align*}\delta-\alpha_2 \in \Sigma D(\mathcal R) \iff \big(X(\mathcal R,\alpha_1+\alpha_2) = + \implies X(\mathcal R,\alpha_1) \in \{0, +\}\big), \end{align*} $$
$$ \begin{align*}\delta-\alpha_1 \in \Sigma D(\mathcal R) \iff \big(X(\mathcal R,\alpha_1+\alpha_2) = + \implies X(\mathcal R,\alpha_2) \in \{0, +\}\big). \end{align*} $$
(Type
$B_2$
): for the Shi region
$\mathcal R_1$
in Figure 4,
$\delta -\alpha _2\notin \Sigma D(\mathcal R_1)$
since the sign type
$X^{\alpha _2}=(X^{\alpha _2}_{\alpha _1},X^{\alpha _2}_{2\alpha _1+\alpha _2},X^{\alpha _2}_{\alpha _1+\alpha _2},X^{\alpha _2}_{\alpha _2})=(0,+,+,-)$
obtained from
$X(\mathcal R_1)$
by replacing
$X(\mathcal R_1,\alpha _1)=+$
by
$0$
, is not admissible; recall that all admissible sign types for type
$B_2$
are given in Figure 4. Observe here, for example, that
$X(\mathcal R_1,\alpha _1+\alpha _2) = +$
, but
$X(\mathcal R_1,\alpha _2) \notin \{0, +\}$
.
Now, for the Shi region
$\mathcal R_2$
in Figure 4,
$\delta -\alpha _2\notin \Sigma D(\mathcal R_2)$
since the sign type
$$ \begin{align*}X^{\alpha_2}=(X^{\alpha_2}_{\alpha_1},X^{\alpha_2}_{2\alpha_1+\alpha_2},X^{\alpha_2}_{\alpha_1+\alpha_2},X^{\alpha_2}_{\alpha_2})=(0,+,-,-)\end{align*} $$
obtained from
$X(\mathcal R_2)$
by replacing
$X(\mathcal R_2,\alpha _1)=+$
by
$0$
is not admissible. Observe here that
$X(\mathcal R_2,\alpha _2+\alpha _1) = +$
, but
$X(\mathcal R_2,\alpha _2) \notin \{0, +\}$
.
More generally, let
$\mathcal R$
be any Shi region in Figure 4 with
$X(\mathcal R,\alpha _2)=+$
, i.e.,
$\delta -\alpha _2\in \Sigma (\mathcal R)$
. It is easy to check case by case that
$\delta -\alpha _1 \in \Sigma D(\mathcal R)$
if and only if for any
$\beta \in \Phi _0^+$
such that
$\alpha _1+\beta \in \Phi _0^+$
and
$X(\mathcal R,\beta ) = +$
we must have
$X(\mathcal R,\beta ) \in \{0, +\}$
. The same remains valid for
$\delta -\alpha _2$
. We have, therefore, in this case the following condition:
$\delta -\alpha _i \in \Sigma D(\mathcal R)$
if and only if
$$ \begin{align*}\big(\beta \in\Phi_0^+ \textrm{ with }\alpha_i+\beta\in \Phi_0^+ \textrm{ and }X(\mathcal R,\alpha_i+\beta) = + \implies X(\mathcal R,\beta) \in \{0, +\}. \end{align*} $$
(Type
$G_2$
): The above condition remains true in this type for any Shi region
$\mathcal R$
(Figure 7); we leave the reader check this assertion on Figure 8. For instance, for the region
$\mathcal R$
indicated in Figure 8,
$\delta -\alpha _1\notin \Sigma D(\mathcal R)$
. Indeed, the sign type
$X^{\alpha _1}=(X^{\alpha _1}_{\alpha _1},X^{\alpha _1}_{3\alpha _1+\alpha _2},X^{\alpha _1}_{2\alpha _1+\alpha _2},X^{\alpha _1}_{\alpha _1+\alpha _2},X^{\alpha _2}_{\alpha _2})=(0,+,+,+,-,-)$
obtained from
$X(\mathcal R)$
by replacing
$X(\mathcal R,\alpha _1)=+$
by
$0$
, is not admissible. Observe here that
$X(\mathcal R,2\alpha _1+\alpha _2) = +$
and
$2\alpha _1+\alpha _2 = \alpha _1+(\alpha _1 +\alpha _2)\in \Phi _0^+$
, but
$X(\mathcal R_1,\alpha _1+\alpha _2) =-\notin \{0, +\}$
.
The Coxeter arrangement of type
$\tilde G_2$
. Each alcove is labeled with its Shi parameterization. The labels
$k(w,\alpha )$
for
$\alpha \in \Phi _0^+$
are indicated in each alcove with the parameterization of the finite root system given at the top of the figure. The long roots are
$ \alpha _2,\ 3\alpha _1+2\alpha _2=s_1(\alpha _2), \textrm { and } 3\alpha _1+\alpha _2=s_2s_1(\alpha _2). $
The shaded region is the dominant region of
$\mathcal A(W,S)$
, which is also the fundamental chamber for the finite Weyl group
$W_0$
.
The Shi arrangement of type
$\tilde G_2$
. Each Shi region is labeled with its admissible sign type. The labels
$X(\mathcal R,\alpha )$
for a Shi region
$\mathcal R$
are indicated in each alcove with the parameterization given at the left-hand side of the figure. The shaded region is the dominant region
$C_\circ $
of
$\mathcal A(W,S)$
. The signs colored in red indicate the descent roots of the corresponding Shi region.
Lemma 5.9 Let
$\mathcal R$
be a Shi region and
$s\in S_0$
such that
$\delta -\alpha _s\in \Sigma (\mathcal R)$
, i.e.,
$X(\mathcal R,\alpha _s)=+$
. Then
$H_{\delta -\alpha _s}$
is a descent wall of
$\mathcal R$
if and only if for any irreducible root subsystem
$\Psi $
of rank
$2$
in
$\Phi _0$
such that
$\alpha _s\in \Psi $
the following condition is verified by
$\mathcal R$
:
-
(⋆) If
$\beta \in \Psi $
with
$\alpha _s+\beta \in \Psi $
and
$X(\mathcal R,\alpha _s+\beta ) = +$
, then
$X(\mathcal R,\beta ) \in \{0, +\}$
.
Proof First, the case of
$\Phi _0$
of rank
$2$
is done in Example 5.8. By Proposition 5.3, we know that
$H_{\delta -\alpha _s}$
is a descent wall of
$\mathcal R$
if and only if
$X^{\alpha _s}$
is admissible. By Theorem 4.9, we know that
$X^{\alpha _s}$
is admissible if and only if for any irreducible root subsystem
$\Psi $
of rank
$2$
in
$\Phi _0$
, the restriction
$(X_\gamma ^{\alpha _s} )_{\gamma \in \Psi ^+}$
of
$X^{\alpha _s}$
to
$\Psi ^+$
is admissible. Since
$X(\mathcal R)$
is admissible and that the only difference between
$X(\mathcal R)$
and
$X^{\alpha _s}$
is that
$X(\mathcal R,\alpha _s)\not = 0$
and
$X^{\alpha _s}_{\alpha _s}=0$
, one has to verify the admissibility of
$(X_\gamma ^{\alpha _s} )_{\gamma \in \Psi ^+}$
only for irreducible root subsystems
$\Psi $
of rank
$2$
in
$\Phi _0$
such that
$\alpha _s\in \Psi $
; it is well known that
$\alpha _s$
is then also a simple root for
$\Psi $
.
But we know that the lemma is true for
$\Psi $
of rank
$2$
, so
$X^{\alpha _s}$
is admissible if and only if
$(X(\mathcal R,\gamma ))_{\gamma \in \Psi ^+}$
satisfies Condition
$(\star )$
for any irreducible root subsystem
$\Psi $
rank
$2$
in
$\Phi _0$
such that
$\alpha _s\in \Psi $
.
6 Dominant Shi regions: the initial case for the proof of Theorem 1.1
Let
$(W,S)$
be an irreducible affine Coxeter system with underlying Weyl group
$W_0$
. In Section 4.3, we explain that it is enough to show that
$L_{\operatorname {Shi}}\subseteq L$
in order to prove Theorem 1.1. The proof is by induction with initial case the case of Shi regions contained in the dominant region. In this section, we explain that any dominant Shi region contains a low element, which follows mainly by the works of Cellini and Papi [Reference Cellini and Papi4–Reference Cellini and Papi6] on ad-nilpotent ideals and Shi’s article on
$\oplus $
sign-types [Reference Shi26].
6.1 Dominant Shi region
Recall from Section 2.1 that the dominant region of the affine Coxeter arrangement
$\mathcal A(W,S)$
is denoted by
$C_\circ $
. A Shi region
$\mathcal R$
is called dominant if
$\mathcal R\subseteq C_\circ $
. An obvious remark is the fact that a Shi region
$\mathcal R$
is dominant if and only if
$X(\mathcal R,\alpha )\in \{+,0\}$
for all
$\alpha \in \Phi _0^+$
.
The Weyl group
$W_0$
is a standard parabolic subgroup of W, since
$W_0$
is generated by
$S_0\subseteq S$
. The set of minimal length coset representatives is
$$ \begin{align*}{}^0W := \{v\in W\mid \ell(sv)>\ell(v), \ \forall s\in S_0\}. \end{align*} $$
The following well-known proposition states that an element is a minimal coset representative for
$W_0\backslash W$
if and only if its corresponding alcove is in the dominant region.
Proposition 6.1 Let
$w\in W$
. We have
$w \in {}^0W$
if and only if
$w\cdot A_\circ \subseteq C_\circ $
.
The inductive step of the proof of Theorem 1.1, which is dealt with in Section 7, is based on the decomposition in minimal length coset representative. Let
$w\in L_{\operatorname {Shi}}$
, then there is
$u\in W_0$
and
$v\in {}^0W$
such that
$w=uv$
and
$\ell (w)=\ell (u)+\ell (v)$
. Since
$L_{\operatorname {Shi}}$
is closed under taking suffixes (Proposition 4.5), then
$v\in L_{\operatorname {Shi}}$
. Therefore, if
$\ell (u)=0$
, i.e.,
$u=e$
, then
$w=v\cdot A_\circ \subseteq C_\circ $
, which is the initial case of our proof by induction.
6.2 Minimal elements in dominant Shi regions
Recall that
$\Psi \subseteq \Phi _0^+$
is an ideal of the root poset
$(\Phi _0^+,\preceq )$
if for all
$\alpha \in \Psi $
and
$\gamma \in \Phi _0^+$
such that
$\alpha \preceq \gamma $
we have
$\gamma \in \Psi $
. By definition of the root poset, this is equivalent to the condition:
$\Psi \subseteq \Phi _0^+$
is an ideal of
$(\Phi _0^+,\preceq )$
if for all
$\alpha \in \Psi $
and
$\beta \in \Phi _0^+$
such that
$\alpha +\beta \in \Phi _0^+$
, then
$\alpha +\beta \in \Psi $
.
Denote by
$\mathcal I(\Phi _0)$
the set of ideals of
$(\Phi _0^+,\preceq )$
and by
$L^0_{\operatorname {Shi}}$
the set of minimal elements in dominant Shi regions. In [Reference Shi26, Theorem 1.4], Shi gave a bijection between dominant Shi regions and
$\mathcal I(\Phi _0)$
. Hence, the number of Shi regions is equal to the number of ideals of the roots poset. We recall here this bijection following the lines of [Reference Cellini and Papi5, Section 4].
It is a general fact that ideals and antichains in
$(\Phi _0^+,\preceq )$
are in natural bijection, which is given by mapping an ideal
$\Psi $
to the set of its minimal elements
$\Psi _{\min }$
. Let
$\Psi \in \mathcal I(\Phi _0)$
, then the corresponding Shi region is
$$ \begin{align*} \mathcal R_\Psi&=&\{x\in C_\circ\mid \langle\beta,x \rangle>1 \textrm{ if } \alpha\preceq \beta \textrm{ for some }\alpha\in \Psi_{\min};\ \langle\beta,x \rangle<1 \textrm{ otherwise }\} .\nonumber \end{align*} $$
In terms of admissible sign types, the bijection is
$$ \begin{align*}\Psi\in \mathcal I(\Phi_0) \longmapsto X(\mathcal R_\Psi,\beta) = \left\{ \begin{array}{cc} +,&\textrm{if } \beta\in \Psi,\\ 0,&\textrm{otherwise.} \end{array} \right. \end{align*} $$
Therefore, by Proposition 4.12, we have
$\Sigma (\mathcal R_\Psi )=\{ \delta -\beta \mid \beta \in \Psi \}.$
Since
$\Psi _{\min }\setminus \{\alpha \}$
is again an antichain for any
$\alpha \in \Psi _{\min }$
, it corresponds to an ideal
$\Psi '$
with
$\Psi ^{\prime }_{\min } = \Psi _{\min }\setminus \{\alpha \}$
. We deduce (Proposition 6.2) that the descent walls of a dominant Shi region correspond, therefore, to the roots in its corresponding antichain.
Proposition 6.2 Let
$\Psi \in \mathcal I(\Phi _0)$
, then
$\Sigma (\mathcal R_\Psi )=\{ \delta -\beta \mid \beta \in \Psi \}$
and
$$ \begin{align*}\Sigma D(\mathcal R_\Psi)= \{\delta-\alpha\mid \alpha\in \Psi_{\inf}\}. \end{align*} $$
We now describe the element
$w_\Psi \in L^0_{\operatorname {Shi}}$
with Shi region
$\mathcal R_\Psi $
.
Theorem 6.3 (Cellini–Papi)
Let
$\Psi \subseteq \Phi _0^+$
be an ideal of the root poset
$(\Phi _0^+,\preceq )$
. There is a unique element
$w_\Psi \in W$
such that
$ N(w_\Psi ) = \operatorname {cone}_\Phi (\{\delta -\alpha \mid \alpha \in \Psi \}). $
Moreover,
$ND_R(w_\Psi )= \{\delta -\alpha \mid \alpha \in \Psi _{\inf }\}$
.
Proof As this result is not stated Per Se in [Reference Cellini and Papi4, Reference Cellini and Papi6], we give, here, a brief proof that points out the relevant results in these articles. In [Reference Cellini and Papi4, Section 2], the authors use the following notations: for
$k\in \mathbb N^*$
, write
$$ \begin{align*}\Psi^k:=\{\alpha_1+\dots +\alpha_k\in\Phi_0^+\mid \alpha_i\in \Psi\} \textrm{ and } \mathbb L_\Psi:=\bigcup_{k\in \mathbb N^*} (k\delta-\Psi^k). \end{align*} $$
Observe that
$\mathbb L_\Psi = \operatorname {cone}_\Phi (\{\delta -\alpha \mid \alpha \in \Psi \})$
. Now, by [Reference Cellini and Papi4, Theorem 2.6], there is a unique
$w_\Psi \in W$
such that
$\mathbb L_\Psi =N(w_\Psi )$
, that is,
$N(w_\Psi )=\operatorname {cone}_\Phi (\{\delta -\alpha \mid \alpha \in \Psi \})$
. The statement
$ND_R(w_\Psi )\supseteq \{\delta -\alpha \mid \alpha \in \Psi _{\inf }\}$
is precisely [Reference Cellini and Papi6, Proposition 3.4], while the reverse inclusion is precisely [Reference Cellini and Papi4, Proposition 2.12(ii)].
Cellini and Papi showed in [Reference Cellini and Papi5] that
$|L_{\operatorname {Shi}}^0|=|\mathcal I(\Phi _0)|={\operatorname {Cat}}(\Phi _0)$
. We obtain, therefore, the following corollary.
Corollary 6.4 Let
$\Psi \in \mathcal I(\Phi _0)$
.
-
(1) The element
$w_\Psi $
in Theorem 6.3 is a low element. -
(2) The Shi region containing
$w_\Psi \cdot A_\circ $
is
$\mathcal R_\Psi $
and is therefore dominant. -
(3) We have
$ND_R(w_\Psi )=\Sigma D(\mathcal R_\Psi )= \{\delta -\alpha \mid \alpha \in \Psi _{\inf }\}$
. -
(4) We have
$|L^0|=|L_{\operatorname {Shi}}^0|=|\mathcal I(\Phi _0)|= {\operatorname {Cat}}(\Phi _0)$
. -
(5) In particular,
$L^0=L_{\operatorname {Shi}}^0=\{w_\Psi \mid \Psi \in \mathcal I(\Phi _0)\}$
.
Proof (i) We have
$w_\Psi \in L$
by definition of low elements since
$\delta -\alpha \in \Sigma $
for all
$\alpha \in \Psi \subseteq \Phi _0^+$
.
(ii) By Theorem 6.3,
$\Sigma (w_\Psi )=N(w_\Psi )\cap \Sigma =\{ \delta -\beta \mid \beta \in \Psi \}$
. Therefore,
$\Sigma (\mathcal R_\Psi )=\Sigma (w_\Psi )$
by Proposition 6.2. Hence,
$w_\Psi \cdot A_\circ \subseteq \mathcal R_\Psi $
by Proposition 4.2(1). Now, by Proposition 6.1,
$w_\Psi \cdot A_\circ $
is dominant, which implies that
$\mathcal R_\Psi $
is also dominant.
(iii) follows immediately from Proposition 6.2 and Theorem 6.3. Let us prove (iv): From (i) and Theorem 6.3(1), there is an injective map from
$\mathcal I(\Phi _0)$
into
$L^0$
. By Proposition 4.4, we have
$|\mathcal I(\Phi _0)|\leq |L^0|\leq |L_{\operatorname {Shi}}^0|$
. We conclude by Theorem 6.3(2) and [Reference Cellini and Papi5]. Then, (v) is a consequence of (i) and (iv).
7 The descent-wall theorem and the inductive case of the proof of Theorem 1.1
In order to handle the inductive case of the proof of Theorem 1.1, we need to prove first that the descent walls of an element in
$L_{\operatorname {Shi}}$
are precisely the descent walls of its corresponding Shi region, which takes the most part of this section. Then we conclude the proof of Theorem 1.1 in Section 7.4.
7.1 The descent-wall theorem
Theorem 7.1 Let
$w\in L_{\operatorname {Shi}}$
with Shi region
$\mathcal R$
, then
$ND_R(w)= \Sigma D(\mathcal R)$
.
The proof of the so-called descent-wall theorem above is based on Proposition 3.7 and on its analog for descent roots of Shi regions.
Proposition 7.2 Let
$w\in L_{\operatorname {Shi}}$
with Shi region
$\mathcal R$
. Let
$s\in D_L(w)\cap S_0$
, and let
$\mathcal R_1$
be the Shi region associated with
$sw \in L_{\operatorname {Shi}}$
. Then
$\Sigma D(\mathcal R_1)=s(\Sigma D(\mathcal R)\setminus \{\alpha _s\})$
.
Before proving Proposition 7.2, we prove the descent-wall theorem. The proof of Proposition 7.2 is given in Section 7.3.
Proof of Theorem 7.1
We decompose
$w=uv$
with
$u\in W_0$
and
$v \in {}^0W$
. We show by induction on
$\ell (u)$
that
$ND_R(w)= \Sigma D(\mathcal R)$
. Notice that
$ND_R(w)\subseteq \Sigma D(\mathcal R)$
by Proposition 5.7.
If
$\ell (u)=0$
, then
$\mathcal R$
is a dominant Shi region. Therefore,
$ND_R(w)= \Sigma D(\mathcal R)$
by Proposition 6.4(iii). Assume now that
$\ell (u)>0$
. Then there is
$s\in D_L(u)\subseteq S_0$
. Therefore,
$\alpha _s\in \Sigma (\mathcal R)=\Sigma (w)$
, but
$\delta -\alpha _s\notin \Sigma (\mathcal R)=\Sigma (w)$
. Set
$w'=(su)v$
and
$\mathcal R'$
its Shi region. By Proposition 4.5, we know that
$w'\in L_{\operatorname {Shi}}$
and
$\ell (w')=\ell (w)-1$
. By induction, we have
$ND_R(w')= \Sigma D(\mathcal R')$
. By Theorem 7.2 and Proposition 3.7, we obtain that
$ \Sigma D(\mathcal R)=\{\alpha _s\}\sqcup s(\Sigma D (\mathcal R'))=\{\alpha _s\}\sqcup s(ND_R(w')) = ND_R(w). $
7.2 Suffix decomposition of some Shi regions
By Proposition 4.2, we know that, for a Shi region
$\mathcal R$
such that
$\alpha _s\in \Sigma (\mathcal R)$
, there is a Shi region
$\mathcal R_1$
such that
$s\cdot \mathcal R_1\subseteq \mathcal R$
; or equivalently
$ \mathcal R_1\subseteq s\cdot \mathcal R$
. In order to prove Proposition 7.2, we need first to describe precisely, in terms of sign types, the image
$s\cdot \mathcal R$
of Shi region
$\mathcal R$
with
$s\in S_0$
.
Proposition 7.3 Let
$w\in L_{\operatorname {Shi}}$
with Shi region
$\mathcal {R}$
, that is,
$w\in \widetilde {\mathcal R}$
. Let
$s \in S_0$
such that
$X(\mathcal {R},\alpha _s) = -$
; so
$\alpha _s\in \Sigma (\mathcal R)$
and
$s\in D_L(w)$
. Denote
$\mathcal R_1$
the Shi region of
$sw\in L_{\operatorname {Shi}}$
, that is,
$sw\in \widetilde {\mathcal R_1}$
. Consider the two following subsets of W:
$$ \begin{align*}\widetilde{\mathcal{R}_0^s} = \{x \in s \widetilde{\mathcal{R}}~|~k(x,\alpha_s) =0\}\ \textrm{ and } \widetilde{\mathcal{R}_+^s }= \{x \in s \widetilde{ \mathcal{R}}~|~k(x,\alpha_s)>0\}. \end{align*} $$
-
(1) We have
$s\widetilde {\mathcal {R}} = \widetilde {\mathcal {R}_0^s} \sqcup \widetilde {\mathcal {R}_+^s}$
. -
(2) If
$k(w,\alpha _s) = -1$
, then
$\widetilde {\mathcal {R}_0^s} \neq \emptyset $
is an equivalence class for
$\sim _{\operatorname {Shi}}$
that corresponds to the Shi region
$\mathcal {R}_0^s =\mathcal R_1$
. Moreover,
$X(\mathcal {R}_1,\alpha _s) = 0$
and
$\widetilde {\mathcal {R}_+^s}$
corresponds to a Shi region
$\mathcal {R}_2$
if it is nonempty. -
(3) If
$k(w,\alpha _s) < -1$
, then
$\widetilde {\mathcal {R}_0^s} = \emptyset $
and
$\widetilde {\mathcal {R}_+^s} \neq \emptyset $
is an equivalence class for
$\sim _{\operatorname {Shi}}$
that corresponds to the Shi region
$\mathcal {R}_+^s=\mathcal R_1.$
Moreover,
$\mathcal R_1=s\cdot \mathcal R$
and
$X(\mathcal {R}_1,\alpha _s) = +$
. -
(4) The signs of the elements of
$s\widetilde {\mathcal {R}}$
can only differ over the position
$\alpha _s$
. In particular,
$X(\mathcal R_1,\gamma )=X(\mathcal R, s(\gamma ))$
for all
$\gamma \in \Phi _0^+\setminus \{\alpha _s\}$
.
See Table 1 or Table 2 for examples in rank
$2$
.
Type
$A_2$
.
Type
$B_2$
.
Proof First, observe that
$\alpha _s\in \Sigma (w)=\Sigma (\mathcal R)$
by Propositions 4.12 and 4.2. In particular,
$s\in D_L(w)$
and
$sw\in L_{\operatorname {Shi}}$
by Proposition 4.5.
(1). Let
$u \in \widetilde {\mathcal {R}}$
, then by Lemma 2.2 and Proposition 2.3, we obtain
$$ \begin{align} k(su,\alpha_s) = k(u,s(\alpha_s)) + k(s,\alpha_s) = k(u,-\alpha_s) - 1 = -(k(u,\alpha_s) +1). \end{align} $$
Moreover, since
$X(\mathcal {R},\alpha _s) = -$
, we have
$k(u,\alpha _s) \leq -1$
. If
$ k(u,\alpha _s) = -1$
, then
$k(su,\alpha _s) =0$
. Now, if
$ k(u,\alpha _s) < -1$
, then
$k(su,\alpha _s)>0$
. Therefore,
$su \in \widetilde {\mathcal {R}_0^s} \sqcup \widetilde {\mathcal {R}_+^s}$
, which implies that
$s \widetilde { \mathcal {R} }\subseteq \widetilde {\mathcal {R}_0^s} \sqcup \widetilde {\mathcal {R}_+^s}$
. The converse inclusion is obvious.
(4). Let
$\gamma \in \Phi _0^+ \setminus \{\alpha _s\}$
and
$u \in \widetilde {\mathcal {R}}$
. By Proposition 2.3, we have
$k(su,\gamma ) = k(u, s(\gamma ))+k(s,\gamma )$
. Since
$\gamma \in \Phi _0^+ \setminus \{\alpha _s\}$
, we have
$s(\gamma ) \in \Phi _0^+\setminus \{\alpha _s\}$
. Therefore, by Lemma 2.2, we have
$k(s,\gamma )=0$
. It follows that
$k(su,\gamma ) = k(u, s(\gamma ))$
. Therefore, by equation (7), we obtain
$X(su,\gamma )=X(u,s(\gamma ))=X(\mathcal R,s(\gamma ))$
, that is, the signs of the elements in
$s\widetilde {\mathcal R}$
can only differ over
$\alpha _s$
. In particular, for
$w\in L_{\operatorname {Shi}}$
and its suffix
$sw\in L_{\operatorname {Shi}}$
, we obtain
$$ \begin{align*}X(\mathcal R_1,\gamma)=X(sw,\gamma)=X(w,s(\gamma))=X(\mathcal R,s(\gamma)), \end{align*} $$
for all
$\gamma \in \Phi _0^+ \setminus \{\alpha _s\}$
.
(2) and (3). By the same arguments as in the proof of (1) applied to
$w\in \widetilde {\mathcal R}$
, we obtain two cases:
$ k(w,\alpha _s) = -1$
and
$ k(w,\alpha _s) < -1$
. The details are similar to the cases (1) and (4) above and are left to the reader.
7.3 Proof of Proposition 7.2
Let
$w\in L_{\operatorname {Shi}}$
with Shi region
$\mathcal R$
. Let
$s\in D_L(w)$
, and let
$\mathcal R_1$
be the Shi region associated with
$sw\in L_{\operatorname {Shi}}$
. Notice that
$X(\mathcal R,\alpha _s)=-$
.
Thanks to Proposition 7.3, we have two cases to consider: either
$s\cdot \mathcal R_1=\mathcal R$
or there is another Shi region
$\mathcal R_2$
distinct from
$\mathcal R_1$
such that
$s\cdot \mathcal R_2\subseteq \mathcal R$
.
The case where
$s\cdot \mathcal R_1=\mathcal R$
is settled by the following lemma.
Lemma 7.4 Let
$\mathcal R$
be a Shi region and
$s\in S_0$
such that
$X(\mathcal R,\alpha _s)=-$
. If a Shi region
$\mathcal R_1$
is such that
$s\cdot \mathcal R_1=\mathcal R$
, then
$\Sigma D(\mathcal R_1)=s(\Sigma D(\mathcal R)\setminus \{\alpha _s\})$
.
Proof Since
$s\cdot \mathcal R_1=\mathcal R$
, the walls of
$\mathcal R_1$
are the image of the walls of
$\mathcal R$
under s. So the corollary follows from the following observations: (1) a wall H separates
$\mathcal R_1$
from
$A_\circ $
if and only if
$s\cdot H$
separates
$s\cdot \mathcal R_1=\mathcal R$
from
$s\cdot A_\circ $
; (2) the only wall separating
$s\cdot A_\circ $
from
$A_\circ $
is
$H_{\alpha _s}$
and so
$A_\circ $
and
$s\cdot A_\circ $
are on the same side of any other hyperplane in the Coxeter arrangement.
Consider now the case for which there is another Shi region
$\mathcal R_2$
distinct from
$\mathcal R_1$
such that
$s\cdot \mathcal R_2\subseteq \mathcal R$
. By Proposition 7.3(1) and (4), we have
$$ \begin{align*}X(\mathcal R_1,\alpha_s)=0\textrm{ and } X(\mathcal R_1,\gamma)=X(\mathcal R, s(\gamma))~\textrm{for all }\gamma\in \Phi_0^+\setminus\{\alpha_s\}. \end{align*} $$
In particular,
$\alpha _s\in \Sigma D(\mathcal R)$
but
$\alpha _s\notin \Sigma D(\mathcal R_1)$
.
For
$\alpha \in \Phi _0^+\setminus \{\alpha _s\}$
, we consider the following sign types:
-
•
$X^{s(\alpha )}=(X^{s(\alpha )}_\gamma )_{\gamma \in \Phi _0^+}$
is defined by
$X^{s(\alpha )}_{s(\alpha )} = 0$
and
$X^{s(\alpha )}_\gamma = X(\mathcal R,\gamma )$
for
$\gamma \in \Phi _0^+\setminus \{s(\alpha )\}$
. -
•
$Y^\alpha =(Y^\alpha _\gamma )_{\gamma \in \Phi _0^+}$
is defined by
$Y^\alpha _\alpha = 0$
and
$Y^\alpha _\gamma = X(\mathcal R_1,\gamma )$
for
$\gamma \in \Phi _0^+\setminus \{\alpha \}$
.
Thanks to Proposition 5.3, we know that
$X^\alpha $
is admissible if and only if either
$\alpha $
or
$\delta -\alpha $
is a descent root of
$\mathcal R$
. Similarly,
$Y^\alpha $
is admissible if and only if either
$\alpha $
or
$\delta -\alpha $
is a descent root of
$\mathcal R_1$
.
Lemma 7.5 Assume that, for any
$\alpha \in \Phi _0^+\setminus \{\alpha _s\}$
, we have
$Y^\alpha $
is admissible if and only if
$X^{s(\alpha )}$
is admissible. Then
$\Sigma D(\mathcal R_1)=s(\Sigma D(\mathcal R)\setminus \{\alpha _s\})$
.
Proof Let
$\alpha \in \Phi _0^+\setminus \{\alpha _s\}$
. Notice that
$s(\alpha ) \in \Phi _0^+\setminus \{\alpha _s\}$
.
We first show that
$\Sigma D(\mathcal R_1)\subseteq s(\Sigma D(\mathcal R)\setminus \{\alpha _s\})$
. Assume first that
$\alpha \in \Sigma D(\mathcal R_1)$
, i.e.,
$X(\mathcal R_1,\alpha )= -$
, by Proposition 5.3. Then
$Y^{\alpha }$
is admissible, so is
$X^{s(\alpha )}$
by assumption. Therefore, either
$s(\alpha )$
or
$\delta -s(\alpha )$
is in
$\Sigma D(\mathcal R)\setminus \{\alpha _s\}$
. Since
$X(\mathcal R,s(\alpha ))=X(\mathcal R_1,\alpha )= -$
, we have
$s(\alpha )\in \Sigma D(\mathcal R)\setminus \{\alpha _s\}$
, by Proposition 5.3 again. Therefore,
$\alpha \in s(\Sigma D(\mathcal R)\setminus \{\alpha _s\})$
. If
$\delta -\alpha \in \Sigma D(\mathcal R_1)$
, then by similar arguments we show that
$\delta -\alpha \in s(\Sigma D(\mathcal R)\setminus \{\alpha _s\})$
. Hence,
$\Sigma D(\mathcal R_1)\subseteq s(\Sigma D(\mathcal R)\setminus \{\alpha _s\})$
.
Assume now that
$\alpha \in s(\Sigma D(\mathcal R)\setminus \{\alpha _s\})$
. Therefore,
$s(\alpha ) \in \Sigma D(\mathcal R)\setminus \{\alpha _s\}$
and
$-=X(\mathcal R,s(\alpha ))=X(\mathcal R_1,\alpha )$
. Hence,
$X^{s(\alpha )}$
is admissible, so is
$Y^\alpha $
by assumption. Then
$\alpha \in \Sigma D(\mathcal R_1)$
. The case of
$\delta -\alpha \in s(\Sigma D(\mathcal R)\setminus \{\alpha _s\})$
is similar. Finally, we get
$\Sigma D(\mathcal R_1)=s(\Sigma D(\mathcal R)\setminus \{\alpha _s\})$
.
Thanks to Lemma 7.5, we only need to show now that, for any
$\alpha \in \Phi _0^+\setminus \{\alpha _s\}$
, we have
$Y^\alpha $
is admissible if and only if
$X^{s(\alpha )}$
is admissible.
Lemma 7.6 Let
$\alpha \in \Phi _0^+\setminus \{\alpha _s\}$
. The following statements are equivalent.
-
(1)
$Y^\alpha $
is admissible if and only if
$X^{s(\alpha )}$
is admissible. -
(2) For any rank
$2$
irreducible root subsystem
$\Psi \subseteq \Phi _0$
such that
$\alpha ,\alpha _s\in \Psi $
, we have
$(Y^\alpha _\gamma )_{\gamma \in \Psi ^+}$
is admissible if and only if
$(X^{s(\alpha )}_\gamma )_{\gamma \in \Psi ^+}$
is admissible.
Proof (i) implies (ii) follows directly from Theorem 4.9. Assume (ii) to be true for all
$\alpha \in \Phi _0^+$
. We describe first four cases that are valid for any rank 2 irreducible rank 2 subsystem
$\Psi $
and any
$\alpha \in \Phi _0^+\setminus \{\alpha _s\}$
. Let
$\Psi $
be a rank
$2$
irreducible root subsystem
$\Psi \subseteq \Phi _0$
and
$\alpha \in \Phi _0^+\setminus \{\alpha _s\}$
.
Case 1. If
$\alpha ,\alpha _s\in \Psi ^+$
, then
$(Y^\alpha _\gamma )_{\gamma \in \Psi ^+}$
is admissible if and only if
$(X^{s(\alpha )}_\gamma )_{\gamma \in \Psi ^+}$
is admissible by assumption.
Case 2. If
$\alpha _s\in \Psi $
and
$\alpha \notin \Psi $
. Then
$s(\Psi )=\Psi $
since
$\Psi $
a root system containing
$\alpha _s$
. Therefore,
$s(\alpha )\notin \Psi $
since
$s(\alpha ) =\alpha -2\langle \alpha ,\alpha _s^\vee \rangle \alpha _s$
. By definition, we have for all
$\gamma \in \Psi \subseteq \Phi _0^+\setminus \{\alpha ,s(\alpha )\}$
:
$$ \begin{align*}Y^\alpha_\gamma = X(\mathcal R_1,\gamma)\textrm{ and }X^{s(\alpha)}_\gamma = X(\mathcal R,\gamma). \end{align*} $$
Therefore, both
$(Y^\alpha _\gamma )_{\gamma \in \Psi ^+}$
and
$(X^{s(\alpha )}_\gamma )_{\gamma \in \Psi ^+}$
are admissible in this case.
Case 3. If
$\alpha \in \Psi $
and
$\alpha _s\notin \Psi $
, then
$s(\Psi ^+)\subseteq \Phi _0^+\setminus \{\alpha _s\}$
is the positive root system associated with root system
$s(\Psi )$
and
$s(\alpha )\in s(\Psi ^+)$
. By definition of
$Y^\alpha $
, we have
$Y^\alpha _\alpha =0$
, and for all
$\gamma \in \Psi ^+\setminus \{\alpha \}$
,
$$ \begin{align*}Y^\alpha_\gamma=X(\mathcal R_1,\gamma) = X(\mathcal R,s(\gamma)). \end{align*} $$
Notice that
$\gamma \in \Psi ^+\setminus \{\alpha \} $
if and only if
$s(\gamma )\in s(\Psi ^+)\setminus \{s(\alpha )\}$
. Now, by definition of
$X^{s(\alpha )}$
, we have
$X^{s(\alpha )}_{s(\alpha )}=0$
and for all
$\nu =s(\gamma )\in s(\Psi ^+)\setminus \{s(\alpha )\}$
, or in other words for all
$\gamma \in \Psi ^+\setminus \{\alpha \} $
, we have
$$ \begin{align*}X^{s(\alpha)}_\nu=X(\mathcal R,\nu)= X(\mathcal R,s(\gamma))=Y^\alpha_\gamma. \end{align*} $$
Therefore, the restriction of
$Y^\alpha $
to
$\Psi $
is admissible if and only if the restriction of
$X^{s(\alpha )}$
to
$s(\Psi )$
is admissible.
Case 4. If
$\alpha \notin \Psi $
and
$\alpha _s\notin \Psi $
, then
$s(\alpha )\notin s(\Psi )$
. By the same line of reasoning than in Case 3, we have, for all
$\gamma \in \Psi ^+\setminus $
,
$$ \begin{align*}Y^\alpha_\gamma=X(\mathcal R_1,\gamma) = X(\mathcal R,s(\gamma)). \end{align*} $$
Moreover, for all
$\nu =s(\gamma )\in s(\Psi ^+)$
, or in other words for all
$\gamma \in \Psi ^+$
, we have
$$ \begin{align*}X^{s(\alpha)}_\nu=X(\mathcal R,\nu)= X(\mathcal R,s(\gamma))=Y^\alpha_\gamma. \end{align*} $$
Therefore, the restriction of
$Y^\alpha $
to
$\Psi $
is admissible if and only if the restriction of
$X^{s(\alpha )}$
to
$s(\Psi )$
is admissible.
Conclusion. By Theorem 4.9, we need to prove that for any rank
$2$
irreducible root subsystem
$\Psi \subseteq \Phi _0$
, we have
$(Y^\alpha _\gamma )_{\gamma \in \Psi ^+}$
is admissible if and only if
$(X^{s(\alpha )}_\gamma )_{\gamma \in \Psi ^+}$
is admissible. Let
$\alpha \in \Phi _0^+\setminus \{\alpha _s\}$
.
Assume first that
$Y^\alpha $
is admissible. By Theorem 4.9, we have to show that the restriction of
$X^{s(\alpha )}$
to any rank
$2$
irreducible root subsystem
$\Psi $
is admissible. Let
$\Psi $
be a rank
$2$
irreducible root subsystem. If
$\alpha _s\in \Psi $
, then by Case 1 the restriction of
$X^{s(\alpha )}$
to
$\Psi $
is admissible since
$Y^\alpha $
is, and by Case 2, the restriction of
$X^{s(\alpha )}$
to
$\Psi $
is always admissible. Assume now that
$\alpha _s\notin \Psi $
.
If
$s(\alpha )\in \Psi $
, then we apply Case 3 with
$s(\Psi )$
and
$\alpha $
. Therefore, the restriction of
$Y^\alpha $
to
$s(\Psi )$
is admissible if and only if the restriction of
$X^{s(\alpha )}$
to
$s(s(\Psi ))=\Psi $
is admissible, which settles that case since
$Y^\alpha $
admissible. If
$s(\alpha )\notin \Psi $
, then we conclude by Case 4 with
$s(\Psi )$
and
$\alpha $
. So
$X^{s(\alpha )}$
is admissible.
The same line of reasoning using Cases 1–4 shows that if
$X^{s(\alpha )}$
then
$Y^\alpha $
is admissible.
Thanks to the above lemmas, Proposition 7.2 is a consequence of the following lemma.
Lemma 7.7 Let
$\Phi _0$
be an irreducible finite crystallographic root system of rank
$2$
, that is,
$\Phi _0$
is of type
$A_2$
,
$B_2$
, or
$G_2$
. Set
$\Delta _0=\{\alpha _1,\alpha _2\}$
and
$S_0=\{s_1,s_2\}$
. Let
$i\in \{1,2\}$
and
$\mathcal R$
be a Shi region such that
$X(\mathcal R,\alpha _i)=-$
. Let
$\mathcal R_1$
,
$\mathcal R_2$
be two distinct Shi regions such that
$s(\mathcal R_1), s(\mathcal R_2)\subseteq \mathcal R$
. Assume that
$X(\mathcal R_1,\alpha _i)=0$
. Let
$\alpha \in \Phi _0^+\setminus \{\alpha _i\}$
, then
$Y^{\alpha }$
is admissible if and only if
$X^{s_i(\alpha )}$
is admissible.
Proof We have
$X(\mathcal R_1,\alpha _i)=0$
,
$X(\mathcal R_2,\alpha _i)=+$
, and
$X(\mathcal R_j,\gamma )=X(\mathcal R, s_i(\gamma ))$
for all
$j=1,2$
and
$\gamma \in \Phi _0^+\setminus \{\alpha _i\}$
by Proposition 7.3(1). We prove this lemma in type
$A_2$
and
$B_2$
and leave the case
$G_2$
to the reader. The proof works as follows. Let
$\alpha \in \Phi _0^+\setminus \{\alpha _i\}$
. We know, thanks to Proposition 5.3, how to identify the descent roots of a Shi region within its sign type (see Examples 5.5 and 5.6). More precisely, in Figures 3 and 4, we have:
-
• a sign
$X(\mathcal R_1,\alpha )$
is colored in red if and only if
$Y^\alpha $
is admissible; -
• equivalently, a sign
$X(\mathcal R,s_i(\alpha ))$
is colored in red if and only if
$X^{s_i(\alpha )}$
is admissible.
So in order to prove our claim and since
$X(\mathcal R_j,\gamma )=X(\mathcal R, s_i(\gamma ))$
for all
$j=1,2$
and
$\gamma \in \Phi _0^+\setminus \{\alpha _i\}$
, it is enough in each case to:
-
(1) list the Shi regions
$\mathcal R$
such that
$s_i\cdot \mathcal R_1, s_i\cdot \mathcal R_2\subseteq \mathcal R$
for two distinct Shi regions
$\mathcal R_1$
,
$\mathcal R_2$
; -
(2) check that, for all
$\beta \in \Phi _0^+\setminus \{\alpha _i\}$
, the sign
$X(\mathcal R_1,\beta )$
is colored in red if and only if
$X(\mathcal R,s_i(\beta ))$
is also colored in red.
The result of this case-by-case analysis, which confirms our statement, is provided in Tables 1 and 2. Recall that
$s_i(\Phi _0^+\setminus \{\alpha _i\})=\Phi _0^+\setminus \{\alpha _i\}$
. These tables are obtained with the help of Figures 3 and 4, with the same notations.
7.4 Conclusion of the proof of Theorem 1.1
In Section 4.3, we explain that it is enough to show that
$L_{\operatorname {Shi}}\subseteq L$
in order to prove Theorem 1.1. We already know that this statement is true in the dominant region:
$L^0_{\operatorname {Shi}}= L^0$
by Corollary 6.4.
We first state the following corollary of the descent-wall theorem: if a hyperplane in
${\operatorname {Shi}}(W,S)$
is parallel to a hyperplane
$H_\alpha $
with
$\alpha \in \Delta _0$
and is also in the basis of the inversion set
$N^1(w)$
for
$w\in L_{\operatorname {Shi}}$
, then this hyperplane must be a descent wall of the alcove of w.
Corollary 7.8 (Of the descent-wall theorem)
Let
$w\in L_{\operatorname {Shi}}$
and
$s\in S_0$
such that
$\delta -\alpha _s\in N^1(w)$
. We have
$\delta -\alpha _s \in ND_R(w)$
.
Proof Let
$\mathcal R$
be the Shi region associated with w. By Theorem 7.1, we know that
$ND_R(w)= \Sigma D(\mathcal R)$
. Assume that
$\delta -\alpha _s \notin \Sigma D(\mathcal R)$
. Then, by Lemma 5.9, there is an irreducible root subsystem
$\Psi $
of rank
$2$
in
$\Phi _0$
such that
$\alpha _s\in \Psi $
and there is
$\beta \in \Psi $
with
$\alpha _s+\beta \in \Psi $
,
$X(\mathcal R,\alpha _s+\beta ) = +$
and
$X(\mathcal R,\beta )=-$
. So
$\delta -(\alpha _s+\beta ),\beta \in N(w)$
by equation (7). Therefore,
$ \delta -\alpha _s = \delta -(\alpha _s+\beta )+\beta , $
which contradicts the fact that
$\delta -\alpha _s\in N^1(w)$
spans a ray of
$\operatorname {cone}(N(w))$
. So
$\delta -\alpha _s \notin \Sigma D(\mathcal R)=ND_R(w)$
.
Proof of Theorem 1.1
Let
$w \in L_{\operatorname {Shi}}$
. We decompose
$w=uv$
with
$u\in W_0$
and
$v \in {}^0W$
. We show by induction on
$\ell (u)$
that
$w\in L$
.
Assume first that
$\ell (u)=0$
, then
$u=e$
and
$w=v\in L_{\operatorname {Shi}}^0=L^0$
by Proposition 6.4. Now assume that
$\ell (u)\geq 1$
. Then there is
$s\in D_L(u)\subseteq D_L(w)\cap S_0$
. So
$w'=u'v=sw\in L_{\operatorname {Shi}}$
by Proposition 4.5. By induction, we have
$sw\in L$
, since
$\ell (u')<\ell (u)$
.
We know that
$\alpha _s\notin N(sw)$
, so
$\alpha _s\notin N^1(sw)$
. Assume
$\delta -\alpha _s\in N^1(sw)$
, then, by Corollary 7.8,
$\delta -\alpha _s\in ND_R(sw)$
. Since
$ND_R(sw)\subseteq s(ND_R(w))$
by Proposition 3.7, we have
$\delta +\alpha _s=s(\delta -\alpha _s) \in ND_R(w)$
. Since
$w\in L_{\operatorname {Shi}}$
, we obtain by Proposition 5.7 that
$ND_R(w)\subseteq \Sigma $
, which contradicts
$\Sigma =\{\alpha ,\delta -\alpha \mid \alpha \in \Phi _0^+\}$
. Therefore,
$\delta -\alpha _s\notin N^1(sw)$
and
$ N^1(sw)\subseteq \Sigma \setminus \{\alpha _s,\delta -\alpha _s\}. $
By Proposition 3.1, we obtain, therefore, that
$s(N^1(sw))\subseteq \Sigma \setminus \{\alpha _s,\delta -\alpha _s\}$
. We conclude by [Reference Dyer and Hohlweg13, Theorem 4.10] that
$$ \begin{align*}N^1(w)\subseteq \{\alpha_s\} \cup s(N^1(sw))\subseteq \{\alpha_s\} \cup(\Sigma\setminus\{\alpha_s,\delta-\alpha_s\})\subseteq \Sigma. \end{align*} $$
In other words,
$w\in L$
.
Acknowledgments
The authors warmly thank Antoine Abram, Balthazar Charles, and Nathan Williams for numerous interesting discussions on the subject. The authors are also grateful to Nathan Williams for having pointed us the results in [Reference Cellini and Papi4, Reference Cellini and Papi5]. The second author thanks François Bergeron for the references on diagonal invariants. Finally, we are indebted to the anonymous referees: their thorough reading of the first version of this article and the comments and suggestions made helped us to considerably improve the content of this article.
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