# Zero fibers of quaternionic quotient singularities

Lien Cartaya  and  Stephen Griffeth

###### Abstract.

We propose a generalization of Haiman’s conjectures on the diagonal
coinvariant rings of real reflection groups to the context of irreducible
quaternionic reflection groups (also known as symplectic reflection groups).
For a reflection group $W$ acting on a quaternionic vector space $V$, by
regarding $V$ as a complex vector space we consider the scheme-theoretic fiber
over zero of the quotient map $\pi:V\to V/W$. For $W$ an irreducible
reflection group of (quaternionic) rank at least $6$, we show that the ring of
functions on this fiber admits a $(g+1)^{n}$-dimensional quotient arising from
an irreducible representation of a symplectic reflection algebra, where
$g=2N/n$ with $N$ the number of reflections in $W$ and
$n=\mathrm{dim}_{\mathbf{H}}(V)$, and we conjecture that this holds in
general. We observe that in fact the degree of the zero fiber is precisely
$g+1$ for the rank one groups (corresponding to the Kleinian singularities).

We thank Vic Reiner and Ulrich Thiel for comments on a preliminary version of
this article. The second author acknowledges the financial support of Fondecyt
Proyecto Regular 1231355.

##  1\. Introduction

###  1.1. Diagonal coinvariant rings

The main purpose of the present paper is to generalize (partly conjecturally)
the results of [Gri4] on the diagonal coinvariant rings of complex reflection
groups to the broader class of quaternionic reflection groups—also known as
_symplectic reflection groups_. That paper was, in turn, inspired by a number
of earlier works. These begin with Haiman’s foundational papers [Hai1],
[Hai2], and [Hai3], where he proved that the dimension of the diagonal
coinvariant ring of the symmetric group $S_{n}$ is $(n+1)^{n-1}$ and
conjectured the lower bound $(h+1)^{n}$ for the dimension of the diagonal
coinvariant ring of an irreducible real reflection group with Coxeter number
$h$ and rank $n$. Gordon proved Haiman’s conjecture in [Gor], and together
with the second author extended this result to complex reflection groups in
[GoGr]. After the appearance of [GoGr], there was no further progress on the
problem of understanding the diagonal coinvariant rings of general reflection
groups for many years, until we made two very surprising discoveries.

First, in [Gri4], one of us observed that the lower bound of [GoGr] may be
further improved to $(g+1)^{n}$, where for an irreducible complex reflection
group the _Coxeter number_ $h$ is

$h=\frac{N+N^{*}}{n},$

with $N$ is the number of reflections, $N^{*}$ the number of reflecting
hyperplanes, and $n$ the rank, and the number $g$ introduced in [Gri4] is

$g=2N/n.$

Evidently $g\geq h$ with equality precisely when every reflection in $W$ has
order $2$ (which occurs in particular when $W$ is real). And second, in [AjGr]
Ajila and the second author observed that a refinement of the strategies
employed in the previous papers explained at least some of the difference
between the dimension of the diagonal coinvariant ring and the lower bound
$(h+1)^{n}$. These two papers together showed that the representation theory
of non-commutative deformations of the quotient map $V\to V/W$ knows far more
about the structure of this map than we had previously presumed. The reasons
for the extent of this knowledge remain mysterious, and our observations here
only deepen the mystery.

In all the papers since [Gor], the strategy for obtaining these lower bounds
is to find an irreducible representation $L$ of a certain deformation of the
ring $\mathbf{C}[V]\rtimes W$ with the property that the space of
$W$-invariants in $L$ is one-dimensional (actually, beginning with Gordon it
has been traditional to look for representations in which the determinant
representation of $W$ occurs exactly once, but upon twisting by the inverse of
the determinant this amounts to the same thing). The dimension of $L$ then
gives the desired lower bound (which turns out, in all the cases in which we
can explicitly compute the correct answer, to be either correct or
mysteriously close to the truth). This is what we will do here, after
formalizing this strategy in 2.9. But the tactics in each case tend to be
quite subtle, due to the difficulty of calculating explicit character formulas
for finite-dimensional representations of the relevant deformations. In
particular, the principal tool applied in [Gor] was the Heckman-Opdam shift
functor; in [GoGr] it was Rouquier’s theorem [Rou] on the uniqueness of
highest weight covers; [Gri4] used Rouquier’s theorem together with the work
[BeCh] of Berest and Chalykh on quasi-invariants; and [AjGr] used the
representation-valued orthogonal polynomials introduced in [Gri2] together
with the results of [Gri1] and [FGM]. In this paper, our main technical inputs
are Crawley-Boevey and Holland’s study [CBH] of the representation theory of
what are nowadays called rank one symplectic reflection algebras. In the end,
that part of [CBH] that we use boils down to a detailed understanding of how
certain reflection functors relate representations of these algebras at
different values of the deformation parameter to one another, and so part of
what we do here might be thought of as directly analogous to Gordon’s reliance
on shift functors. In the remainder of this introduction we give precise
definitions and statements of our results.

###  1.2. The zero fiber ring

Let $F$ be a field, let $V$ be a finite-dimensional $F$-vector space, and let
$W\subseteq\mathrm{GL}(V)$ be a finite group. The ring of functions on the
scheme $V/W$ is, by definition, the ring $F[V]^{W}$ of $W$-invariants in the
ring $F[V]$ of polynomial functions on $V$, and the inclusion
$F[V]^{W}\hookrightarrow F[V]$ corresponds to the projection $\pi:V\to V/W$.
The most interesting fiber of this projection is the fiber $\pi^{-1}(0)$ over
$0$, which we call the _zero-fiber_. It is the subscheme of $V$ corresponding
to the quotient $F[\pi^{-1}(0)]=F[V]/I$ of $F[V]$, where $I$ is the ideal of
$F[V]$ generated by all $f-f(0)$ for $f\in F[V]^{W}$. When $W$ is a reflection
group acting in $V$, it has been traditional to call this ring the
_coinvariant ring_ of $W$, which conflicts with the use of the terminology
_coinvariant_ for the largest quotient object on which the group acts
trivially. Therefore we take this opportunity to refer to this ring in the
more general context of an arbitrary linear group as the _zero-fiber ring_.

The _degree_ $\delta=\delta(W)$ of the zero-fiber is the dimension of the
zero-fiber ring. It is a very interesting and subtle invariant of the linear
group $W$. When the base field is of characteristic zero, it is not hard to
see that we have the bounds

$d\leq\delta(W)\leq{n+d-1\choose d-1}\quad\hbox{where $d=|W|$ and
$n=\mathrm{dim}(V)$.}$

These bounds are moreover sharp in the sense that for given $d$ and $n$ there
is a group $W$ of order $d$ acting on a vector space of dimension $n$ that
achieves each of them. Namely, the lower bound is achieved precisely when $W$
is a reflection group, and the upper bound is achieved exactly when $W$ is a
group of scalar matrices. Thus there is nothing more to say without singling
out some class of groups with more structure. Here we propose that the one
interesting class of groups to study is the class of _quaternionic reflection
groups_. See, for instance, [McK3] (finite subgroups of the quaternions) and
[GNS] (finite subgroups of $\mathrm{SL}_{3}(\mathbf{C})$) for earlier studies
of this problem.

###  1.3. Quaternionic reflection groups

We consider the division rings
$\mathbf{R}\subseteq\mathbf{C}\subseteq\mathbf{H}$ of real numbers, complex
numbers, and quaternions. Let $D$ be one of $\mathbf{R}$, $\mathbf{C}$, or
$\mathbf{H}$. Given a finite-dimensional right $D$-vector space $V$ and a
finite subgroup $W\leq\mathrm{GL}(V)$, a _reflection_ in $W$ is an element
$r\in W$ such that $\mathrm{codim}_{D}(\mathrm{fix}_{V}(r))=1$. The group $W$
is a _reflection group_ if it is generated by the set $R$ of reflections it
contains. If $D=\mathbf{R}$ or $\mathbf{C}$ and $W$ is a reflection group
acting in the vector space $V$, then $W\otimes 1$ is a reflection group acting
in $V\otimes_{D}\mathbf{H}$. Thus each real or complex reflection group may be
regarded as a quaternionic reflection group via extension of scalars. We will
refer to quaternionic reflection groups (including those that arise via
extension of scalars from real or complex reflection groups) alternatively as
_symplectic reflection groups_ ; in the next paragraph we briefly explain the
reason for this terminology. For now we note that according to Cohen [Coh],
the irreducible quaternionic reflection groups may be usefully divided into
two classes, the primitive and imprimitive groups. There are two infinite
families of imprimitive groups, and a primitive group is either the symmetric
group in its irreducible reflection representation, a member of an infinite
family of quaternionic-primitive but complex-imprimitive groups that exists
only in rank $2$, or one of a finite list of exceptional groups. In this paper
we are concerned mainly with the infinite family of irreducible imprimitive
groups, which are certain normal subgroups of wreath products of the symmetric
group with finite subgroups of $\mathbf{H}^{\times}$.

Let $W$ be a quaternionic reflection group acting on $V$. There exists a
$W$-invariant positive-definite Hermitian form $(\cdot,\cdot):V\times
V\to\mathbf{H}$. Now $\mathbf{H}$ is the (right) $\mathbf{C}$-vector space
with basis $1,j$. The form $(\cdot,\cdot)$ may therefore be written uniquely
as

$(v_{1},v_{2})=\langle v_{1},v_{2}\rangle^{\prime}+j\langle
v_{1},v_{2}\rangle$

for certain $\mathbf{C}$-valued functions $\langle\cdot,\cdot\rangle$ and
$\langle\cdot,\cdot\rangle^{\prime}$. One checks that in fact
$\langle\cdot,\cdot\rangle$ is a $W$-invariant non-degenerate
$\mathbf{C}$-biliinear alternating form on the $\mathbf{C}$-vector space $V$,
and that $\langle\cdot,\cdot\rangle^{\prime}$ is a $W$-invariant positive
definite Hermitian form on the $\mathbf{C}$-vector space $V$, which are
moreover related by the formula

$\langle v_{1},v_{2}\rangle^{\prime}=\overline{\langle v_{1},v_{2}j\rangle}.$

Thus each quaternionic reflection group $W$ produces, upon restriction of
scalars to $\mathbf{C}$, a symplectic $\mathbf{C}$-vector space $V$ on which
$W$ acts by symplectic transformations, and such that the resulting group is
generated by elements with fixed-space of complex codimension $2$ (these are
sometimes known as _symplectic reflections_). It is this $\mathbf{C}$-vector
space $V$ together with the quotient map $\pi:V\to V/W$ that we will study.

###  1.4. Results and conjectures on the zero-fiber ring of a quaternionic
reflection group

######  Theorem 1.1.

Let $W$ be either an irreducible imprimitive quaternionic reflection group of
rank at least $3$ or the quaternionification of an irreducible complex
reflection group acting on an $n$-dimensional vector space $V$. Suppose $W$
contains $N$ reflections and let $g=2N/n$. Then there exists a quotient of the
zero-fiber ring of dimension $(g+1)^{n}$. Moreover, if $W$ is a quaternionic
group of rank $1$ then the zero fiber ring is of dimension exactly
$g+1=2|W|-1$.

We remark that in case $W$ is the quaternionification of a compex reflection
group, the theorem follows from the results of [Gri4]. Therefore we are
reduced to studying the case of imprimitive groups of rank at least $3$, which
we do using the classification and a case by case check. Of course a more
satisfying explanation (for instance generalizing what is done in [Gri4])
would be very desirable. We state our hope as a conjecture (which is a
quaternionic version of the conjecture of Haiman [Hai1] on the diagonal
coinvariant rings of real reflection groups; case-by-case inspection shows
that the number $g$ appearing here is an integer):

######  Conjecture 1.2.

Let $W$ be an irreducible quaternionic reflection group containing $N$
reflections and acting in the $n$-dimensional quaternionic vector space $V$.
Put $g=2N/n$. Then there is a $(g+1)^{n}$-dimensional quotient ring of the
zero-fiber ring of $W$.

The results of [EtMo] imply that the quotients we construct here arise from
irreducible representations that exist at generic points in a certain
hyperplane in the parameter space for the symplectic reflection algebra; the
same phenomenon occurs for the representations constructed in [Gri4]. So one
might optimistically conjecture that this happens for general quaternionic
reflection groups. One might additionally hope that these representations
produce an extension of some of the constructions from [ARR] to the case of
quaternionic reflection groups.

There is a second phenomena which we would like to understand: for $W$ a
complex reflection group, the space of semi-invariants for the determinant
representation in the non-commutative fiber constructed in [GoGr] is of
dimension equal to the $W$-Catalan number (these numbers were first introduced
in [BeRe], who conjectured the result later proved by [GoGr]; surprisingly,
this same dimension occurs also for the larger non-commutative fiber from
[Gri4]). In these cases, there is an additional internal grading on the
representations involved, and both the dimensions and their $q$-analogs factor
in a nice way. In 3.10, we observe a similar phenomenon for the (ungraded)
dimensions of certain spaces of semi-invariants. It would be interesting to
investigate if this phenomenon persists for other classes of quaternionic
reflection groups, and if conjectures along the lines of those from [Stu]
might be formulated.

##  2\. Preliminaries

###  2.1. Quaternions

We write $\mathbf{H}$ for the ring of quaternions, which is the
$\mathbf{R}$-algebra with basis $1,i,j,k$ and multiplication determined by the
rules

$i^{2}=j^{2}=k^{2}=-1=ijk.$

The _conjugate_ of a quaternion is written as

$\overline{a+bi+cj+dk}=a-bi-cj-dk\quad\hbox{for $a,b,c,d\in\mathbf{R}$,}$

and this defines an $\mathbf{R}$-linear anti-automorphism of $\mathbf{H}$. The
_norm_ $|z|$ of $z\in\mathbf{H}$ is then determined by
$|z|^{2}=\overline{z}z\in\mathbf{R}$. We write $\mathbf{H}^{n}$ for the right
$\mathbf{H}$-vector space of column vectors of length $n$ with entries in
$\mathbf{H}$, and thus identify $\mathrm{End}_{\mathbf{H}}(\mathbf{H}^{n})$
with the matrix ring $\mathrm{Mat}_{n\times n}(\mathbf{H})$ of $n$ by $n$
matrices with entries in $\mathbf{H}$, acting on $\mathbf{H}^{n}$ by left
multiplication. We will write $\mathrm{GL}_{n}(\mathbf{H})$ for the group of
invertible elements of this matrix ring, and define the _quaternionic unitary
group_

$\mathrm{U}_{n}(\mathbf{H})=\\{g\in\mathrm{GL}_{n}(\mathbf{H})\ |\
\overline{g}^{t}=g^{-1}\\},$

where for a matrix $g$ we write $\overline{g}$ for the matrix obtained by
conjugating all the entries of $g$ and $\overline{g}^{t}$ for its transpose.
In particular, the group $\mathrm{U}_{1}(\mathbf{H})\leq\mathbf{H}^{\times}$
is the group of quatenions of norm $1$, and contains each finite subgroup
$\Gamma$ of $\mathbf{H}^{\times}$. In general, the group
$\mathrm{U}_{n}(\mathbf{H})$ consists of the elements of
$\mathrm{GL}_{n}(\mathbf{H})$ preserving the positive definite Hermitian form

$(x,y)=\sum_{p=1}^{n}\overline{x_{p}}y_{p},\quad\hbox{where
$x,y\in\mathbf{H}^{n}$.}$

###  2.2. Quaternionic reflection groups

Each finite subgroup $W$ of $\mathrm{GL}_{n}(\mathbf{H})$ is conjugate to a
finite subgroup of the unitary group $\mathrm{U}_{n}(\mathbf{H})$. Given a
finite subgroup $W\leq\mathrm{U}_{n}(\mathbf{H})$, we write

$R=\\{r\in W\ |\ \mathrm{dim}_{\mathbf{H}}(\mathrm{fix}(r))=n-1\\}$

for the set of _reflections_ it contains. The group $W$ is a _quaternionic
reflection group_ if it is generated by $R$. For instance, each complex
reflection group produces a quaternionic reflection group via extension of
scalars. Given an irreducible quaternionic reflection group we define $N=|R|$
and $g=2N/n$, an analog of the Coxeter number of $W$ (for the
quaternionification of a complex reflection group with reflections of order
greater than two, $g$ is strictly larger than the usual Coxeter number $h$).
Cohen [Coh] has classified the irreducible quaternionic reflection groups, and
examining the classification shows that $g$ is always an integer. It would be
interesting to have a uniform explanation for this fact.

###  2.3. The finite subgroups of $\mathbf{H}^{\times}$ and their McKay graphs

Let $\Gamma$ be a finite subgroup of $\mathbf{H}^{\times}$. Identifying
$\mathbf{H}$ with $\mathbf{C}^{2}$, the group $\Gamma$ is identified with a
finite subgroup of $\mathrm{SL}_{2}(\mathbf{C})$. Its _McKay graph_ is the
graph whose vertex set $I$ is the set of isomorphism classes of irreducible
$\mathbf{C}\Gamma$-modules, with $d$ edges connecting $i$ and $j$ if $j$
appears with multiplicity $d$ in $i\otimes\mathbf{C}^{2}$ (equivalently, if
$i$ appears with multiplicity $d$ in $j\otimes\mathbf{C}^{2}$). McKay (see for
example [McK1] and the announcement [McK2]) observed that the graphs arising
in this fashion are precisely the affine Dynkin diagrams of type $ADE$: the
cyclic subgroups produce type $A$, the binary dihedral groups produce type
$D$, and the binary platonic groups produce type $E$. Writing
$K(\mathbf{C}\Gamma)$ for the Grothendieck group of the category of finite-
dimensional $\mathbf{C}\Gamma$ modules, we then have

$K(\mathbf{C}\Gamma)\cong\mathbf{Z}^{I},$

identifying $K(\mathbf{C}\Gamma)$ with the root lattice $\mathbf{Z}^{I}$
associated with the Dynkin diagram of $\Gamma$. Thus given a
$\mathbf{C}\Gamma$-module $M$, its _character_ is the class
$\mathrm{ch}(M)\in\mathbf{Z}_{\geq 0}^{I}\subseteq\mathbf{Z}^{I}$ of $M$ in
the Grothendieck group, which we may also visualize as a labeling of the
Dynkin diagram by non-negative integers. In particular, the character of the
regular representation of $\Gamma$ is the fundamental imaginary root $\delta$.

###  2.4. The root system

In accordance with our identification of $\mathbf{Z}^{I}$ with the root
lattice, we will sometimes write $\alpha_{i}$ for the simple root associated
with the $i$th vertex of $I$, which has coefficient $1$ on $i$ and $0$ on each
$j\neq i$. With this convention,

$\delta=\sum_{i\in I}n_{i}\alpha_{i},$

where $n_{i}=\mathrm{dim}_{\mathbf{C}}(i)$. The root system associated with
the Dynkin diagram may then be described as follows: the roots are those
elements of the form $n\delta+\alpha$, where $n\in\mathbf{Z}$ and $\alpha$ is
a root of the finite root system associated with the Dynkin diagram obtained
by deleting the vertex $0$ corresponding to the trivial representation of
$\Gamma$. The positive roots are those of the form $n\delta+\alpha$, where
$n\geq 0$, $\alpha$ is a finite root, and $\alpha$ is positive if $n=0$.

###  2.5. The groups $W_{n}(\Gamma,\Delta)$

We now fix a positive integer $n$, a finite subgroup $\Gamma$ of
$\mathbf{H}^{\times}$, and a normal subgroup $\Delta\leq\Gamma$ such that
$\Gamma/\Delta$ is abelian. We will define a certain subgroup
$W_{n}(\Gamma,\Delta)\leq\mathrm{U}_{n}(\mathbf{H})$ associated with these
data. Given $\gamma\in\Gamma$ and an integer $1\leq p\leq n$, we write
$\gamma^{(p)}$ for the diagonal matrix with $\gamma$ in position $(p,p)$ and
with $1$ in every other diagonal position. We identify the symmetric group
$S_{n}$ with the subgroup of $\mathrm{U}_{n}(\mathbf{H})$ consisting of
permutation matrices (those with exactly one $1$ in each row and each column,
and $0$’s in the other positions).

Next we put

$W_{n}(\Gamma)=\\{\gamma_{1}^{(1)}\gamma_{2}^{(2)}\cdots\gamma_{n}^{(n)}w\ |\
\gamma_{1},\gamma_{2},\dots,\gamma_{n}\in\Gamma,\ w\in
S_{n}\\}\leq\mathrm{U}_{n}(\mathbf{H})\\}.$

Thus for $n=1$ we have $W_{1}(\Gamma)=\Gamma$, and for $\Gamma=\\{1\\}$ we
have $W_{n}(\Gamma)=S_{n}$. In general, $W_{n}(\Gamma)$ consists of matrices
with exactly one non-zero entry in each row and each column, and so that the
non-zero entries are elements of $\Gamma$. Finally we define

$W_{n}(\Gamma,\Delta)=\\{\gamma_{1}^{(1)}\gamma_{2}^{(2)}\cdots\gamma_{n}^{(n)}w\in
W_{n}(\Gamma)\ |\ \gamma_{1}\gamma_{2}\cdots\gamma_{n}\in\Delta\\}.$

We note that in the definition of $W_{n}(\Gamma,\Delta)$, the order of the
factors in the product $\gamma_{1}\gamma_{2}\cdots\gamma_{n}$ is irrelevant
since $\Gamma/\Delta$ is abelian. Thus $W_{n}(\Gamma,\Delta)$ consists of $n$
by $n$ quaternionic matrices with exactly one non-zero entry in each row and
each column, and so that the non-zero entries are elements of $\Gamma$ with
product in $\Delta$. When $n=1$ we have
$W_{1}(\Gamma,\Delta)=W_{1}(\Delta)=\Delta$ so we only obtain a new group for
$n>1$.

###  2.6. Reflections and some numerology for $W_{n}(\Gamma,\Delta)$

Each group $W_{n}(\Gamma,\Delta)$ is generated by the set $R$ of reflections
it contains. For $W_{n}(\Gamma)$, this set $R$ consists of two sorts of
elements:

  * (a)

For each $\gamma\in\Gamma$ and each pair $1\leq p<q\leq n$ of integers, the
element

$\gamma^{(p)}(pq)(\gamma^{(p)})^{-1},$

where $(pq)$ is the transposition matrix interchanging the $p$th and $q$th
canonical basis vectors of $\mathbf{H}^{n}$ and leaving the others fixed, and

  * (b)

for each $\gamma\in\Gamma\setminus\\{1\\}$ and each integer $1\leq p\leq n$,
the element $\gamma^{(p)}$.

Therefore $W_{n}(\Gamma)$ contains

$N={n\choose 2}|\Gamma|+n(|\Gamma|-1)$

reflections. Of these, the reflections that belong to $W_{n}(\Gamma,\Delta)$
are all those of type (a), together with the reflections of type (b) such that
$\gamma\in\Delta$. Hence $W_{n}(\Gamma,\Delta)$ contains

(2.1) $N={n\choose 2}|\Gamma|+n(|\Delta|-1)$

reflections, and in particular we have

(2.2) $g=2N/n=(n-1)|\Gamma|+2(|\Delta|-1).$

###  2.7. Semi-direct product algebra

Let $F$ be a field and let $A$ be an $F$-algebra equipped with an action by
automorphisms of a group $G$. The _semi-direct product algebra_ (or _twisted
group ring_ or _smash product algebra_) $A\rtimes G$ is

$A\otimes_{F}FG\quad\text{with multiplication given by}\quad(a_{1}\otimes
g_{1})(a_{2}\otimes g_{2})=a_{1}g_{1}(a_{2})\otimes g_{1}g_{2}$

for $a_{1},a_{2}\in A$ and $g_{1},g_{2}\in G$. For instance, given a field
$F$, an $F$-vector space $V$, and a linear group $W\leq\mathrm{GL}(V)$, the
group $W$ acts by automorphisms on the tensor algebra $T(V)$ of $V$, as well
as on various related algebras such as the algebra $F[V]$, so we may form the
associated semi-direct product algebras.

###  2.8. Deformations

In this subsection and the next two, we will work with an arbitrary field $F$,
a finite-dimensional $F$-vector space $V$, and a finite subgroup
$W\leq\mathrm{GL}(V)$. Our applications will all involve $F=\mathbf{C}$, but
here we present the simple facts we will need in their natural generality.

Let $A$ be a (unital, associative) $F$-algebra. A (non-negative, increasing,
exhaustive) _filtration_ on $A$ is a sequence $(A^{\leq
d})_{d\in\mathbf{Z}_{\geq 0}}$ of subspaces of $A$ indexed by non-negative
integers $d\in\mathbf{Z}_{\geq 0}$ such that

$1\in A^{\leq 0},\quad A^{\leq d}\subseteq A^{\leq e}\quad\hbox{if $d\leq
e$,}\quad A^{\leq d}A^{\leq e}\subseteq A^{\leq d+e}\quad\text{and}\quad
A=\bigcup_{d=0}^{\infty}A^{\leq d}.$

The _associated graded algebra_ of $A$ with respect to this filtration is

$\mathrm{gr}(A)=\bigoplus_{d=0}^{\infty}A^{\leq d}/A^{\leq d-1},$

where $A^{\leq-1}=0$. The multiplication is induced by the multiplication of
$A$.

A _deformation_ of $F[V]\rtimes W$ is a filtered $F$-algebra $A$ equipped with
an isomorphism of graded $F$-algebras $\phi:F[V]\rtimes W\to\mathrm{gr}(A)$.
We note that upon restricting to the degree $0$ subalgebras, the map $\phi$
induces an isomorphism from $FW$ to $A^{\leq 0}$, so in particular $FW$ may be
identified with the subalgebra of degree $0$ elements of $A$. Assuming that
the order of $W$ is invertible in $F$, the _spherical subalgebra_ of a
deformation $A$ of $F[V]\rtimes W$ is the idempotent slice subalgebra $eAe$,
where

$e=\frac{1}{|W|}\sum_{w\in W}w$

is the symmetrizing idempotent. If we think of the inclusion $F[V]^{W}\cong
e(F[V]\rtimes W)e\subseteq F[V]\rtimes W$ as being a non-commutative version
of the quotient map $V\to V/W$, then the inclusion $eAe\subseteq A$ should
likewise be regarded as a non-commutative version of this quotient map. Taking
this philosophy seriously leads to the definitions in the next paragraph.

###  2.9. Non-commutative points and fibers

Assuming as above that the order $|W|$ of $W$ is invertible in $F$, a _non-
commutative point_ of $V/W$ is a one-dimensional representation of $eAe$ for
some deformation $A$ of $F[V]\rtimes W$. A _non-commutative fiber_ of $V\to
V/W$ is a representation $M$ of some deformation $A$ of $F[V]\rtimes W$ with
the properties

  * (a)

$\mathrm{dim}_{F}(M^{W})=1$, and

  * (b)

$M$ is generated by $M^{W}$ as an $A$-module.

Given a non-commutative fiber $M$ we define a filtration $M^{\leq d}$ on $M$
by

$M^{\leq d}=A^{\leq d}M^{W}.$

Each non-commutative point $N$ produces a non-commutative fiber
$M=Ae\otimes_{eAe}N$, and conversely each non-commutative fiber $M$ produces a
non-commutative point $N=eM\cong M^{W}$. It seems likely that the
classification problem for non-commutative points and fibers will be very
interesting and challenging (for instance, by analogy with the same problem
for finite $W$-algebras; see e.g. [Los]).

The next lemma explains the utility of non-commutative fibers in the context
of the zero-fiber ring.

######  Lemma 2.1.

Let $M$ be a non-commutative fiber. Then $F[V]^{W}_{>0}$ acts by zero on
$\mathrm{gr}(M)$, which is therefore a quotient of $F[\pi^{-1}(0)]$. In
particular, the degree of the zero fiber is at least the dimension of $M$.

###### Proof.

First we observe that, since $M^{W}$ generates $M$ as an $A$-module, the
filtration $M^{\leq d}$ is exhaustive and hence the dimension of
$\mathrm{gr}(M)$ is equal to the dimension of $M$. By construction,
$\mathrm{gr}(M)$ is generated as a $F[V]$-module by its degree $0$ subspace:
given $\overline{m}\in\mathrm{gr}_{d}(M)=M^{\leq d}/M^{\leq d-1}$ there are
$a\in A^{\leq d}$ and $m_{0}\in M^{W}$ with $m=am_{0}$, implying

$\overline{m}=\overline{a}m_{0}\in(F[V]\rtimes
W)\mathrm{gr}_{0}(M)=F[V]\mathrm{gr}_{0}(M).$

Next, observe that the unique occurrence of the trivial representation of $W$
in $\mathrm{gr}(M)$ lies in degree $0$. This implies that each positive degree
$W$-invariant polynomial acts by $0$ on the degree $0$ part of
$\mathrm{gr}(M)$ and hence on all of $\mathrm{gr}(M)$. Hence $\mathrm{gr}(M)$
is a quotient of the zero-fiber ring $F[\pi^{-1}(0)]$. ∎

###  2.10. Normal subgroups

Next suppose $W\lhd N\leq\mathrm{GL}(V)$ is a pair consisting of two finite
linear groups with $W$ normal in $N$; we have in mind here the pair
$W_{n}(\Gamma,\Delta)\lhd W_{n}(\Gamma)$. The group $N$ then stabilizes the
set of positive-degree $W$-invariants in $F[V]$ and hence acts by
automorphisms on the zero-fiber ring $F[\pi^{-1}(0)]$ for $W$. We have the
following analog of the previous lemma.

######  Lemma 2.2.

Let $A$ be a deformation of $F[V]\rtimes N$ and let $M$ be an $A$-module with
the properties

  * (a)

$\mathrm{dim}_{F}(M^{W})=1$, and

  * (b)

$M$ is generated by $M^{W}$ as an $A$-module.

Then $F[V]^{W}_{>0}$ acts by zero on $\mathrm{gr}(M)$, which is therefore a
quotient of $F[\pi^{-1}(0)]$. In particular, the degree of the zero fiber is
at least the dimension of $M$.

###### Proof.

The proof is the same, once we observe that

$(F[V]\rtimes N)\mathrm{gr}_{0}(M)=F[V]\mathrm{gr}_{0}(M),$

since $N$ acts by some linear character on $\mathrm{gr}_{0}(M)=M^{W}$. ∎

###  2.11. The symplectic reflection algebra of rank $1$

Here we define the deformation of $\mathbf{C}[x,y]\rtimes\Gamma$ introduced in
[CBH], also known as the _symplectic reflection algebra_ (as introduced in
more generality in [EtGi]) of $\Gamma$. Thus let
$\Gamma\subseteq\mathrm{SL}_{2}(\mathbf{C})$ be a finite subgroup, and choose
linear functions $x,y$ on $V$ with $\langle x,y\rangle=1$, where
$\langle\cdot,\cdot\rangle$ is the symplectic form on $V^{*}$ that is dual to
the standard symplectic form on $V$. We have $\mathbf{C}[V]=\mathbf{C}[x,y]$.
We write $I$ for an index set for the irreducible $\mathbf{C}\Gamma$-modules
(thus as above $I$ is the set of vertices of the Dynkin diagram of $\Gamma$),
and for $i\in I$ let $e_{i}\in Z(\mathbf{C}\Gamma)$ be the corresponding
central primitive idempotent. We let $c=(c_{i})_{i\in I}\in\mathbf{C}^{I}$ be
a collection of complex numbers indexed by $I$ and put

$H_{c}(\Gamma,V)=\mathbf{C}\langle x,y\rangle\rtimes\Gamma/(xy-yx=\sum_{i\in
I}c_{i}e_{i}),$

where $\mathbf{C}\langle x,y\rangle$ is the tensor algebra of $V^{*}$, on
which $\Gamma$ acts by automorphisms. The _PBW theorem_ (see e.g. [Gri1],
Theorem 2.1) for $H_{c}(\Gamma,V)$ states that the monomials
$x^{a}y^{b}\gamma$ for $a,b\in\mathbf{Z}_{\geq 0}$ and $\gamma\in\Gamma$ are a
$\mathbf{C}$-basis of $H_{c}(\Gamma,V)$. Equipping $H_{c}(\Gamma,V)$ with the
filtration for which $x$ and $y$ have degree $1$ and elements of $\Gamma$ have
degree $0$, this implies that $H_{c}(\Gamma,V)$ is a deformation of
$\mathbf{C}[x,y]\rtimes\Gamma$.

###  2.12. A deformation of $\mathbf{C}[V^{\times n}]\rtimes W_{n}(\Gamma)$

We have $V^{\times n}=\mathbf{H}^{n}$, which we regard as a
$\mathbf{C}$-vector space on which $W_{n}(\Gamma)$ acts $\mathbf{C}$-linearly
by restriction of scalars. Fixing a parameter $c$ as above, we form the tensor
product algebra $H_{c}(\Gamma,V)^{\otimes n}$. This algebra is equipped with
the permutation action of $S_{n}$ on the tensor factors, and we form the semi-
direct product algebra

$H_{c}(\Gamma,V)^{\otimes n}\rtimes S_{n}.$

The tensor product filtration on $H_{c}(\Gamma,V)^{\otimes n}$ extends to a
filtration on $H_{c}(\Gamma,V)^{\otimes n}\rtimes S_{n}$ by placing $S_{n}$ in
degree $0$, and with this filtration the algebra $H_{c}(\Gamma,V)^{\otimes
n}\rtimes S_{n}$ is a deformation of $\mathbf{C}[V^{\times n}]\rtimes
W_{n}(\Gamma)$. It is a special case of the symplectic reflection algebra of
type $W_{n}(\Gamma)$ (see definition 2.1 and the beginning of section 3 of
[EtMo]).

###  2.13. Representations

Given an $H_{c}(\Gamma,V)$-module $M$ and a $\mathbf{C}S_{n}$-module $N$, the
vector space $M^{\otimes n}\otimes N$ is a $H_{c}(\Gamma,V)^{\otimes n}\rtimes
S_{n}$-module.

######  Lemma 2.3.

If $M$ and $N$ are irreducible finite-dimensional modules for
$H_{c}(\Gamma,V)$ and $\mathbf{C}S_{n}$, respectively, then $M^{\otimes
n}\otimes N$ is an irreducible finite-dimensional $H_{c}(\Gamma,V)^{\otimes
n}\rtimes S_{n}$-module.

###### Proof.

The proof of this is very nearly standard, but we indicate it briefly here for
the reader’s convenience. It suffices to show that the canonical map

$H_{c}(\Gamma,V)^{\otimes n}\rtimes
S_{n}\longrightarrow\mathrm{End}_{\mathbf{C}}(M^{\otimes n}\otimes N)$

is surjective. But this follows from the surjectivity of the maps
$H_{c}(\Gamma,V)\to\mathrm{End}_{\mathbf{C}}(M)$,
$\mathbf{C}S_{n}\to\mathrm{End}_{\mathbf{C}}(N)$, and
$\mathrm{End}_{\mathbf{C}}(M)^{\otimes
n}\otimes\mathrm{End}_{\mathbf{C}}(N)\to\mathrm{End}_{\mathbf{C}}(M^{\otimes
n}\otimes N)$, upon noting that given $w\in S_{n}$ its action on $M^{\otimes
n}$ may be achieved by some element of $H_{c}(\Gamma,V)^{\otimes n}$. ∎

###  2.14. Finite-dimensional representations of $H_{c}(\Gamma,V)$.

Let $M$ be a finite-dimensional $H_{c}(\Gamma,V)$-module. As above, its
_character_ $\mathrm{ch}(M)$ the element of the Grothendieck ring
$K_{0}(\mathbf{C}\Gamma)\cong\mathbf{Z}^{I}$ represented by its class as a
$\mathbf{C}\Gamma$-module.

Writing $\alpha\cdot c=\sum_{i\in I}c_{i}d_{i}$ if $\alpha=\sum_{i\in
I}d_{i}\alpha_{i}$ for the usual dot product and with the notation above for
the root system $R$ associated with the Dynkin diagram of $\Gamma$, we define

$R_{c}=\\{\alpha\in R\ |\ \alpha\cdot c=0\\}$

and let $\Sigma_{c}$ be the set of minimal positive elements of $R_{c}$. Then
Corollary 3.5 and Theorem 7.4 of [CBH] imply:

######  Theorem 2.4.

The correspondence $M\mapsto\mathrm{ch}(M)$ defines a bijection from the set
of isomorphism classes of finite-dimensional simple $H_{c}(\Gamma,V)$-modules
to the set $\Sigma_{c}$.

As a corollary:

######  Corollary 2.5.

Let $\alpha$ be a positive real root. For generic elements $c$ of the
hyperplane defined by $c\cdot\alpha=0$, there is a unique simple finite-
dimensional $H_{c}(\Gamma,V)$-module $M$, and we have $\mathrm{ch}(M)=\alpha$.

The corollary provides a large supply of non-commutative fibers. The largest
character of such a non-commutative fiber is given by
$\mathrm{ch}(M)=\delta+\phi$, where $\delta$ is the fundamental imaginary root
and $\phi$ is the highest root of the finite root system associated with
$\Gamma$. The dimension of this $M$ is precisely

$\mathrm{dim}(M)=2|\Gamma|-1=g+1.$

This shows that the dimension of the zero-fiber for $\Gamma$ is at least
$g+1$. In the next section, we will observe that in fact this dimension is
equal to $g+1$, and also generalize the preceding construction to the groups
$W_{n}(\Gamma,\Delta)$.

##  3\. Proof of the main results

###  3.1. Outline

Here we prove Theorem 1.1. We will begin with the case of a finite subgroup
$\Gamma\leq\mathrm{SL}_{2}(\mathbf{C})$, proving that in this case the zero
fiber ring is of dimension precisely $g+1$ and hence that the non-commutative
fiber we constructed above is in fact a deformation of it. We then consider
the groups $W_{n}(\Gamma,\Delta)\lhd W_{n}(\Gamma)$.

###  3.2. Finite subgroups of $\mathrm{SL}_{2}$

Let $V=\mathbf{C}^{2}$ and let $\Gamma\subseteq\mathrm{SL}(V)$ be a finite
subgroup. The quotient space $V/\Gamma$ is known as a _Kleinian singularity_
(or alternatively, as a _du Val singularity_ or _simple surface singularity_).
The fibers of the quotient map are all reduced of cardinality $|\Gamma|$
except the zero-fiber. We will prove

######  Theorem 3.1.

The degree of the zero-fiber of the map $V\to V/\Gamma$ is $2|\Gamma|-1=g+1$.

The theorem is of course easy to deduce from the work of McKay [McK3] on the
graded character of the zero-fiber ring. We present a proof here for the
reader’s convenience; our interest in this particular formulation is the
existence of a a single irreducible non-commutative fiber with precisely the
dimension of the zero-fiber ring. Additionally, the proof we give provides an
explicit Groebner basis for the ideal generated by the positive degree
$\Gamma$-invariants.

The proof is case by case. In each case we will find a set $S$ of polynomials
in the ideal generated by the $\Gamma$-invariant polynomials of positive
degree such that the quotient of $\mathbf{C}[x,y]$ by the ideal generated by
the initial terms of the elements of $S$ is of dimension $2|\Gamma|-1$. This
gives the desired upper bound on the dimension of the zero-fiber ring, and
when combined with the lower bound that follows from our previous construction
of a non-commutative fiber of dimension $g+1$, establishes that the set $S$
is, in each case, a Groebner basis of $I$. Throughout we use the lexicographic
order on monomials, so that $x^{a}y^{b}$ comes before $x^{c}y^{d}$ if $a>c$ or
if $a=c$ and $b>d$.

###  3.3. Cyclic groups

Let $\zeta$ be a primitive $\ell$th root of $1$ and let $\Gamma$ be the
subgroup of $\mathrm{SL}_{2}(\mathbf{C})$ generated by the matrix

$w=\left(\begin{matrix}\zeta&0\\\ 0&\zeta^{-1}\end{matrix}\right).$

The action of $w$ on the ring $\mathbf{C}[x,y]$ of polynomial functions on
$\mathbf{C}^{2}$ is given by

$w\cdot x=\zeta^{-1}x\quad\text{and}\quad w\cdot y=\zeta y.$

It follows that the polynomials $x^{\ell},xy$ and $y^{\ell}$ are
$\Gamma$-invariant. The quotient by the ideal they generated is spanned by the
monomials $1,x,\dots,x^{\ell-1},y,y^{2},\dots,y^{\ell-1}$ and hence has
dimension at most $2\ell-1=2|\Gamma|-1$.

###  3.4. Binary dihedral groups $\mathrm{BD}_{2n}$

We write $\zeta$ for a primitive $2n$th root of $1$, and let

$w_{1}=\left(\begin{matrix}\zeta&0\\\
0&\zeta^{-1}\end{matrix}\right)\quad\text{and}\quad
w_{2}=\left(\begin{matrix}0&i\\\ i&0\end{matrix}\right).$

We consider the polynomials

$\phi_{1}=x^{n}+y^{n},\quad\phi_{2}=x^{n}-y^{n},\quad\text{and}\quad\phi_{3}=xy.$

Direct calculation shows that these are semi-invariants with

$w_{1}\cdot\phi_{1}=-\phi_{1},\ w_{2}\cdot\phi_{1}=(-i)^{n}\phi_{1},\quad
w_{1}\cdot\phi_{2}=-\phi_{2},\ w_{2}\cdot\phi_{2}=-(-i)^{n}\phi_{2},\quad
w_{1}\cdot\phi_{3}=\phi_{3},\ \text{and}\ w_{2}\cdot\phi_{3}=-\phi_{3}.$

Hence if $n$ is even then the polynomials

$F_{1}=\phi_{3}^{2},\quad F_{2}=\phi_{2}^{2},\quad\text{and}\quad
F_{3}=\phi_{1}\phi_{2}\phi_{3}$

are $\Gamma$-invariant, while if $n$ is odd then the polynomials

$G_{1}=\phi_{3}^{2},\quad G_{2}=\phi_{1}\phi_{2},\quad\text{and}\quad
G_{3}=\phi_{2}^{2}\phi_{3}$

are $\Gamma$-invariant.

We first assume $n$ is even, and in particular $n\geq 2$. We have

$F_{1}=x^{2}y^{2},\quad F_{2}=x^{2n}-2x^{n}y^{n}+y^{2n},\quad\text{and}\quad
F_{3}=xy(x^{2n}-y^{2n}).$

Hence the polynomial

$y^{2n+2}=y^{2}F_{2}-(x^{2n-2}-2x^{n-2}y^{n})F_{1}$

belongs to $I$. Likewise

$xy^{2n+1}=\frac{1}{2}\left(xyF_{2}-F_{3}+2x^{n-1}y^{n-1}F_{1}\right)$

belongs to $I$. The quotient by the ideal $J$ generated by the leading terms
of the set

$S=\\{F_{1},F_{2},y^{2n+2},xy^{2n+1}\\}$

is spanned by

$\\{1,x,\dots,x^{2n-1},y,y^{2},\dots,y^{2n+1},xy,x^{2}y,\dots,x^{2n-1}y,xy^{2},\dots,xy^{2n}\\}$

and is hence of dimension at most

$2n+(2n+1)+2n-1+2n-1=8n-1=2|\Gamma|-1.$

Next we assume $n$ is odd. The invariant polynomials here are

$G_{1}=x^{2}y^{2},\quad G_{2}=x^{2n}-y^{2n},\quad\text{and}\quad
G_{3}=xy(x^{2n}-2x^{n}y^{2}+y^{2n}).$

Since

$y^{2n+2}=x^{2n-2}G_{1}-y^{2}G_{2}$

we have $y^{2n+2}\in I$. And since

$xy^{2n+1}=\frac{1}{2}\left(G_{3}-2x^{n-1}y^{n-1}G_{1}-xyG_{2}\right)$

we have $xy^{2n+1}\in I$. Thus again the desired dimension is bounded by
$8n-1=2|\Gamma|-1$.

###  3.5. The binary tetrahedral group

Here we follow [Dol], starting on page 8, and consider the binary tetrahedral
group $\Gamma$ generated by the matrices

$w_{1}=\left(\begin{matrix}i&0\\\ 0&-i\end{matrix}\right),\quad
w_{2}=\left(\begin{matrix}0&i\\\ i&0\end{matrix}\right),\quad\text{and}\quad
w_{3}=\frac{1}{1-i}\left(\begin{matrix}1&i\\\ 1&-i\end{matrix}\right).$

Direct calculation shows that

$\phi_{1}=xy(x^{4}-y^{4}),\quad\phi_{2}=x^{4}+2\sqrt{-3}x^{2}y^{2}+y^{4},\quad\text{and}\quad\phi_{3}=x^{4}-2\sqrt{-3}x^{2}y^{2}+y^{4}$

are semi-invariants for $\Gamma$, and then that

$f_{1}=\phi_{1}=xy(x^{4}-y^{4}),\quad
f_{2}=\phi_{2}\phi_{3}=x^{8}+14x^{4}y^{4}+y^{8}\quad\text{and}\quad
f_{3}=\frac{1}{2}(\phi_{2}^{3}+\phi_{3}^{3})=x^{12}-33x^{8}y^{4}-33x^{4}y^{8}+y^{12}$

are invariants for $\Gamma$.

We define

$g_{1}=yf_{2}-x^{3}f_{1}=15x^{4}y^{5}+y^{9}\quad\text{and}\quad
g_{2}=xg_{1}-15y^{4}f_{1}=xy^{9},$

so that $g_{1},g_{2}\in I$, and then put

$h=f_{3}+(47y^{4}-x^{4})f_{2}=624x^{4}y^{8}+48y^{12}.$

Finally, defining

$g_{3}=5h-208y^{3}g_{1}=32y^{12}\in I$

we have obtained the desired set

$S=\\{f_{1},f_{2},g_{1},g_{2},g_{3}\\}$

whose initial terms

$\\{x^{5}y,x^{8},x^{4}y^{5},xy^{9},y^{12}\\}$

generate an ideal of codimension $47=2|\Gamma|-1$.

###  3.6. The binary octahedral group

Here we follow [Dol], starting on page 9, and consider the binary octahedral
group $\Gamma$ generated by the matrices

$w_{1}=\left(\begin{matrix}e^{2\pi i/8}&0\\\ 0&e^{-2\pi
i/8}\end{matrix}\right),\quad w_{2}=\left(\begin{matrix}0&i\\\
i&0\end{matrix}\right),\quad\text{and}\quad
w_{3}=\frac{1}{1-i}\left(\begin{matrix}1&i\\\ 1&-i\end{matrix}\right).$

Since the binary tetrahedral group is a normal subgroup of $\Gamma$, it is not
surprising that its invariants

$\phi_{1}=xy(x^{4}-y^{4}),\quad\phi_{2}=x^{8}+14x^{4}y^{4}+y^{8},\quad\text{and}\quad\phi_{3}=x^{12}-33x^{8}y^{4}-33x^{4}y^{8}+y^{12}$

are semi-invariants for $\Gamma$. Direct calculation then shows that

$f_{1}=\phi_{1}^{2}=x^{10}y^{2}-2x^{6}y^{6}+x^{2}y^{10},\quad
f_{2}=\phi_{2}=x^{8}+14x^{4}y^{4}+y^{8},\quad\text{and}\quad
f_{3}=\phi_{1}\phi_{3}=x^{17}y-34x^{13}y^{5}+34x^{5}y^{13}-xy^{17}$

are $\Gamma$ invariants.

We define

$g_{1}=\frac{1}{16}(x^{2}y^{2}f_{2}-f_{1})=x^{6}y^{6}\in I$

and then

$g_{2}=y^{6}f_{2}-x^{2}g_{2}=14x^{4}y^{10}+y^{14}\in I\quad\text{and}\quad
g_{3}=x^{2}y^{6}(x^{8}+14x^{4}y^{4}+y^{8})-(x^{4}+14y^{4})g_{2}=x^{2}y^{14}\in
I.$

Next we observe that, working modulo $g_{2}=x^{6}y^{6}$, we have

$y^{10}(x^{8}+14x^{4}+y^{8})=14x^{4}y^{14}+y^{18}$

and, still working modulo $g_{2}=x^{6}y^{6}$,

$14x^{4}y^{14}+y^{18}-14x^{2}y^{4}f_{1}=y^{18}\in I.$

Thus we define $g_{4}=y^{18}\in I$. Finally we observe that

$g_{5}=(7x^{9}y-336x^{5}y^{5}+41xy^{9})f_{2}+4656x^{3}y^{3}g_{2}-7f_{3}=48xy^{17}\in
I.$

Hence the set

$S=\\{f_{2},g_{1},g_{2},g_{3},g_{4},g_{5}\\}$

is contained in $I$, with initial terms the set

$\\{x^{8},x^{6}y^{6},x^{4}y^{10},x^{2}y^{14},xy^{17},y^{18}\\}.$

It follows that the degree of the zero fiber here is at most $95=2|\Gamma|-1$.

###  3.7. The binary icosahedral group

Here we follow [Dol], starting on page 9. We take $\zeta=e^{2\pi i/5}$ a
primitive $5$th root of unity, and let $\Gamma$ to be the group generated by

$w_{1}=\left(\begin{matrix}e^{2\pi i/10}&0\\\ 0&e^{-2\pi
i/10}\end{matrix}\right),\quad w_{2}=\left(\begin{matrix}0&i\\\
i&0\end{matrix}\right),\quad\text{and}\quad
w_{3}=\frac{1}{\sqrt{5}}\left(\begin{matrix}\zeta-\zeta^{4}&\zeta^{2}-\zeta^{3}\\\
\zeta^{2}-\zeta^{3}&-\zeta+\zeta^{4}\end{matrix}\right).$

This group is perfect, so its semi-invariants are invariants, and as in [Dol]
we observe that the polynomials

$f_{1}=xy(x^{10}+11x^{5}y^{5}-y^{10}),\quad
f_{2}=-(x^{20}+y^{20})+228(x^{15}y^{5}-x^{5}y^{15})-494x^{10}y^{10},$

and

$f_{3}=x^{30}+y^{30}+522(x^{25}y^{5}-x^{5}y^{25})-10005(x^{20}y^{10}+x^{10}y^{20})$

are semi-invariants (and hence invariants). We put

$g_{1}=(x^{9}-239x^{4}y^{5})f_{1}+yf_{2}=-3124x^{10}y^{11}+11x^{5}y^{16}-y^{21}\in
I$

and

$g_{2}=(x^{10}-239x^{5}y^{5}+3124y^{10})f_{1}+xyf_{2}=34375x^{6}y^{16}-3125xy^{21}\in
I.$

Next we set

$g_{3}=\frac{1}{140}\left(3124^{2}(y^{6}f_{2}+x^{9}y^{5}f_{1})+(3124\cdot
239x^{5}-1543751y^{5})g_{1}\right)=-16020500x^{5}y^{21}-58683y^{26}$

and then

$g_{4}=\frac{-1}{52081300000}(16020500y^{5}g_{2}+34375xg_{3})=xy^{26}.$

Finally we set

$h_{1}=f_{3}+x^{10}f_{2}-(750x^{14}y^{4}-18749x^{9}y^{9})f_{1}=206761x^{15}y^{15}-28755x^{10}y^{20}-522x^{5}y^{25}+y^{30}\in
I,$
$h_{2}=3124h_{1}+206761x^{5}y^{4}g_{1}=-87556200x^{10}y^{20}-1837490x^{5}y^{25}+3124y^{30}\in
I,$

and observe that modulo $g_{2}$ and $g_{3}$, the polynomial $h$ is a non-zero
multiple of $g_{5}=y^{30}$, which is therefore in $I$. Thus the set

$S=\\{f_{1},f_{2},g_{1},g_{2},g_{3},g_{4},g_{5}\\}$

is a subset of $I$ with leading terms

$\\{x^{11}y,x^{20},x^{10}y^{11},x^{6}y^{16},x^{5}y^{21},xy^{26},y^{30}\\}.$

It follows as above that the dimension of the zero fiber is at most
$239=2|\Gamma|-1$. This completes the proof of Theorem 3.1, and hence Theorem
1.1 in case $n=1$.

###  3.8. Spaces of semi-invariants

We begin with a very general lemma, whose proof follows easily from Clifford
theory.

######  Lemma 3.2.

Let $G$ be a finite group, let $N\lhd G$ be a normal subgroup such that $G/N$
is abelian, let $\chi:G\to\mathbf{C}^{\times}$ be a linear character, and let
$M$ be an irreducible $\mathbf{C}G$-module such that the restriction of $M$ to
$N$ contains a summand isomorphic to the restriction of $\chi$ to $N$. Then
$M$ is one-dimensional, and hence given by a linear character
$\psi:G\to\mathbf{C}^{\times}$ with restriction to $N$ equal to $\chi$
restricted to $N$.

###### Proof.

By Clifford theory, the restriction of $M$ to $N$ is a sum of conjugates of
$\chi$ restricted to $N$. But since $\chi$ is a character of $G$, it is
conjugation-invariant, so the restriction of $M$ to $N$ is a sum of copies of
$\chi$, implying that $N$ acts via $\chi$ on $M$. Hence $N$ acts trivially on
$M\otimes\chi^{-1}$, which is therefore an irreducible $G/N$ module. Since
$G/N$ is abelian, it follows that $M$ is one-dimensional. ∎

Now fix a pair $(\chi,\eta)$ consisting of a linear character $\chi$ of
$\Gamma$ and a linear character $\eta$ of $S_{n}$. This pair determines a
linear character $(\chi,\eta)$ of $W_{n}(\Gamma)$ by the rule

$(\chi,\eta)(\gamma_{1}^{(1)}\cdots\gamma_{n}^{(n)}w)=\eta(w)\cdot\prod_{i=1}^{n}\chi(\gamma_{i}),$

and hence upon restriction to $W_{n}(\Gamma,\Delta)$, a linear character of
$W_{n}(\Gamma,\Delta)$. Different pairs $(\chi,\eta)$ define different
characters of $W_{n}(\Gamma)$, and all linear characters of $W_{n}(\Gamma)$
are of this form. Two pairs $(\chi,\eta)$ and $(\chi^{\prime},\eta^{\prime})$
restrict to the same character of $W_{n}(\Gamma,\Delta)$ if and only if
$\eta=\eta^{\prime}$ and the restrictions of $\chi$ and $\chi^{\prime}$ to
$\Delta$ are equal. Moreover, if $n\geq 3$ then all characters of
$W_{n}(\Gamma,\Delta)$ are of this form. The following lemma is the key input
in order to apply Lemmas 2.1 and 2.2. In what follows, for an isoclass $\mu$
of representations of a finite group $G$ and a $G$-module $M$, we write
$M^{\mu}$ for the sum of all submodules of $M$ of type $\mu$, and we write
$G^{\vee}$ for the group of linear characters of $G$.

######  Lemma 3.3.

With notation as above, given a $\mathbf{C}\Gamma$-module $L$, the space of
$(\chi,\eta)$-relative invariants for the action of $W_{n}(\Gamma,\Delta)$ on
$L^{\otimes n}\otimes\mathrm{det}$ is

$(L^{\otimes
n}\otimes\mathrm{det})^{(\chi,\eta)}=\begin{cases}\bigoplus_{\begin{subarray}{c}\psi\in\Gamma^{\vee}\\\
\psi|_{\Delta}=\chi\end{subarray}}\Lambda^{n}(L^{\psi})\quad\hbox{if
$\eta=\mathrm{triv}$, and}\\\
\bigoplus_{\begin{subarray}{c}\psi\in\Gamma^{\vee}\\\
\psi|_{\Delta}=\chi\end{subarray}}\mathrm{Sym}^{n}(L^{\psi})\quad\hbox{if
$\eta=\mathrm{det}$.}\end{cases}$

###### Proof.

We put

$\Gamma^{\times n}_{\Delta}=\\{(\gamma_{1},\dots,\gamma_{n})\in\Gamma^{\times
n}\ |\ \gamma_{1}\gamma_{2}\cdots\gamma_{n}=1\ \mathrm{mod}\ \Delta\\},$

which is a normal subgroup of $\Gamma^{\times n}$. For a representation $L$ of
$W_{n}(\Gamma)$ we compute the $W_{n}(\Gamma,\Delta)$ semi-invariants in
$L^{\otimes n}\otimes\mathrm{det}$ as follows: first we compute the
$\Gamma^{\times n}_{\Delta}$ semi-invariants, and then take $S_{n}$ invariants
or anti-invariants as appropriate. Now as a $G=\Gamma^{\times n}$-module,
$L^{\otimes n}$ decomposes as a sum of irreducibles of the form
$L^{\mu_{1}}\otimes\cdots\otimes L^{\mu_{n}}$ for various irreducible
$\Gamma$-submodules $L^{\mu_{i}}$ of $L$. By applying the previous lemma with
$N=\Gamma^{\times n}_{\Delta}$ and the character of $G$ induced by $\chi$, the
only ones that can contribute the the semi-invariants we are after must have
all $L^{\mu_{i}}$ one-dimensional. But now one checks using the definition of
$N$ that in fact each factor must correspond to the same linear character
$\psi$ with restriction to $\Delta$ equal to that of $\chi$. Finally, by
taking $S_{n}$-invariants or anti-invariants, as the case may be, we obtain
the result. ∎

###  3.9. Proof of Theorem 1.1

We have already proved the assertions of the theorem in case $n=1$. For $n\geq
2$ we begin with the case $W_{n}(\Gamma)$, in order to illustrate the ideas
without the case-by-case considerations that will be necessary for the groups
$W_{n}(\Gamma,\Delta)$. Using Corollary 2.5 we choose a parameter
$c\in\mathbf{C}^{I}$ such that there is an irreducible
$H_{c}(\Gamma,V)$-module $L$ with character $\mathrm{ch}(L)=n\delta+\phi$. By
Lemma 2.3, the representation $L^{\otimes n}\otimes\mathrm{det}$ of
$H_{c}(\Gamma,V)^{\otimes n}\rtimes S_{n}$ is irreducible. By Lemma 3.3, the
space of $W_{n}(\Gamma)$-invariants in this representation is one-dimensional,
so it is a non-commutative fiber. By Lemma 2.1, its associated graded module
is a quotient ring of the zero-fiber ring, and since its dimension is

$\mathrm{dim}_{\mathbf{C}}\left(L^{\otimes
n}\otimes\mathrm{det}\right)=\mathrm{dim}(L)^{n}=\left(n|\Gamma|+|\Gamma|-1\right)^{n}=(g+1)^{n},$

we have completed the proof in this case.

Now we handle the case of $W_{n}(\Gamma,\Delta)$. We may assume $\Gamma$ is
not abelian, since in that case the results of [Gri4] apply. By Lemma 3.3, we
search for an irreducible $H_{c}(\Gamma,\mathbf{C}^{2})$-module $L$ with
character as large as possible subject to the condition that

(3.1) $\mathrm{dim}(L^{\chi})\leq n\quad\hbox{for all
$\chi\in(\Gamma/\Delta)^{\vee}$, with equality for exactly one such
$\chi\in(\Gamma/\Delta)^{\vee}$.}$

Recalling that the character of any $L$ must be a positive root and hence of
the form $m\delta+\alpha$ for $m\geq 0$ and $\alpha$ a root which is positive
if $m=0$, we see that a maximal choice of $L$ has the form
$\mathrm{ch}(L)=(n-1)\delta+\alpha$ for a positive root $\alpha$ with the
property that $\alpha_{i}=1$ for exactly one index $i$ corresponding to a
linear character of $\Gamma/\Delta$. (We could also take $L$ with
$\mathrm{ch}(L)=n\delta-\beta$ for a positive root $\beta$ such that
$\beta_{i}=1$ for all $i$ corresponding to non-trivial characters of
$\Gamma/\Delta$). For $L$ with character $(n-1)\delta+\alpha$, its dimension
is

$\mathrm{dim}(L)=\sum k_{i}n_{i}+(n-1)|\Gamma|,$

so the equality $\mathrm{dim}(L)=g+1$ is equivalent to

$\sum k_{i}n_{i}=2|\Delta|-1.$

It remains to show that in each case we can choose $\alpha$ a finite root such
that this holds, and so that there is precisely one character $\chi$ of
$\Gamma/\Delta$ such that $\alpha_{i}=1$. In fact, we will look for $\alpha$ a
maximal positive root with this latter property, and observe that each such
$\alpha$ satisfies the former.

The simplest case is that in which $|\Gamma/\Delta|=2$. In this case the
largest possible choice $\alpha=\phi$ works, so that

$\mathrm{ch}(L)=(n-1)\delta+\phi.$

The remaining cases are the commutator subgroup $\Delta$ of index $3$ in the
binary tetrahedral group (of type $E_{6}^{(1)}$), and the commutator subgroup
(which is of order $n$, or equivalenty index $4$) of the binary dihedral group
of order $4n$. We now complete the proof of Theorem 1.1 by specifying, in each
case, a particular $\alpha$ with the property (3.1). For the reader’s
convenience we also label the Dynkin diagram with the integers $n_{i}$ giving
the coefficients of $\delta$ on the simple roots.

$\delta\ \text{for type}\ E_{6}^{(1)}\quad\leavevmode\hbox to119.23pt{\vbox
to71.48pt{\pgfpicture\makeatletter\hbox{\hskip 7.49968pt\lower-16.33905pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }
{{}} {{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.82861pt}{0.0pt}\pgfsys@curveto{2.82861pt}{1.56221pt}{1.56221pt}{2.82861pt}{0.0pt}{2.82861pt}\pgfsys@curveto{-1.56221pt}{2.82861pt}{-2.82861pt}{1.56221pt}{-2.82861pt}{0.0pt}\pgfsys@curveto{-2.82861pt}{-1.56221pt}{-1.56221pt}{-2.82861pt}{0.0pt}{-2.82861pt}\pgfsys@curveto{1.56221pt}{-2.82861pt}{2.82861pt}{-1.56221pt}{2.82861pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.16667pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{28.88586pt}{0.0pt}\pgfsys@curveto{28.88586pt}{1.56221pt}{27.61946pt}{2.82861pt}{26.05725pt}{2.82861pt}\pgfsys@curveto{24.49504pt}{2.82861pt}{23.22864pt}{1.56221pt}{23.22864pt}{0.0pt}\pgfsys@curveto{23.22864pt}{-1.56221pt}{24.49504pt}{-2.82861pt}{26.05725pt}{-2.82861pt}\pgfsys@curveto{27.61946pt}{-2.82861pt}{28.88586pt}{-1.56221pt}{28.88586pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{26.05725pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.05725pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{7.02861pt}{0.0pt}\pgfsys@lineto{19.02864pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.89058pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{54.94312pt}{0.0pt}\pgfsys@curveto{54.94312pt}{1.56221pt}{53.67671pt}{2.82861pt}{52.1145pt}{2.82861pt}\pgfsys@curveto{50.55229pt}{2.82861pt}{49.28589pt}{1.56221pt}{49.28589pt}{0.0pt}\pgfsys@curveto{49.28589pt}{-1.56221pt}{50.55229pt}{-2.82861pt}{52.1145pt}{-2.82861pt}\pgfsys@curveto{53.67671pt}{-2.82861pt}{54.94312pt}{-1.56221pt}{54.94312pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{52.1145pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{52.1145pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{33.08586pt}{0.0pt}\pgfsys@lineto{45.08589pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{47.94783pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$3$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{81.00037pt}{0.0pt}\pgfsys@curveto{81.00037pt}{1.56221pt}{79.73396pt}{2.82861pt}{78.17175pt}{2.82861pt}\pgfsys@curveto{76.60954pt}{2.82861pt}{75.34314pt}{1.56221pt}{75.34314pt}{0.0pt}\pgfsys@curveto{75.34314pt}{-1.56221pt}{76.60954pt}{-2.82861pt}{78.17175pt}{-2.82861pt}\pgfsys@curveto{79.73396pt}{-2.82861pt}{81.00037pt}{-1.56221pt}{81.00037pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{78.17175pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{78.17175pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{59.14311pt}{0.0pt}\pgfsys@lineto{71.14314pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{74.00508pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{107.05762pt}{0.0pt}\pgfsys@curveto{107.05762pt}{1.56221pt}{105.79121pt}{2.82861pt}{104.229pt}{2.82861pt}\pgfsys@curveto{102.6668pt}{2.82861pt}{101.40039pt}{1.56221pt}{101.40039pt}{0.0pt}\pgfsys@curveto{101.40039pt}{-1.56221pt}{102.6668pt}{-2.82861pt}{104.229pt}{-2.82861pt}\pgfsys@curveto{105.79121pt}{-2.82861pt}{107.05762pt}{-1.56221pt}{107.05762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{104.229pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{104.229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{85.20036pt}{0.0pt}\pgfsys@lineto{97.2004pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{100.06233pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@beginscope\pgfsys@invoke{ } {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{54.94312pt}{26.05725pt}\pgfsys@curveto{54.94312pt}{27.61946pt}{53.67671pt}{28.88586pt}{52.1145pt}{28.88586pt}\pgfsys@curveto{50.55229pt}{28.88586pt}{49.28589pt}{27.61946pt}{49.28589pt}{26.05725pt}\pgfsys@curveto{49.28589pt}{24.49504pt}{50.55229pt}{23.22864pt}{52.1145pt}{23.22864pt}\pgfsys@curveto{53.67671pt}{23.22864pt}{54.94312pt}{24.49504pt}{54.94312pt}{26.05725pt}\pgfsys@closepath\pgfsys@moveto{52.1145pt}{26.05725pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{52.1145pt}{26.05725pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{52.1145pt}{7.02861pt}\pgfsys@lineto{52.1145pt}{19.02864pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{57.78905pt}{13.93828pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{54.94312pt}{52.1145pt}\pgfsys@curveto{54.94312pt}{53.67671pt}{53.67671pt}{54.94312pt}{52.1145pt}{54.94312pt}\pgfsys@curveto{50.55229pt}{54.94312pt}{49.28589pt}{53.67671pt}{49.28589pt}{52.1145pt}\pgfsys@curveto{49.28589pt}{50.55229pt}{50.55229pt}{49.28589pt}{52.1145pt}{49.28589pt}\pgfsys@curveto{53.67671pt}{49.28589pt}{54.94312pt}{50.55229pt}{54.94312pt}{52.1145pt}\pgfsys@closepath\pgfsys@moveto{52.1145pt}{52.1145pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{52.1145pt}{52.1145pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{52.1145pt}{33.08586pt}\pgfsys@lineto{52.1145pt}{45.08589pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{57.78905pt}{39.99553pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\text{and}\
\alpha\ \text{is}\quad\leavevmode\hbox to119.23pt{\vbox
to71.48pt{\pgfpicture\makeatletter\hbox{\hskip 7.49968pt\lower-16.33905pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }
{{}} {{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.82861pt}{0.0pt}\pgfsys@curveto{2.82861pt}{1.56221pt}{1.56221pt}{2.82861pt}{0.0pt}{2.82861pt}\pgfsys@curveto{-1.56221pt}{2.82861pt}{-2.82861pt}{1.56221pt}{-2.82861pt}{0.0pt}\pgfsys@curveto{-2.82861pt}{-1.56221pt}{-1.56221pt}{-2.82861pt}{0.0pt}{-2.82861pt}\pgfsys@curveto{1.56221pt}{-2.82861pt}{2.82861pt}{-1.56221pt}{2.82861pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.16667pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{28.88586pt}{0.0pt}\pgfsys@curveto{28.88586pt}{1.56221pt}{27.61946pt}{2.82861pt}{26.05725pt}{2.82861pt}\pgfsys@curveto{24.49504pt}{2.82861pt}{23.22864pt}{1.56221pt}{23.22864pt}{0.0pt}\pgfsys@curveto{23.22864pt}{-1.56221pt}{24.49504pt}{-2.82861pt}{26.05725pt}{-2.82861pt}\pgfsys@curveto{27.61946pt}{-2.82861pt}{28.88586pt}{-1.56221pt}{28.88586pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{26.05725pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.05725pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{7.02861pt}{0.0pt}\pgfsys@lineto{19.02864pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.89058pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{54.94312pt}{0.0pt}\pgfsys@curveto{54.94312pt}{1.56221pt}{53.67671pt}{2.82861pt}{52.1145pt}{2.82861pt}\pgfsys@curveto{50.55229pt}{2.82861pt}{49.28589pt}{1.56221pt}{49.28589pt}{0.0pt}\pgfsys@curveto{49.28589pt}{-1.56221pt}{50.55229pt}{-2.82861pt}{52.1145pt}{-2.82861pt}\pgfsys@curveto{53.67671pt}{-2.82861pt}{54.94312pt}{-1.56221pt}{54.94312pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{52.1145pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{52.1145pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{33.08586pt}{0.0pt}\pgfsys@lineto{45.08589pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{47.94783pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{81.00037pt}{0.0pt}\pgfsys@curveto{81.00037pt}{1.56221pt}{79.73396pt}{2.82861pt}{78.17175pt}{2.82861pt}\pgfsys@curveto{76.60954pt}{2.82861pt}{75.34314pt}{1.56221pt}{75.34314pt}{0.0pt}\pgfsys@curveto{75.34314pt}{-1.56221pt}{76.60954pt}{-2.82861pt}{78.17175pt}{-2.82861pt}\pgfsys@curveto{79.73396pt}{-2.82861pt}{81.00037pt}{-1.56221pt}{81.00037pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{78.17175pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{78.17175pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{59.14311pt}{0.0pt}\pgfsys@lineto{71.14314pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{74.00508pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{107.05762pt}{0.0pt}\pgfsys@curveto{107.05762pt}{1.56221pt}{105.79121pt}{2.82861pt}{104.229pt}{2.82861pt}\pgfsys@curveto{102.6668pt}{2.82861pt}{101.40039pt}{1.56221pt}{101.40039pt}{0.0pt}\pgfsys@curveto{101.40039pt}{-1.56221pt}{102.6668pt}{-2.82861pt}{104.229pt}{-2.82861pt}\pgfsys@curveto{105.79121pt}{-2.82861pt}{107.05762pt}{-1.56221pt}{107.05762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{104.229pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{104.229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{85.20036pt}{0.0pt}\pgfsys@lineto{97.2004pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{100.06233pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$0$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@beginscope\pgfsys@invoke{ } {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{54.94312pt}{26.05725pt}\pgfsys@curveto{54.94312pt}{27.61946pt}{53.67671pt}{28.88586pt}{52.1145pt}{28.88586pt}\pgfsys@curveto{50.55229pt}{28.88586pt}{49.28589pt}{27.61946pt}{49.28589pt}{26.05725pt}\pgfsys@curveto{49.28589pt}{24.49504pt}{50.55229pt}{23.22864pt}{52.1145pt}{23.22864pt}\pgfsys@curveto{53.67671pt}{23.22864pt}{54.94312pt}{24.49504pt}{54.94312pt}{26.05725pt}\pgfsys@closepath\pgfsys@moveto{52.1145pt}{26.05725pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{52.1145pt}{26.05725pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{52.1145pt}{7.02861pt}\pgfsys@lineto{52.1145pt}{19.02864pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{57.78905pt}{13.93828pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{54.94312pt}{52.1145pt}\pgfsys@curveto{54.94312pt}{53.67671pt}{53.67671pt}{54.94312pt}{52.1145pt}{54.94312pt}\pgfsys@curveto{50.55229pt}{54.94312pt}{49.28589pt}{53.67671pt}{49.28589pt}{52.1145pt}\pgfsys@curveto{49.28589pt}{50.55229pt}{50.55229pt}{49.28589pt}{52.1145pt}{49.28589pt}\pgfsys@curveto{53.67671pt}{49.28589pt}{54.94312pt}{50.55229pt}{54.94312pt}{52.1145pt}\pgfsys@closepath\pgfsys@moveto{52.1145pt}{52.1145pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{52.1145pt}{52.1145pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{52.1145pt}{33.08586pt}\pgfsys@lineto{52.1145pt}{45.08589pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{57.78905pt}{39.99553pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$0$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ $\delta\ \text{for
type}\ D_{n+2}^{(1)}\quad\leavevmode\hbox to154.14pt{\vbox
to45.42pt{\pgfpicture\makeatletter\hbox{\hskip 7.49968pt\lower-16.33905pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }
{{}} {{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.82861pt}{0.0pt}\pgfsys@curveto{2.82861pt}{1.56221pt}{1.56221pt}{2.82861pt}{0.0pt}{2.82861pt}\pgfsys@curveto{-1.56221pt}{2.82861pt}{-2.82861pt}{1.56221pt}{-2.82861pt}{0.0pt}\pgfsys@curveto{-2.82861pt}{-1.56221pt}{-1.56221pt}{-2.82861pt}{0.0pt}{-2.82861pt}\pgfsys@curveto{1.56221pt}{-2.82861pt}{2.82861pt}{-1.56221pt}{2.82861pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.16667pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{28.88586pt}{0.0pt}\pgfsys@curveto{28.88586pt}{1.56221pt}{27.61946pt}{2.82861pt}{26.05725pt}{2.82861pt}\pgfsys@curveto{24.49504pt}{2.82861pt}{23.22864pt}{1.56221pt}{23.22864pt}{0.0pt}\pgfsys@curveto{23.22864pt}{-1.56221pt}{24.49504pt}{-2.82861pt}{26.05725pt}{-2.82861pt}\pgfsys@curveto{27.61946pt}{-2.82861pt}{28.88586pt}{-1.56221pt}{28.88586pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{26.05725pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.05725pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{7.02861pt}{0.0pt}\pgfsys@lineto{19.02864pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.89058pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{54.94312pt}{0.0pt}\pgfsys@curveto{54.94312pt}{1.56221pt}{53.67671pt}{2.82861pt}{52.1145pt}{2.82861pt}\pgfsys@curveto{50.55229pt}{2.82861pt}{49.28589pt}{1.56221pt}{49.28589pt}{0.0pt}\pgfsys@curveto{49.28589pt}{-1.56221pt}{50.55229pt}{-2.82861pt}{52.1145pt}{-2.82861pt}\pgfsys@curveto{53.67671pt}{-2.82861pt}{54.94312pt}{-1.56221pt}{54.94312pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{52.1145pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{52.1145pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{33.08586pt}{0.0pt}\pgfsys@lineto{45.08589pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{47.94783pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}
{{{{}}}}{{}}{{}}{}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{78.85071pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\ldots}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{59.14311pt}{0.0pt}\pgfsys@lineto{71.14314pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}
{{{{}}}}{{}}{{}}{} {{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{115.91553pt}{0.0pt}\pgfsys@curveto{115.91553pt}{1.56221pt}{114.64912pt}{2.82861pt}{113.08691pt}{2.82861pt}\pgfsys@curveto{111.5247pt}{2.82861pt}{110.2583pt}{1.56221pt}{110.2583pt}{0.0pt}\pgfsys@curveto{110.2583pt}{-1.56221pt}{111.5247pt}{-2.82861pt}{113.08691pt}{-2.82861pt}\pgfsys@curveto{114.64912pt}{-2.82861pt}{115.91553pt}{-1.56221pt}{115.91553pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{113.08691pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{113.08691pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{94.05827pt}{0.0pt}\pgfsys@lineto{106.0583pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{108.92024pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{141.97278pt}{0.0pt}\pgfsys@curveto{141.97278pt}{1.56221pt}{140.70638pt}{2.82861pt}{139.14417pt}{2.82861pt}\pgfsys@curveto{137.58195pt}{2.82861pt}{136.31555pt}{1.56221pt}{136.31555pt}{0.0pt}\pgfsys@curveto{136.31555pt}{-1.56221pt}{137.58195pt}{-2.82861pt}{139.14417pt}{-2.82861pt}\pgfsys@curveto{140.70638pt}{-2.82861pt}{141.97278pt}{-1.56221pt}{141.97278pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{139.14417pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{139.14417pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{120.11552pt}{0.0pt}\pgfsys@lineto{132.11555pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{134.9775pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@beginscope\pgfsys@invoke{ } {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{28.88586pt}{26.05725pt}\pgfsys@curveto{28.88586pt}{27.61946pt}{27.61946pt}{28.88586pt}{26.05725pt}{28.88586pt}\pgfsys@curveto{24.49504pt}{28.88586pt}{23.22864pt}{27.61946pt}{23.22864pt}{26.05725pt}\pgfsys@curveto{23.22864pt}{24.49504pt}{24.49504pt}{23.22864pt}{26.05725pt}{23.22864pt}\pgfsys@curveto{27.61946pt}{23.22864pt}{28.88586pt}{24.49504pt}{28.88586pt}{26.05725pt}\pgfsys@closepath\pgfsys@moveto{26.05725pt}{26.05725pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.05725pt}{26.05725pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{26.05725pt}{7.02861pt}\pgfsys@lineto{26.05725pt}{19.02864pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.7318pt}{13.93828pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} ;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@beginscope\pgfsys@invoke{ } {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{115.91553pt}{26.05725pt}\pgfsys@curveto{115.91553pt}{27.61946pt}{114.64912pt}{28.88586pt}{113.08691pt}{28.88586pt}\pgfsys@curveto{111.5247pt}{28.88586pt}{110.2583pt}{27.61946pt}{110.2583pt}{26.05725pt}\pgfsys@curveto{110.2583pt}{24.49504pt}{111.5247pt}{23.22864pt}{113.08691pt}{23.22864pt}\pgfsys@curveto{114.64912pt}{23.22864pt}{115.91553pt}{24.49504pt}{115.91553pt}{26.05725pt}\pgfsys@closepath\pgfsys@moveto{113.08691pt}{26.05725pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{113.08691pt}{26.05725pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{113.08691pt}{7.02861pt}\pgfsys@lineto{113.08691pt}{19.02864pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{118.76146pt}{13.93828pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} ;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\text{and}\
\alpha\ \text{is}\quad\leavevmode\hbox to154.14pt{\vbox
to45.42pt{\pgfpicture\makeatletter\hbox{\hskip 7.49968pt\lower-16.33905pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }
{{}} {{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.82861pt}{0.0pt}\pgfsys@curveto{2.82861pt}{1.56221pt}{1.56221pt}{2.82861pt}{0.0pt}{2.82861pt}\pgfsys@curveto{-1.56221pt}{2.82861pt}{-2.82861pt}{1.56221pt}{-2.82861pt}{0.0pt}\pgfsys@curveto{-2.82861pt}{-1.56221pt}{-1.56221pt}{-2.82861pt}{0.0pt}{-2.82861pt}\pgfsys@curveto{1.56221pt}{-2.82861pt}{2.82861pt}{-1.56221pt}{2.82861pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.16667pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{28.88586pt}{0.0pt}\pgfsys@curveto{28.88586pt}{1.56221pt}{27.61946pt}{2.82861pt}{26.05725pt}{2.82861pt}\pgfsys@curveto{24.49504pt}{2.82861pt}{23.22864pt}{1.56221pt}{23.22864pt}{0.0pt}\pgfsys@curveto{23.22864pt}{-1.56221pt}{24.49504pt}{-2.82861pt}{26.05725pt}{-2.82861pt}\pgfsys@curveto{27.61946pt}{-2.82861pt}{28.88586pt}{-1.56221pt}{28.88586pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{26.05725pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.05725pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{7.02861pt}{0.0pt}\pgfsys@lineto{19.02864pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.89058pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{54.94312pt}{0.0pt}\pgfsys@curveto{54.94312pt}{1.56221pt}{53.67671pt}{2.82861pt}{52.1145pt}{2.82861pt}\pgfsys@curveto{50.55229pt}{2.82861pt}{49.28589pt}{1.56221pt}{49.28589pt}{0.0pt}\pgfsys@curveto{49.28589pt}{-1.56221pt}{50.55229pt}{-2.82861pt}{52.1145pt}{-2.82861pt}\pgfsys@curveto{53.67671pt}{-2.82861pt}{54.94312pt}{-1.56221pt}{54.94312pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{52.1145pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{52.1145pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{33.08586pt}{0.0pt}\pgfsys@lineto{45.08589pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{47.94783pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}
{{{{}}}}{{}}{{}}{}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{78.85071pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\ldots}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{59.14311pt}{0.0pt}\pgfsys@lineto{71.14314pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}
{{{{}}}}{{}}{{}}{} {{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{115.91553pt}{0.0pt}\pgfsys@curveto{115.91553pt}{1.56221pt}{114.64912pt}{2.82861pt}{113.08691pt}{2.82861pt}\pgfsys@curveto{111.5247pt}{2.82861pt}{110.2583pt}{1.56221pt}{110.2583pt}{0.0pt}\pgfsys@curveto{110.2583pt}{-1.56221pt}{111.5247pt}{-2.82861pt}{113.08691pt}{-2.82861pt}\pgfsys@curveto{114.64912pt}{-2.82861pt}{115.91553pt}{-1.56221pt}{115.91553pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{113.08691pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{113.08691pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{94.05827pt}{0.0pt}\pgfsys@lineto{106.0583pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{108.92024pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{141.97278pt}{0.0pt}\pgfsys@curveto{141.97278pt}{1.56221pt}{140.70638pt}{2.82861pt}{139.14417pt}{2.82861pt}\pgfsys@curveto{137.58195pt}{2.82861pt}{136.31555pt}{1.56221pt}{136.31555pt}{0.0pt}\pgfsys@curveto{136.31555pt}{-1.56221pt}{137.58195pt}{-2.82861pt}{139.14417pt}{-2.82861pt}\pgfsys@curveto{140.70638pt}{-2.82861pt}{141.97278pt}{-1.56221pt}{141.97278pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{139.14417pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{139.14417pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{120.11552pt}{0.0pt}\pgfsys@lineto{132.11555pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{134.9775pt}{-13.00604pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$0$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@beginscope\pgfsys@invoke{ } {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{28.88586pt}{26.05725pt}\pgfsys@curveto{28.88586pt}{27.61946pt}{27.61946pt}{28.88586pt}{26.05725pt}{28.88586pt}\pgfsys@curveto{24.49504pt}{28.88586pt}{23.22864pt}{27.61946pt}{23.22864pt}{26.05725pt}\pgfsys@curveto{23.22864pt}{24.49504pt}{24.49504pt}{23.22864pt}{26.05725pt}{23.22864pt}\pgfsys@curveto{27.61946pt}{23.22864pt}{28.88586pt}{24.49504pt}{28.88586pt}{26.05725pt}\pgfsys@closepath\pgfsys@moveto{26.05725pt}{26.05725pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.05725pt}{26.05725pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{26.05725pt}{7.02861pt}\pgfsys@lineto{26.05725pt}{19.02864pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.7318pt}{13.93828pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$0$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} ;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@beginscope\pgfsys@invoke{ } {{}} {{{{}}}}{{}}{{}}{}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{115.91553pt}{26.05725pt}\pgfsys@curveto{115.91553pt}{27.61946pt}{114.64912pt}{28.88586pt}{113.08691pt}{28.88586pt}\pgfsys@curveto{111.5247pt}{28.88586pt}{110.2583pt}{27.61946pt}{110.2583pt}{26.05725pt}\pgfsys@curveto{110.2583pt}{24.49504pt}{111.5247pt}{23.22864pt}{113.08691pt}{23.22864pt}\pgfsys@curveto{114.64912pt}{23.22864pt}{115.91553pt}{24.49504pt}{115.91553pt}{26.05725pt}\pgfsys@closepath\pgfsys@moveto{113.08691pt}{26.05725pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{113.08691pt}{26.05725pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{113.08691pt}{7.02861pt}\pgfsys@lineto{113.08691pt}{19.02864pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{118.76146pt}{13.93828pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$0$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} ;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

###  3.10. Catalan numbers for $W_{n}(\Gamma,\Delta)$?

As mentioned in the introduction, it is natural to wonder if there are
reasonable product formulas for the dimensions of the spaces of
$(\chi,\mathrm{det})$ semi-invariants appearing in our non-commutative fibers,
as happens for complex reflection groups. Even though the statement of Lemma
3.3 seems _a priori_ discouraging, as it contains direct sums, one may observe
by a case-by-case analysis that in fact the dimension always factors into a
product of $n$ linear functions of $n$ divided by $n!$. We have collected the
results of this straightforward calculation in the next table. We note that
they do not depend on $\Gamma$ except through its relationship to $\Delta$
(though presumably their graded versions do).

$\begin{array}[]{c|c}\hline\cr\text{The pair}\ (\Gamma,\Delta)\ \text{and the
conditions on}\ \chi&\text{The dimension of}\ (L^{\otimes
n}\otimes\mathrm{det})^{(\chi,\mathrm{det})}\\\ \hline\cr&\\\
\Gamma=\Delta,\quad\chi\neq 1&\frac{2n(2n-1)(2n-2)\cdots(n+2)(n+1)}{n!}\\\
&\\\ \hline\cr&\\\
\Gamma=\Delta,\quad\chi=1&\frac{(2n-1)(2n-2)\cdots(n+1)n}{n!}\\\ &\\\
\hline\cr&\\\ |\Gamma:\Delta|=2,\quad\chi|_{\Delta}\neq
1&\frac{(4n-2)(2n-2)(2n-3)\cdots(n+1)n}{n!}\\\ &\\\ \hline\cr&\\\
|\Gamma:\Delta|=2,\quad\chi|_{\Delta}=1&\frac{(3n-2)(2n-2)(2n-3)\cdots(n+1)n}{n!}\\\
&\\\ \hline\cr&\\\ \Gamma\ \text{of type}\ E_{6}^{(1)},\
|\Gamma:\Delta|=3&\frac{(4n-3)(2n-2)(2n-3)\cdots(n+1)n}{n!}\\\ &\\\
\hline\cr&\\\ \Gamma\ \text{of type}\ D_{m+2}^{(1)},\
|\Gamma:\Delta|=4&\frac{(5n-4)(2n-2)(2n-3)\cdots(n+1)n}{n!}\\\ &\\\
\hline\cr\end{array}$

The results for $(\chi,\mathrm{triv})$ semi-invariants in the cases not
covered above are less interesting, but we present them here:

$\begin{array}[]{c|c}\hline\cr\text{The pair}\ (\Gamma,\Delta)\ \text{and the
conditions on}\ \chi&\text{The dimension of}\ (L^{\otimes
n}\otimes\mathrm{det})^{(\chi,\mathrm{triv})}\\\ \hline\cr&\\\
\Gamma=\Delta,\quad\chi\neq 1&n+1\\\ &\\\ \hline\cr&\\\
|\Gamma:\Delta|=2,\quad\chi|_{\Delta}\neq 1&2\\\ &\\\ \hline\cr\end{array}$

## References

  * [AjGr] C. Ajlia and S. Griffeth, _Representation theory and the diagonal coinvariant ring of the type B Weyl group_ , J. Pure Appl. Algebra, 228 (2024), no.6, Paper No. 107592.
  * [ARR] D. Armstrong, V. Reiner, and B. Rhoades, _Parking spaces_ , Adv. Math., 269 (2015), 647–706.
  * [BeCh] Y. Berest and O. Chalykh, _Quasi-invariants of complex reflection groups_ , Compos. Math., 147 (2011), no.3, 965–1002.
  * [BeRe] D. Bessis and V. Reiner, _Cyclic sieving of noncrossing partitions for complex reflection groups_ , Ann. Comb., 15 (2011), no. 2, 197–222.
  * [CBH] W. Crawley-Boevey and M. Holland, _Non-commutative deformations of Kleinian singularities_ , Duke Math. J., 92 (1998), no.3, 605–635.
  * [Coh] A. Cohen, _Finite quaternionic reflection groups_ , J. Algebra, 64 (1980), no.2, 293–324.
  * [Dol] I. Dolgachev, _McKay correspondence_ , course notes available at https://dept.math.lsa.umich.edu/ idolga/McKaybook.pdf
  * [EtGi] P. Etingof and V. Ginzburg, _Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism_ , Invent. Math., 147 (2002), no.2, 243–348.
  * [EtMo] P. Etingof and S. Montarani, _Finite dimensional representations of symplectic reflection algebras associated to wreath products_ , Represent. Theory, 9 (2005), 457–467.
  * [FGM] S. Fishel, S. Griffeth, and E. Manosalva, _Unitary representations of the Cherednik algebra: $V^{*}$-homology_, Math. Z., 299 (2021), no.3-4, 2215–2255.
  * [GNS] Y. Gomi, I. Nakamura, and K. Shinoda, _Coinvariant algebras of finite subgroups of $\mathrm{SL}(3,\mathbf{C})$_, Canad. J. Math., 56 (2004), no. 3, 495–528
  * [Gor] I. Gordon, _On the quotient ring by diagonal invariants_ , Invent. Math. 153 (2003), no. 3, 503–518.
  * [GoGr] I. Gordon and S. Griffeth, _Catalan numbers for complex reflection groups_ , Amer. J. Math., 134 (2012), no.6, 1491–1502.
  * [Gri1] S. Griffeth, _Towards a combinatorial representation theory for the rational Cherednik algebra of type $G(r,p,n)$_, Proc. Edinb. Math. Soc. (2), 53 (2010), no.2, 419–445.
  * [Gri2] S. Griffeth, _Orthogonal functions generalizing Jack polynomials_ , Trans. Amer. Math. Soc., 362 (2010), no.11, 6131–6157.
  * [Gri3] S. Griffeth, _Unitary representations of cyclotomic rational Cherednik algebras_ , J. Algebra, 512 (2018), 310–356.
  * [Gri4] S. Griffeth, _The diagonal coinvariant ring of a complex reflection group_ , Algebra Number Theory, 17 (2023), no.11, 2033–2053.
  * [Hai1] M. Haiman, _Conjectures on the quotient ring by diagonal invariants_ , J. Algebraic Combin., 3 (1994), no.1, 17–76.
  * [Hai2] M. Haiman, _Hilbert schemes, polygraphs and the Macdonald positivity conjecture_ , J. Amer. Math. Soc., 14 (2001), no.4, 941–1006.
  * [Hai3] M. Haiman, _Vanishing theorems and character formulas for the Hilbert scheme of points in the plane_ , Invent. Math., 149 (2002), no.2, 371–407.
  * [Los] I. Losev, _$1$ -dimensional representations and parabolic induction for $W$-algebras_, Adv. Math., 226 (2011), no.6, 4841–4883.
  * [McK1] J. McKay, _Graphs, singularities, and finite groups_ , The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), pp. 183–186, Proc. Sympos. Pure Math., 37, American Mathematical Society, Providence, RI, 1980.
  * [McK2] J. McKay, _Cartan matrices, finite groups of quaternions, and Kleinian singularities_ , Proc. Amer. Math. Soc., 81 (1981), no.1, 153–154.
  * [McK3] J. McKay, _Semi-affine Coxeter-Dynkin graphs and $G\subseteq\mathrm{SU}_{2}(\mathbf{C})$_, Dedicated to H. S. M. Coxeter on the occasion of his 90th birthday, Canad. J. Math., 51 (1999), no.6, 1226–1229.
  * [Rou] R. Rouquier, _$q$ -Schur algebras and complex reflection groups_, Mosc. Math. J., 8 (2008), no.1, 119–158, 184.
  * [Stu] C. Stump, _$q,t$ -Fuß-Catalan numbers for finite reflection groups_, J. Algebraic Combin., 32 (2010), no. 1, 67–97.
  * [Val] R. Vale, _Rational Cherednik algebras and diagonal coinvariants of $G(m,p,n)$_, J. Algebra, 311 (2007), no.1, 231–250.