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# Classification of affine normal $SL_{2}$-varieties with a dense orbit

Andres Fernandez Herrero and Rodrigo Horruitiner

###### Abstract

In this note we give a classification of all affine normal $SL_{2}$-varieties
containing an open dense orbit over an algebraically closed field of
characteristic zero. Such a classification was first obtained by Popov in
[Pop73]. Here we provide an alternative approach.

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Affine pointed $SL_{2}$-varieties with a dense orbit
  4. 4 Classification of affine spherical $SL_{2}$-varieties
  5. 5 Classification of affine normal $SL_{2}$-varieties with an open dense orbit
  6. A Some lemmas about binomial sums

##  1 Introduction

In this note we give a classification of affine normal $SL_{2}$-varieties with
a dense open orbit over an algebraically closed field of characteristic zero.
Our proof is based on elementary representation theory and the Hilbert-Mumford
criterion.

An affine algebraic normal $SL_{2}$-variety $X$ with a dense orbit is
determined by a subalgebra $k[X]$ of the coordinate ring $k[SL_{2}]$ that is
stable under the $SL_{2}$ action. This, in turn, is determined by an
“admissible” subalgebra $k[X]^{\bar{U}}$ of $k[SL_{2}]^{\bar{U}}$ of
invariants under the lower-triangular unipotent subgroup $\bar{U}$ of $SL_{2}$
(see Definition 3.1).

The question is then to determine which subalgebras are admissible. In section
3 we study the multiplicative structure of the left regular representation
$k[SL_{2}]$ and give a criterion for admissibility (Proposition 3.6). This
criterion is a central ingredient in the proofs of the results, and it has a
straightforward graphical interpretation inside the lattice of monomials in
$k[SL_{2}]^{\bar{U}}$.

If $k[X]^{\bar{U}}$ has dimension 1, then $X$ is spherical (Proposition 5.2).
In section 4 we use the Hilbert-Mumford criterion (Lemma 4.4) and the
admissibility criterion to obtain the classification in the case when $X$ is
spherical (Theorem 4.10).

In section 5 we proceed to the case when $k[X]^{\bar{U}}$ has dimension 2. In
this case we use the admissibility criterion to show that $X$ admits an extra
torus action induced by the right regular representation on $k[SL_{2}]$ (Lemma
5.4). This allows us to prove the main theorem of this paper:

######  Theorem.

5.10 Let $X$ be an affine normal $SL_{2}$-variety with an open dense orbit.
Then $X$ is $SL_{2}$-isomorphic to exactly one of the following

  1. (1)

A homogenenous space $SL_{2}/H$, where $H$ is as in Proposition 5.1.

  2. (2)

(See Definition 4.8) The spherical variety
$SL_{2}/\\!/(\mu_{f}\ltimes\overline{U})=Spec\left(R_{\infty}^{(f)}\right)$
for a unique positive integer $f$.

  3. (3)

(See Definition 5.7) $Spec\left(R^{(f)}_{q}\right)$ for a unique positive
integer $f$ and a unique rational number $q\geq 1$ . In this case the
stabilizer of the dense orbit is $\mu_{f}$.

This classification is in fact complete in the case where $X$ has dimension
$2$ without the assumption of normality. However, when $X$ has dimension $3$
it is not. See [Pop73, §4] for examples.

The results presented here are known, and were first proved by Popov in
[Pop73]. Popov considers an equivariant linear embedding $X\subset V$ ([Pop73,
Theorem 2]), and then proceeds to deduce intrinsic invariants of $X$ from
invariants of the embedding ([Pop73, Theorem 4]). See also Kraft’s book
[Kra85, III.4] for a shorter proof in the spirit of Popov’s argument. Another
approach to this classification can be found in the work of Luna and Vust
[LV83, §9], which uses their more general theory of spherical embeddings.

Our approach, in contrast to the previous ones, is based on the combinatorics
of the monomials of $k[X]^{\bar{U}}$ under the multiplicative structure of the
left regular representation. This lends itself to a straightforward graphical
interpretation of the results, and of the classifying invariants $(q,f)$
inside the lattice of monomials in $k[SL_{2}]^{\bar{U}}$. Even though the
results are known, we hope that someone will find this new presentation to be
useful.

See the work of Batyrev and Haddad [BH08] for applications of Theorem 5.10
above and further study of these varieties. In fact [BH08] gives an explicit
categorical quotient description of the $3$-dimensional $SL_{2}$-varieties
appearing in the classification. This is used to compute their Cox rings and
to construct $SL_{2}$-equivariant flips as GIT quotients.

Acknowledgements. We would like to thank Dan Halpern-Leistner for asking the
question that lead to the classification result, and Allen Knutson for making
us aware of the Luna-Vust theory. We would also like to thank Victor Batyrev.
from whom we learned about the results in [BH08].

##  2 Notation

Let $k$ be an algebraically closed field of characteristic $0$. All schemes
will be assumed to be defined over $k$. In this note we study varieties $X$
equipped with a left action of the algebraic group $SL_{2}$. We will
abbreviate this by saying that $X$ is a $SL_{2}$-variety.

We denote by $k[SL_{2}]$ the coordinate ring of the affine algebraic group
$SL_{2}$. We think of a generic matrix in $SL_{2}$ as having entries
$\begin{bmatrix}a&b\\\ c&d\end{bmatrix}$. Then we have
$k[SL_{2}]=k[a,b,c,d]/(ad-bc-1)$. The $k$-vector space $k[SL_{2}]$ acquires
the structure of a $SL_{2}\times SL_{2}$ representation. The first copy of
$SL_{2}$ acts via the left regular representation, while the second copy of
$SL_{2}$ acts via the right regular representation. We will employ the
superscript $(-)^{op}$ whenever we are thinking of the second copy of
$SL_{2}$. For example a $SL_{2}^{op}$-subrepresentation will be a subspace
that is stable with respect to the right regular action.

We will denote by $\overline{B}$ the Borel subgroup in $SL_{2}$ consisting of
lower triangular matrices. We let $T$ denote the maximal torus
$T\subset\overline{B}$ defined by

$T=\left\\{\begin{bmatrix}t^{-1}&0\\\
0&t\end{bmatrix}\right\\}\,\cong\mathbb{G}_{m}$

We will write $\overline{U}$ for the unipotent radical of $\overline{B}$.

##  3 Affine pointed $SL_{2}$-varieties with a dense orbit

In this section we describe a correspondence between affine normal pointed
$SL_{2}$-varieties with a dense orbit and certain class of subalgebras of the
polynomial ring $k[a,b]$. We call this class of subalgebras “admissible”.

Let $X$ be an affine normal $SL_{2}$-variety. Let $R$ denote the coordinate
ring of $X$. Choose $p\in X(k)$. We can define a map $\phi:SL_{2}\rightarrow
X$ given by $\phi(g)=g\cdot p$. Suppose further that the $SL_{2}$-orbit of $p$
is dense in $X$. This means that $\phi$ is dominant, which implies that we can
view $R$ as a subalgebra of $k[SL_{2}]$.

We can therefore understand such pointed $SL_{2}$-varieties once we determine
the (left) $SL_{2}$-stable, normal, finitely generated subalgebras of
$k[SL_{2}]$. By highest weight theory, these are determined by their
$\overline{U}$-invariants.

Note that the $\overline{U}$-invariants of the left regular representation are
$k[SL_{2}]^{\overline{U}}=k[a,b]$. Here the torus $T$ acts with weight $1$ on
both $a$ and $b$. This motivates the following definition.

######  Definition 3.1.

Let $A$ be a subalgebra of $k[a,b]$. We say that $A$ is admissible if

  1. (a)

$A$ is homogeneous for the grading by total degree (i.e. it is $T$-stable).

  2. (b)

$A$ is normal and finitely generated.

  3. (c)

Let $V_{A}$ denote the $SL_{2}$-subrepresentation of $k[SL_{2}]$ generated by
$A$. Then $V_{A}$ is closed under multiplication (i.e. it is a subalgebra of
$k[SL_{2}]$).

Now we state the promised correspondence.

######  Lemma 3.2.

The map $A\mapsto V_{A}$ sets up a bijection between admissible subalgebras
and $SL_{2}$-stable, normal, finitely generated $k$-subalgebras of the
coordinate ring $k[SL_{2}]$.

###### Proof.

Let’s define an inverse map. For any $SL_{2}$-stable subalgebra $R$ of
$k[SL_{2}]$, we define $A_{R}\vcentcolon=R^{\overline{U}}=R\cap k[a,b]$. This
is a subalgebra of $k[a,b]$. It is $T$-stable because $R$ is $T$-stable. By
highest weight theory, any representation of $SL_{2}$ is generated by its
subspace of $\overline{U}$-invariants. So we get $V_{A_{R}}=R$.

This establishes a bijection between $SL_{2}$-stable subalgebras of
$k[SL_{2}]$ and subalgebras of $k[a,b]$ satisfying conditions (a) and (c) in
Definition 3.1. In order to conclude the proof of the lemma, it suffices to
show that $R$ is finitely generated and normal if and only if $A_{R}$ is
finitely generated and normal.

Suppose that $R$ is finitely generated. Then $R^{\overline{U}}$ is finitely
generated by [Per][Thm. 4.1.12]. Conversely, suppose that $A_{R}$ is finitely
generated. Let $\\{v_{i}\\}_{i\in I}$ be a finite set of generators of
$A_{R}$. Let $W_{\\{v_{i}\\}}$ denote the finite dimensional
$SL_{2}$-submodule of $k[SL_{2}]$ generated by the $v_{i}$. Then we have that
the algebra $k[W_{\\{v_{i}\\}}]$ generated by $W_{\\{v_{i}\\}}$ is a left
$SL_{2}$-stable subalgebra of $R$. By assumption on the set $\\{v_{i}\\}$, we
know that $k[W_{\\{v_{i}\\}}]$ contains $A_{R}$. Therefore, we must have
$k[W_{\\{v_{i}\\}}]=R$ by highest weight theory. So $R$ is finitely generated.

The fact that $R$ is normal if and only if $A_{R}$ is normal follows from the
argument in [Per][Prop. 4.1.16 (ii)]. ∎

The rest of this section is dedicated to proving Proposition 3.6. This is a
criterion for determining when condition (c) in Definition 3.1 is satisfied.

We will need to understand the multiplicative structure of the left regular
representation $k[SL_{2}]$. Consider the coordinate ring $W=k[a,b,c,d]$ of the
vector space of $2\times 2$ matrices. $SL_{2}$ acts on this ring by
precomposing with the inverse of multiplication on the left. We have
$W=\bigoplus_{i,j\in\mathbb{N}}W^{i,j}$ where

$W^{i,j}=Sym^{i}\left(ka\oplus kc\right)\otimes Sym^{j}\left(kb\oplus
kd\right)$

By representation theory of $SL_{2}$, we know that $W^{(i,j)}$ breaks up into
a direct sum of irreducibles

$W^{i,j}=\bigoplus_{s=0}^{min(i,j)}L^{i,j}_{i+j-2s}$

Here $L^{i,j}_{i+j-2s}$ has highest weight $i+j-2s$ with respect to the Borel
$\overline{B}$. It is apparent that $L^{i,j}_{i+j}$ has highest weight vector
$a^{i}b^{j}$. In order to determine the highest weight vectors for the other
summands, we will do a trick with the determinant. Note that multiplication by
the invariant polynomial $ad-bc$ induces an injective map
$W^{i,j}\hookrightarrow W^{i+1,j+1}$. By the decomposition above, this must
induce isomorphisms $L^{i,j}_{i+j-2s}\xrightarrow{\sim}L^{i+1,j+1}_{i+j-2s}$.
We conclude by using this map that $L^{i,j}_{i+j-2s}$ has highest weight
vector $(ad-bc)^{s}a^{i-s}b^{j-s}$.

Thanks to the weight vectors obtained just above, we can describe explicitly
the highest weight vectors of a tensor product of abstract irreducible
representations of $SL_{2}$. But before we do that we need some notation.

######  Definition 3.3.

Let $V$ be a representation of $SL_{2}$ and let $v\in V$ be nonzero weight
vector with weight $n$. Then for every $e\in\mathbb{N}$, we define
$\langle-e\rangle v$ to be the unique vectors determined by the equation

$\begin{bmatrix}1&-x\\\ 0&1\end{bmatrix}\cdot
v=\sum_{e\in\mathbb{N}}\binom{n}{e}\,x^{e}\,\langle-e\rangle v$

In other words, $\langle-d\rangle v$ is the component of weight $n-2e$ in the
vector
$\begin{bmatrix}1&-1\\\ 0&1\end{bmatrix}\cdot\left(\binom{n}{e}^{-1}v\right)$

######  Proposition 3.4.

Let $V^{1}$, $V^{2}$ be two irreducible representations of $SL_{2}$ with
highest weight $n_{1}$ and $n_{2}$ respectively. Let $v^{1}$ and $v^{2}$ be
highest weight vectors of $V^{1}$ and $V^{2}$ respectively. The tensor product
breaks up as a direct sum of irreducible representations

$V^{1}\otimes V^{2}=\bigoplus_{s=0}^{min(n_{1},n_{2})}V_{n_{1}+n_{2}-2s}$

where $V_{n_{1}+n_{2}-2s}$ is an irreducible representation of highest weight
$n_{1}+n_{2}-2s$ and highest weight vector

$v_{s}=\sum_{e=0}^{s}(-1)^{e}\binom{s}{e}\langle-e\rangle v_{1}\otimes\langle
e-s\rangle v_{2}$

###### Proof.

It suffices to prove this theorem for a specific realization of the
irreducible representations $V^{1}$ and $V^{2}$. For example we can use
$V^{1}\cong Sym^{n_{1}}\left(ka\oplus kc\right)$ and $V^{2}\cong
Sym^{n_{2}}\left(kb\oplus kd\right)$. We can take $v^{1}=a^{n_{1}}$ and
$v^{2}=b^{n_{2}}$. Now one can check that the vector $v_{s}$ described in this
proposition reduces to $(ad-bc)^{s}a^{n_{1}-s}b^{n_{2}-s}$. We have seen in
our discussion above that this is indeed the required highest weight vector. ∎

######  Corollary 3.5.

Let $p_{1}=\sum_{i=0}^{n_{1}}z_{1,i}a^{i}b^{n_{1}-i}$ be a homogeneous
polynomial of total degree $n_{1}$ in $k[a,b]$. Similarly, let
$p_{2}=\sum_{j=0}^{n_{2}}z_{2,j}a^{j}b^{n_{2}-j}$ be a homogeneous polynomial
of degree $n_{2}$. Let $V^{p_{1}}$ (resp. $V^{p_{2}}$) be the irreducible
subrepresentation of $W=k[a,b,c,d]$ with highest weight vector $p_{1}$ (resp.
$p_{2}$). Then the tensor product decomposes into a direct sum

$V^{p_{1}}\otimes
V^{p_{2}}=\bigoplus_{s=0}^{min(n_{2},n_{2})}V^{p_{1},p_{2}}_{n_{1}+n_{2}-2s}$

where $V^{p_{1},p_{2}}_{n_{1}+n_{2}-2s}$ is an irreducible representation of
highest weight $n_{1}+n_{2}-2s$.

The image of a highest weight vector $v^{p_{1},p_{2}}_{s}$ of
$V^{p_{1},p_{2}}_{n_{1}+n_{2}-2s}$ under the multiplication map
$m:V^{p_{1}}\otimes V^{p_{2}}\rightarrow W$ is

$m(v^{p_{1},p_{2}}_{s})=\sum_{\alpha=s}^{n_{1}+n_{2}-s}y_{\alpha,s}^{p_{1},p_{2}}\,(ad-
bc)^{s}\cdot a^{\alpha-s}\cdot b^{n_{1}+n_{2}-\alpha-s}$

where $y_{\alpha,s}^{p_{1},p_{2}}$ is given by

$y_{\alpha,s}^{p_{1},p_{2}}=\sum_{i+j=\alpha}\sum_{e=0}^{s}(-1)^{e}\frac{\binom{s}{e}}{\binom{n_{1}}{e}\binom{n_{2}}{s-e}}\binom{n_{1}-i}{e}\binom{n_{2}-j}{s-e}z_{1,i}z_{2,j}$

###### Proof.

We know that $m(v^{p_{1},p_{2}}_{s})$ is a linear combination of highest
weight vectors in $W$, because multiplication is $SL_{2}$-equivariant. We have
seen that the highest weight vectors in $W$ of weight $n_{1}+n_{2}-2s$ and
total degree $n_{1}+n_{2}$ in the variables $a,b,c,d$ are of the form $(ad-
bc)^{s}\cdot a^{\alpha-s}\cdot b^{n_{1}+n_{2}-\alpha-s}$ for some
$s\leq\alpha\leq n_{1}+n_{2}-s$. So we only need to determine the coefficients
$y_{\alpha,s}^{p_{1},p_{2}}$. This can be done by using the formula in
Proposition 3.4, expanding with the binomial theorem and looking at the
coefficient of $a^{\alpha}\cdot d^{s}\cdot b^{n_{1}+n_{2}-\alpha-s}$. ∎

We conclude this section with a useful criterion to determine if a subalgebra
of $k[a,b]$ is admissible.

######  Proposition 3.6.

Let $A\subset k[a,b]$ be a normal, finitely generated homogeneous subalgebra.
Then $A$ is admissible if and only if for all pairs of homogeneous polynomials
$p_{1}=\sum_{i=0}^{n_{1}}z_{1,i}a^{i}b^{n_{1}-i}$ and
$p_{2}=\sum_{j=0}^{n_{2}}z_{2,j}a^{j}b^{n_{2}-j}$ in $A$, all of the
polynomials $w^{p_{1},p_{2}}_{s}$ defined for $0\leq s\leq min(n_{1},n_{2})$
by

$w^{p_{1},p_{2}}_{s}=\sum_{\alpha=s}^{n_{1}+n_{2}-s}y_{\alpha,s}^{p_{1},p_{2}}\,a^{\alpha-s}\,b^{n_{1}+n_{2}-\alpha-s}$

with

$y_{\alpha,s}^{p_{1},p_{2}}=\sum_{i+j=\alpha}\sum_{e=0}^{s}(-1)^{e}\frac{\binom{s}{e}}{\binom{n_{1}}{e}\binom{n_{2}}{s-e}}\binom{n_{1}-i}{e}\binom{n_{2}-j}{s-e}z_{1,i}z_{2,j}$

also belong to the subalgebra $A$.

###### Proof.

Let $V_{A}$ denote the $SL_{2}$ subrepresentation of $k[SL_{2}]$ generated by
$A$. We can see that $V_{A}$ is closed under multiplication if and only if for
all pairs of homogeneous polynomials $(p_{1},p_{2})$ we have that all of the
highest weight vectors of the product $V^{p_{1}}\cdot V^{p_{2}}$ are also in
$A$. By Proposition 3.5 these highest weight vectors are just the polynomials
$w^{p_{1},p_{2}}_{s}$ (recall that $ad-bc=1$ in $k[SL_{2}]$). ∎

$u$$v$$p_{1}$$p_{2}$$p_{1}p_{2}=w_{0}^{p_{1},p_{2}}$$w_{1}^{p_{1},p_{2}}$$w_{2}^{p_{1},p_{2}}$$w_{3}^{p_{1},p_{2}}$

Figure 1: Admissibility criterion for two homogeneous polynomials
$p_{1}=ab^{2}+b^{3}$, $p_{2}=a^{2}b^{2}+ab^{3}+b^{4}$

.

This criterion has a simple graphical interpretation. Give two polynomials
$p_{1}$ and $p_{2}$ in $A$, we may consider their Newton polygons (i.e. the
convex hull of points $(u,v)\in\mathbb{N}^{2}$ corresponding to monomials
$a^{u}b^{v}$ in their support) and draw a diagram as in Figure 1 above.

As long as the coefficients $y_{\alpha,s}^{p_{1},p_{2}}$ are nonzero, we can
deduce that some elements of convenient form must be present in the algebra
$A$ starting from some known elements $p_{1}$, $p_{2}$ (for example,
$w_{3}^{p_{1},p_{2}}=y^{p_{1},p_{2}}_{3,3}b$ in the image).

##  4 Classification of affine spherical $SL_{2}$-varieties

For this section, we will put an additional assumption on our affine pointed
$SL_{2}$-variety $(X,p)$. Recall that we can use the point $p$ to view $X$ as
$Spec(R)$ for some $SL_{2}$-stable subalgebra $R\subset k[SL_{2}]$.

######  Lemma 4.1.

Let $X=Spec(R)$ be an affine normal pointed variety as above. Then the
following statements are equivalent.

  1. (1)

The orbit $\overline{B}\cdot p$ is dense.

  2. (2)

Every $T$-weight space of the algebra of invariants $R^{\overline{U}}=R\cap
k[a,b]$ is one-dimensional.

###### Proof.

See [Per][Lemma 4.2.2]. ∎

As a consequence we get the following proposition.

######  Proposition 4.2.

The correspondence $A\mapsto V_{A}$ defines a bijection between the following
two sets

  1. (a)

Admissible subalgebras $A$ such that for all $n\in\mathbb{N}$ the $k$-space of
homogeneous polynomials of degree $n$ in $A$ has dimension at most $1$.

  2. (b)

Affine normal pointed $SL_{2}$-varieties $X=Spec(V_{A})$ such that
$\overline{B}\cdot p$ is dense.

###### Proof.

This is an immediate consequence of Lemma 3.2 and Lemma 4.1. ∎

######  Definition 4.3.

An affine normal $SL_{2}$-variety $X$ is called spherical if it has a
$\overline{B}$-orbit that is dense.

In order to classify spherical varieties, we will need a couple of lemmas.
Recall that a one-parameter subgroup of $SL_{2}$ is a homomorphism of
algebraic groups $\gamma:\mathbb{G}_{m}\rightarrow SL_{2}$. Let
$\mathbb{G}_{m}\hookrightarrow\mathbb{A}^{1}$ be the usual open immersion.
Given a morphism $\phi:\mathbb{G}_{m}\rightarrow X$ we say that
$\text{lim}_{t\to 0}\,\phi(t)$ exists if the map $\phi$ extends to a morphism
$\widetilde{\phi}:\mathbb{A}^{1}\rightarrow X$. If this is the case, we will
use the notation $\text{lim}_{t\to 0}\,\phi(t)=\widetilde{\phi}(0)$.

######  Lemma 4.4.

Let $X$ be an affine $SL_{2}$-variety and let $p\in X(k)$. Suppose that the
orbit $SL_{2}\cdot p$ is open dense in $X$. For any closed point $x\in
X(k)\setminus(SL_{2}(k)\cdot p)$, there exists a nonconstant one-parameter
subgroup $\gamma:\mathbb{G}_{m}\rightarrow SL_{2}$ such that $\text{lim}_{t\to
0}\,\gamma(t)\cdot p$ is contained in the closure of the orbit $SL_{2}\cdot
x$.

###### Proof.

This is a version of the Hilbert-Mumford criterion. It follows from a
modification of the argument in [MFK94][pg.53-54] after embedding $X$ into the
total space of a linear representation of $SL_{2}$. ∎

Let $\gamma:\mathbb{G}_{m}\rightarrow SL_{2}$ be a nontrivial one-parameter
subgroup. Recall that there is a Borel subgroup $B_{\gamma}$ associated to
$\gamma$. It is defined by

$B_{\gamma}=\left\\{\,g\in SL_{2}\,\,\mid\,\,\text{lim}_{t\to
0}\,\gamma(t)\,g\,\gamma(t)^{-1}\;\;\text{exists in $SL_{2}$}\,\right\\}$

The image $\text{Im}\,\gamma$ is a maximal torus inside $B_{\gamma}$. The
dominant weights of $B_{\gamma}$ are the characters of $\text{Im}\,\gamma$
that pullback to a nonnegative weight of $\mathbb{G}_{m}$ under the
homomorphism $\gamma$.

For the next lemma, we let $(X,p)$ be an affine pointed variety such that
$SL_{2}\cdot p$ is dense. As usual, we view the coordinate ring $R$ as a
$SL_{2}$-stable subalgebra of $k[SL_{2}]$. Recall that we use the supercript
$(-)^{op}$ to denote the right regular action.

######  Lemma 4.5.

Let $\gamma:\mathbb{G}_{m}\rightarrow SL_{2}$ be a non-constant one-parameter
subgroup. Let $B_{\gamma}$ denote the corresponding Borel subgroup. Then the
following statements are equivalent

  1. (1)

The limit $\text{lim}_{t\to 0}\,\gamma(t)\cdot p$ exists in $X$

  2. (2)

The subalgebra $R\subset k[SL_{2}]$ is contained in the sum of those
$T^{op}$-weight spaces of $k[SL_{2}]$ that are dominant for the Borel
$B_{\gamma}^{op}$.

###### Proof.

Consider the composition

$\displaystyle\varphi:\;\;SL_{2}\times\mathbb{G}_{m}\;\;\xrightarrow{m}\;\;SL_{2}\;\;\longrightarrow\;\;X$
$\displaystyle(g,t)\;\;\;\;\mapsto\;\;\;gt\;\;\mapsto\;\;\;\;g\gamma(t)\cdot
p$

At the level of coordinate rings, we have

$\varphi^{*}:R\;\hookrightarrow\;k[SL_{2}]\;\xrightarrow{m^{*}}\;k[SL_{2}]\otimes
k[T,T^{-1}]$

Notice that $\mathbb{G}_{m}$ acts by multiplication on the right on
$k[SL_{2}]$ via the homomorphism $\gamma$. The corresponding comodule map is
$m^{*}$ above. If $f\in R$, we can break $f$ into a sum of weight vectors
$f=\sum_{n\in\mathbb{Z}}f_{n}$ with $f_{n}\in k[Sl_{2}]$ having weight $n$. By
definition we have

$\varphi^{*}(f)=\sum_{n\in\mathbb{Z}}f_{n}\otimes T^{n}$

Assume that $\text{lim}_{t\to 0}\,\gamma(t)\cdot p$ exists. By definition,
this means that the map
$\varphi|_{\\{\text{id}\\}\times\mathbb{G}_{m}}:\mathbb{G}_{m}\rightarrow X$
extends to a map $\widetilde{\varphi}_{0}:\mathbb{A}^{1}\rightarrow X$.
Therefore, we get a $G$-equivariant map

$\displaystyle\widetilde{\varphi}:\;\;G\times\mathbb{A}^{1}\;\;\;\longrightarrow\;\;X$
$\displaystyle(g,t)\;\;\;\mapsto\;\;\;g\cdot\widetilde{\varphi}_{0}(t)$

By construction this map agrees with $\varphi$ on $G\times\mathbb{G}_{m}$. We
conclude that the map $\varphi^{*}:R\rightarrow k[SL_{2}]\otimes k[T,T^{-1}]$
factors through the subring $k[SL_{2}]\otimes k[T]\subset k[SL_{2}]\otimes
k[T,T^{-1}]$. This means that for all $f\in R$ we have $f_{n}=0$ for $n<0$. In
other words, $f$ belongs to the sum of weight spaces of the right regular
representation $k[SL_{2}]$ that are dominant for $B_{\gamma}$.

Conversely, suppose that for any $f\in R$ we have $f_{n}=0$ for all $n<0$.
This implies that the morphism $\varphi:G\times\mathbb{G}_{m}\rightarrow X$
extends to to a morphism $\widetilde{\varphi}:G\times\mathbb{A}^{1}\rightarrow
X$. We can restrict $\widetilde{\varphi}$ to $\\{id\\}\times\mathbb{A}^{1}$ in
order to deduce that the limit $\text{lim}_{t\to 0}\,\gamma(t)\cdot p$ exists.
∎

The following is the main step in classifying affine spherical
$SL_{2}$-varieties.

######  Proposition 4.6.

Let $X$ be an affine spherical $SL_{2}$-variety. Then $X$ is
$SL_{2}$-isomorphic to one of the following

  1. (a)

A homogeneous space $G/H$ for some closed algebraic subgroup $H\subset G$.

  2. (b)

$Spec\left(V_{k[b^{f}]}\right)$ for some positive integer $f$.

  3. (c)

$Spec\left(V_{k[ab]}\right)$.

  4. (d)

$Spec\left(V_{k[(ab)^{2}]}\right)$.

###### Proof.

Choose a point $p$ in the dense $\overline{B}$-orbit. This allows us to view
$X$ as $Spec(R)$ for some $SL_{2}$-stable subalgebra $R\subset k[SL_{2}]$.

Assume that $X$ is not a homogeneous space. By the Hilbert-Mumford criterion
(Lemma 4.4) there exists a nontrivial one-parameter subgroup
$\gamma:\mathbb{G}_{m}\rightarrow SL_{2}$ such that $\text{lim}_{t\to
0}\,\gamma(t)\cdot p$ exists. Let $B_{\gamma}$ be the corresponding Borel
subgroup. Notice that right multiplying the algebra $R$ by an element $g\in
SL_{2}(k)$ establishes an $SL_{2}$-isomorphism
$Spec(R)\xrightarrow{\sim}Spec(R\cdot g)$. By Lemma 4.5, we have that
$\text{lim}_{t\to 0}\,g^{-1}\gamma(t)g\cdot p$ exists in $Spec(R\cdot g)$. The
corresponding Borel subgroup will be $g^{-1}B_{\gamma}g$. By choosing an
appropriate element $g\in SL_{2}(k)$ we can assume without loss of generality
that $B_{\gamma}=\overline{B}$.

Let $A=R\cap k[a,b]$ be the associated admissible subalgebra of $k[a,b]$. If
$A$ is the field of constants $k$ then we have $R=k$. So the corresponding
variety $X=Spec(k)$ is the homogeneous space $SL_{2}/SL_{2}$, which is not the
case by assumption. Therefore $A$ contains some homogeneous polynomial with
positive degree in $k[a,b]$.

Let $p$ denote a homogeneous polynomial of smallest positive degree $n$ in
$A$. We claim that $A=k[p]$. Indeed let $S$ denote the set of homogeneous
polynomials in $A$. It follows from the multiplicity condition $(2)$ in Lemma
4.1 that all the weight spaces of the localization $A[S^{-1}]$ have dimension
$1$. Let $l\mathbb{Z}\subset\mathbb{Z}$ denote the subgroup of weights of
$A[S^{-1}]$, where $l$ is a positive integer. If $l=n$ then multiplicity one
implies that $A[S^{-1}]=k[p^{\pm 1}]$. Therefore $A=k[p]$. If $l\neq n$ then
$l$ is a proper divisor of $n$. Let $p^{\prime}$ be an element of $A[S^{-1}]$
of weight $l$. By multiplicity one for the weight spaces, we must have
$(p^{\prime})^{\frac{n}{l}}\in k\,p$. Since $A$ is normal, this implies
$p^{\prime}\in A$. So we get a contradiction to the minimality of the degree
$n$. This concludes the proof of the claim $A=k[p]$.

Suppose that $p=\sum_{i=0}^{n}z_{i}a^{i}b^{n-i}$. By Lemma 4.5, $p$ must be
contained in a sum of $\overline{B}^{op}$-dominant weight spaces. Concretely,
this means that $z_{i}=0$ whenever $n-i<i$. Let us denote by $m$ the largest
index such that $z_{m}\neq 0$. We must have $m\leq\lfloor\frac{n}{2}\rfloor$.

Let’s apply the criterion in Proposition 3.6 to the pair $(p,p)$. For each
$0\leq s\leq n$, we want to determine whether the polymomial $w^{p,p}_{s}$ is
in the subalgebra $A=k[p]$. Recall that

$w^{p,p}_{s}=\sum_{\alpha=s}^{2n-s}y_{\alpha,s}^{p,p}\,a^{\alpha-s}\cdot
b^{n_{1}+n_{2}-\alpha-s}$

with

$y_{\alpha,s}^{p,p}=\sum_{i+j=\alpha}\sum_{e=0}^{s}(-1)^{e}\frac{\binom{s}{e}}{\binom{n}{e}\binom{n}{s-e}}\binom{n-i}{e}\binom{n-j}{s-e}z_{i}z_{j}$

Note that $y_{\alpha,s}^{p,p}=0$ for all $\alpha>2m$. This is because
$z_{i}=0$ whenever $i>m$. This means that for $s=2m$ we have a single term

$w^{p,p}_{2m}=y_{2m,2m}^{p,p}\,b^{2n-4m}$

with

$y_{2m,2m}^{p,p}=2z_{m}^{2}\cdot\sum_{e=0}^{2m}(-1)^{e}\frac{\binom{2m}{e}}{\binom{n}{e}\binom{n}{2m-e}}\binom{n-m}{e}\binom{n-m}{2m-e}$

We have $z_{m}\neq 0$ by assumption. Lemma A.1 (with $i=m$ and $j=n-m$)
implies that $y_{2m,2m}^{p,p}\neq 0$. This means that $b^{2n-4m}$ must be a
constant multiple of a power of $p$. We have four possible cases.

  1. (C1)

$n\neq 2m$. Then $b^{2n-4m}$ is not a scalar. This means that $p$ can be taken
to be a power of $b$. So $A=k[b^{f}]$ for some positive integer $f$. Therefore
$X$ satisfies $(b)$ in the proposition.

  2. (C2)

$n=2$ and $m=1$. Then up to multiplying $p$ by a scalar, we can write
$p=ab+zb^{2}$ for some $z\in k$. Right multiplication by the element
$\begin{bmatrix}1&0\\\ -z&1\end{bmatrix}$ establishes a $SL_{2}$-isomorphism
between $k[ab-zb^{2}]$ and the algebra $k[ab]$. This means that $X$ satisfies
$(c)$ in the proposition.

  3. (C3)

$n=4$ and $m=2$. Then $p$ can be chosen to be of the form
$p=a^{2}b^{2}+z_{1}ab^{3}+z_{0}b^{4}$. We can right multiply by
$\begin{bmatrix}1&0\\\ -\frac{1}{2}z_{1}&1\end{bmatrix}$ in order reduce to
$z_{1}=0$. So $A=k[p]=k[a^{2}b^{2}+z_{0}b^{4}]$. We claim that this algebra is
admissible only if $z_{0}=0$. Indeed, the vector $w^{p,p}_{2}$ has a single
nonzero summand in this case

$w^{p,p}_{2}=y_{4,2}^{p,p}a^{2}b^{2}$

with

$\displaystyle
y_{2,2}^{p,p}=\sum_{e=0}^{2}(-1)^{e}\frac{\binom{2}{e}}{\binom{4}{e}\binom{4}{2-e}}\binom{2}{e}\binom{4}{2-e}z_{0}+\sum_{e=0}^{2}(-1)^{e}\frac{\binom{2}{e}}{\binom{4}{e}\binom{4}{2-e}}\binom{4}{e}\binom{2}{2-e}z_{0}$

This simplifies to

$y_{2,2}^{p,p}=\frac{1}{3}z_{0}$

So we have $w^{p,p}_{2}=\frac{1}{3}z_{0}a^{2}b^{2}$. This can’t be in $A=k[p]$
unless $z_{0}=0$.

We conclude that $A=k[(ab)^{2}]$. Therefore $X$ satisfies $(d)$ in the
proposition.

  4. (C4)

$n=2m$ and $m>2$. We will show that this case can’t actually happen.

We can write $p=z_{m}(ab)^{m}+\sum_{i=0}^{m-1}z_{i}a^{i}b^{2m-i}$ with
$z_{m}\neq 0$. Let’s look at the highest weight vector $w^{p,p}_{2}$. Notice
that the degree of $w^{p,p}_{2}$ is $4m-4$. Since $m>2$, the only way
$w^{p,p}_{2}$ can be a multiple of a power of $p$ is if $w^{p,p}_{2}=0$ (by
degree count). This means in particular that $y_{2m,2}^{p,p}$ must be $0$.
Since $z_{i}=0$ for $i>m$, the expression for $y_{2m,2}^{p,p}$ simplifies to

$\displaystyle
y_{2m,2}^{p,p}=\sum_{e=0}^{2}(-1)^{e}\frac{\binom{2}{e}}{\binom{2m}{e}\binom{2m}{2-e}}\binom{m}{e}\binom{m}{2-e}z_{m}^{2}$

By Lemma A.2 with $i=0$, we have

$y_{2m,2}^{p,p}=z_{m}^{2}\cdot\frac{-1}{2m-1}$

So we see that $y_{2m,2}^{p,p}=0$ implies that $z_{m}=0$, a contradiction.

∎

$u$$v$$p$$p^{2}=w_{0}^{p,p}$$w_{2}^{p,p}$

$u$$v$$p$$p^{2}=w_{0}^{p,p}$$w_{2m}^{p,p}=b^{f}$

Figure 2: Application of the criterion in Proposition 3.6 to $(p,p)$ in the
proof of Proposition 4.6. By Lemma 4.5, $p$ lies above the dotted line as
shown. Cases (C4) on the left and (C1) on the right.

The following proposition shows that cases $(b)$, $(c)$ and $(d)$ can actually
arise.

######  Proposition 4.7.

For any positive integer $f$, the algebra $k[b^{f}]$ is admissible. The
algebras $k[(ab)]$ and $k(ab)^{2}]$ are also admissible.

###### Proof.

Set $R_{f}=k[b^{f},db^{f-1},d^{2}b^{f-2},...,d^{f}]$ inside $k[SL_{2}]$. Note
that $R_{f}$ is $SL_{2}$-stable. We have $k[b^{f}]=R_{f}\cap k[a,b]$. This
shows that $V_{k[b^{f}]}=R_{f}$ in part $(b)$ of the previous proposition.

It can be checked that we have $k[ab]=k[ab,ad,bc,cd]\,\cap\,k[a,b]$. So in
part $(c)$ we have $V_{k[ab]}=k[ab,ad,bc,cd]$.

Finally, one can check in a similar way that $V_{k[(ab)^{2}]}$ is the
subalgebra
$k[a^{2}b^{2},ab(ad+bc),cd(ad+bc),c^{2}d^{2}]$ inside $k[SL_{2}]$. ∎

######  Definition 4.8.

Let $f$ be a positive integer. We define $R_{\infty}^{(f)}$ to be the
$SL_{2}$-stable subalgebra of $k[SL_{2}]$ generated by $k[b^{f}]$. More
concretely,

$R_{\infty}^{(f)}=k[b^{f},db^{f-1},d^{2}b^{f-2},...,d^{f}]$

Note that $R_{\infty}^{(f)}$ is the subalgebra of
$(\mu_{f}\ltimes\overline{U})^{op}$-invariants
$k[SL_{2}]^{(\mu_{f}\ltimes\overline{U})^{op}}$. Here
$\mu_{f}\ltimes\overline{U}\subset T\ltimes\overline{U}=\overline{B}$.

Now it remains to classify the affine spherical homogeneous spaces.

######  Proposition 4.9.

Up to isomorphism, the affine spherical homogeneous spaces for $SL_{2}$ are

  1. (a)

$SL_{2}/SL_{2}=Spec(k)$.

  2. (b)

$SL_{2}/T$. This is isomorphic to $Spec\left(V_{k[ab]}\right)$.

  3. (c)

$SL_{2}/N_{T}$. This isomorphic to $Spec\left(V_{k[(ab)^{2}}\right)$.

###### Proof.

By Matsushima’s criterion [Ric77] the homogeneous space $SL_{2}/H$ is affine
if and only if $H$ is reductive. If $SL_{2}/H$ has a dense
$\overline{B}$-orbit, then by dimension count we must have $\text{dim}\,H\geq
1$. By the classification of connected reductive groups, we must have
$\text{dim}H=1\text{ or }3$. If $\text{dim}\,H=3$, then $H=SL_{2}$. If
$\text{dim}\,H=1$, then the neutral component of $H$ must be a torus. By
conjugating we can assume that the neutral component of $H$ is $T$. So we must
have either $H=T$ or $H$ is the normalizer of the torus
$N_{T}=\mathbb{Z}/2\mathbb{Z}\ltimes T$.

The explicit description of the algebras follows from taking the subalgebra of
$H^{op}$-invariants of $k[a,b]$ in each case. ∎

######  Theorem 4.10.

Every affine spherical $SL_{2}$-variety is $SL_{2}$-isomorphic to exactly one
of the following

  1. (a)

$SL_{2}/SL_{2}=Spec(k)$.

  2. (b)

$SL_{2}/T$.

  3. (c)

$SL_{2}/N_{T}$.

  4. (d)

(See Definition 4.8)
$SL_{2}/\\!/(\mu_{f}\ltimes\overline{U})=Spec\left(R_{\infty}^{(f)}\right)$
for a unique positive integer $f$.

###### Proof.

The fact that this list is exhaustive follows from Proposition 4.6 and
Proposition 4.9. We just need to show that no two of these spherical varieties
are $SL_{2}$-isomorphic. $SL_{2}/SL_{2}$ is the only one that has dimension
$0$, so it can’t be isomorphic to the others.

In order to study the other ones, we can look at the algebras of
$\overline{U}$-invariants. Recall by Proposition 4.7

  1. $k[SL_{2}/T]^{\overline{U}}\cong k[ab]$.

  2. $k[SL_{2}/N_{T}]^{\overline{U}}\cong k[(ab)^{2}]$.

  3. $\left(R_{\infty}^{(f)}\right)^{\overline{U}}\cong k[b^{f}]$.

These are isomorphisms as algebras with a left $T$-action. Note that most of
these algebras are not isomorphic even as $T$-representations. In fact, the
only ones that are isomorphic as $T$-representations are $k[b^{2}]\cong k[ab]$
and $k[b^{4}]\cong k[(ab)^{2}]$. We are therefore reduced to showing that
$SL_{2}/T\ncong Spec\left(R_{\infty}^{(2)}\right)$ and $SL_{2}/N_{T}\ncong
Spec\left(R_{\infty}^{(4)}\right)$ as $SL_{2}$-varieties.

$R_{\infty}^{(2)}=k[b^{2},bd,d^{2}]$ is isomorphic as a ring to
$k[x,y,z]/(y^{2}-xz)$. Note that this ring has a singularity at the origin
$x=y=z=0$. So $Spec\left(R_{\infty}^{(2)}\right)$ is not smooth, and therefore
it can’t be isomorphic to $SL_{2}/T$.

A similar reasoning shows that $Spec\left(R_{\infty}^{(4)}\right)$ is not
smooth, and so it can’t possibly be isomorphic to $SL_{2}/N_{T}$. ∎

######  Remark 4.11.

$Spec\left(R_{\infty}^{(f)}\right)$ will have a singular point for $f>1$.
Indeed, $Spec\left(R_{\infty}^{(f)}\right)$ is isomorphic to the affine cone
over the degree $f$ Veronese embedding of $\mathbb{P}^{1}$ into
$\mathbb{P}^{f}$. For $f>1$, this will be an affine normal surface with a
singular point (the cone point).

##  5 Classification of affine normal $SL_{2}$-varieties with an open dense
orbit

In this section we will apply similar techniques to classify all
$SL_{2}$-isomorphism classes of affine normal $SL_{2}$-varieties with an open
dense orbit. We start by dealing with the homogeneous ones.

######  Proposition 5.1.

Let $X=SL_{2}/H$ be an affine homogeneous $SL_{2}$-variety. Then up to
$SL_{2}$-isomorphism, exactly one of the following holds

  1. (a)

$X$ is spherical. We have either $H=T$, $H=N_{T}$ or $H=SL_{2}$.

  2. (b)

$(TypeA)$ $H$ is the $f$-torsion subgroup $\mu_{f}\subset T$ for some positive
integer $f$.

  3. (c)

$(TypeD)$ $H$ is the subgroup $\mathbb{Z}/2\mathbb{Z}\ltimes\mu_{f}$ inside
$N_{T}$.

  4. (d)

$(E_{6})$ $H\subset SL_{2}$ is the binary tetrahedral group $\mathbb{T}$.

  5. (e)

$(E_{7})$ $H\subset SL_{2}$ is the binary octahedral group $\mathbb{O}$.

  6. (f)

$(E_{8})$ $H\subset SL_{2}$ is the binary icosahedral group $\mathbb{I}$.

###### Proof.

If $\text{dim}\,H\geq 1$ then we are in case $(a)$ by the proof of Proposition
4.9. If $\text{dim}\,H=0$, then $H$ is a finite subgroup of $SL_{2}$. We are
reduced to classifying finite subgroups of $SL_{2}$ up to conjugation. Such
classification is well known, see [Spr77][4.4] for a treatment. The
classification includes exactly the cases $(b)$ through $(f)$ above. ∎

For completeness, let’s recall explicit descriptions of the exceptional cases
$(d)$, $(e)$ and $(f)$ above. See [Slo80][§6.1] or [Spr77][4.4] for details.

  1. $(E_{6})$

$\mathbb{T}$ is generated by the group
$\mathbb{Z}/2\mathbb{Z}\ltimes\mu_{2}\subset N_{T}$ and the matrix
$\frac{1}{\sqrt{2}}\begin{bmatrix}\epsilon^{7}&\epsilon^{7}\\\
\epsilon^{5}&\epsilon\end{bmatrix}$. Here $\epsilon$ is a primitive $8$-th
root of unity.

  2. $(E_{7})$

$\mathbb{O}$ is generated by $\mathbb{T}$ and the matrix
$\begin{bmatrix}\epsilon&0\\\ 0&\epsilon^{7}\end{bmatrix}$, where $\epsilon$
is the same $8$th root of unity as above.

  3. $(E_{8})$

$\mathbb{I}$ is generated by the matrices $-\begin{bmatrix}\eta^{3}&0\\\
0&\eta^{2}\end{bmatrix}$ and
$\frac{1}{\eta^{2}-\eta^{3}}\begin{bmatrix}\eta+\eta^{4}&1\\\
1&-\eta-\eta^{4}\end{bmatrix}$, where $\eta$ is a primitive $5$th root of
unity.

The admissible subalgebra $A\subset k[a,b]$ in each homogeneous case $SL/H$
can be computed as the ring of right invariants $k[a,b]^{H^{op}}$ (we take the
distinguished point $p$ to be the identity). In Proposition 4.9 we have
already described $A$ when $\text{dim}\,H\geq 1$. See [Spr77][Ch. 4] or
[Dol][Ch. 1] for details on the computation of these algebras when $H$ is a
finite group.

It remains to classify the nonhomogeneous $SL_{2}$-varieties. We will split
the classification into two cases, depending on the dimension of
$k[X]^{\overline{U}}$.

######  Proposition 5.2.

Let $X$ be an affine normal $SL_{2}$-variety with an open dense orbit. Assume
that $X$ is not a homogeneous $SL_{2}$-variety. Suppose that the ring
$k[X]^{\overline{U}}$ has dimension 1. Then $X$ is spherical. In particular
$X$ is as in part $(d)$ of Theorem 4.10.

###### Proof.

Choose a closed point $p$ in the dense open orbit of $X$. This allows us to
view $X$ as $Spec(R)$ for some left $SL_{2}$-subalgebra $R\subset k[SL_{2}]$.
Since $X$ is not a homogeneous $SL_{2}$-variety, there is some closed point in
the complement of the open dense orbit. By the Hilbert-Mumford criterion 4.4,
we know that there exists a nonconstant one-parameter subgroup $\gamma$ such
that $\text{lim}_{t\to 0}\,\gamma(t)\cdot p$ exists in $Spec(R)$. After right
multiplying by and element of $SL_{2}(k)$, we can assume that
$B_{\gamma}=\overline{B}$.

Let $A=R^{\overline{U}}=R\cap k[a,b]$ be the corresponding admissible algebra.
By assumption, $\text{dim}\,A=1$. Therefore $A\neq k$. This means that there
exist some homogeneous polynomial $p\in A$ with positive degree. If $k[p]=A$,
then $X$ is spherical by Proposition 4.2.

Assume that $A\neq k[p]$. We claim that $b^{f}\in A$ for some $f\geq 1$. Up to
a scalar multiple, we can write
$p=a^{m}b^{n-m}+\sum_{i=0}^{m-1}z_{i}a^{i}b^{n-i}$ for some $m\leq n$. Since
$B_{\gamma}=\overline{B}$, we conclude by Lemma 4.5 that
$m\leq\left\lfloor\frac{n}{2}\right\rfloor$. If $m\neq\frac{n}{2}$, then the
argument in Proposition 4.6 Case (C1) shows that the element $w^{p,p}_{2m}$ is
of the form $zb^{f}$ for some $z\in k\setminus\\{0\\}$ and $f\geq 1$. So we
would be done with the claim in this case. Suppose therefore that
$m=\frac{n}{2}$.

Since $k[p]\neq A$, there is a nonzero homogeneous polynomial $h\in A\setminus
k[p]$. We can write
$h=a^{s}b^{t-s}+\sum_{i=0}^{s-1}z^{\prime}_{i}a^{i}b^{t-i}$. If
$s\neq\frac{t}{2}$, then by the same reasoning as above we can conclude that
$b^{f}\in A$ for some $f\geq 1$. Otherwise, we can look at the nonzero
polynomial $p^{t}-h^{n}$. Notice that this is a homogeneous polynomial of
degree $nt$. The highest power of the variable $a$ must be strictly smaller
than $\frac{nt}{2}$ because we are cancelling the monomials with the highest
$a$-exponents in $p^{t}$ and $h^{m}$. Therefore, we can just apply the
argument of 4.6, Case (C1) to deduce that $b^{f}\in A$ for some $f\geq 1$.

Let $f$ denote the minimal positive integer such that $b^{f}\in A$ (i.e.
$k[b^{f}]=A\cap k[b]$). Since $\text{dim}\,A=1$, the localization
$A\otimes_{k[b^{f}]}k(b^{f})$ must be integral over the field $k(b^{f})$. Note
that $A\otimes_{k[b^{f}]}k(b^{f})\subset k(b)[a]$. The integral closure of the
field $k(b^{f})$ in the polynomial ring $k(b)[a]$ is just $k(b)$. Therefore we
must have $A\subset k(b)$. This implies that $A=A\cap k[b]=k[b^{f}]$. It
follows again that $X$ is spherical. ∎

We are left with the case of dimension 2. We start with a useful lemma.

######  Lemma 5.3.

Let $X$ be an affine variety equipped with a left action of a unipotent group
$U$. Then we have equality between $U$-invariants
$\text{Frac}(k[X]^{U})=\text{Frac}(k[X])^{U}$.

###### Proof.

See the argument in [Per][4.1.16 (i)]. ∎

For the following lemma, recall that the choice of a closed point $X$ in the
open dense orbit allows us to view $X=Spec(R)$ for some left $SL_{2}$-stable
subalgebra $R\subset k[SL_{2}]$.

######  Lemma 5.4.

Let $X$ be an affine normal $SL_{2}$-variety with an open dense orbit. Assume
that $X$ is not a homogeneous $SL_{2}$-variety. Suppose that the ring
$k[X]^{\overline{U}}$ has dimension 2. Then $X$ admits a right $T$-action
commuting with the $SL_{2}$-action.

More precisely, there exists a closed point $p\in X$ in the open dense orbit
such that the corresponding $SL_{2}$-stable subalgebra $R\subset k[SL_{2}]$ is
$T^{op}$-stable.

###### Proof.

Let’s start by choosing a closed point $p$ in the open dense orbit of $X$. Let
$R$ be the $SL_{2}$-stable subalgebra of $k[SL_{2}]$ such that $X=Spec(R)$.
Since $X$ is not a homogeneous $SL_{2}$-variety, there is some closed point in
the complement of the open dense orbit. By the Hilbert-Mumford criterion 4.4,
we know that there exists a nonconstant one-parameter subgroup $\gamma$ such
that $\text{lim}_{t\to 0}\,\gamma(t)\cdot p$ exists in $Spec(R)$. After right
multiplying by and element of $SL_{2}(k)$, we can assume that
$B_{\gamma}=\overline{B}$.

Let $A=R^{\overline{U}}=R\cap k[a,b]$ be the corresponding admissible
subalgebra of $k[a,b]$. We just need to show that $A$ is right $T$-stable.
This is because the $T^{op}$-action commutes with the left $SL_{2}$-action. So
if $A$ is $T^{op}$-stable, then its $SL_{2}$-span $R$ will also be
$T^{op}$-stable.

It suffices to prove that for every homogeneous polynomial $p\in A$, all of
the $T^{op}$-weight components of $p$ are also in $A$.

We first claim that there exists some positive integer $f$ such that $b^{f}\in
A$. Let $p\in A$ be a homogeneous polynomial of degree $n$. Up to a scalar
multiple, we can write $p=a^{m}b^{n-m}+\sum_{i=0}^{m-1}z_{i}a^{i}b^{n-i}$ for
some $m\leq n$. Since $B_{\gamma}=\overline{B}$, we conclude by Lemma 4.5 that
$m\leq\left\lfloor\frac{n}{2}\right\rfloor$. If $m\neq\frac{n}{2}$, then the
argument in Proposition 4.6 Case (C1) shows that the element $w^{p,p}_{2m}$ is
of the form $zb^{f}$ for some $z\in k\setminus\\{0\\}$ and $f\geq 1$. So we
would be done in this case. Suppose therefore that $m=\frac{n}{2}$. Consider
the subalgebra $k[p]\subset A$. This inclusion can’t be an equality because
$A$ has dimension $2$ by assumption. This means that there is a nonzero
homogeneous polynomial $h\in A\setminus k[p]$. We can write
$h=a^{s}b^{t-s}+\sum_{i=0}^{s-1}z^{\prime}_{i}a^{i}b^{t-i}$. If
$s\neq\frac{t}{2}$, then by the same reasoning as above we can conclude that
$b^{f}\in A$ for some $f\geq 1$. Otherwise, we can look at the nonzero
polynomial $p^{t}-h^{n}$. Notice that this is a homogeneous polynomial of
degree $nt$. The highest power of the variable $a$ must be strictly smaller
than $\frac{nt}{2}$ because we are cancelling the monomials with the highest
$a$-exponents in $p^{t}$ and $h^{m}$. Therefore, we can just apply the
argument of 4.6, Case (C1) to deduce that $b^{f}\in A$ for some $f\geq 1$.

Now let $f$ be the smallest positive integer such that $b^{f}\in A$. Let
$\text{Stab}_{p}$ denote the stabilizer of the distinguished point $p$. Note
that the algebra $R$ must be invariant under (i.e. pointwise fixed by) the
right action of $\text{Stab}_{p}$. This in particular implies that $A$ is
invariant under the right action of $\text{Stab}_{p}$. Since $b^{f}\in A$,
this means that $b^{f}$ is fixed by the right action of $\text{Stab}_{p}$. It
can be checked that the right stabilizer of $b^{f}$ is the subgroup
$\mu_{f}\ltimes\overline{U}$ in $\overline{B}=T\ltimes\overline{U}$. Note that
we must have $\text{dim}\,\text{Stab}=0$. Otherwise
$\overline{U}\subset\text{Stab}_{p}$, and so we would have

$A\subset k[a,b]^{\text{Stab}_{p}^{op}}\subset
k[a,b]^{\overline{U}^{op}}=k[b]$

This is a contradiction to the assumption that $\text{dim}\,A=2$. Therefore
$\text{Stab}_{p}$ is a finite subgroup of $\mu_{f}\ltimes\overline{U}$. We can
use an element of the connected group $\overline{B}$ to conjugate
$\text{Stab}_{p}$ to a subgroup of $T$. So after right multiplying our algebra
$R$ by an element of $\overline{B}$, we can assume without loss of generality
that $\text{Stab}_{p}$ is a finite subgroup of $T$. We still know that
$\text{Stab}_{p}$ fixes $b^{f}$, and therefore
$\text{Stab}_{p}\subset\mu_{f}$.

We claim that in fact $\text{Stab}_{p}=\mu_{f}$. Say
$\text{Stab}_{p}=\mu_{e}\subset\mu_{f}$ for some $e$ dividing $f$. Note that
$Spec(R)$ contains $SL_{2}/\text{Stab}_{p}$ as an open subscheme. The variety
$SL_{2}/\text{Stab}_{p}$ is $Spec$ of the algebra of right
$\mu_{e}$-invariants $k[SL_{2}]^{\mu_{e}^{op}}$. Now we can use Lemma 5.3 in
order to get the following chain of equalities.

$\text{Frac}(A)=\text{Frac}(R^{\overline{U}})=\text{Frac}(R)^{\overline{U}}=\text{Frac}\left(k[SL_{2}]^{\mu_{e}^{op}}\right)^{\overline{U}}=\text{Frac}\left((k[SL_{2}]^{\mu_{e}^{op}})^{\overline{U}}\right)$

Since the left and right $SL_{2}$-actions commute, we can exchange the order
of taking right and left invariants. This way we get

$\displaystyle\text{Frac}(A)=\text{Frac}\left((k[SL_{2}]^{\overline{U}})^{\mu_{e}^{op}}\right)=\text{Frac}\left(k[a,b]^{\mu_{e}^{op}}\right)=\text{Frac}(k[a^{e},b^{e},ab])$

In particular $b^{e}\in\text{Frac}(A)$. Since $A$ is normal and $b^{f}\in A$,
we must have $b^{e}\in A$. By minimality of $f$ we have $e=f$. This concludes
the proof that $\text{Stab}_{p}=\mu_{f}$. Since $A$ is fixed by the right
action of $\text{Stab}_{p}$, we must have $A\subset
k[a,b]^{\mu_{f}^{op}}=k[a^{f},b^{f},ab]$.

We are now ready to finish the proof of the proposition. Let $p$ be a
homogeneous polynomial in $A$. We will distinguish two cases.

  1. (C1)

$f$ is odd. Since $A\subset k[a^{f},b^{f},ab]$, we can write

$p=a^{m}b^{n-m}+\sum_{i=1}^{\left\lfloor\frac{m}{f}\right\rfloor}z_{i}\,a^{m-if}\,b^{n-m+if}$

for some $z_{i}\in k$. The $T^{op}$-weight components of $p$ are all the
monomial summands $z_{i}a^{m-if}b^{n-m+if}$ and $a^{m}b^{n-m}$ in the sum
above. We claim that we actually have $a^{m-if}b^{n-m+if}\in A$ for all $0\leq
i\leq\left\lfloor\frac{m}{f}\right\rfloor$. This would imply that all
$T^{op}$-weight components are in $A$, thus concluding our proof.

Let’s prove this last claim. We will assume without loss of generality that
$m\geq f$, otherwise the claim is obvious. Let us set
$p_{2}\vcentcolon=b^{2f}$. We know that $b^{2f}\in A$ by our work above. Let’s
look at the highest weight vector

$w^{p,p_{2}}_{f}=\sum_{\alpha=f}^{n+f}y_{\alpha,f}^{p,p_{2}}\,a^{\alpha-f}b^{n+f-\alpha}$

Notice that $y_{\alpha,f}^{p,p_{2}}=0$ for all $\alpha>m$. This is because in
the sum for $y_{\alpha,f}^{p,p_{2}}$ there are no notrivial pairs $(i,j)$ with
$i+j>m$. So we actually have

$w^{p,p_{2}}_{f}=\sum_{\alpha=f}^{m}y_{\alpha,f}^{p,p_{2}}\,a^{\alpha-f}b^{n+f-\alpha}$

The coefficient $y_{m,f}^{p,p_{2}}$ can be checked to be

$y_{m,f}^{p,p_{2}}=\sum_{e=0}^{f}(-1)^{e}\frac{\binom{f}{e}}{\binom{n}{e}\binom{2f}{f-e}}\binom{n-m}{e}\binom{2f}{f-e}$

This can be simplified to

$y_{m,f}^{p,p_{2}}=\sum_{e=0}^{f}(-1)^{e}\frac{\binom{f}{e}\binom{n-m}{e}}{\binom{n}{e}}$

By Lemma A.3, we have $y_{m,f}^{p,p_{2}}\neq 0$. Therefore, we can divide by
$y_{m,f}^{p,p_{2}}$ and write

$\frac{1}{y_{m,f}^{p,p_{2}}}w_{f}^{p,p_{2}}=a^{m-f}b^{n-m+f}+\sum_{i=2}^{\left\lfloor\frac{m}{f}\right\rfloor}\frac{y_{m-i+1,f}^{p,p_{2}}}{y_{m,f}^{p,p_{2}}}a^{m-if}b^{n-m+if}$

Let us call $q_{0}\vcentcolon=p$ and
$q_{1}\vcentcolon=\frac{1}{y_{m,f}^{p,p_{2}}}w_{f}^{p,p_{2}}$. For every
$0\leq j\leq m$, we can define recursively
$q_{j}\vcentcolon=\frac{1}{y_{m-j+1,f}^{q_{j-1},\,p_{2}}}w^{q_{j-1},\,p_{2}}$.
The computation above shows that we can write

$q_{j}=a^{m-jf}b^{n-m+jf}+\sum_{i=j+1}^{\left\lfloor\frac{m}{f}\right\rfloor}z_{j,i}a^{m-if}b^{n-m+if}$

for some coefficients $z_{j,i}\in k$. By assumption $A$ is admissible, so
$q_{j}\in A$ for all $j$. We will now show by descending induction that
$a^{m-if}b^{n-m+if}\in A$ for all $0\leq
i\leq\left\lfloor\frac{m}{f}\right\rfloor$. For the base case we have
$q_{\left\lfloor\frac{m}{f}\right\rfloor}=b^{n-m+\left\lfloor\frac{m}{f}\right\rfloor
f}\in A$. Let $0\leq j\leq\left\lfloor\frac{m}{f}\right\rfloor$. Assume by
induction that we know that $a^{m-if}b^{n-m+if}\in A$ for all $j+1\leq
i\leq\left\lfloor\frac{m}{f}\right\rfloor$. Then, we can write the monomial
$a^{m-jf}b^{n-m+jf}$ as

$a^{m-jf}b^{n-m+jf}=q_{j}-\sum_{i=j+1}^{\left\lfloor\frac{m}{f}\right\rfloor}z_{j,i}a^{m-if}b^{n-m+if}$

Since all of the summands in the right-hand side are in $A$, we conclude that
$a^{m-jf}b^{n-m+jf}\in A$. So we are done by induction.

  2. (C2)

$f$ is even. Since $A\subset k[a^{f},b^{f},ab]$, we can write

$p=a^{m}b^{n-m}+\sum_{i=1}^{\left\lfloor\frac{2m}{f}\right\rfloor}z_{i}\,a^{m-i\frac{f}{2}}\,b^{n-m+i\frac{f}{2}}$

for some $z_{i}\in k$. The $T^{op}$-weight components of $p$ are all the
monomial summands $z_{i}a^{m-i\frac{f}{2}}b^{n-m+i\frac{f}{2}}$ and
$a^{m}b^{n-m}$ in the sum above. We can do the same argument as in case $(C1)$
but replacing every instance of $f$ by $\frac{f}{2}$ (e.g. $p_{2}=b^{f}$ and
we look at the highest weight vector $w^{p,p_{2}}_{\frac{f}{2}}$).

∎

$u$$v$$p$$pb^{2f}=w_{0}^{p,p_{2}}$$w_{f}^{p,p_{2}}$ Figure 3: Application of
the criterion in Proposition 3.6 to $(p,b^{2f})$ in the proof of Lemma 5.4,
case (C1). The darker grid represents the sublattice induced by $f=3$.

######  Definition 5.5.

Let $q\geq 1$ be a rational number. We define $S_{q}$ to be the subalgebra of
$k[a,b]$ generated by all monomials $a^{i}b^{j}$ satisfying $j\geq qi$. We
will denote by $V_{S_{q}}$ the left $SL_{2}$-subrepresentation of $k[SL_{2}]$
spanned by $S_{q}$.

######  Lemma 5.6.

The algebra $S_{q}$ is admissible in the sense of Definition 3.1.

###### Proof.

$Spec(S_{q})$ is a toric variety for the torus $(T\times T^{op})/\mu_{2}$,
where we are embbeding $\mu_{2}$ diagonally. Motivated by this, we will use
the notation for toric varieties as in [CLS11].

Let $\Sigma_{q}$ be the semigroup in $\mathbb{N}^{2}\subset\mathbb{Z}^{2}$
defined by

$\Sigma_{q}=\left\\{(i,j)\in\mathbb{N}^{2}\;\mid\;a^{i}b^{j}\in
S_{q}\right\\}$

Equivalently

$\Sigma_{q}=\left\\{(i,j)\;\mid\;j\geq qi\right\\}$

Notice that this semigroup is saturated in $\mathbb{Z}^{2}$. Actually it
corresponds to the rational cone $\mathbb{R}^{\geq 0}(0,1)+\mathbb{R}^{\geq
0}(1,q)$ inside the vector space
$\mathbb{R}^{2}=\mathbb{Z}^{2}\otimes\mathbb{R}$. By [CLS11][Thm. 1.3.5] and
[CLS11][Prop. 1.2.17], the algebra $S_{q}=k[\Sigma_{q}]$ is normal and
finitely generated.

In order to prove condition $(c)$ in the definition of admissible algebra, we
can use Proposition 3.6. In this case it suffices to check the condition when
$p_{1},p_{2}$ are monomials in $a,b$. Let $p_{1}=a^{i}b^{j}$ and
$p_{2}=a^{l}b^{f}$ be monomials in $S_{q}$. We need to look at

$w^{p_{1},p_{2}}_{s}=\sum_{\alpha=s}^{i+j+l+f-s}y_{\alpha,s}^{p_{1},p_{2}}a^{\alpha-s}b^{i+j+l+f-\alpha-s}$

Notice that in this case $y_{\alpha,s}=0$ unless $\alpha=i+l$, because there
is only one nontrivial coefficient in both $p_{1}$ and $p_{2}$. So

$w^{p_{1},p_{2}}_{s}=y_{i+l,s}^{p_{1},p_{2}}\,a^{i+l-s}\,b^{j+f-s}$

In order to show that this is in $S_{q}$, it suffices to show that $j+f-s\geq
q\cdot(i+l-s)$. This follows from combining the inequalities $j\geq qi$,
$f\geq ql$ and $-s\geq-qs$ (the last one because $q\geq 1$). ∎

######  Definition 5.7.

For any positive integer $f$ and any rational number $q\geq 1$, we will denote
by $S_{q}^{(f)}$ the subalgebra of $k[SL_{2}]$ defined by
$S_{q}^{(f)}\vcentcolon=S_{q}\cap k[a^{f},b^{f},ab]$. We will denote by
$R_{q}^{(f)}$ the left $SL_{2}$-subrepresentation of $k[SL_{2}]$ spanned by
$S_{q}^{(f)}$.

######  Lemma 5.8.

  1. (a)

For every positive integer $f$ and rational number $q\geq 1$, the algebra
$S_{q}^{(f)}$ is admissible. In particular $R^{(f)}_{q}$ is a subalgebra of
$k[SL_{2}]$.

  2. (b)

Let $f_{1},f_{2}$ be positive integers and let $q_{1},q_{2}\geq 1$ be rational
numbers. If $f_{1}\neq f_{2}$ or $q_{1}\neq q_{2}$, then the varieties
$Spec\left(R_{q_{1}}^{(f_{1})}\right)$ and
$Spec\left(R_{q_{2}}^{(f_{2})}\right)$ are not $SL_{2}$-isomorphic.

###### Proof.

  1. (a)

Notice that we have $S_{q}^{(f)}=\left(S_{q}\right)^{\mu_{f}^{op}}$ by
definition. By Lemma 5.6, the algebra $S_{q}$ is finitely generated and
normal. By [NS02], this implies that the algebra of invariants
$\left(S_{q}\right)^{\mu_{f}^{op}}$ is finitely generated. Also $S_{q}$ being
normal implies that $\left(S_{q}\right)^{\mu_{f}^{op}}$ is normal.

We are left to prove property $(c)$ in Definition 3.1. This amounts to showing
that $R_{q}^{(f)}$ is an algebra. But notice that
$R_{q}^{(f)}=\left(V_{S_{q}}\right)^{\mu_{f}^{op}}$. By Lemma 5.6 we know that
$V_{S_{q}}$ is closed under multiplication. We conclude that the
$\mu_{f}^{op}$-invariants $\left(V_{S_{q}}\right)^{\mu_{f}^{op}}$ are also
closed under multiplication, as desired.

  2. (b)

It follows from the argument in the proof of Lemma 5.4 that the stabilizer of
the open dense orbit in $Spec\left(R_{q_{1}}^{(f_{1})}\right)$ (resp.
$Spec\left(R_{q_{2}}^{(f_{2})}\right)$) is $\mu_{f_{1}}$ (resp.
$\mu_{f_{2}}$). If $f_{1}\neq f_{2}$, this shows that
$Spec\left(R_{q_{1}}^{(f_{1})}\right)$ and
$Spec\left(R_{q_{2}}^{(f_{2})}\right)$ can’t possibly be $SL_{2}$-isomorphic.
Assume $f_{1}=f_{2}=f$. We can look at the algebras of
$\overline{U}$-invariants

$\left(R_{q_{1}}^{(f)}\right)^{\overline{U}}=S_{q_{1}}^{(f)}$
$\left(R_{q_{2}}^{(f)}\right)^{\overline{U}}=S_{q_{2}}^{(f)}$

Suppose without loss of generality that $q_{1}>q_{2}$. Then we have
$S_{q_{1}}^{(f)}\subsetneq S_{q_{2}}^{(f)}$. This implies that
$S_{q_{1}}^{(f)}$ and $S_{q_{2}}^{(f)}$ are not isomorphic even as left
$T$-representations. This is because for $n>>0$, the dimension of the
$T$-weight space $\left(S_{q_{1}}^{(f)}\right)_{n}$ must be strictly smaller
than the dimension of the $T$-weight space $\left(S_{q_{2}}^{(f)}\right)_{n}$.

∎

######  Proposition 5.9.

Let $X$ be an affine normal $SL_{2}$-variety with an open dense orbit. Assume
that $X$ is not a homogeneous $SL_{2}$-variety. Suppose that the ring
$k[X]^{\overline{U}}$ has dimension 2. Then there exists a unique positive
integer $f$ and a unique rational number $q\geq 1$ such that $X$ is
$SL_{2}$-isomorphic to $Spec\left(R_{q}^{(f)}\right)$.

###### Proof.

By Lemma 5.4, we can choose a closed point $p$ in the dense orbit such that

  1. (a)

$\text{lim}_{t\to 0}\,\gamma(t)\cdot p$ exists in $X$ for some $\gamma$ with
$B_{\gamma}=\overline{B}$.

  2. (b)

The stabilizer of $p$ is the $f$-torsion subgroup $\mu_{f}\subset T$.

We can use $p$ to view $X$ as $Spec(R)$ for some $SL_{2}$-stable algebra
$R\subset k[SL_{2}]$. Let $A=R\cap k[a,b]$ be the corresponding admissible
subalgebra of $k[a,b]$. It suffices to show that $A=S_{q}^{(f)}$ for some
$q\geq 0$. By Lemma 5.4, $A$ is generated by $T\times T^{op}$-weights in
$k[a,b]$. These are just the monomials in $k[a,b]$, so $A$ is determined by a
semigroup on the set of monomials. Let $\Sigma_{A}$ denote the semigroup of
$\mathbb{N}^{2}$ given by

$\Sigma_{A}\vcentcolon=\left\\{(i,j)\in\mathbb{N}^{2}\;\mid\;a^{i}b^{j}\in
A\right\\}$

Also, let $\Sigma^{(f)}$ be the semigroup of $\mathbb{N}^{2}$ determined by
$k[a^{f},b^{f},ab]$

$\Sigma^{(f)}\vcentcolon=\left\\{(i,j)\in\mathbb{N}^{2}\;\mid\;a^{i}b^{j}\in
k[a^{f},b^{f},ab]\right\\}=\left\\{(i,j)\in\mathbb{N}^{2}\;\mid\;f\,\,|\,\,(j-i)\right\\}$

We have seen during the proof of Lemma 5.4 that there exists a positive
integer $f$ such that $A\subset k[a^{f},b^{f},ab]$ and that

$\text{Frac}(A)=k(a,b)^{\mu_{f}^{op}}=k(a^{f},b^{f},ab)$

This means that $\Sigma_{A}\subset\Sigma^{(f)}$ and
$\mathbb{Z}\Sigma_{A}=\mathbb{Z}\Sigma^{(f)}$. Since $A$ is normal, the proof
of [CLS11][Thm. 1.3.5] implies that $A$ is saturated in
$\mathbb{Z}\Sigma^{(f)}$. Also, $\Sigma_{A}$ is generated by a finite set of
monomial generators of $A$. Since $\Sigma_{A}$ is finitely generated and
saturated in $\mathbb{Z}\Sigma^{(f)}$, we must have that
$\Sigma_{A}=C\cap\Sigma^{(f)}$ for some rational cone $C$ on the vector space
$\mathbb{R}^{2}=\mathbb{Z}^{2}\otimes\mathbb{R}$. In two dimensions, such a
cone must be of the form $\mathbb{R}^{\geq 0}v_{1}+\mathbb{R}^{\geq 0}v_{2}$
for two (possibly equal) lattice vectors $v_{1},v_{2}\in\mathbb{Z}^{2}$. Since
$\mathbb{Z}\Sigma_{A}=\mathbb{Z}\Sigma^{(f)}$, this means that $v_{1}$ and
$v_{2}$ must be linearly independent.

By definition, we have $\Sigma_{A}\subset\mathbb{N}^{2}$. Since
$B_{\gamma}=\overline{B}$, Lemma 4.5 implies that $i\leq j$ for all
$(i,j)\in\Sigma_{A}$. So the cone $C$ must be contained in the cone
$\mathbb{R}^{\geq 0}(0,1)+\mathbb{R}^{\geq 0}(1,1)$. By the proof of Lemma
5.4, we have $b^{f}\in A$. This means that $(0,1)\in C$, and so one of the
vectors can be taken to be $v_{1}=(0,1)$. The other linearly independent
vector is of the form $v_{2}=(n_{1},n_{2})$ for some positive integers
$n_{1}\leq n_{2}$. So the cone is $C=\mathbb{R}^{\geq
0}(0,1)+\mathbb{R}^{2}(n_{1},n_{2})$. We conclude that

$A=k[\Sigma_{A}]=k[C\cap\Sigma^{(f)}]=S_{\frac{n_{2}}{n_{1}}}^{(f)}$

Uniqueness follows from part $(b)$ of Lemma 5.8. ∎

Now we can put everything together to get a classification of all affine
normal $SL_{2}$-varieties with an open dense orbit.

######  Theorem 5.10.

Let $X$ be an affine normal $SL_{2}$-variety with an open dense orbit. Then
$X$ is $SL_{2}$-isomorphic to exactly one of the following

  1. (1)

A homogenenous space $SL_{2}/H$, where $H$ is as in Proposition 5.1.

  2. (2)

(See Definition 4.8) The spherical variety
$SL_{2}/\\!/(\mu_{f}\ltimes\overline{U})=Spec\left(R_{\infty}^{(f)}\right)$
for a unique positive integer $f$.

  3. (3)

(See Definition 5.7) $Spec\left(R^{(f)}_{q}\right)$ for a unique positive
integer $f$ and a unique rational number $q\geq 1$ . In this case the
stabilizer of the dense orbit is $\mu_{f}$.

###### Proof.

This follows by combining Proposition 5.1, Proposition 5.2 and Proposition
5.9. ∎

We end this section by describing explicitly the subalgebra
$R_{q}^{(f)}\subset k[SL_{2}]$. Let $\\{a^{i_{m}}b^{j_{m}}\\}_{m\in I}$ be a
finite set of monomial generators of $S_{q}^{(f)}$. This set can be obtained
in practice by imitating the proof of Gordan’s lemma [CLS11][Prop. 1.2.17]. In
other words, we look at $\Sigma^{(f)}\cap\Delta$, where $\Delta$ is a
fundamental domain for the discrete subgroup of $\mathbb{R}^{2}$ generated by
$(0,1)$ and $(n_{1},n_{2})$. Here $n_{1},n_{2}$ are some positive integers
such that $\frac{n_{2}}{n_{1}}=q$ and $n_{2}-n_{1}$ is divisible by $f$.

For each $m\in I$, let us denote by $V^{(m)}$ the left
$SL_{2}$-subrepresentation of $k[SL_{2}]$ spanned by the highest weight vector
$a^{i_{m}}b^{j_{m}}$. A simple $SL_{2}$ computation shows that $V^{(m)}$ has a
basis $\left(v_{s}^{(m)}\right)_{s=0}^{i_{m}+j_{m}}$, where

$v_{s}^{(m)}=\sum_{i+j=s}\binom{i_{n}}{i}\binom{j_{m}}{j}\,a^{i_{m}-i}\,c^{i}\,b^{j_{m}-j}\,d^{j}$

By the proof of the finite generation statement in Lemma 3.2, the
corresponding $SL_{2}$-stable subalgebra $R_{q}^{(f)}$ is generated as a
$k$-algebra by the elements $v_{s}^{(m)}$ for $m\in I$ and $0\leq s\leq
i_{m}+j_{m}$.

##  Appendix A Some lemmas about binomial sums

In this appendix we collect some lemmas on binomial sums that we use for our
classification results.

######  Lemma A.1.

Let $i,j$ be nonnegative integers with $i\leq j$. Then, we have

$\sum_{e=0}^{2i}(-1)^{e}\frac{\binom{2i}{e}}{\binom{i+j}{e}\binom{i+j}{2i-e}}\binom{j}{e}\binom{j}{2i-e}=(-1)^{i}\,\frac{\binom{2i}{i}}{\binom{i+j}{i}^{2}}\,$

In particular this sum is never $0$.

###### Proof.

We can use the identity

$\frac{\binom{j}{e}\binom{j}{2i-e}}{\binom{i+j}{e}\binom{i+j}{2i-e}}=\frac{1}{\binom{i+j}{i}^{2}}\binom{j+i-e}{i}\binom{j-i+e}{i}$

to write the sum as

$\frac{1}{\binom{i+j}{i}^{2}}\cdot\sum_{e=0}^{2i}(-1)^{e}\binom{2i}{e}\binom{j+i-e}{i}\binom{j-i+e}{i}$

We are reduced to showing the identity

$\sum_{e=0}^{2i}(-1)^{e}\binom{2i}{e}\binom{j+i-e}{i}\binom{j-i+e}{i}=(-1)^{i}\binom{2i}{i}$

Let’s first show that the left hand side does not depend on $j$. Let us fix
$i$ and view the expression as a polynomial in $j$. We can expand

$\binom{j+i-e}{i}\binom{j-i+e}{i}=\sum_{s=0}^{2i}p_{s}(e)\,j^{2i-s}$

where $p_{s}(e)$ is a polynomial on $e$ with coefficients in $\mathbb{Q}[i]$.
Replacing this in the sum above, we get a polynomial in $j$ given by

$\sum_{s=0}^{2i}\left(\sum_{e=0}^{2i}(-1)^{e}\binom{2i}{e}\,p_{s}(e)\right)\,j^{2i-s}$

In order to show that the sum does not depend on $j$, it suffices to show that
$\sum_{e=0}^{2i}(-1)^{e}\binom{2i}{e}p_{s}(e)=0$ for $s<2i$. Notice that
$p_{s}(e)$ has degree at most $s$ in the variable $e$. So we can write

$p_{s}=\sum_{f=0}^{s}q_{s,f}(i)\,e^{f}$

for some $q_{s,f}\in\mathbb{Q}[i]$. We can use this to get

$\sum_{e=0}^{2i}(-1)^{e}\binom{2i}{e}\,p_{s}(e)=\sum_{f=0}^{s}q_{s,f}(i)\sum_{e=0}^{2i}(-1)^{e}\binom{2i}{e}e^{f}$

Therefore we are reduced to showing that
$\sum_{e=0}^{2i}(-1)^{e}\binom{2i}{e}e^{f}=0$ for $f\leq s<2i$. In order to
show this, we use the binomial theorem. Notice that the binomial theorem
implies that

$\sum_{e=0}^{2i}x^{e}\binom{2i}{e}e^{f}=\left(x\frac{d}{dx}\right)^{f}(1+x)^{2i}$

We can plug in $x=-1$ in the left hand side to get
$\sum_{e=0}^{2i}(-1)^{e}\binom{2i}{e}e^{f}$. On the other hand, if we plug in
$x=-1$ in the right hand side we get $0$ (one can see this by using the
product rule and $f<2i$).

We conclude that the sum

$\sum_{e=0}^{2i}(-1)^{e}\binom{2i}{e}\binom{j+i-e}{i}\binom{j-i+e}{i}$

does not depend on $j$. So we can evaluate by plugging in any value of $j$ we
wish. If we plug in $j=i$, then we see that the only nonzero summand occurs
when $e=i$. Hence we get

$\displaystyle\sum_{e=0}^{2i}(-1)^{e}\binom{2i}{e}\binom{j+i-e}{i}\binom{j-i+e}{i}\xlongequal{j=i}\sum_{e=0}^{2i}(-1)^{e}\binom{2i}{e}\binom{2i-e}{i}\binom{e}{i}=(-1)^{i}$

as desired. ∎

######  Lemma A.2.

Let $m$ be a positive integer and let $0\leq i\leq m$. Then we have

$\sum_{e=0}^{2}(-1)^{e}\frac{\binom{2}{e}}{\binom{2m}{e}\binom{2m}{2-e}}\binom{m}{e}\binom{m+i}{2-e}=\frac{mi^{2}-m^{2}}{2m^{2}\cdot(2m-1)}$

###### Proof.

This is just tedious algebra. One needs to sum three terms and operate with
the fractions to arrive at the expression in the right-hand side. ∎

The following sum is needed for some of the arguments in the last section.

######  Lemma A.3.

Let $f,m,n$ be positive integers with $f\leq m\leq n$. Then

$\sum_{e=0}^{f}(-1)^{e}\frac{\binom{f}{e}\binom{n-m}{e}}{\binom{n}{e}}=\frac{\binom{m}{f}}{\binom{n}{f}}$

In particular, this sum is not $0$.

###### Proof.

Fix the positive integer $f$. Let’s interpret $n,m$ as variables. For
psychological purposes, let’s rename $n=x$ and $m=y$. The identity amounts to
an equality of rational functions in the variables $x,y$

$\sum_{e=0}^{f}(-1)^{e}\binom{f}{e}\frac{\prod_{i=0}^{e-1}(x-y-i)}{\prod_{j=0}^{e-1}(y-i)}\;=\;\frac{\prod_{i=0}^{e-1}(y-i)}{\prod_{i=0}^{e-1}(x-i)}$

We can rewrite this using the Pochhammer symbol to simplify notation

$\sum_{e=0}^{f}\frac{(-f)_{e}\,(y-x)_{e}}{n!(-x)_{e}}=\frac{(-y)_{f}}{(-x)_{f}}$

The left-hand side is just the finite hypergeometric function
${}_{2}F_{1}(-f,\,y-x;\,-x;\,1)$ (evaluated at $z=1$). The last equality of
rational functions above is a direct consequence of the Chu-Vandermonde
identity [AAR99][Cor. 2.2.3]. ∎

## References

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Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, New
York 14853, USA

E-mail address<EMAIL_ADDRESS>
E-mail address<EMAIL_ADDRESS>


The Markov chain $\Gamma_{S,S^{\prime}}$ on $\Sigma_{m}$ is aperiodic for any
$m\leq D/4$.

###### Proof.

Given two sets $S,S^{\prime}\in\Sigma_{m}$, we are going to show that the
Markov chain can connect $S$ to $S^{\prime}$ with paths of any length $l\geq
l_{\rm min}$, with $l_{\rm min}$ some finite integer. To do so, let us
introduce a third state $S^{\prime\prime}$ with the property that all
bitstrings $z\in S^{\prime\prime}$ have zeros at positions $i-1$ and $i+1$,
for some $i$ (this limits the size of the subset to $m\leq D/4$). Since the
chain is irreducible, we can find a sequence of $l_{1}$ gates whose product
maps $S$ to $S^{\prime\prime}$ and another sequence of $l_{2}$ gates whose
product maps $S^{\prime\prime}$ to $S^{\prime}$. Then, inserting a gate
$u_{i11}$ between these two sequences $\iota$ times produces a path from $S$
to $S^{\prime}$ of length $l_{1}+\iota+l_{2}$. This can take any value larger
than $l_{\rm min}\equiv l_{1}+l_{2}$. ∎

Reversibility, irreducibility and aperiodicity imply that the Markov chain has
the uniform distribution $\pi_{m}$ as its unique steady state Levin and Peres
(2017). This proves that our 3-qubit automaton gate set equilibrates to the
uniform distribution on all subset states with $|S|=m\leq D/4$.

Furthermore, it is easy to see that three-qubit gates are necessary to obtain
the above result. Indeed two-qubit automaton gates (generated by NOT and CNOT)
are affine transformations, in the sense that for any such gate $u$ we have
$u\ket{\mathbf{z}}=\ket{A_{u}\mathbf{z}\oplus\mathbf{b}_{u}}$, with $A_{u}$ a
binary matrix and $\mathbf{b}_{u}$ a bitstring. Any transformations that are
not affine cannot be generated by two-qubit automaton gates. As it turns out,
the gate set comprising NOT and CNOT gates satisfies irreducibility up to
$m=3$; an example of two subsets that cannot be connected with $m=4$ and $N=3$
is given by

$S=\begin{bmatrix}0&1&1\\\ 0&1&0\\\ 0&0&1\\\ 0&0&0\end{bmatrix},\quad
S^{\prime}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\ 0&0&0\end{bmatrix}$ (86)

(each row is a bitstring $\mathbf{z}$ in the set). It is easy to see that the
sum (modulo 2) of bitstrings in each set, $\mathbf{z}_{\rm tot}$, transforms
under automaton gate $u$ according to $\mathbf{z}_{\rm tot}\mapsto
A_{u}\mathbf{z}_{\rm tot}$ (the additive term $\mathbf{b}_{u}$ appears 4 times
and thus cancels). But $S$ has $\mathbf{z}_{\rm tot}=(0,0,0)$ while
$S^{\prime}$ has $\mathbf{z}_{\rm tot}=(1,1,1)$, so any sequence of $u$’s
mapping $S$ to $S^{\prime}$ must be such that $\prod_{u}A_{u}$ annihilates
$(1,1,1)$. This is incompatible with a reversible computation (e.g. it would
imply a collision between strings $000$ and $111$).

Another way of arriving at a similar conclusion is that pseudoentanglement
requires magic Grewal  _et al._ (2023), and two-qubit automaton gate sets
happen to also be Clifford. Thus if we start from a stabilizer subset state
(e.g. $\ket{0}^{\otimes N-k}\otimes\ket{+}^{\otimes k}$) we cannot produce a
pseudoentangled ensemble. However, for $m\leq 3$, it is instead possible to
show that the gate set comprising NOT and CNOT yields irreducible Markov
chains. This is perhaps related to the fact that mimicking the 3 lowest
moments of the Haar measure does not require magic (as Clifford circuits form
3-designs on qubits).

##  Appendix F Upper bound on relaxation time with local gate set

In Sec. IV.3 we study the relaxation time of the Markov chains induced by the
local automaton gate set
$\mathcal{G}=\\{\textsf{C}_{i-1,a}\textsf{C}_{i+1,b}{X}_{i}\\}$ and, based on
numerical observations, we conjecture the scaling $t_{\rm
rel}^{(m)}=\Theta(N)$, independent of $m$ (Conjecture 1). While the single-
particle spectrum, solved analytically in App. D.1, gives a lower bound
$t_{\rm rel}^{(m)}\geq\Omega(N)$, an upper bound is harder to obtain
analytically. Here we prove the upper bound

$t_{\rm rel}^{(m)}\leq O(mN^{5}).$ (87)

We believe this is a very loose upper bound (again Conjecture 1 corresponds to
$t_{\rm rel}^{(m)}\leq O(N)$) and it would be interesting to identify
different techniques to tighten it.

Our proof technique is based on the comparison method Levin and Peres (2017),
which provides a bound

$t_{\rm rel}\leq B\tilde{t}_{\rm rel}$ (88)

on the relaxation time $t_{\rm rel}$ of a Markov chain $\Gamma$ in terms of
the relaxation time $\tilde{t}_{\rm rel}$ of another Markov chain
$\tilde{\Gamma}$ on the same space. To define the coefficient $B$, we first
need to introduce some background and notation. We consider a graph $(V,E)$
with vertices $V$ and edges $E$; the reversible Markov chain $\Gamma$ allows
transitions between any two vertices connected by an edge, with probability
$\Gamma_{e_{1}e_{2}}$ where $(e_{1},e_{2})=e\in E$ denotes the edge. We also
have another graph with the same vertices but different edges,
$(V,\tilde{E})$, with a Markov chain $\tilde{\Gamma}$ allowing transitions
along the $\tilde{E}$ edges. We assume that, for any edge
$\tilde{e}=(x,y)\in\tilde{E}$ in the second graph, one can find a path
$\gamma_{xy}$ along the first graph $(V,E)$ connecting $x$ to $y$.
$|\gamma_{xy}|$ denotes the length of that path. We say that $e\in\gamma_{xy}$
if the path traverses edge $e\in E$. With this notation in place, we can write
down the expression for the coefficient $B$ in the comparison bound:

$B=\max_{e\in E}\frac{1}{\Gamma_{e_{1}e_{2}}}\sum_{x,y:\gamma_{xy}\ni
e}\tilde{\Gamma}_{xy}\left|\gamma_{xy}\right|.$ (89)

Our strategy is to use our Markov chain (induced by the local automaton gate
set $\mathcal{G}$) as $\Gamma$ and use the Markov chain induced by ‘simple
permutations’ Gowers (1996); Hoory  _et al._ (2005) as $\tilde{\Gamma}$.
‘Simple permutations’ are defined as the gate set
$\tilde{\mathcal{G}}=\\{v_{ijk,\sigma}:\,i,j,k\in[N],\,\sigma\in S_{8}\\}$,
where $v_{ijk,\sigma}$ acts as a 3-qubit automaton gate141414In ‘width-2’
simple permutations Hoory  _et al._ (2005); Brodsky and Hoory (2008), which we
use here to establish our upper bound, $\sigma$ is restricted to act as a flip
of bit $i$ controlled by an arbitrary Boolean function of bits $j,k$. The
Toffoli gate falls in this category, with the Boolean function being AND.
(associated to a permutation $\sigma\in S_{8}$) on qubits $i,j,k$. Note that
the choice of qubits $i,j,k$ is also arbitrary, in particular they need not be
consecutive. So ‘simple permutations’ are 3-local but not geometrically local.
‘Simple permutations’ have been studied in the probability literature Gowers
(1996); Hoory  _et al._ (2005) due to their good mixing properties, and
several bounds on their relaxation and mixing times are known. For our
purpose, we will need the following fact:

######  Fact F.1 (relaxation time from simple permutations).

‘Simple permutations’ $\tilde{\mathcal{G}}$ induce a Markov chain on
$\Sigma_{m}$ whose relaxation time is bounded above as

$\tilde{t}_{\rm rel}^{(m)}\leq O(mN^{2}).$ (90)

###### Proof.

Ref. Brodsky and Hoory (2008) considers the random walk induced by ‘simple
permutations’ on ordered $m$-tuples, $\mathbf{z}=(z_{1},\dots z_{m})$, rather
than sets $S=\\{z_{1},\dots z_{m}\\}\in\Sigma_{m}$. It shows that the former
has relaxation time $\tilde{t}_{\rm rel}^{(m,\text{ord.})}\leq O(mN^{2})$. We
will show that

$\tilde{t}_{\rm rel}^{(m)}\leq\tilde{t}_{\rm rel}^{(m,\text{ord.})}$ (91)

(the walk on ordered $m$-tuples relaxes no faster than that on subsets), which
implies Eq. (90).

Consider an eigenfunction $f$ of the Markov chain on subsets:

$\frac{1}{|\tilde{\mathcal{G}}|}\sum_{v\in\tilde{\mathcal{G}}}f(v(S))=\lambda
f(S).$ (92)

Let us define a function $g$ on the space of ordered $m$-tuples by setting
$g(\mathbf{z})=f({\rm set}(\mathbf{z}))$, where ${\rm
set}(\mathbf{z})\in\Sigma_{m}$ is the unordered set of elements in the
$m$-tuple $\mathbf{z}$. We can easily check that $g$ is an eigenfunction of
the Markov chain on ordered $m$-tuples with the same eigenvalue $\lambda$:

$\displaystyle\frac{1}{|\tilde{\mathcal{G}}|}\sum_{v\in\tilde{\mathcal{G}}}g(v(\mathbf{z}))$
$\displaystyle=\frac{1}{|\tilde{\mathcal{G}}|}\sum_{v\in\tilde{\mathcal{G}}}f({\rm
set}(v(\mathbf{z})))$
$\displaystyle=\frac{1}{|\tilde{\mathcal{G}}|}\sum_{v\in\tilde{\mathcal{G}}}f(v({\rm
set}(\mathbf{z})))$ $\displaystyle=\lambda f({\rm set}(\mathbf{z}))=\lambda
g(\mathbf{z}).$ (93)

Thus the spectrum of the Markov chain on subsets is included within the
spectrum of the Markov chain on ordered tuples. Recalling that the relaxation
time is the inverse spectral gap, Eq. (91) follows. ∎

Now that we have an upper bound on $\tilde{t}_{\rm rel}^{(m)}$, it remains to
upper-bound the coefficient $B$ in the comparison bound, Eq. (89). We break it
down into three bounds:

######  Fact F.2.

For any edge $e\in E$, we have

$\frac{1}{\Gamma_{e}}\leq O(N).$ (94)

###### Proof.

Follows from the fact that $\Gamma_{S,S^{\prime}}\geq
1/|\mathcal{G}|=\frac{1}{4N}$ (at least one gate $u\in\mathcal{G}$ must
connect $S$ to $S^{\prime}$). ∎

######  Fact F.3.

For any two subsets $S$, $S^{\prime}$ we have

$|\gamma_{S,S^{\prime}}|\leq O(N).$ (95)

###### Proof.

For any two sets $S$, $S^{\prime}$ connected by a simple permutation
$v_{ijk,\sigma}\in\tilde{\mathcal{G}}$, we can choose a path of local
automaton gates $u\in\mathcal{G}$ that implements a sequence of SWAPs to bring
qubits $i,j,k$ to consecutive positions (this takes $O(N)$ gates), implement
the permutation $\sigma$ (this takes $O(1)$ gates), and then return the three
qubits to positions $i,j,k$ ($O(N)$ gates again). In all this path has length
$|\gamma_{S,S^{\prime}}|\leq O(N)$. ∎

######  Fact F.4.

For any edge $e\in E$ we have

$\sum_{S,S^{\prime}:\,\gamma_{S,S^{\prime}}\ni
e}\tilde{\Gamma}_{S,S^{\prime}}\leq O(N).$ (96)

###### Proof.

We can write $S^{\prime}$ as $v(S)$ for some gate $v\in\tilde{\mathcal{G}}$.
There may be more than one way to do this; to account for it, we introduce the
multiplicity
$\mu_{S,S^{\prime}}\equiv|\\{v\in\tilde{\mathcal{G}}:\,S^{\prime}=v(S)\\}|$.
We obtain

$\sum_{\begin{subarray}{c}S,S^{\prime}\in\Sigma_{m}:\\\
\gamma_{S,S^{\prime}}\ni
e\end{subarray}}\tilde{\Gamma}_{S,S^{\prime}}=\sum_{\begin{subarray}{c}v\in\tilde{\mathcal{G}},S\in\Sigma_{m}:\\\
\gamma_{S,v(S)}\ni
e\end{subarray}}\frac{\tilde{\Gamma}_{S,v(S)}}{\mu_{S,v(S)}}$ (97)

where the factor $1/\mu_{S,v(S)}$ accounts for the over-counting of terms when
switching summation variables from $S^{\prime}$ to $v$. Now, by definition
$\tilde{\Gamma}_{S,S^{\prime}}=\mu_{S,S^{\prime}}/|\tilde{\mathcal{G}}|$
(fraction of the gate set that maps $S$ to $S^{\prime}$), which gives

$\sum_{\begin{subarray}{c}S,S^{\prime}\in\Sigma_{m}:\\\
\gamma_{S,S^{\prime}}\ni
e\end{subarray}}\tilde{\Gamma}_{S,S^{\prime}}=\frac{1}{|\tilde{\mathcal{G}}|}|\\{(v,S):\,\gamma_{S,v(S)}\ni
e\\}|.$ (98)

Our choice of paths $\gamma_{S,S^{\prime}}$ (see proof of Eq. (95) above)
depends only on the gate $v$ (or more precisely, on one of the
$\mu_{S,S^{\prime}}$ possible gates that map $S$ to $v(S)$, chosen
arbitrarily): we decompose $v=u_{l}u_{l-1}\cdots u_{1}$ into a sequence of
$l\leq O(N)$ local automaton gates, $u_{i}\in\mathcal{G}$. The condition
$\gamma_{S,v(S)}\ni e$ thus requires that $e_{1}=u_{i}\cdots u_{1}(S)$ and
$e_{2}=u_{i+1}(e_{1})$ for some $i\in[l]$; equivalently, it requires
$S=u_{1}\cdots u_{i}(e_{1})$. Therefore, given $e$ and $v$, there are $O(N)$
choices for $S$, indexed by the integer $i$. The $|\tilde{\mathcal{G}}|$
choices for $v$ cancel out the denominator in Eq. (98), and we obtain the
desired bound. ∎

Putting it all together, Fact F.1 bounds the relaxation time in the ‘simple
permutation’ Markov chain as $\tilde{t}_{\rm rel}^{(m)}\leq O(mN^{2})$; Facts
F.2, F.3, and F.4 together yield the comparison coefficient $B\leq O(N^{3})$;
plugging these into the comparison bound (Eq. (88)) gives the desired result
(Eq. (90)).

We close with two observations. First, since our best lower bound on the
mixing time (for general $m$) is $t_{\rm mix}^{(m)}\geq\tilde{\Omega}(mN)$,
this upper bound on $t_{\rm rel}^{(m)}$ is not sufficient to separate the
relaxation and mixing times. Second, using the relaxation time, we can also
bound the mixing time above by Eq. 28. This gives the (likely loose) upper
bound $t_{\rm mix}^{(m)}\leq O(m^{2}N^{6})$.

##  Appendix G Upper bound on pseudothermalization time for subset states

In this Appendix we prove the upper bound Eq. (44) on the pseudothermalization
time for subset states, Sec. V. We start from the key result, Eq. (38-39),
which relates pseudothermalization to the equilibration of the Markov chains
$\Phi_{m+\delta\leftarrow K}[p_{t}]$ on $\Sigma_{m+\delta}$, for
$\delta=0,\dots m$.

Let us decompose the matrix $\mathcal{M}^{(m)}_{p_{t}}$ based on the values of
the difference between subsets, $\delta=|S^{\prime}\setminus
S^{\prime\prime}|$:

$\displaystyle\mathcal{M}^{(m)}_{p_{t}}$
$\displaystyle=\sum_{\delta=0}^{m}O_{\delta},$ (99) $\displaystyle O_{\delta}$
$\displaystyle=\binom{K}{m}^{-1}\binom{K}{m+\delta}\times$
$\displaystyle\quad\times\sum_{\begin{subarray}{c}S^{\prime},S^{\prime\prime}\in\Sigma_{m}:\\\
|S^{\prime}\setminus
S^{\prime\prime}|=\delta\end{subarray}}\ket{S^{\prime}}f^{(m+\delta)}_{p_{t}}(S^{\prime}\cup
S^{\prime\prime})\bra{S^{\prime\prime}},$ (100)

where $f^{(m+\delta)}_{p_{t}}$ is defined in Eq. (40). The diagonal term
$O_{0}$ provides a lower bound: $\|\mathcal{M}^{(m)}_{p_{t}}\|\geq\|O_{0}\|$,
which is used to derive the lower bound on the prethermalization time, Eq.
(43). To get an upper bound, we need to consider the off-diagonal terms as
well.

We have

$\left\|\mathcal{M}_{p_{t}}^{(m)}\right\|_{{\rm
tr}}\leq\sum_{\delta=0}^{m}\|O_{\delta}\|_{{\rm tr}}.$ (101)

We will upper-bound each $\|O_{\delta}\|_{\rm tr}$ by the sum of the absolute
values of its elements: for any operator $A$, one has

$\|A\|_{\rm tr}=\|\sum_{ij}A_{ij}\outerproduct{i}{j}\|_{\rm
tr}\leq\sum_{ij}\|A_{ij}\outerproduct{i}{j}\|_{\rm tr}=\sum_{ij}|A_{ij}|.$
(102)

For each $\delta$, the set of possible entries is give by $f_{m+\delta}(S)$,
with $S\in\Sigma_{m+\delta}$. Each entry appears multiple times: there are
$\binom{m+\delta}{m}\binom{m}{m-\delta}$ ways of choosing
$S^{\prime},S^{\prime\prime}\in\Sigma_{m}$ such that $|S^{\prime}\cup
S^{\prime\prime}|=m+\delta$. Therefore,

$\displaystyle\|O_{\delta}\|_{\rm tr}$
$\displaystyle\leq\sum_{S\in\Sigma_{m+\delta}}\binom{m+\delta}{m}\binom{m}{m-\delta}\binom{K}{m}^{-1}\binom{K}{m+\delta}|f^{(m+\delta)}_{p_{t}}(S)|$
$\displaystyle\leq\binom{m}{\delta}\binom{K-m}{\delta}\sum_{S\in\Sigma_{m+\delta}}|\Phi_{m+\delta\leftarrow
K}[p_{t}](S)-\pi_{m+\delta}(S)|$
$\displaystyle=\binom{m}{\delta}\binom{K-m}{\delta}\Delta_{m+\delta},$ (103)

where in the second line we combined the various binomial coefficients and in
the last line we recognized the total variation distance:

$\Delta_{m+\delta}=\|\Phi_{m+\delta\leftarrow K}[p_{t}]-\pi_{m+\delta}\|_{\rm
tv}.$ (104)

Now, using the results of Appendix B, and specifically Fact B.4, we have that
$\Delta_{m+\delta}$ is non-decreasing in $\delta$. It follows that

$\displaystyle\|\mathcal{M}^{(m)}_{p_{t}}\|_{\rm tr}$
$\displaystyle\leq\Delta_{2m}\sum_{\delta=0}^{m}\binom{m}{\delta}\binom{K-m}{\delta}=\binom{K}{m}\Delta_{2m}.$
(105)

Thus, to make the LHS smaller than $\epsilon$, the Markov chain on
$\Sigma_{2m}$ must equilibrate to within total variation error
$\Delta_{2m}\leq\binom{K}{m}^{-1}\epsilon$. Defining the mixing time $t_{\rm
mix}(\epsilon)$ based on a total variation threshold $\epsilon$ (see Sec.
II.2), one has the following general inequality:

$t_{{\rm mix}}(2^{-l-1})\leq lt_{{\rm mix}}(1/4).$ (106)

It follows then that

$\displaystyle t_{\rm PTherm}^{(m)}\leq\left[\log_{2}\binom{K}{m}\right]t_{\rm
mix}^{(2m)}\leq m\log_{2}(K)t_{\rm mix}^{(2m)}.$ (107)

Finally, invoking the upper bound on the mixing time (conditional on
Conjecture 1) gives

$\displaystyle t_{\rm PTherm}^{(m)}\leq O(\log(K)m^{2}N^{2}).$ (108)

In the case of $K=\exp({\rm polylog}(N))$ (where the entropy of the random
subset ensemble is polylogarithmic) this reduces to

$\displaystyle t_{\rm PTherm}^{(m)}\leq\tilde{O}(m^{2}N^{2}),$ (109)

where the symbol $\tilde{O}$ represents an upper bound up to $\log(N)$
factors.