# Group Theory, Irreducible Representations, and Tensor Products

and How to Use them to Build Equivariant Polynomials

Mario Geiger

Swiss Equivariant Learning Workshop

Polynomials

1. Define properly what is an equivariant polynomial
2. Give the tools to create them

Examples

1. Spherical Harmonics

intro

polynomials

examples

intro

polynomials

examples

intro

polynomials

examples

# Group and Representations

"what are the operations"

"how they compose"

intro

polynomials

examples

# Group and Representations

"what are the operations"

"how they compose"

"vector spaces on which the action of the group is defined"

intro

polynomials

examples

# Group and Representations

"what are the operations"

"how they compose"

"vector spaces on which the action of the group is defined"

rotations, parity, (translations)

scalars, vectors, pseudovectors, ...

intro

polynomials

examples

# Group and Representations

Group $$G$$

• $$\text{identity} \in G$$
• associativity $$g (hk) = (gh)k$$
• inverse $$g^{-1} \in G$$

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

intro

polynomials

examples

# Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group $$G$$

• $$\text{identity} \in G$$
• associativity $$g (hk) = (gh)k$$
• inverse $$g^{-1} \in G$$

intro

polynomials

examples

# Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group $$G$$

• $$\text{identity} \in G$$
• associativity $$g (hk) = (gh)k$$
• inverse $$g^{-1} \in G$$

intro

polynomials

examples

# Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group $$G$$

• $$\text{identity} \in G$$
• associativity $$g (hk) = (gh)k$$
• inverse $$g^{-1} \in G$$

intro

polynomials

examples

Representation $$D(g, x)$$

• $$g\in G$$, $$x \in V$$
• Linear $$D(g, x+y) = D(g,x) + D(g,y)$$
• Follow the structure of the group
$$D(gh,x) = D(g, D(h,x))$$

# Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group $$G$$

• $$\text{identity} \in G$$
• associativity $$g (hk) = (gh)k$$
• inverse $$g^{-1} \in G$$

intro

polynomials

examples

Representation $$D(g, x)$$

• $$g\in G$$, $$x \in V$$
• Linear $$D(g, x+y) = D(g,x) + D(g,y)$$
• Follow the structure of the group
$$D(gh,x) = D(g, D(h,x))$$

# Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group $$G$$

• $$\text{identity} \in G$$
• associativity $$g (hk) = (gh)k$$
• inverse $$g^{-1} \in G$$

intro

polynomials

examples

Representation $$D(g, x)$$

• $$g\in G$$, $$x \in V$$
• Linear $$D(g, x+y) = D(g,x) + D(g,y)$$
• Follow the structure of the group
$$D(gh,x) = D(g, D(h,x))$$

# Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group $$G$$

• $$\text{identity} \in G$$
• associativity $$g (hk) = (gh)k$$
• inverse $$g^{-1} \in G$$

intro

polynomials

examples

Representation $$D(g, x)$$

• $$g\in G$$, $$x \in V$$
• Linear $$D(g, x+y) = D(g,x) + D(g,y)$$
• Follow the structure of the group
$$D(gh,x) = D(g, D(h,x))$$

# Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group $$G$$

• $$\text{identity} \in G$$
• associativity $$g (hk) = (gh)k$$
• inverse $$g^{-1} \in G$$

intro

polynomials

examples

Representation $$D(g, x)$$

• $$g\in G$$, $$x \in V$$
• Linear $$D(g, x+y) = D(g,x) + D(g,y)$$
• Follow the structure of the group
$$D(gh,x) = D(g, D(h,x))$$

Equivalent notation $$D(g) x$$

• $$D(g) : V\to V$$
• $$D(g) \in \mathbb{R}^{d\times d}$$
• $$D(gh) = D(g) D(h)$$

# Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group $$G$$

• $$\text{identity} \in G$$
• associativity $$g (hk) = (gh)k$$
• inverse $$g^{-1} \in G$$

intro

polynomials

examples

Representation $$D(g, x)$$

• $$g\in G$$, $$x \in V$$
• Linear $$D(g, x+y) = D(g,x) + D(g,y)$$
• Follow the structure of the group
$$D(gh,x) = D(g, D(h,x))$$

# Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Equivalent notation $$D(g) x$$

• $$D(g) : V\to V$$
• $$D(g) \in \mathbb{R}^{d\times d}$$
• $$D(gh) = D(g) D(h)$$

Group $$G$$

• $$\text{identity} \in G$$
• associativity $$g (hk) = (gh)k$$
• inverse $$g^{-1} \in G$$

intro

polynomials

examples

Representation $$D(g, x)$$

• $$g\in G$$, $$x \in V$$
• Linear $$D(g, x+y) = D(g,x) + D(g,y)$$
• Follow the structure of the group
$$D(gh,x) = D(g, D(h,x))$$

# Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Equivalent notation $$D(g) x$$

• $$D(g) : V\to V$$
• $$D(g) \in \mathbb{R}^{d\times d}$$
• $$D(gh) = D(g) D(h)$$

Group $$G$$

• $$\text{identity} \in G$$
• associativity $$g (hk) = (gh)k$$
• inverse $$g^{-1} \in G$$

intro

polynomials

examples

Representation $$D(g, x)$$

• $$g\in G$$, $$x \in V$$
• Linear $$D(g, x+y) = D(g,x) + D(g,y)$$
• Follow the structure of the group
$$D(gh,x) = D(g, D(h,x))$$

# Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Equivalent notation $$D(g) x$$

• $$D(g) : V\to V$$
• $$D(g) \in \mathbb{R}^{d\times d}$$
• $$D(gh) = D(g) D(h)$$

Group $$G$$

• $$\text{identity} \in G$$
• associativity $$g (hk) = (gh)k$$
• inverse $$g^{-1} \in G$$

intro

polynomials

examples

# Examples of representations

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9$$

intro

polynomials

examples

# Examples of representations

3 scalars

Representations are like data types

It tells you how to interpret the data...

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9$$

intro

polynomials

examples

# Examples of representations

3 scalars

Representations are like data types

It tells you how to interpret the data with respect to the $$G$$!

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9$$

intro

polynomials

examples

# Examples of representations

3 scalars

Representations are like data types

It tells you how to interpret the data with respect to the $$G$$!

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9$$

Knowing that $$a_1, a_2, a_3$$ are scalars tells you that they are not affected by a rotation of your system

intro

polynomials

examples

# Examples of representations

3 scalars

a vector

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9$$

Representations are like data types

It tells you how to interpret the data with respect to the $$G$$!

intro

polynomials

examples

# Examples of representations

3 scalars

a vector

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9$$

If the system is rotated, the 3 components of the vector change together!

Representations are like data types

It tells you how to interpret the data with respect to the $$G$$!

intro

polynomials

examples

# Examples of representations

3 scalars

a vector

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9$$

a vector

The two vectors transforms indepentently

Representations are like data types

It tells you how to interpret the data with respect to the $$G$$!

intro

polynomials

examples

# Examples of representations

3 scalars

a vector

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9$$

a vector

Representations are like data types

It tells you how to interpret the data with respect to the $$G$$!

system rotated by $$g$$

$$D(g)\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}$$

intro

polynomials

examples

# Examples of representations

3 scalars

a vector

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9$$

a vector

Representations are like data types

It tells you how to interpret the data with respect to the $$G$$!

system rotated by $$g$$

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}$$

intro

polynomials

examples

# Examples of representations

3 scalars

a vector

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9$$

a vector

Representations are like data types

It tells you how to interpret the data with respect to the $$G$$!

system rotated by $$g$$

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}$$

intro

polynomials

examples

# Examples of representations

3 scalars

a vector

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9$$

a vector

Representations are like data types

It tells you how to interpret the data with respect to the $$G$$!

system rotated by $$g$$

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}$$

intro

polynomials

examples

# Examples of representations

3 scalars

a vector

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9$$

a vector

Representations are like data types

It tells you how to interpret the data with respect to the $$G$$!

system rotated by $$g$$

$$\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}$$

intro

polynomials

examples

# Equivariance

$$V$$

$$V'$$

intro

polynomials

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# Equivariance

$$V$$

$$V'$$

$$D(g)$$

$$D'(g)$$

$$V$$

$$V'$$

intro

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examples

# Equivariance

$$V$$

$$V'$$

$$D(g)$$

$$D'(g)$$

$$V$$

$$V'$$

$$f$$

intro

polynomials

examples

# Equivariance

$$V$$

$$V'$$

$$D(g)$$

$$D'(g)$$

$$V$$

$$V'$$

$$f$$

$$f$$

intro

polynomials

examples

# Equivariance

$$V$$

$$V'$$

$$D(g)$$

$$D'(g)$$

$$V$$

$$V'$$

$$f$$

$$f$$

$$f(D(g) x)$$

intro

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# Equivariance

$$V$$

$$V'$$

$$D(g)$$

$$D'(g)$$

$$V$$

$$V'$$

$$f$$

$$f$$

$$f(D(g) x)$$

$$D'(g) f(x)$$

intro

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examples

# Equivariance

$$V$$

$$V'$$

$$D(g)$$

$$D'(g)$$

$$V$$

$$V'$$

$$f$$

$$f$$

$$f(D(g) x)$$

$$D'(g) f(x)$$

$$=$$

intro

polynomials

examples

# Polynomials

Most simple yet powerful functions

intro

polynomials

examples

# Polynomials

Most simple yet powerful functions

Be able to create polynomials compatible with rotations is a powerful tool

intro

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# Polynomials

$$x \mapsto x^2 + 2(x-4)$$

intro

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# Polynomials

$$x \mapsto x^2 + 2(x-4)$$

$$\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto x^2 + 2(y-z)x$$

intro

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# Polynomials

$$x \mapsto x^2 + 2(x-4)$$

$$\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto x^2 + 2(y-z)x$$

$$\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto \begin{bmatrix} x^2 + 2(y-z) \\ z^4 + 100 x y z\end{bmatrix}$$

intro

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# Equivariant Polynomials

$$P(D(g) x) = D'(g) P(x)$$

$$\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto \begin{bmatrix} x^2 + 2(y-z) \\ z^4 + 100 x y z\end{bmatrix}$$

😟

Not equivariant

$$\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto \begin{bmatrix} x^2 + y^2 + z^2 \end{bmatrix}$$

😌

Equivariant

intro

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examples

# 3 Tools to Make Equivariant Polynomials

• 🔨Composition $$\circ$$
• 🔧Addition $$+$$
• 🔩Multiplication $$\otimes$$

intro

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# 🔨Composition

two equivariant functions

$$f: V_1 \rightarrow V_2$$

$$h: V_2 \rightarrow V_3$$

$$h\circ f$$ is equivariant!! 😊

$$h(f(D_1(g) x)) = h(D_2(g) f(x)) = D_3(g) h(f(x))$$

intro

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examples

two equivariant functions

$$f: V_1 \rightarrow V_3$$

$$h: V_2 \rightarrow V_3$$

$$h + f$$ is equivariant!! 😊

$$f(D_1(g) x) + h(D_2(g)x) = D_3(g) (f(x) + h(x))$$

intro

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examples

# 🔩Multiplication

$$\begin{bmatrix} {\color{red} x_1}\\{\color{red} x_2}\\{\color{red} x_3}\end{bmatrix}$$

$$\begin{bmatrix} {\color{blue} y_1}\\{\color{blue} y_2}\\{\color{blue} y_3}\\{\color{blue} y_4}\\{\color{blue} y_5} \end{bmatrix}$$

transforming with $$D(g)$$

transforming with $$D'(g)$$

$$= \begin{bmatrix} x_1y_1 & x_1y_2 & x_1y_3 & x_1y_4 & x_1y_5 \\ x_2y_1 & x_2y_2 & x_2y_3 & x_2y_4 & x_2y_5 \\ x_3y_1 & x_3y_2 & x_3y_3 & x_3y_4 & x_3y_5 \end{bmatrix}$$

intro

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# 🔩Multiplication

$$\otimes$$

$$= \begin{bmatrix}{\color{red} x_1} {\color{blue} y_1}&{\color{red} x_1} {\color{blue} y_2}&{\color{red} x_1} {\color{blue} y_3}&{\color{red} x_1} {\color{blue} y_4}&{\color{red} x_1} {\color{blue} y_5}\\{\color{red} x_2} {\color{blue} y_1}&{\color{red} x_2} {\color{blue} y_2}&{\color{red} x_2} {\color{blue} y_3}&{\color{red} x_2} {\color{blue} y_4}&{\color{red} x_2} {\color{blue} y_5}\\{\color{red} x_3} {\color{blue} y_1}&{\color{red} x_3} {\color{blue} y_2}&{\color{red} x_3} {\color{blue} y_3}&{\color{red} x_3} {\color{blue} y_4}&{\color{red} x_3} {\color{blue} y_5}\end{bmatrix}$$

Tensor Product

transforms with $$D(g) \otimes D'(g)$$ 😊

$$\dim( D \otimes D' ) = \dim( D ) \dim( D' )$$

$$\begin{bmatrix} {\color{red} x_1}\\{\color{red} x_2}\\{\color{red} x_3}\end{bmatrix}$$

$$\begin{bmatrix} {\color{blue} y_1}\\{\color{blue} y_2}\\{\color{blue} y_3}\\{\color{blue} y_4}\\{\color{blue} y_5} \end{bmatrix}$$

intro

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$$\text{vector} \otimes \text{vector} \otimes \text{vector} \otimes \dots$$

tensors of shape $$3\times3\times3\times\dots$$

Cartesian Tensors

The grow in dimension is problematic

Luckily the tensor product is reducible

# Reducible representations

intro

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examples

$$D$$ defined on $$V$$

is reducible if

$$\exists W \subset V$$

such that

$$D|_W$$ is a representation

# Reducible representations

### Famous Example

$$\begin{bmatrix}{\color{red} x_1} {\color{blue} x_2}&{\color{red} x_1} {\color{blue} y_2}&{\color{red} x_1} {\color{blue} z_2}\\{\color{red} y_1} {\color{blue} x_2}&{\color{red} y_1} {\color{blue} y_2}&{\color{red} y_1} {\color{blue} z_2}\\{\color{red} z_1} {\color{blue} x_2}&{\color{red} z_1} {\color{blue} y_2}&{\color{red} z_1} {\color{blue} z_2}\end{bmatrix}$$

$$D$$ defined on $$V$$

is reducible if

$$\exists W \subset V$$

such that

$$D|_W$$ is a representation

$$\begin{bmatrix} {\color{red} x_1}\\{\color{red} y_1}\\{\color{red} z_1}\end{bmatrix}\otimes\begin{bmatrix} {\color{blue} x_2}\\{\color{blue} y_2}\\{\color{blue} z_2}\end{bmatrix} =$$

intro

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examples

intro

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# Reducible representations

$${\color{red}x_1}{\color{blue}x_2} + {\color{red}y_1}{\color{blue}y_2} + {\color{red}z_1} {\color{blue}z_2}$$

$$\begin{bmatrix}c ( {\color{red}x_1} {\color{blue}z_2} + {\color{red}z_1} {\color{blue}x_2} ) \\ c ( {\color{red}x_1} {\color{blue}y_2} + {\color{red}y_1} {\color{blue}x_2} ) \\ 2 {\color{red}y_1} {\color{blue}y_2} - {\color{red}x_1} {\color{blue}x_2} - {\color{red}z_1} {\color{blue}z_2} \\ c ( {\color{red}y_1} {\color{blue}z_2} + {\color{red}z_1} {\color{blue}y_2} ) \\ c ( {\color{red}z_1} {\color{blue}z_2} - {\color{red}x_1} {\color{blue}x_2} ) \\\end{bmatrix}$$

$$\begin{bmatrix}{\color{red}y_1}{\color{blue}z_2}-{\color{red}z_1} {\color{blue}y_2}\\ {\color{red}z_1}{\color{blue}x_2}-{\color{red}x_1}{\color{blue}z_2}\\ {\color{red}x_1}{\color{blue}y_2}-{\color{red}y_1}{\color{blue}x_2}\end{bmatrix}$$

$$\begin{bmatrix}{\color{red} x_1} {\color{blue} x_2}&{\color{red} x_1} {\color{blue} y_2}&{\color{red} x_1} {\color{blue} z_2}\\{\color{red} y_1} {\color{blue} x_2}&{\color{red} y_1} {\color{blue} y_2}&{\color{red} y_1} {\color{blue} z_2}\\{\color{red} z_1} {\color{blue} x_2}&{\color{red} z_1} {\color{blue} y_2}&{\color{red} z_1} {\color{blue} z_2}\end{bmatrix}$$

$$3\times3=1+3+5$$

These can be seen as polynomials!

$$D$$ is reducible if

$$\exists W \subset V$$

such that

$$D|_W$$ is a representation

intro

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# Irreducible representations

For the group of rotations ($$SO(3)$$)

They are index by $$L=0, 1, 2, \dots$$

Of dimension $$2L+1$$

 L=0 d=1 scalar, s orbital L=1 d=3 vector, p orbital L=2 d=5 d orbital ...

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# Irreducible representations

For the group of rotations + parity ($$O(3)$$)

They are index by $$L=0, 1, 2, \dots$$

and $$p=\pm 1$$

Of dimension $$2L+1$$

Even

Odd

 L=0 d=1 scalar L=1 d=3 speudo vector L=2 d=5 ...
 L=0 d=1 pseudo scalar L=1 d=3 vector L=2 d=5 ...

# 🔩Multiplication

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$$L_1 \otimes L_2 = |L_1-L_2| \oplus \dots \oplus (L_1+L_2)$$

generalization of $$3\times3=1+3+5$$

• $$2 \otimes 1 = 1 \oplus 2 \oplus 3$$
• $$2 \otimes 2 = 0\oplus 1 \oplus 2 \oplus 3 \oplus 4$$

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# 🔩Multiplication

Composition of equivariant polynomials ✅

Tensor Product of equivariant polynomials ✅

$$L_1 \otimes L_2 = |L_1-L_2| \oplus \dots \oplus (L_1+L_2)$$

p2(p1(x, y), z)
p1(x, y) + p2(w, z)
from e3nn import o3

p = o3.FullTensorProduct("1o", "1o")

p(x, y)

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# Spherical Harmonics

Representation of highest order ($$l$$) in $$\vec x \otimes \vec x \otimes \dots$$

# Conclusion

• Group and representations
• 😊 Equivariance $$f({\color{purple}D(g)} x) = {\color{darkgreen}D'(g)} f(x)$$
• 🔨Composition
• Polynomials
•  🔩 Tensor Product  $$L_1 \otimes L_2 = |L_1-L_2| \oplus \dots \oplus (L_1+L_2)$$
• ​Examples
• ​Spherical Harmonics

Thank you for listening!

By Mario Geiger

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