This question is heavily related to one of my favorite relations between geometry and representation theory. Consider simple Lie algebras of the following types:

• $A_n$
• $D_n$
• $E_6$
• $E_7$
• $E_8$

Then these Dynkin diagrams correspond to all possible finite subgroups of $SL_2.$ The relation is given by special classes of isolated surface singularities known as Kleinian or Du Val singularities. These arise as follows. Given a finite subgroup $G \subset SL_2,$ we have an action of $G$ on $\mathbb{C}^2$ with no fixed points other than the origin. If we then look at the geometric quotient $\mathbb{C}^2/G$ corresponding to $G$-invariant polynomials in $\mathbb{C}[x,y],$ it is generated by three polynomials $f_1, f_2, f_3$ which are related by a homogeneous polynomial $g$ of degree 3 such that $g(f_1, f_2, f_3) = 0.$ We can then identify $\mathbb{C}^2/G$ with the hypersurface $\{ g = 0 \} \subset \mathbb{C}^3.$

The resulting hypersurfaces have the following equations (with corresponding subgroup):

• $A_n: x^{n+1} + y^2 + z^2$ (cyclic)
• $D_n: x^{n-1} + xy^2 + z^2$ (dihedral)
• $E_6: x^4 + y^3 + z^2$ (tetrahedral)
• $E_7: x^3y + y^3 + z^2$ (octahedral)
• $E_8: x^5 + y^3 + z^2$ (icosahedral)

The Dynkin diagram enter as follows. Each of these surfaces can be resolved through a finite number of blow-ups, and the exceptional fiber in the resolution will consist of a copy of $\mathbb{P}^1$ for each node of the Dynkin diagram, each of which is joined to the point of another $\mathbb{P}^1$ if there is a corresponding edge in the Dynkin diagram connecting the two nodes (so in the cyclic case, it's just a chain of $\mathbb{P}^1$'s).

Lastly, there is a neat connection between this and Springer theory that goes as follows. Let $\mathcal{N}$ denote the nilpotent cone of a Lie algebra of one of the types listed above, and let $\mathcal{O}$ denote the subregular orbit. Then $\mathcal{O}$ has codimension two in $\mathcal{N}$ and hence the corresponding Kostant/Slodowy slice is a surface in $\mathcal{N}.$ It then turns out that this surface is one of the surface singularities listed above, and that the corresponding Springer fiber of a subregular element is isomorphic to the exceptional fiber in the resolution of the surface mentioned above. So the Springer resolution encodes the information of the successive blow-ups of these surfaces.

A few good references:

Milnor, Singular Points of Complex Hypersurfaces

Dimca, Singularities and Topology of Hypersurfaces

Slodowy, Simple Singularities and Simple Algebraic Groups

This question is heavily related to one of my favorite relations between geometry and representation theory. Consider simple Lie algebras of the following types:

• $A_n$
• $D_n$
• $E_6$
• $E_7$
• $E_8$

Then these Dynkin diagrams correspond to all possible finite subgroups of $SL_2.$ The relation is given by special classes of isolated surface singularities known as Kleinian or Du Val singularities. These arise as follows. Given a finite subgroup $G \subset SL_2,$ we have an action of $G$ on $\mathbb{C}^2$ with no fixed points other than the origin. If we then look at the geometric quotient $\mathbb{C}^2/G$ corresponding to $G$-invariant polynomials in $\mathbb{C}[x,y],$ it is generated by three homogeneous polynomials $f_1, f_2, f_3$ which are related by a weighted homogeneous polynomial $g$ of degree 3 such that $g(f_1, f_2, f_3) = 0.$ We can then identify $\mathbb{C}^2/G$ with the hypersurface $\{ g = 0 \} \subset \mathbb{C}^3.$

The resulting hypersurfaces have the following equations (with corresponding subgroup):

• $A_n: x^{n+1} + y^2 + z^2$ (cyclic)
• $D_n: x^{n-1} + xy^2 + z^2$ (dihedral)
• $E_6: x^4 + y^3 + z^2$ (tetrahedral)
• $E_7: x^3y + y^3 + z^2$ (octahedral)
• $E_8: x^5 + y^3 + z^2$ (icosahedral)

The Dynkin diagram enter as follows. Each of these surfaces can be resolved through a finite number of blow-ups, and the exceptional fiber in the resolution will consist of a copy of $\mathbb{P}^1$ for each node of the Dynkin diagram, each of which is joined to the point of another $\mathbb{P}^1$ if there is a corresponding edge in the Dynkin diagram connecting the two nodes (so in the cyclic case, it's just a chain of $\mathbb{P}^1$'s).

Lastly, there is a neat connection between this and Springer theory that goes as follows. Let $\mathcal{N}$ denote the nilpotent cone of a Lie algebra of one of the types listed above, and let $\mathcal{O}$ denote the subregular orbit. Then $\mathcal{O}$ has codimension two in $\mathcal{N}$ and hence the corresponding Kostant/Slodowy slice is a surface in $\mathcal{N}.$ It then turns out that this surface is one of the surface singularities listed above, and that the corresponding Springer fiber of a subregular element is isomorphic to the exceptional fiber in the resolution of the surface mentioned above. So the Springer resolution encodes the information of the successive blow-ups of these surfaces.

A few good references:

Milnor, Singular Points of Complex Hypersurfaces

Dimca, Singularities and Topology of Hypersurfaces

Slodowy, Simple Singularities and Simple Algebraic Groups

This question is heavily related to one of my favorite relations between geometry and representation theory. Consider simple Lie algebras of the following types:

• $A_n$
• $D_n$
• $E_6$
• $E_7$
• $E_8$

Then these Dynkin diagrams correspond to all possible finite subgroups of $SL_2.$ The relation is given by special classes of isolated surface singularities known as Kleinian or Du Val singularities. These arise as follows. Given a finite subgroup $G \subset SL_2,$ we have an action of $G$ on $\mathbb{C}^2$ with no fixed points other than the origin. If we then look at the geometric quotient $\mathbb{C}^2/G$ corresponding to $G$-invariant polynomials in $\mathbb{C}[x,y],$ it is generated by three polynomials $f_1, f_2, f_3$ which are related by a homogeneous polynomial $g$ of degree 3 such that $g(f_1, f_2, f_3) = 0.$ We can then identify $\mathbb{C}^2/G$ with the hypersurface $\{ f = 0 \} \subset \mathbb{C}^3.$

The resulting hypersurfaces have the following equations (with corresponding subgroup):

• $A_n: x^{n+1} + y^2 + z^2$ (cyclic)
• $D_n: x^{n-1} + xy^2 + z^2$ (dihedral)
• $E_6: x^4 + y^3 + z^2$ (tetrahedral)
• $E_7: x^3y + y^3 + z^2$ (octahedral)
• $E_8: x^5 + y^3 + z^2$ (icosahedral)

The Dynkin diagram enter as follows. Each of these surfaces can be resolved through a finite number of blow-ups, and the exceptional fiber in the resolution will consist of a copy of $\mathbb{P}^1$ for each node of the Dynkin diagram, each of which is joined to the point of another $\mathbb{P}^1$ if there is a corresponding edge in the Dynkin diagram connecting the two nodes (so in the cyclic case, it's just a chain of $\mathbb{P}^1$'s).

Lastly, there is a neat connection between this and Springer theory that goes as follows. Let $\mathcal{N}$ denote the nilpotent cone of a Lie algebra of one of the types listed above, and let $\mathcal{O}$ denote the subregular orbit. Then $\mathcal{O}$ has codimension two in $\mathcal{N}$ and hence the corresponding Kostant/Slodowy slice is a surface in $\mathcal{N}.$ It then turns out that this surface is one of the surface singularities listed above, and that the corresponding Springer fiber of a subregular element is isomorphic to the exceptional fiber in the resolution of the surface mentioned above. So the Springer resolution encodes the information of the successive blow-ups of these surfaces.

A few good references:

Milnor, Singular Points of Complex Hypersurfaces

Dimca, Singularities and Topology of Hypersurfaces

Slodowy, Simple Singularities and Simple Algebraic Groups

This question is heavily related to one of my favorite relations between geometry and representation theory. Consider simple Lie algebras of the following types:

• $A_n$
• $D_n$
• $E_6$
• $E_7$
• $E_8$

Then these Dynkin diagrams correspond to all possible finite subgroups of $SL_2.$ The relation is given by special classes of isolated surface singularities known as Kleinian or Du Val singularities. These arise as follows. Given a finite subgroup $G \subset SL_2,$ we have an action of $G$ on $\mathbb{C}^2$ with no fixed points other than the origin. If we then look at the geometric quotient $\mathbb{C}^2/G$ corresponding to $G$-invariant polynomials in $\mathbb{C}[x,y],$ it is generated by three polynomials $f_1, f_2, f_3$ which are related by a homogeneous polynomial $g$ of degree 3 such that $g(f_1, f_2, f_3) = 0.$ We can then identify $\mathbb{C}^2/G$ with the hypersurface $\{ g = 0 \} \subset \mathbb{C}^3.$

The resulting hypersurfaces have the following equations (with corresponding subgroup):

• $A_n: x^{n+1} + y^2 + z^2$ (cyclic)
• $D_n: x^{n-1} + xy^2 + z^2$ (dihedral)
• $E_6: x^4 + y^3 + z^2$ (tetrahedral)
• $E_7: x^3y + y^3 + z^2$ (octahedral)
• $E_8: x^5 + y^3 + z^2$ (icosahedral)

The Dynkin diagram enter as follows. Each of these surfaces can be resolved through a finite number of blow-ups, and the exceptional fiber in the resolution will consist of a copy of $\mathbb{P}^1$ for each node of the Dynkin diagram, each of which is joined to the point of another $\mathbb{P}^1$ if there is a corresponding edge in the Dynkin diagram connecting the two nodes (so in the cyclic case, it's just a chain of $\mathbb{P}^1$'s).

Lastly, there is a neat connection between this and Springer theory that goes as follows. Let $\mathcal{N}$ denote the nilpotent cone of a Lie algebra of one of the types listed above, and let $\mathcal{O}$ denote the subregular orbit. Then $\mathcal{O}$ has codimension two in $\mathcal{N}$ and hence the corresponding Kostant/Slodowy slice is a surface in $\mathcal{N}.$ It then turns out that this surface is one of the surface singularities listed above, and that the corresponding Springer fiber of a subregular element is isomorphic to the exceptional fiber in the resolution of the surface mentioned above. So the Springer resolution encodes the information of the successive blow-ups of these surfaces.

A few good references:

Milnor, Singular Points of Complex Hypersurfaces

Dimca, Singularities and Topology of Hypersurfaces

Slodowy, Simple Singularities and Simple Algebraic Groups