Fano Plane

This list of configurations comes from https://en.wikipedia.org/wiki/Fano\_plane#Configurations

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{\displaystyle Fano={1 \over 168}(a\_{1}^{7}+21a\_{1}^{3}a\_{2}^{2}+42a\_{1}a\_{2}a\_{4}+56a\_{1}a\_{3}^{2}+48a\_{7}).}

## There are 7 points

* There are 7 points, and 24 symmetries fixing any point.

**How many points are in a Fano plane ?** The cycle index for Fano acting on points is

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{\displaystyle {1 \over |Fano|}\sum \_{\sigma \in Fano}fix(\sigma ).a^{\sigma }}

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{\displaystyle {Fano \over X}={1 \over 24}\left(a\_{1}^{6}+9a\_{1}^{2}a\_{2}^{2}+6a\_{2}a\_{4}+8a\_{3}^{2}\right)=6T7}

## 42 ordered pairs, 21 unordered pairs

* There are 42 ordered pairs of points, and each may be mapped by a symmetry onto any other ordered pair. For any ordered pair there are 4 symmetries fixing it.

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{\displaystyle {Fano \over X.X}=a\_{1}.{1 \over 4}(a\_{1}^{4}+3a\_{2}^{2})=X.Klein}

* Correspondingly, there are 21 unordered pairs of points, each of which may be mapped by a symmetry onto any other unordered pair. For any unordered pair there are 8 symmetries fixing it.

How many 2-sets are in a Fano plane ? The cycle index for Fano acting on 2-sets is

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{\displaystyle {Fano \over {\{X,X\}}}=X.Square=1T1.4T3}

## 21 flags + 84 pointed triangles

* There are 21 flags consisting of a line and a point on that line. Each flag corresponds to the unordered pair of the other two points on the same line. For each flag, 8 different symmetries keep it fixed.
* There are 84 ways of specifying a triangle together with one point on that triangle, each of which has two symmetries fixing it.

How many X.2-sets are in a Fano plane**? The cycle index for Fano acting on 2-sets is**

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{\displaystyle {1 \over |Fano|}\sum \_{\sigma \in Fano}fix(\sigma ).a^{\sigma }}

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{\displaystyle {Fano \over {X.\{X,X\}}}={1 \over 8}(a\_{1}^{4}+3a\_{2}^{2}+2a\_{1}^{2}a\_{2}+2a\_{4})+a\_{1}^{2}.{1 \over 2}(a\_{1}^{2}+a\_{2})=Square+X.X.Ens\_{2}}

## 7 quadrangles, (7 lines) + 28 triangles

* There are 7 lines, and 24 symmetries fixing any line.
* There are 28 triangles, which correspond one-for-one with the 28 bitangents of a quartic (Manivel 2006). For each triangle there are six symmetries fixing it, one for each permutation of the points within the triangle.
* There are 7 ways of selecting a quadrangle of four (unordered) points no three of which are collinear, and 24 symmetries that fix any such quadrangle. These four points form the complement of a line, which is the diagonal line of the quadrangle.

**How many 3-sets are in a Fano plane ?** The cycle index for Fano acting on 3-sets is

1

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{\displaystyle {1 \over |Fano|}\sum \_{\sigma \in Fano}fix(\sigma ).a^{\sigma }}

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{\displaystyle {Fano \over {\{X,X,X\}}}={a\_{1}^{4}+3a\_{2}^{2}+6a\_{1}^{2}a\_{2}+8a\_{1}a\_{3}+6a\_{4} \over 24}+a\_{1}.{a\_{1}^{3}+3a\_{1}a\_{2}+2a\_{3} \over 6}=2{b\_{1}^{3}+3b\_{1}b\_{2}+2b\_{3} \over 6}}

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{\displaystyle =Ens\_{4}+X.Ens\_{3}=4T5+1T1.3T2}

## 168 labeled triangles

* This means that there are 168 labeled triangles fixed only by the identity collineation

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{\displaystyle {Fano \over {X.X.X}}=Klein+X.X.X.X}

## 28 anti-flags

There are 28 ways of selecting a point and a line that are not incident to each other (an anti-flag), and six ways of permuting the Fano plane while keeping an anti-flag fixed. For every non-incident point-line pair (p,l), the three points that are unequal to p and that do not belong to l form a triangle, and for every triangle there is a unique way of grouping the remaining four points into an anti-flag.

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{\displaystyle {Fano \over {X.\{X,X,X\}}}={a\_{1}^{3}+3a\_{1}a\_{2}+2a\_{3} \over 6}+{a\_{1}^{3}+3a\_{1}a\_{2}+2a\_{3} \over 6}+{a\_{1}^{3}+a\_{1}a\_{2} \over 2}=Ens\_{3}+Ens\_{3}+X.Ens\_{2}}

## 28 hexagons

* There are 28 ways of specifying a hexagon in which no three consecutive vertices lie on a line, and six symmetries fixing any such hexagon.

TO DO

## 84 pentagons

* There are 84 ways of specifying a pentagon in which no three consecutive vertices lie on a line, and two symmetries fixing any pentagon.

## etc

### **(X,X,X)**

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{\displaystyle {Fano \over {(X,X,X)}}={1 \over 12}(a\_{1}^{4}+3a\_{2}^{2}+8a\_{1}a\_{3})+a\_{1}.{1 \over 3}(a\_{1}^{3}+2a\_{3})=Alt\_{4}+X.Alt\_{3}}

### **[X,X,X,X]**

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{\displaystyle {Fano \over {[X,X,X,X]}}=2.a\_{1}.{1 \over 2}(a\_{1}^{2}+a\_{2})=X.Ens\_{2}+X.Ens\_{2}}

### **Klein**

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{\displaystyle {Fano \over Klein}=2.a\_{1}^{3}=X.X.X+X.X.X}

### **(X,X,X,X)**

42 cyclic quadrangles

There are 42 ways of selecting a quadrangle of four cyclically ordered points no three of which are collinear, and four symmetries that fix any such ordered quadrangle. For each not-oriented quadrangle there are two cyclic orders.

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{\displaystyle {Fano \over {(X,X,X,X)}}=a\_{1}{a\_{1}^{2}+a\_{2} \over 2}+a\_{1}^{3}=X.Ens\_{2}+X.X.X}

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{\displaystyle =a\_{1}^{3}b\_{1}^{4}+a\_{1}{a\_{1}^{2}b1^{4}+a1^{3}b\_{2}^{2}+2a\_{2}b\_{4} \over 4}+=Y.Y.Y.Y+C\_{4}(Y)}

### **{X,X,X,X}**

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{\displaystyle {Fano \over \{X,X,X,X\}}={1 \over 6}(a\_{1}^{3}+3.a\_{1}a\_{2}+2.a\_{3})+{1 \over 6}(a\_{1}^{3}+3.a\_{1}a\_{2}+2.a\_{3})=Ens\_{3}+Ens\_{3}=3T2+3T2}