Triple ratio

In the mathematics is Tripelverhältnis (engl. Triple ratio ) is an invariant of linear algebra that the cross ratio of the projective geometry generalized and in particular in the representation theory of surface groups is important.

Flags, generic triples

→ Main article : Flag (mathematics)

Let it be a -dimensional vector space . A complete flag is a sequence of subspaces with ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle (E_{0},E_{1},\ldots ,E_{n})}$

${\displaystyle E_{0}\subsetneq E_{1}\subsetneq \ldots \subsetneq E_{n}}$

and for , in particular and . ${\displaystyle \dim E_{i}=i}$ ${\displaystyle i=0,\ldots ,n}$ ${\displaystyle E_{0}=0}$ ${\displaystyle E_{n}=V}$

A triple of complete flags is called generic if all occurring subspaces are transverse to each other, which is a sufficient condition for this ${\displaystyle (E,F,G)}$

${\displaystyle E_{a}\cap F_{b}\cap G_{c}=0\ \forall \ a+b+c=n}$.

definition

Let be a generic triple of complete flags of a -dimensional vector space . We fix an isomorphism and thus also an isomorphism . ${\displaystyle (E,F,G)}$ ${\displaystyle n}$ ${\displaystyle V}$ ${\displaystyle V\simeq \mathbb {R} ^{n}}$ ${\displaystyle \Lambda ^{n}V\simeq \mathbb {R} }$

For each we choose elements ${\displaystyle i=1,\ldots ,n}$

${\displaystyle e^{i}\in \Lambda ^{i}E_{i},f^{i}\in \Lambda ^{i}F_{i},g^{i}\in \Lambda ^{i}G_{i}}$.

(Because of this, these elements are unique except for multiplication by real numbers other than zero.) We also denote the images of these elements with . ${\displaystyle \Lambda ^{i}E_{i}\simeq \mathbb {R} ,\Lambda ^{i}F_{i}\simeq \mathbb {R} ,\Lambda ^{i}G_{i}\simeq \mathbb {R} }$ ${\displaystyle \Lambda ^{i}V}$ ${\displaystyle e^{i},f^{i},g^{i}}$

Be positive, whole numbers with . The (a, b, c) triple ratio of the generic triple of complete flags is defined by the formula ${\displaystyle a,b,c}$ ${\displaystyle a+b+c=n}$ ${\displaystyle (E,F,G)}$

${\displaystyle T_{abc}(E,F,G):={\frac {e^{a+1}\wedge f^{b}\wedge g^{c-1}}{e^{a-1}\wedge f^{b}\wedge g^{c+1}}}{\frac {e^{a}\wedge f^{b-1}\wedge g^{c+1}}{e^{a}\wedge f^{b+1}\wedge g^{c-1}}}{\frac {e^{a-1}\wedge f^{b+1}\wedge g^{c}}{e^{a+1}\wedge f^{b-1}\wedge g^{c}}}\in \mathbb {R} }$.

The six wedge products are each elements of , assuming genericity that they are all non-zero. Note that these are only clearly defined except for multiplication with real numbers, but that every element occurs equally in the numerator and denominator and is therefore well-defined. ${\displaystyle \Lambda ^{n}V=\mathbb {R} }$ ${\displaystyle e^{i}\in \Lambda ^{i}E_{i},f^{i}\in \Lambda ^{i}F_{i},g^{i}\in \Lambda ^{i}G_{i}}$ ${\displaystyle T_{abc}(E,F,G)}$

Geometric interpretation for n = 3

The triple ratio of three flags in is the double ratio of the four projective straight lines after identifying the set of projective straight lines in with a projective straight line . ${\displaystyle (A,a),(B,b),(C,c)}$ ${\displaystyle P^{2}}$ ${\displaystyle a,AB,A(b\cap c),AC}$ ${\displaystyle P^{2}}$ ${\displaystyle P^{1}}$

In particular:

• the triple ratio is −1 if and only if either the straight lines have a common point ( theorem of Ceva ) or the points lie on a straight line ( theorem of Menelaus ) or both. ${\displaystyle A(b\cap c),B(c\cap a),C(a\cap b)}$ ${\displaystyle A,B,C}$
• the triple ratio is positive if and only if the triangle ABC is inscribed in the triangle . ${\displaystyle (a\cap b)(b\cap c)(c\cap a)}$

Complete invariant

The triple ratio is a complete invariant of generic triples with base changes : ${\displaystyle A\in GL(V)}$

Theorem (Fock-Goncharov): For two generic triples of complete flags and then and there is a linear mapping with ${\displaystyle (E,F,G)}$ ${\displaystyle (E^{\prime },F^{\prime },G^{\prime })}$ ${\displaystyle A\in GL(V)}$

${\displaystyle AE_{i}=E_{i}^{\prime },AF_{i}=F_{i}^{\prime },AG_{i}=G_{i}^{\prime },i=0,\ldots ,n}$,

if

${\displaystyle T_{a,b,c}(E,F,G)=T_{abc}(E^{\prime },F^{\prime },G^{\prime })}$

holds for all triples of positive, whole numbers with . ${\displaystyle a,b,c}$ ${\displaystyle a+b+c=n}$

literature

• Fock- Goncharov : Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. No. 103: 1-211 (2006). pdf
• Bonahon -Dreyer: Parametrizing Hitchin components pdf

Individual evidence

1. ↑ A detailed proof can be found in:

   Yuichi Kabaya: On Fock-Goncharov coordinates of the once-punctured torus groups pdf