Word groups
In this short section we introduce objects that nevertheless plays a vital role in understanding paths on quivers. These are word groups that describe possible path words, but divorced from any particular path in a quiver.
Notation
We write the word group on a set of cardinals as \(\bindCards{\wordGroupSymbol }{\card{\card{c}_1},\card{\card{c}_2},\ellipsis ,\card{\card{c}_{\sym{n}}}}\).
The word group of a quiver \(\quiver{Q}\), written \(\wordGroup{\quiver{Q}}\), is just the word group on the cardinals of that quiver.
If we wish only to specify how many cardinals are present, we will write this as \(\wordGroup{\quiver{\sym{n}}}\).
If it is clear from context what cardinals we are talking about, we'll just write \(\wordGroupSymbol\).
Elements
A group elements \(\elemOf{\groupElement{ \omega }}{\wordGroupSymbol }\) is a word, which is a finite sequence of (possibly inverted) cardinals. For example:
\[ \bindCards{\wordGroupSymbol }{\rform{\card{r}},\bform{\card{b}}} = \list{\word{1},\word{\rform{\card{r}}},\word{\rform{\ncard{r}}},\word{\bform{\card{b}}},\word{\bform{\ncard{b}}},\word{\rform{\card{r}}}{\rform{\card{r}}},\word{\rform{\ncard{r}}}{\rform{\ncard{r}}},\word{\rform{\card{r}}}{\bform{\card{b}}},\word{\rform{\card{r}}}{\bform{\ncard{b}}},\word{\rform{\ncard{r}}}{\bform{\card{b}}},\word{\rform{\ncard{r}}}{\bform{\ncard{b}}},\word{\bform{\card{b}}}{\rform{\card{r}}},\word{\bform{\card{b}}}{\rform{\ncard{r}}},\word{\bform{\ncard{b}}}{\rform{\card{r}}},\word{\bform{\ncard{b}}}{\rform{\ncard{r}}},\word{\bform{\card{b}}}{\bform{\card{b}}},\word{\bform{\ncard{b}}}{\bform{\ncard{b}}},\word{\rform{\card{r}}}{\rform{\card{r}}}{\rform{\card{r}}},\word{\rform{\card{r}}}{\rform{\card{r}}}{\bform{\card{b}}},\ellipsis } \]We reserve the symbol \(\card{1}\) to refer to the group identity, the empty word consisting of zero cardinals, which would otherwise be hard to indicate textually, since it naturally be written as a blank space.
The number of words is obviously infinite if there is at least one cardinal.
Reduced form
The words are subject to the identity that we can rewrite any subword (any contiguous subsequence of cardinals) according to \(\concat{\card{c} \inverted{\card{c}}} = \concat{\inverted{\card{c}} \card{c}} = \card{1}\). Removing such adjacent inverses is called reduction.
We typically prefer to write these words in reduced form, so that the word \(\word{\gform{\card{g}}}{\rform{\card{r}}}{\rform{\inverted{\card{r}}}}{\bform{\card{b}}}\) is reduced to \(\word{\gform{\card{g}}}{\bform{\card{b}}}\).
Concatenation
The group multiplication of two words \(\elemOf{\groupElement{ \upsilon },\groupElement{ \omega }}{\wordGroupSymbol }\) is simply their concatenation, which we will write by putting words next to each other with a small gap: \(\concat{\groupElement{ \upsilon }\,\groupElement{ \omega }}\).
For the the case \(\groupElement{ \upsilon } = \word{\gform{\card{g}}}{\bform{\card{b}}}{\rform{\ncard{r}}},\groupElement{ \omega } = \word{\rform{\card{r}}}\), this looks like \(\concat{\groupElement{ \upsilon }\,\groupElement{ \omega }} = \concat{\word{\gform{\card{g}}}{\bform{\card{b}}}{\rform{\ncard{r}}}\,\word{\rform{\card{r}}}}\), which reduces to \(\word{\gform{\card{g}}}{\bform{\card{b}}}\).
Notice that if there is more than one cardinal, the group is not Abelian, since \(\concat{\groupElement{ \upsilon }\,\groupElement{ \omega }} \neq \concat{\groupElement{ \omega }\,\groupElement{ \upsilon }}\) in general. The order of cardinals in a word matters!
Here we list some concatenations of words in \(\bindCards{\wordGroupSymbol }{\rform{\card{r}},\gform{\card{g}},\bform{\card{b}}}\):
\[ \begin{csarray}{rlcl}{abeeb} \word{\card{1}} & \word{\card{1}} & = & \word{\card{1}}\\ \word{\card{1}} & \word{\rform{\card{r}}} & = & \word{\rform{\card{r}}}\\ \word{\rform{\card{r}}} & \word{\rform{\ncard{r}}} & = & \word{\card{1}}\\ \word{\rform{\card{r}}}{\bform{\card{b}}} & \word{\gform{\card{g}}} & = & \word{\rform{\card{r}}}{\bform{\card{b}}}{\gform{\card{g}}}\\ \word{\rform{\card{r}}}{\rform{\card{r}}}{\bform{\card{b}}} & \word{\bform{\ncard{b}}}{\gform{\card{g}}}{\gform{\card{g}}} & = & \word{\rform{\card{r}}}{\rform{\card{r}}}{\gform{\card{g}}}{\gform{\card{g}}} \end{csarray} \]Inverses
To invert a word we reverse its individual letters and invert them.
Here we list a small number of examples of words from \(\wordGroup{\quiver{Q}}\) for \(\cardinalList(\quiver{Q}) = \{\rform{\card{r}},\bform{\card{b}}\}\), side-by-side with their inverses:
\[ \begin{csarray}{rlrlrlrl}{aeieieiei} \groupElement{ \omega } & \groupInverse{\groupElement{ \omega }} & \groupElement{ \omega } & \groupInverse{\groupElement{ \omega }} & \groupElement{ \omega } & \groupInverse{\groupElement{ \omega }} & \groupElement{ \omega } & \groupInverse{\groupElement{ \omega }}\\ \word{\card{1}} & \word{\card{1}} & \word{\rform{\card{r}}} & \word{\rform{\ncard{r}}} & \word{\rform{\card{r}}}{\rform{\card{r}}} & \word{\rform{\ncard{r}}}{\rform{\ncard{r}}} & \word{\rform{\card{r}}}{\gform{\card{g}}}{\bform{\card{b}}} & \word{\bform{\ncard{b}}}{\gform{\ncard{g}}}{\rform{\ncard{r}}}\\ & & \word{\gform{\card{g}}} & \word{\gform{\ncard{g}}} & \word{\rform{\card{r}}}{\gform{\card{g}}} & \word{\gform{\ncard{g}}}{\rform{\ncard{r}}} & & \\ & & \word{\bform{\card{b}}} & \word{\bform{\ncard{b}}} & \word{\gform{\card{g}}}{\bform{\ncard{b}}} & \word{\bform{\card{b}}}{\gform{\ncard{g}}} & & \end{csarray} \]Being free
Beyond the identity \(\concat{\card{c} \inverted{\card{c}}} = \concat{\inverted{\card{c}} \card{c}} = \card{1}\), which reflects in some sense the most generic property of a group, we do not impose any further relations (hence the term "free"). This implies that if two elements of \(\wordGroupSymbol\) "look different" in reduced form – that is, they contain a difference sequence of cardinals – they are different elements of the group.
Relationship to paths
As the name suggests, the path word of a path in \(\quiver{Q}\) is an element of the word group \(\wordGroup{\quiver{Q}}\), and path composition (when defined) will yield a path whose word is the concatenation (the group operation of \(\wordGroupSymbol\)) of the path words:
\[ \wordOf(\pathCompose{\path{P_1}}{\path{P_2}}) = \concat{\wordOf(\path{P_1})\,\wordOf(\path{P_2})} \]A choice of vertex and a word will uniquely identify a path in a quiver, if one exists. Due to the local uniqueness property, there cannot be more than one path starting at a given vertex that posses a given word. However, there may be zero paths starting at a vertex with a given word.
It should be obvious that \(\functionSignature{\wordOf}{\pathGroupoid{\quiver{Q}}}{\wordGroup{\quiver{Q}}}\) is a groupoid homomorphism from the path groupoid to the word group (as \(\wordGroup{\quiver{Q}}\) is a group it is naturally also a groupoid).