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 \( \bindCards{\wordGroupSymbol }{\card{c_1},\card{c_2},\elSy,\card{c_{\sym{n}}}} \).
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:
\[ \bindCards{\wordGroupSymbol }{\reFo{\card{r}},\blFo{\card{b}}} = \list{\word{1},\word{\reFo{\card{r}}},\word{\reFo{\ncard{r}}},\word{\blFo{\card{b}}},\word{\blFo{\ncard{b}}},\word{\reFo{\card{r}}}{\reFo{\card{r}}},\word{\reFo{\ncard{r}}}{\reFo{\ncard{r}}},\word{\reFo{\card{r}}}{\blFo{\card{b}}},\word{\reFo{\card{r}}}{\blFo{\ncard{b}}},\word{\reFo{\ncard{r}}}{\blFo{\card{b}}},\word{\reFo{\ncard{r}}}{\blFo{\ncard{b}}},\word{\blFo{\card{b}}}{\reFo{\card{r}}},\word{\blFo{\card{b}}}{\reFo{\ncard{r}}},\word{\blFo{\ncard{b}}}{\reFo{\card{r}}},\word{\blFo{\ncard{b}}}{\reFo{\ncard{r}}},\word{\blFo{\card{b}}}{\blFo{\card{b}}},\word{\blFo{\ncard{b}}}{\blFo{\ncard{b}}},\word{\reFo{\card{r}}}{\reFo{\card{r}}}{\reFo{\card{r}}},\word{\reFo{\card{r}}}{\reFo{\card{r}}}{\blFo{\card{b}}},\elSy} \]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 \( \word{\grFo{\card{g}}}{\reFo{\card{r}}}{\reFo{\inverted{\card{r}}}}{\blFo{\card{b}}} \) is reduced to \( \word{\grFo{\card{g}}}{\blFo{\card{b}}} \).
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 \( \groupElement{ \upsilon } = \word{\grFo{\card{g}}}{\blFo{\card{b}}}{\reFo{\ncard{r}}},\groupElement{ \omega } = \word{\reFo{\card{r}}} \), this looks like \( \concat{\groupElement{ \upsilon }\,\groupElement{ \omega }} = \concat{\word{\grFo{\card{g}}}{\blFo{\card{b}}}{\reFo{\ncard{r}}}\,\word{\reFo{\card{r}}}} \), which reduces to \( \word{\grFo{\card{g}}}{\blFo{\card{b}}} \).
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 \( \bindCards{\wordGroupSymbol }{\reFo{\card{r}},\grFo{\card{g}},\blFo{\card{b}}} \):
\[ \begin{csarray}{rlcl}{abee} \word{\card{1}} & \word{\card{1}} & = & \word{\card{1}}\\ \word{\card{1}} & \word{\reFo{\card{r}}} & = & \word{\reFo{\card{r}}}\\ \word{\reFo{\card{r}}} & \word{\reFo{\ncard{r}}} & = & \word{\card{1}}\\ \word{\reFo{\card{r}}}{\blFo{\card{b}}} & \word{\grFo{\card{g}}} & = & \word{\reFo{\card{r}}}{\blFo{\card{b}}}{\grFo{\card{g}}}\\ \word{\reFo{\card{r}}}{\reFo{\card{r}}}{\blFo{\card{b}}} & \word{\blFo{\ncard{b}}}{\grFo{\card{g}}}{\grFo{\card{g}}} & = & \word{\reFo{\card{r}}}{\reFo{\card{r}}}{\grFo{\card{g}}}{\grFo{\card{g}}} \end{csarray} \]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 \( \cardinalList(\quiver{Q}) = \{\reFo{\card{r}},\blFo{\card{b}}\} \), side-by-side with their inverses:
\[ \begin{csarray}{rlrlrlrl}{aeieieie} \groupElement{ \omega } & \groupInverse{\groupElement{ \omega }} & \groupElement{ \omega } & \groupInverse{\groupElement{ \omega }} & \groupElement{ \omega } & \groupInverse{\groupElement{ \omega }} & \groupElement{ \omega } & \groupInverse{\groupElement{ \omega }}\\ \word{\card{1}} & \word{\card{1}} & \word{\reFo{\card{r}}} & \word{\reFo{\ncard{r}}} & \word{\reFo{\card{r}}}{\reFo{\card{r}}} & \word{\reFo{\ncard{r}}}{\reFo{\ncard{r}}} & \word{\reFo{\card{r}}}{\grFo{\card{g}}}{\blFo{\card{b}}} & \word{\blFo{\ncard{b}}}{\grFo{\ncard{g}}}{\reFo{\ncard{r}}}\\ & & \word{\grFo{\card{g}}} & \word{\grFo{\ncard{g}}} & \word{\reFo{\card{r}}}{\grFo{\card{g}}} & \word{\grFo{\ncard{g}}}{\reFo{\ncard{r}}} & & \\ & & \word{\blFo{\card{b}}} & \word{\blFo{\ncard{b}}} & \word{\grFo{\card{g}}}{\blFo{\ncard{b}}} & \word{\blFo{\card{b}}}{\grFo{\ncard{g}}} & & \end{csarray} \]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).